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+[    {        "title": "1310.6117v1.Magnetization_Characteristic_of_Ferromagnetic_Thin_Strip_by_Measuring_Anisotropic_Magnetoresistance_and_Ferromagnetic_Resonance.pdf",        "content": "\t\r \n1\t\r \t\r Magnetization Characteristic of Ferromagnetic Thin Strip by Measuring Anisotropic Magnetoresistance and Ferromagnetic Resonance Ziqian Wang, Guolin Yu, Xinzhi Liu, Bo Zhang, Xiaoshuang Chen and Wei Lu National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai, 200083, China  Abstract The magnetization characteristic in a permalloy thin strip is investigated by electrically measuring the anisotropic magnetoresistance and ferromagnetic resonance in in-plane and out-of-plane configurations. Our results indicate that the magnetization vector can rotate in the film plane as well as out of the film plane by changing the intensity of external magnetic field of certain direction. The magnetization characteristic can be explained by considering demagnetization and magnetic anisotropy. Our method can be used to obtain the demagnetization factor, saturated magnetic moment and the magnetic anisotropy.  Key words: A. Permalloy Thin Strips; D. Anisotropic Magnetoresistance; D. Demagnetization; D. Magnetic Anisotropy.  \t\r \n2\t\r \t\r   1. Introduction Anisotropic magnetoresistance (AMR) effect, which is resulted from the anisotropy of spin-orbit interaction in ferromagnetic materials [1,2], was first discovered by Thomson in 1857 [3]. The effect bears an essential role for both scientific perspectives and technological applications [4-8]. AMR is manifested in the dependence of the resistivity on the angle between current and magnetization direction [1,2], and is given by 20- sinAARR RRθ=+.     (1) Here R0 represents the resistance while the magnetization M is perpendicular to the induced current, RA is the decrement resistance, and θ is the angle of magnetization M with respect to current. In ferromagnetic devices of certain structures, M is parallel to the effective magnetic field Heff, including external field Hex, anisotropy built-in field Ha and demagnetization field Hd. Researchers can obtain these parameters through finding θ.   The purpose of this article is to obtain magnetization characteristic of a ferromagnetic thin strip through AMR measurement. Ferromagnetic resonance (FMR) detection is considered as an ancillary method to \t\r \n3\t\r \t\r determine the magnetization based on fitting the measured FMR dispersion curves via Kittel’s theory [9].   2. Material and Methods The present work is performed on a Ni80Fe20 (Permalloy, Py) thin film deposited on a 5×6 mm2 GaAs single crystal substrate. This polycrystalline structured Py film is patterned to a stripe shape by photolithography and lift off techniques. The dimensions of our sample is: length=2400µm, width=200µm and thickness=50nm.  The Py strip is fixed on a rotatable holder by an adjustable wedge. The external magnetic field, represented as (),,ex x y zHH H H=v, encloses an intersection angle α with the long-axis of the strip, and β represents the dip of the wedge. Accurate α and β are recorded by a readout on the holder and a goniometer. AMR is measured by detecting the resistance between two electrodes at each side of the strip’s length, as illustrated in Fig. 1(a). In FMR measurement, modulated microwaves are propagating to the holder normally through a rectangular waveguide of X band, while the modulation frequency is 5.37 kHz. The FMR signals are also electrically measured in field-swept mode by using a lock-in amplifier connecting those two electrodes via gold bonding wires and coaxial cables.  \t\r \n4\t\r \t\r The coordinate system we select in this article is demonstrated in Fig. 1(b).  The long axis of the thin strip is set as z-axis, the direction perpendicular to the strip’s plane is defined as y-axis, and the strip lies in the xz-plane. ΦH, ΦM are recorded as the misalignments of Hex and M with respect to xz-plane. The in-plane components of Hex and M encloses the angles ΘH, ΘM with z-axis, respectively. Hence ΘH, ΦH, α and β follows the relations as()1tan tan cosHαβ−Θ=and()1sin sin sinHαβ−Φ=.  In section 3.1, AMR and FMR measurements in weak Hex are illustrated, and both of Ha and Hd are taken into account. We will show how M rotates from parallel to Ha via only changing the magnitude of Hex generated by an electromagnet at room temperature. The detailed process of obtaining the demagnetization factors and magnetization through AMR and FMR experiments is also introduced in this subsection. For further investigating, AMR in out-of-plane configuration in stronger Hex is discussed in section 3.2. In this subsection, the Py thin strip is placed in Hex produced by a cryomagnet, which is carried out at liquid helium temperature.   3. Results and Discussion 3.1. Weak External Field Condition It is noted as “weak field condition” when the magnitude of Hex is \t\r \n5\t\r \t\r smaller than 2000 Oe. The measured AMR and FMR data of this Py thin strip in in-plane and out-of-plane configurations are shown in Fig. 2. Meanwhile, the demagnetization coefficient and M are obtained via fitting  these experimental results by selecting appropriate models. Fig. 2(a) shows the measured sheet resistance R versus Hex at different ΘH in in-plane magnetized configuration. It is illustrated that R reaches its maximum at Hex=0, hence M is parallel, or anti-parallel, to z-axis as a higher resistance state without external magnetic field. The increment of R achieved by decreasing the magnitude of Hex implies M’s rotation from parallel with Hex to z-axis, as demonstrated in Eq. (1). The direction of M at zero-field state is caused by Ha, the anisotropic built-in field. Ha is along with z-axis because of the lowest free energy for thin strip structure in this direction.   Here we assume Ha as a static filed, which is parallel to z-axis and is expressed as ()0, 0,aaHH=v. The effect of other important factor on the rotation of M is Hd, the demagnetization field. Hd depends on the direction and magnitude of M, it is written by(),,dx x x y y y z z zHN M N M N M=−v, where Nxx, Nyy, and Nzz are the demagnetization factors in x, y and z directions. The demagnetization factors satisfy the correlation of 1xx yy zzNNN++=for SI and 4xx yy zzNNNπ++=for CGS [11]. Accordingly, the yielded relations \t\r \n6\t\r \t\r between Μ and Ηex are: ,cos sin , sin , cos cos ,cos sin , sin , cos cosy yy yx xx x z axy zx ex H H x ex H z ex H HxM M y M zM MHN MHN M HHMM MHH HH HHMM MM MM−−+===Φ Θ = Φ =Φ Θ=Φ Θ = Φ =Φ Θvv vvv v    (2)  The demagnetization field along z-axis is neglected since Nzz is much smaller than Nxx and Nyy in thin strip structure, as shown in Fig. 1(b). It is difficult to provide analytical solutions for Eqs. (2), however getting numerical solutions is not a hard task. We use the numerical method to fit the experiment data. In in-plane configuration under weak field condition, ΦH=ΦM=0 and Hy=0, and Eqs. (2) are transformed as, sin sin cossin cosex H xx M ex H aMMHN MHHMMΘ− Θ Θ+=ΘΘvv vvv .        (3) Considering ΦH=ΦM=0, we have θ=ΘH. Taking the numerical results of Eq. (3) into Eq. (1), Ha and NxxMx are obtained. R0 and RA can be recorded directly through Fig. 1 as R0=87.7 Ω and Ra=1.5Ω. Other fitted results are Ha=1.95Oe and NxxMx=5.1Oe.  The demagnetization factor Nyy is significantly larger than Nxx according to the thin strip structure. If Hex consists of y-component, M should be misaligned away to Hex, which causes the surface magnetic charges in each side of xz-plane [10]. These surface magnetic charges generate a demagnetization field inside the sample with its direction \t\r \n7\t\r \t\r opposite to the y-component of Hex, and finally prohibit the misalignment of M. According to the shape of our sample, the demagnetization field generated by a very slight y-component of M can offset the y-component of Hex. Thus, only the xz-component of Hex is worth to be considered, the motion of M according to changing Hex is almost in-plane, and we have, cos sin sin cos cos,sin cos.ex H H xx M ex H H aMMMHN M HHMMθΦΘ − Θ ΦΘ +=ΘΘΘ≈vv vvv        (4)  The larger out-of-plane component of Hex can be provided by increasing the dip angle β of the wedge. Comparating to the in-plane Hex, larger out-of-plane Hex is needed for obtaining the same Heff and θ, as showing in Fig. 2(b).  The calculated magnitudes of M is obtained by fitting FMR experiment. The dispersion curves in different configurations are recorded in Fig. 2(c). For an in-plane M assisted with microwave in a magnetic field H, the resonant frequency fr for the rf signal is given by()()2ry y z z x x z zfH N N M H N N Mπγ⎡⎤=+ − + −⎡⎤⎣⎦⎣⎦[9].\t\r In our sample, we have eff ex d aHH H H H==+ +vv v and 0181 GHz/Tγµ= for Py, here µ0 is the permeability of vacuum. The Hex’s range is from 1000Oe to 1800Oe, such amplitudes are much smaller than ΝyyM, and consequently a very slight ΦH can generate large enough Hd to overcome the y-component of \t\r \n8\t\r \t\r Hex unless the in-plane component of Hex is significantly smaller than its y-component. For the schematic of β= 0, 45°and 70° in Fig. 2(c) for Hex larger than 1000Oe, NxxMx and Ha are ignorable, and the simplified expression of dispersion curve is given by: ()2c o s c o sre x H y y e x HfH N M Hπγ=Φ + Φ.                          (5)  In weak field condition, since Hex’s y-component is offseted by Hd, higher Hex is needed in order to achieve high enough resonant Heff for larger β while the assisted microwave’s frequency is fixed. Eq. (5) is used to fit the electrically detected FMR dispersion curves in Fig. 2(c) at different β.  The fitted magnetization is M=10750Oe, and the demagnetization factor along with x-axis is calculated as Nxx=0.00047.  3.2. Strong External Field Condition According to Eq. (2), larger Hex may provide y-component for Heff, and the direction of M can be tilted away from xz-plane. In terms of ΦM≠0 for this situation, Eq. (1) would be revised as, ()22 2 20-s i n c o s c o t c o sAA M M M MRR RR=+ Φ + Θ Θ Φ.                (6) Here ΦM and ΘM are deduced from Eqs. (2). The magnetization characteristic of Py thin strip under stronger Hex is investigated in liquid helium temperature around 4.2K. Although R0, RA and M vary at different temperatures, AMR feature of this Py thin strip in low \t\r \n9\t\r \t\r temperature and Hex up to 5 T evolves as the prediction of Eqs. (2), seeing in Fig. 3.  The movement of M from z-axis to Hex is separated by two steps, the in-plane magnetization as investigated in the former paragraphs, and the out-of-plane magnetization while is Hex high enough. The movement of M can be illustrated by an approximate picture. ΦM≈0 in the first step, M rotates rapidly from ΘM=0 to ΘM= ΘH. The process is displaced in the embedded picture of Fig. 3. In the second step, larger xz-component of Hex only keeps ΘM= ΘH. However, ΦM increases with stronger y-component of Hex, as shown in Fig. 3.   Meanwhile, we should indicate that the two-step magnetic movement does not exist in every applied out-of-plane Hex. Taking ΦH=90° as an instance, the movement of M only includes out-of-plane step in yz-plane because Hex contributes no x-component field to the effective field.   4. Conclusions In summary, we have demonstrated how the magnetization characteristic of a ferromagnetic thin strip changes in different external magnetic field based on the AMR and FMR measurements by considering demagnetization and magnetic anisotropy. It is shown that \t\r \n10\t\r \t\r the magnetization vector can rotate in the film plane as well as out of the film plane by sweeping the intensity of external magnetic field, while the direction of external field is fixed. The out-of-plane AMR’s low-temperature and high-field features are also well explained.  Our method can be used to obtain the demagnetization factor, saturated magnetic moment and the magnetic anisotropy.  Acknowledgement   The work is supported by the State Key Program for Basic Research of China (2013CB632705, 2011CB922004), the National Natural Science Foundation of China (10990104).            \t\r \n11\t\r \t\r References: [1] T. R. McGuire and R. I. Potter, IEEE Trans. Magn. 11, 1018 (1975). [2] R. C. O’Handley, Modern Magnetic Materials: Principles and Applications (Wiley, New York, 2000), p. 573. [3] W. Thomson, Proc. R. Soc. London 8, 546 (1857) [4] R. E. Camley, B. V. Mcgrath, Y. Khivintsev, and Z. Celinski, Phys. Rev. B, 78, 024425 (2008). [5] M. P. Kostylev, G. Gubbiotti, J.-G. Hu, G. Carlotti, T. Ono, and R. L. Stamps, Phys. Rev. B, 76, 054422 (2007). [6] A. V. Chumak, A. A. Serga, B. Hillebrands, G. A. Melkov, V. Tiberkevich, and A. N. Slavin, Phys. Rev. B, 79, 014405 (2009). [7] Pawel. Buczek, Arthur. ernst, and Leonid. M. Sandratskii, Phys. Rev. Lett, 105, 097205 (2010). [8] A. Taroni, A. Bergman, L. Bergqvist, J. Hellsvik, and O. Eriksson, Phys. Rev. Lett, 107, 037202 (2011). [9] C. Kittel, Phys. Rev. 78, 266 (1950) [10] Masanori. Kobayashi, and Yoshifumi. Ishikawa, IEEE Trans on Magn, Vol.28, pp1810 (1992). [11] C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc., New York, 1986) 6th edition.   \t\r \n12\t\r \t\r  \n Figure 1. (a) The schematic of our experiment and (b) the coordinate system used in this article. The rf signal and Lock-In amplifier are only applied in FMR measurement. The demagnetization field along x and y direction are also notified as -NxxMx and -NyyMy.    \n\t\r \n13\t\r \t\r  \n Figure 2. The experimental (colored dots) and fitted (solid lines) results for (a). in-plane magnetized AMR at different ΘH, (b). AMR of in-plane and out-of-plane magnetizations for ΘH=90° at different ΦH, and (c). the resonant frequency of FMR with respect to external field for both in-plane and out-of-plane configurations at different β, here α is fixed at 45°.  \n\t\r \n14\t\r \t\r  \n Figure 3. AMR measurement in stronger Hex at 4.2K. α and β are set as 45° and 30°, respectively. Inset: detailed sheet resistance of this thin strip at lower external field.  \n"    },    {        "title": "1401.1672v1.Dynamic_exchange_via_spin_currents_in_acoustic_and_optical_modes_of_ferromagnetic_resonance_in_spin_valve_structures.pdf",        "content": "1 \n Dynamic exchange via spin currents in acoustic and optical modes of \nferromagnetic resonance in spin -valve structures  \n \nA.A. Timopheev1, Yu.G. Pogorelov2, S. Cardoso3, P.P. Freitas3, G.N. Kakazei2,4, N.A . Sobolev1 \n1Departamento de Física and  I3N, Universidade de Aveiro, 3810 -193 Aveiro, Portugal  \n2IFIMUP and IN -Institute of Nanoscience and Nanotechnology, Departamento de Física e Astronomia, \nUniversidade do Porto, 4169 -007 Porto, Portugal  \n3INESC -MN and IN -Institute of Nanoscience and Nanotechn ology, 1000 -029 Lisbon, Portugal  \n4Institute of Magnetism, NAS of Ukraine, 03142 Kiev, Ukraine  \n     \ne-mail:  andreyt@ua.pt  \nTwo ferromagnetic layer s magnetically decoupled by a thick normal metal spacer layer can be, \nnevertheless, dynamically coupled via spin currents emitted by the spin -pump and absorbed through the \nspin-torque effects at the neighboring interfaces. A decrease of damping in both layers due to a partial \ncompensation of the angular momentum leakage in e ach layer was previously observed at the coincidence \nof the two ferromagnetic resonances. In case of non -zero magnetic coupling, such a dynamic exchange \nwill depend on the mutual precession of the magnetic moments in the layers. A difference in the linewid th \nof the resonance peaks is expe cted for the acoustic and optical regimes of precession. However, the \ninterlayer coupling hybridizes the resonance responses of the layers and therefore can also change their \nlinewidths. The interplay between the two mechan isms has never been considered before. In the present \nwork, the joint influence of the hybridization and non -local damping on the linewidth has been studied in \nweakly coupled NiFe/CoFe/Cu/CoFe/MnIr spin -valve multilayers. It has been found that the dynamic  \nexchange by spin currents is different in the optical and acoustic modes, and this difference is dependent \non the interlayer coupling strength. In contrast to the acoustic precession mode, the dynamic exchange in \nthe optical mode works as an additional da mping source. A simulation in the framework of the Landau -\nLifshitz -Gilbert formalism for two ferromagnetic layers coupled magnetically and by spin currents has \nbeen done to separate the effects of the non -local damping from the resonance modes hybridizatio n. In \nour samples both mechanisms bring about linewidth changes of the same order of magnitude, but lead to \na distinctly different angular behavior. The obtained results are relevant  for a broad class of coupled \nmagnetic multilayers  with ballistic regime o f the spin transport . \n    \n1. Introduction  \nSpin current, a flow of angular momentum , is a basic concept in spintronics  and spin caloritronics [1, \n2]. Spin current generation is experimentally accessible via spin pumping [3-5], spin Seebek effect [6], spin \nHall effect [7, 8] and acoustic wave propagation in the case of magnetic insulators [9]. The spin -orbit \ninteraction plays a fundamental role in these effects. The presence of a spin current in a normal metal 2 \n (NM) or semiconductor can be detected by the inverse spin Hall effect [10-12] or as a change of the \neffective damping in an adjacent ferromagnetic (FM) layer [3-5]. The latter effect allows one to alter the \nswitching field of the FM layer and even sustain  a stable precession in it [13-15]. It is hard to overestimate \nthe fundamental and practical importance of the issues emerging from the investigation  of the spin \ncurrents.  \nA precessing magnetic moment in a FM layer acts as a spin battery [16] injecting a pu re spin current in \na neighboring NM layer through the FM/NM interface. This spin current can then  return to  the NM/FM \ninterface bringing the carried angular momentum  back to the precessing spins of the FM layer. Depending \non the spin -orbit interaction stre ngth and the layer thickness, the normal metal will absorb a certain part \nof the angular momentum flow via the spin -flip relaxation processes. Thus, the backflow through the \nNM/FM interface will be always weaker than the direct flow, which results in an enhanced  precession  \ndamping  [3-5]. The spin diffusion length of the normal metal and the spin mixing  interface conductance \ncan be evaluated in this way [3-5, 17]. \nAn interesting result has been obtained for a FM/NM/FM trilayer [18] having non -identical FM layers. \nThe asymmetry provided different angular dependences of the ferromagnetic resonance (FMR) fields of \nthe FM layers. When the external magnetic field was directed at an angle for which the FMR peak \npositions coincide, a narrowing of both resonances w as observed. The explanation of this effect is that, \nfor the case of separately precessing FM layers, the spin current generated in  a precessing FM layer is \nabsorbed in the other, non -resonating FM layer, which causes , in a full analogy to the written  abov e, a \ndamping enhancement,  while for the case of a mutual resonant precession this spin current leakage is \npartially compensated by the spin current from the other FM layer. In this experiment , the NM spacer was \nthin enough for the spin current to be consid erable at the second NM/FM interface, but thick enough to \nexclude any possible magnetic coupling between the FM layers.  \nIndeed, the magnetic coupling between two FM layers complicates the analysis of the spin -current -\ninduced non -local damping. If the coupling is strong enough, the resonance response of the system is \nrepresented by the collective acoustic and optical modes w hich are the in -phase and out -of-phase mutual \nprecession modes in the  FM layers. There is no separate precession in such a regime – the precession in \none layer drags the magnetic moment in the other one. Moreover, the linewidths of the resonance peaks \nare dependent o n the field separation betwe en them, and usually these parameters are angular dependent. \nAnd finally, the interaction fundamentally forbids the peaks to have a crossing point, i.e. the anticrossing \nis a characteristic feature here. The stronger the interlayer coupling, the larger is the anticrossing \nseparation between  the modes. From this point of view, the difference of damping  for the acoustic and \noptical modes in  a FM/NM/FM trilayer as a result of a dynamic spin currents exchange, theoreticall y \npredicted by Kim and Chappert [19], seems to be experimentally unachievable. Nevertheless , in several \nrecent papers [20-22] experimental observations of this effect  have been already claimed.  There  is, 3 \n however, a full ignorance of the fact that the FMR p eaks hybridization will also influence the linewidth \neven if a separate measurement of the precession in each layer can be done.  \nMotivated by this, we have performed  a comprehensive  study of weakly coupled spin -valve (SV) \nmultilayers, where the hybridizati on is weak and the layers behave almost independently, conserving at \nthe same time the main features of the acoustic and optical modes of the collective magnetic response. \nOne important objective is to separate the hybridization -induced change of the FMR linewidth from the \nspin-current -induced one and to check in this way the difference between the spin -current -induced \ndamping in the optical and acoustic regime s of precession. W e present an  experimental study of the FMR \nin NiFe/CoFe/Cu/CoFe/MnIr SV  multilayers  conducted using a standard X -band EPR spectrometer. Our \nstudy is acc ompanied by a simulation of the microwave absorption in such a magnetically coupled system \nin the presence of dynamical exchange by spin currents  in the framework of the Landau -Lifshitz -Gilbert \nformalism.  \n \n2. Experimental details  \nFMR was measured  at room temp erature using a Bruker ESP 300E E SR spectrometer at a microwave \nfrequency  of 9.67 GHz. The f irst derivative of the microwave absorp tion by the magnetic field  was \nregistered. For each sample, a series of in-plane FMR spectra were collected for different ang les of  the \nmagnetic field in the film plane with respect to the internal exchange bias  field. Each FMR spectrum , \nexperimentally measured or simulated,  was fitted by Lorentzian functions to obtain  angular dependences \nof the resonance field and linewidth.  The least -squares method was employed.  \nThe SV multilayers were grown by the ion -beam deposition in a Nordiko  3000 system.  The cobalt -\niron fixed layer is exchange coupled to the MnIr  antiferromagnet (AF), the free layer is a bilayer \ncomposed of a permalloy  and a cobalt -iron sublayers, and the copper spacer separates the free and fixed \nlayers. Two series of samples were used in the study:  \n1) Glass / Ta(30 Å) / Ni 80Fe20(30 Å) / Co 80Fe20(25 Å) / Cu ( dCu) / Co 80Fe20(25 Å) / Mn 82Ir18(80 Å) / \nTa(30 Å) – the average thickness of the copper spacer , dCu, varies from 17 to 28 Å in 1  Å steps.  \n2) Glass / Ta(30 Å) / Ni80Fe20(56 ‒ dF) / Co 80Fe20(dF) / Cu(22 Å) / Co 80Fe20(25 Å) / Mn 82Ir18(80 Å) / \nTa(50 Å)  – the relative thicknesses of the permalloy and cobal -iron sublayers var y within the \n56 Å thick free layer by setting the parameter dF to 8, 16, 24, 32 and 40  Å. \nAdditionally, separate free layers ( Glass / Ta(30 Å) / Ni80Fe20(56 ‒ dF) / Co 80Fe20(dF) / Cu(22 Å) / \nTa(50 Å)) of the first and second series were grown to serve as reference samples.  \nThe first series was already studied in Refs. [23, 24]. It has been shown that the samples with tCu > 16 Å \nare in the weak coupling regime, and the main interlayer coupling mechanism here is  Néel’s “orange -\npeel” magnetostatic interaction  [25]. When the copper spacer thickness grows from 17 to 28 Å, the 4 \n interlayer coupling energy is reduced from 1.1×102 erg/cm2 to 4×103 erg/cm2, which corresponds to a \nvariation of the effective interaction field on the free layer from  17 to 6 Oe.   \nThe second series has a fixed metallic spacer thickness, tCu = 22 Å, while the free layer effective \nmagnetization, 4π Meff, determined by the Kittel formula, gradually varies from 15 kG to 8.5 kG.  In this \nway the angular dependence of the free layer resonance field can be  vertically  shifted  with respect to that \nof the fixed layer . \n \n3. Simulation of the microwave absorption spectrum  \nA SV is considered as a system of two cou pled FM layers  consisting of a free and a fixed layer with \nthe thicknesses d1, d2, volume saturation magnetization s Ms1, Ms2, and in -plane uniaxial magnetic \nanisotropy constants K1, K2, respectively. The e xchange coupling of the fixe d layer to the AF  layer with \nthe interface coupling energy Eex is defined by a unidirectional anisotropy with the effective field \nex 2 s2E d M\n. The e asy axes of all three anisotropies lay in  the sample plane and have the same direction \nalong the magnetic field applied at annealing. The m agnetizations in  both layers are assumed  to be  \nuniform, thus the bilayer magnetic state is completel y described by the unit vectors \nˆˆ,12mm  of their \ninstantaneous directions. The layers are coupled by the Heisenberg exchange interaction , Eic.  \nThen the magnetic energy density per unit area of the considered system can be written as:  \n \n  \n\n22\ntot 1 s1 1 ext s1 mw s1\n22\ns2 2 ext s2\n2 icex\nmw s2\n2 s2ˆˆ ˆ ˆ ˆ ˆ ˆ 2\nˆ ˆ ˆ ˆ ˆ 2\nˆˆ .ˆˆˆ  U d M K H M h M\nM K H M\ndEEhMdM\n    \n   \n1 1 1 0 1 1\n2 2 2 0\n12\n2 2 2m ·n m ·û m ·h m ·h\nm ·n m ·û m ·h\nm ·m\nm ·h m ·û  (1) \nThere are also included four unit vectors determining the spatial orientation of the effective fields:  the \neasy axis \nˆûn  of the uniaxial and unidirectional anisotropies  (here \nˆn  is the normal to the multilayer \nplane) , the direction \nˆ\n0h  of the external magnetic field Hext, and the direction \n1ˆh  of the microwave \nmagnetic  field hmw.  \nThe spin -pump / spin -sink mechanism  in our SVs is considered as follows. The CoFe/Cu and \nCu/CoFe interfaces are assumed to be identical and t o give rise to an effective spin mixing conductance in \nthe FM1/NM/FM2 structure characterized  by the parameter  AFNF [26] which is in  a gener ic case \ndependent on the relative magnetization orienta tions in the layers, \nˆˆ,12mm . Since the copper spacer is \nmuch thinner than the  the spin -diffusion length ( λsd ~ 0.4 µm at T = 300  K), the transfer of the angular \nmomentum from one FM layer to the other occurs in a purely ballistic regime, i.e. the spin current emitted 5 \n at the first CoFe /Cu interface is fully absorbed at the second Cu/CoFe interface . The spin current \nbackflow is not considered separately: it just renormalizes the parameter  AFNF. The spin -pump  / spin-\ntorque induced damping \nsp  for each layer is influenced by its thickness, saturation magnetization and g-\nfactor.  The dynamics of such a structure can be described by a system of coupled Landau -Lifshitz -Gilbert \nequations with additional spin -pump / spin -torque induced Gilbert -like damping  terms [5]: \n \n1 eff sp\ntot\neff\nFNF\nsp B\nsˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ,\n1,ˆ\n,4\n, 1,2,\n.i\niis\ni\nit t t t\nU\ndM\nAgdM\nij\nij  \n                   \n\n\n\nm m m mm H m m m\nHmii\ni\nij i i i\ni i i i j\ni  (2) \nThe microwave field, \nmwˆjthe1h , is linear ly polarized and directed along the multilayer \nnormal, \n1ˆˆ||hn , while the external static magnetic field lies in the film plane,  \n0ˆˆhn , making an angle h \nwith the system’s easy axis \nû . A linear response of the system relates to small angle deviations from the \nequilibrium, \n2ˆ ˆ ˆ ˆ ,    1 1 2 1 1 2 2mδm , m δm δm m , δm m . The  complex vectors \njte12δm , δm  \ncan be found from a linear 4 ×4 system by Eqs. (2) linearized near the equilibrium . This system is too \ncomplicated for an analytical treatment but easily solved numerically using  a standard  desktop computer. \nA certain simplification can be achieved using spherical coordinates. The microwave absorption is \nproportional to the imaginary part of the microwave susceptibility in the direction of the  microwave field : \n \n 1 s1 2 s2 212\n1 2 Cu mw 1 2ˆˆ\nIm()d M d M dd\nd d d h d d        1 1 1δm h δm h . (3) \nTo treat the volume microwa ve susceptibility of a SV, the metallic  spacer width, dCu, was added in Eq. \n(3). Then a full cycle of calculations in each simulation consists of : i) finding the equilibrium orientation \nof the magnetic moments by the minimization of Eq. (1) ; ii) numerical solution of Eq. (2) linearized near \nthe equilibrium; iii) combining the obtained precession amplitudes in the volume susceptibility by Eq. (3) . \nThe separate susceptibility of each layer can be obtained if the thickness of the other layer is set to zero at \nthe last step of calculations. This can be useful in the analysis of experimental data obtained by the \nelement -specific X -ray magnetic circular dichroism, time -resolved Kerr microscopy and other techniques \nallowing to separately measure the microwave responses  of the layers [22, 27, 28]. 6 \n The m agnetic parameters in our simulations were set in accordance  to the experiment. In the studied \nsamples , the in -plane effective fields of the free and fixed layers are several times lower than the \nresonance field of the free layer ( Hres > 600 Oe), whose FMR linewidth will be the main discussion  issue  \nin the present paper. This implies that at the free layer’s resonance conditions the magnetic fi eld almost \naligns both magnetic moments. Thus, the dynamic exchange via spin currents will be considered in the \ncollinear regime , and the parameter AFNF is assumed to be independent of the in -plane magnetic field \norientation.  \n \n4. General features of the FM R in both SV series  \nThe d ynamic s of two coupled FM layers can be described in terms of acoustic and optical modes, a \nhybridized response of the system to the exciting microwave field. These modes are the in-phase and out -\nof-phase mutual precession of the magnetic moments in the FM layers.  The acoustic mode bears averaged \nmagnetic parameters of the system, while the optical one gives information about the system’s \nasymmetry. The interlayer coupling shifts the optical mode away from the acoustic one, therefor e, the \ncoupling strength can be determined if the other effective fields in the system are known. However, this is \na strong coupling regime which has few similarities with the FMR of standard SV multilayers, including \nthe samples used in this study, where the effective inte rlayer coupling does not exceed  several tens of \nOersted.   \n \n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0fixed layerfree layerspin valve\n  '', arb. units\nHext, kOeh= 0o,\nEic=10-2erg/cm2,\nhmw= 1 Oe,\n/2 = 9.67 GHz.\n \nFig. 1.  FMR spectrum calculated for a SV in the weak coupling regime (top curve). The m iddle and \nbottom curves show separated responses from the free and fixed layers in the SV. The layer  parameters \ncorrespond to the  first series of SVs: d1 = 5.5nm, α1 = 0.012, Ms1 = 1155 emu/cm3, K1 = 5.7103 \nerg/cm3; d2 = 2.5 nm, α2 = 0.055, Ms2 = 1175 emu/cm3, K2 = 1.7104 erg/cm3, Eex = 0.094 erg/cm2 and \nEic = 0.01 erg/cm2. \n 7 \n The samples  under study are in a weak coupling regime provided by Néel’s “orange -peel” \nmagnetostatic interaction [25]. The determined effective interlayer coupling field, acting from one layer to \nanother, is in the 10 to 30 Oe range for both layers [24] in all sample s of the two series. The main \ninteraction effect is a constant decrease of the resonance field in each laye r. This and other  related effects \nare thoroughly discussed in Ref.  [24].  \nTo support the ideology of the weak coupling regime, a simulation of the microwave response has \nbeen done using a parameter  set for the first series and the interlayer coupling stren gth Eic = 0.01 erg/cm2. \nThe spin -pump  / spin-sink mechanism was switched off: αsp1 = αsp2 = 0. Fig. 1 shows a typical microwave \nabsorption spectrum of a SV multilayer and respective separated responses  of each layer in it. The \nmagnetic moments are precessing almost independently , and therefore each peak can be associated with \nthe precession of the magnetization in a specific layer . The a symmetry of the thicknesses, damping  \nparameter s and magnetizations is clearly manifested  in these spectra. A fixed layer  with half the thickness \nof the free one  is much easier dragged by the precessing free layer. However , the inverse effect, a drag of \nthe free layer by the reson ance precession in the fixe d layer, is not so pronounced: only a small \nasymm etry on the wings of the free layer peak is observed. A four times strong er damping, mainly that \ndue to the contact with an antiferromagnet [24], produces a much lower precession amplitude of the fixed \nlayer. The situation gets even worse because the free layer is twice  as thick as the fixed one, t hus, the \neffective interlayer coupling field, acting on the free layer from the preces sing fixed layer, is about two \ntimes  lower. Leaping ahead, it is evid ent that the spin -pump  / spin-torque effect will be more pronounced \nin the free layer . \nA very important feature is that, despite the almost independent precession of the layers, an optical -\nlike and acoustic -like behavior is still present in the dynamics. A precessing layer drags the magnetization \nof the other layer either in the “in -phase” or in the “out -of-phase” regime . For the case of \nferromagnetically coupled layers, the optical mode (an out -of-phase mutual precession) has , in a given \nmagnetic field,  a higher precession frequen cy than the acoustic mode (an in -phase mutual precession). \nTherefore , the optical mode will be observe d, at a given microwave frequency,  in lower r esonance fields. \nA specifics of the first sample series is that, for the parallel and antiparallel orientatio ns of Hext (φh = 0º \nand 180 º), the resonance field of the fixed layer is respectively lower (~  300 Oe) or higher (~  1000 Oe) \nthan th at of the free layer (~  700 Oe in both cases). As seen from Fig. 1 , this brings  about  an interesting \nbehavior: the precession of the free layer in the parallel Hext (Hext > 0, φh = 0º) drags the fixed layer “in -\nphase”, while in the antiparallel orientation ( Hext < 0, φh = 0º) it drags the fixed layer “out -of-phase”, i.e. \nin the optical mode.   \nIt is evident that the switching between the acoustic and optical “drag” regimes would disappear with \nthe fixed layer resonance peak being  below that of the free layer. This justifies our choice of the sample \nseries: a variation of the interlayer coupling in the first series sho uld influence the intensity of the dragged 8 \n precession, while varying the effective magnetization of the free layer in the second series will tune the \nresonance field of the free layer with respect to that of the fixed one.  \nFig. 2  shows the evolution of the angular dependences of the resonance field in both series . The \ngeneral properties of the samples are as follow s. The effective field of unidirectional anisotropy for the \nfixed layer is about 300 Oe , and it is the main in -plane anisotropic contribution he re. The free layer has a \nweak in -plane unidirectional anisotropy of 5 to 20 Oe, var ying with the NiFe/CoFe composition. The \nmagnetic parameters of the free layer are less fluctuating than those of the fixed one since the former  is \nthicker and always deposi ted on the same surface. The i ncreased roughness of the fixed layer also \nstrongly influe nces the AF /FM interface, giving rise to fluctuations not only of the fixed layer ’s effective \nmagnetization but also of the exchange bias coupling. It is hard  as well  to prepare reference samples for \nthe fixed layer . Our previous investigation ha s shown that a separately deposited fixed layer has \nconsiderably different magnetic parameters [24]. The s trong angular variation of the resonance field and \nthe direct contact wi th the AF has also a strong  influence  on the angular dependence of the linewidth even \nin a separately deposited fixed layer. Moreover, as the linewidth is extracted using the least -squares \nmethod, the accuracy of the fitti ng for the low -intensity peak stemming from the fixed layer will be m uch \nlower than for the free layer . Due to  these reasons and the asymmetry discussed above, the following  \ndiscussion of the experimental results is mostly focused on the linewidth, ∆Hfr, of the free -layer -related  \npeak and on its angular dependence, ∆Hfr(φh). \n \n0 60 120 1800,40,60,81,0\nfixed layer's peaks\n  Hres, kOe\nh, deg. dCu= 28 A\n dCu= 24 A\n dCu= 21 A\n dCu= 17 Afree layer's peaks\n   \n0 60 120 1800,20,40,60,81,0\nfree layer's \npeaks\nfixed layer's\n peaks\n  Hres, kOe\nh, deg.Ni80Fe20/Co80Fe20:\n 48 A /   8A\n 40 A / 16A\n 32 A / 24A\n 16 A / 40A  \nFig. 2.  Angular dependences of the FMR peaks for the free and fixed layer: the first series where the \ninterlayer coupling strength is varied by gradual ly changing the metal  spacer thickness dCu (left panel); \nthe second series where the mean FMR  field of the free layer  is varied by gradual ly changing the free \nlayer effective magnetization, Ms1 (right panel).  \n \n5. Analysis of angular dependences  9 \n Additional reference samples  which completely duplicate the free layer and the next nearest \nnonmagnetic layers in each SV sample  have been grown and used as a reference in the analysis of the \nangular dependences of the free layer FMR linewidth, ∆Hfr(φh). It has been found that < ∆Hfr> (averaged \nover the whole φh range) of each reference sample is at least 20% lower than < ∆Hfr> in the corresponding \nSV sample. However, the increased damping in the presence of a second FM layer (i.e. fixed layer) \ncannot be uniquely associ ated with the spin -pump  / spin -sink mechanism [5, 26], because a non -zero \ninterlayer coupling causes a hybridization of the resonance modes. Though the layers are weakly  coupled, \neach layer’s resonance mode bears a small portion of the magnetic behavior of the layer coupled to it. As \nthe free layer’s damping parameter is several times lower than the fixed -layer -related  one, the observed \nFMR line broadening in the SV can have both origins , and it demands a quantitative analysis. At the s ame \ntime, the shape of the ∆Hfr(φh) dependence in the SV samples deserves  additional attention .  \n \n0 60 120 18064728088\n  \nHfr, Oe\nh, deg. dCu=28A,\n dCu=24A,\n dCu=21A,\n dCu=17A.  \nreference sample\n   \n16 18 20 22 24 26 28481216Relative step height ,  %\ndCu, Å0.011 0.0088 0.0076 0.0064 0.0052 0.01 Eic, erg/cm2\n0.004\n72747678808284\n<Hfr >, Oe  \nFig. 3.  Linewidth of the free layer in the first sample series.  Left panel: The angular dependence for \ndifferent copper spacer thicknesses. The reference sample  curve  does not show step s. Right panel: The \nrelative step height  and mean linewidth versus the interlayer coupling strength . \n \nFig. 3  shows experimental results obtained on the first series of s amples, where the interlayer \ncoupling has been gradually tuned by changing the copper spacer thickness. The reference layer does not \nshow any noticeable ∆Hfr(φh) dependence . In contrast, a step -like shape of  the ∆Hfr(φh) dependence has \nbeen observed in all  SVs. A noticeable growth of ∆Hfr is observed for the antiparal lel orientation of the \nmagnetic field (90 º < φh < 270 º). The transition from the weaker  damped to the stronger  damped regime is \nquite smooth and occurs within the angular range where the fixed layer peak crosses the free layer’s one \n(see Fig. 2 ). The relative step height in the ∆Hfr(φh) dependence has been found to decrease w ith \nincreasing  copper spacer  thickness , dCu. In other words, with decreasing interlayer coupling, assumed to \nbe the only parameter influenc ing the free layer in this series, the observed step height  also decreases. As \nseen from Fig. 3 , the relative step height monotonously decreases from 12% to  4% with decreasing \ninterlayer coupling. It should be noted that, among the other extracted SV parameters analyzed as a 10 \n function of dCu, this one has the smoothest  dependence. As an example, we show the thickness \ndependence of < ∆Hfr> averaged over the whole [0,  360º] range of angles ( Fig. 3 ). Though the scattering \nof experimental points is several times higher, this parameter also shows a tendency to decrease, whose \nnature is hard to identify at present. A degree of resonance modes hybridization, weaken ing with \ndecreasing interlayer coupling, seems to be the most probable source of this effect.  The free layer \nresonance precession drags the magnetic moment of the fixed layer , and this could be itself an additional \nsource of increased linewidth. A more det ailed discussion o f a simultaneous influence of hybridization \nand spin -pump  / spin-sink effects on the linewidth will be given in the next Section.  \n0 60 120 180556065707580859095\nNi80Fe20/Co80Fe20:  48 A /   8A,  40 A / 16A,\n 32 A / 24A,  24 A / 32A,  16 A / 40A.\n  \nH, Oe\nh, Deg.reference samples\n \nFig. 4.  Angular dependences of the linewidth for the free layer in the second sa mple series and in the \nrespective reference samples.  \nIn the second SV series , an increase of the < ∆Hfr> parameter in comparison with the reference layers \nis also clearly seen (see Fig. 4 ). At the same time, the observed step -like ∆Hfr(φh) dependence has \nrevealed  additional features. The step from the weaker  damped to strong er damped regime is shifted to \nhigher angles as the mean resonance field of the free layer get s higher. The observed shift completely \nmatches that of the crossing angle, i.e., the angle  where the resonances of the free and fixed layers \ncoincide (see Fig. 2 ). The most important feature is the absence of step -like behavior in  the ∆Hfr(φh) \ndependence for the sample with  the Ni80Fe20(48 Å) / Co 80Fe20(8 Å) free layer. Fig. 2 shows that the \nresonances are not crossing there at all: the free layer’s resonance field is always higher than the  fixed \nlayer’s one.  \nAs compared to the first series, there are also additional peculiarities in the ∆Hfr(φh) dependences, \ndistorting the step -like shape. Th ey, however, are linked to the intrinsic  angular dependence of ∆Hfr of a \nconcrete free layer. An analysis of the reference samples shows that the increase of the Co 80Fe20 / Ni80Fe20 \nthickness ratio causes a noticeable increase in the angular variation of ∆Hfr. Also a considerable variation \nof the damping parameter is observed in the reference samples , however , of a nonsystematic character . \nThese intrinsic features, as seen from Fig. 4 , are conserved also in the SV samples.  \n 11 \n  \nThus, the observed experimental results can be res umed as follows. When the fixed layer resonance \nfield is higher than the free layer’s one, the linewidth of the free layer  peak, ∆H fr, get s larger. The \nrespective angular dependence,  ∆H fr(φh), shows a step -like shape with the threshold an gular position \ncorresponding to the crossing region of the free and fixed layer resonances. The step height decreas es \nwith decreas ing interlayer coupling strength. This effect is absent in the reference samples containing \nonly the free layer , as well as it  disappears in the SVs where the resonances of the free and fixed layers do \nnot cross.  \n \n6. Hybridization versus non -local damping  \nTo clarify the interpretation of the experiment, a series of in -plane FMR spectra w ere simulated as a \nfunction of the in -plane magnetic field direction φh employing the formalism described in Sec. 3 . The \nsimulated spectr a display the resonance peaks by the free and fixed layer (as shown, e.g., in Fig. 1 ). By \nfitting a set of overlapping Lor entzians to the simulated spectrum, the resonance peaks’ parameters were \ndeduced. Then the angular dependence of the linewidth of the free layer, ∆Hfr(φh), was analyzed. For the \nfirst sample series, the layer  parameters and coupling were determined in our previous work  [24] on \nexactly the same samples. For the second series, these parameters were chosen to reproduce the \nexperiment as close as possible , and the interlayer coupling was fixed to Eic = 0.01 erg/cm2 in all SV s. \nFluctuating parameters of the fixe d layer and a slight variation of the internal damping of the free layer \n0 60 120 18066697281848790\n00 - SP\nIC - SP\nIC - 00\n  Hfr, Oe\nh, deg.00 - 00 \nFig. 5.  Simulated angular dependences of the free layer’s FMR linewidth. Four different regimes are \nshown: “00 -00”: Eic=  0 and \nFNFA  = 0; “IC -00”: Eic = 0.01 erg/cm2 and \nFNFA  = 10; “00 -SP”: Eic = 0 \nand \nFNFA  = 1.11015 cm‒2; “IC -SP”: Eic = 0.01 erg/cm2 and \nFNFA  = 1.11015 cm‒2. The layer  \nparameters refer to the first series of SVs, as they are already listed in the caption to Fig. 1 . 12 \n noted in the experiment were ignored in the simulation. In both series, the effective spin -mixing \nconductance for the whole FM/NM/FM structure is assumed  to be \nFNFA  = 1.11015 cm2 (which is \nslightly lower than in case of a single Co/Cu interface ~ 1.41015 cm2 [26]), in units of e2/h. \nRelative contributions of the hybridization and spin -pump  / spin-sink effects to the linewidth of a \nweakly coupled SV system are the central  object of th e present  investigation. Referring to a SV from the \nfirst series, we have done four different simulations (see Fig. 5 ) of the ∆Hfr(φh) dependences. First , both \nthe interlayer coupling (IC) and the spin mixing conductivity (SP) were set to zero (the “00 -00” curve). \nThis has demonstrated that the fitting procedure  correctly extracts the linewidth , and the free layer’s ∆Hfr \ndoes not depend on the peaks separa tion between the free and fixed layers (when fully uncoupled). It has \nbeen found that a small increase of ∆Hfr is observed in the crossing region. This increase, however, is \nlower than 0.3%, thus being at least one order of magnitude lower than the other f actors relevant for the \n∆Hfr(φh) dependence , both in the experiment and simulation. Therefore , this factor was ignored in the \nabove experimental data and will be omitted  in the further  consideration s.  \nThe next simulation has been made with Eic = 0.01 erg/ cm2 and \nFNFA  = 0 (the “IC -00” curve). In this \ncase, a noticeable increase (~  7%) in ∆Hfr is observed in the crossing region. This effect can be only \nattributed to an enhanced hybridization of the resonance peaks in th is region. When increasing the \nlinewidth of the free layer peak, the hybridizat ion also makes the fixed layer  peak narrower. The \ndependence of the hybridization degree on the distance between  the resonance peaks is also responsible \nfor the fact that  the ∆Hfr value for the antiparallel orientation ( φh = 180 º) is slightly higher ( by ~ 1.3%) \nthan that for the parallel orientation ( φh = 0º). As seen from Fig. 2 , the resonance peaks are indeed closer \nto each other in the antiparallel orientation . It is worth noti ng that the shape of the ∆Hfr(φh) dependence is \nquite different from the exp erimentally observed step -like profil e.  \nA pure spin -pump / spin -sink regime has been set in the next simulation, i.e. with Eic = 0 and \nFNFA  = \n1.11015 cm‒2. The corresponding ∆Hfr(φh) dependence is labeled “00 -SP”. In comparison with the \npreviously discussed regime, ∆Hfr is depressed (by ~  2%) in the crossing region. This effect was observed \nexperimentally in a FM/NM/FM system and has been interpreted as a p artial compensation of the spin \ncurrent leakage which occurs when  both FM layers are in resonance precession [5] and thus emit the spin \ncurrents. Without discussing t his in details, we note only two points: i) due to the considerably  thicker \nFM layers in our SVs , the observed effect is much weaker than in the above mentioned paper  [5]. Since \nthe spin torque effect is of interfacial origin, its influence scales with the  inverse layer thickness; ii) the \nspin-pump  / spin-sink and hybridi zation effect s work in the opposite  senses  in the crossing region.  \n \n 13 \n \n0.000 0.013 0.0266080100120\n0FNFA\n    h=00\n   h=1800\n  \nHfr, Oe\nEic, erg/cm2\n15 21.1 10 cmFNFA  \nFig. 6.  Linewidth of the free layer in the parallel and antiparallel orientation versus the interlayer \ncoupling strength simulated through spin conductiv ity (and without it). The layer  parameters are set \nfor the first series of SVs, as they are already listed in the caption to Fig. 1 . \nThe last simulation, labeled “IC -SP”, shows a simultaneous action of the interlayer coupling and sp in-\npump / spin -sink effect,  i.e. Eic = 0.01 erg/cm2 and \nFNFA  = 1.11015 cm‒2. As seen from Fig. 5 , there is a \ngood agreement with the experiment. The step size in the ∆Hfr(φh) dependence is ~  8%, also very close to \nthe experimental values. In the parallel orientation ( φh = 0º), the ∆Hfr value  is almost the same as in the \ncrossing region for the case of the pure spin -pump  / spin-sink effect. This means that a partial \ncompensation of the spin current leakage takes place in the whole range of angles for the acoustic regime \nof precession ( ‒90º < φh < 90º). On the contrary , in t he optical regime ( ‒110º > φh > 110 º) the free layer \nsuffers additional damping, absent in the previously discussed “00 -SP” simulation. The explanation is as \nfollow s. The p recession can be  geometrically separated in a transversal and a longitudinal component of \nmagnetization with respect to its equilibrium orientation. The c onservation of angular momentum allows \nthe same separation for the generated spin current. For a small -angle precession, the transversal \ncomponent of magnetization (  sin(θprec)) is larger than the longitudinal one ( sin2(θprec/2)). The \ntransversal part varies in time, while the longitudinal does not (at least in the linear response \napproximation , neglecting, e. g., a possible nutation). The i mportance of the time -dependent  transversal \npart of the spin current has been recently sho wn in Ref.  [29]. Both components are transferred by the spin \ncurrent from one FM layer to the other. In the acoustic precession mode (a s well as in the crossing point \nfor the “00 -SP” case), the transversal component of the spin current from the second layer is in -phase \nwith the transversal part of that from the first layer. Therefore , the spin current absorbed at the interface \nshould act in an “anti -damping” manne r. On the contrary, in the optical pr ecession regi me the transversal \ncomponent of the absorbed spin current is out -of-phase with the magnetic moment precession , and \ntherefore an extra damping occurs. An increase of the non-local damping in the optical precession regime \nin a magnetically  coupled FM/NM/FM trilayer has been predicted by Kim in Ref.  [19]. Probably this \neffect was observed in several  papers [20-22]. However , its interpretation in these papers fully ignores the \nhybridization of resonance modes , and therefore it is hard to draw  some clear  conclusions.  14 \n The weak interlayer coupling and an almost symmetri cal position of the free layer  peak with respect \nto the fixed one in the first SV series play an important role  in the non -local damping effect. Fig. 6  shows \nthe calculated ∆Hfr parameter versus the interlayer coupling strength for φh = 0º and φh = 180 º, with and \nwithout spin -pump  / spin-sink effect. It is seen that, for Eic < 0.013 erg/cm2, the increase of ∆Hfr occurs  \nmerely due to the non-local damping effect, while for a stronger coupling  the hybridization takes a \ncomparable role , and the se two contributions are hardly separable in a real experiment . From this \nsimulation  it is also seen that the dynamic exchang e via spin currents is quite different in the optical and \nacoustic precession modes . The i ncrease of ∆Hfr due to increasing hybridization is suppressed in the \nacoustic mode ( φh = 0º) by “anti -damping”, i.e., in-phase interaction between the transversal components \nof magnetization and the absorbed spin current. On the  contrary, in the  optical  precession  mode  (φh = \n180º) the effect of non -local damping is considerably enhanced, as the transversal components of the \nprecessing magnetization and of the absorbed spin current are out -of-phase.  \n \n0 60 120 1807580859095\n2\n15 2\nFNF0.01erg/cm ,\n1.1 10 cm .icE\nA \n\nMs1 = 1600 emu/cm3Ms1 = 1200 emu/cm3Ms1 = 1000 emu/cm3Ms1 = 800 emu/cm3 \n \nHfr, Oe\nh, deg.Ms1 = 700 emu/cm3\n   \n0 60 120 180666870727476\n  \nHfr, Oe\nh, deg. Ms1 = 700 emu/cm3\n Ms1 = 800 emu/cm3\n Ms1 = 1000 emu/cm3\n Ms1 = 1200 emu/cm3\n Ms1 = 1600 emu/cm3\n2\nFNF0.01erg/cm ,\n0.icE\nA\n  \nFig. 7.  Angular behavior of the linewidth in the second series of SVs, with a gradual variation of the \neffective magnetization of the free layer , simulated considering the  spin conductivity and without it. For \nthe red and black curves, the fixed layer resonance does  not cross that of the free layer  anymore . The \nparameters set is the same as for the first se ries and with Ms2 = 1525 emu/cm3 and Eex = 0.12 erg/cm2. \n \nA simulation of the ∆Hfr(φh) dependence in the second series, where the effective magnetization of \nthe free layer,  Ms1, is gradually changed, complete s the discussion. A comparison of the simulation ( Fig. \n7) with the experiment ( Fig. 2 , right panel) allows one to conclude that the effects of non -local  damping \nare also clearly seen here. First, when the free layer ’s saturation magnetization is such low that the fixed \nlayer pe ak does not cros s the free layer resonance, and therefor e, the precessing free layer drags the fixed \nlayer always in -phase (ac oustic mode), a characteristic step -like feature in the ∆Hfr(φh) dependence \ndisappears. I n these regime, the calculated ∆Hfr(φh) dependences are fundament ally different , \nirrespectively of whether the spi n conductivity exist s in the system or not. For the case of \nFNFA  = 0, the 15 \n fixed layer  peak approaching the free layer  one at φh = 180 º induces an enhanced hybridization , and ∆Hfr \ngrow s, while for \nFNFA  = 1.11015 cm‒2 the enhanced hybridization is fully suppressed by the described \nabove “anti -damping” feature of the acoustical mode of precession in the presence of spin conductivity. A \ndecrease of ∆Hfr is observed when the fixed layer peak is approaching. T he closer i s the fixed layer  \nresonance  to the free  layer  one, the higher is the precession amplitude in the fixed layer , and thus the \nhigher is the generated spin current. Therefore , a decrease of ∆Hfr is observed. Another distinct feature of \nthe non -local damping is  a continuous growth of the low -angle part of the ∆Hfr(φh) dependence (which \ncorresponds to the acoustical  precession mode) with decreas ing Ms1. As Ms1 decreases, all effective fields \narising from the interface , as well as the spin torque emerging from the absorbed spin current , will \nincrease. For the case of zero spin conductivity, the low -angle part of the ∆Hfr(φh) dependence remains \nalways the same. Both these features are clearly seen in  the experiment ( Fig. 2 , right panel ). \n \n7. Conclusions  \nIn-plane angular dependences of the free layer���s FMR linewidth have been studied in two series of \nspin-valve  multilayers , where the free and fixed layers are weakly coupled by N éel’s “orange peel” \nmagneto static interaction. In the first series, the interlayer coupling strength was  varied by changing the \nmetal  spacer thickness, while in the second series the in -plane resonance field of the free layer was tuned \nby changing the Ni 80Fe20/Co 80Fe20 thickness rat io.  \nThe main experimental results are as follow s. The a ngular dependence of the linewidth of the free \nlayer displays a characteristic step -like feature. When the resonance field of the fixed layer is higher than \nthat of the free layer , the damping increase s. The transition from the weakly  damped to strongly  damped \nregime occurs in the angular region of the peaks crossing. The reference  samples, containing only a free \nlayer and an adjacent nonmagnetic layer, do not show such a behavior. Similarly, no step is observed in \nthe samples from the second series , where the fixed layer  peak does not cross that of the free layer at all. \nThe step size decreases with decreasing interlayer coupling strength .  \nA comparison with simulation s has shown th at the observed effect is due to the non -local damping \neffect. In the weakly coupled regime, the hybridization of the resonance peaks is low , and each peak can \nbe attributed to the resonance precession of a particular layer. At the same time, due to a non -zero \nmagnetic coupling, the resonant precession in one layer induces a small correlated precession (“drag”) in \nthe other one. Depending on the relative fi eld position of the free layer  resonance peak with respect to the \nfixed one, the fixed layer  magnetic moment is “dragged” either in the acoustic -like (“in -phase” precession \nin both layers) or optical -like (“out -of-phase” mutual precession) regime. Therefore, varying the in -plane \nangle between the external magnetic field and the exchange bias field  and chan ging in this way the \nrelative peaks field position , one can switch between these two regimes. In case of ballistic regime of spin \ntransport, a dditionally to the time -independent longitudinal component, the spin current generated by the 16 \n dragged fixed layer has a time -varying transversal component, being “in-phase” or “out -of-phase” with \nthe time -varying transversal component of the free layer’s precessing magnetization. The resulting spin -\ntorque effect on the free layer will be either of “anti -damping” or “e xtra-damping” type, experimentally \nobservable as an additional increase/decrease of the linewidth in the antiparallel/parallel orientation. It is \nworth noting  that the acoustic regime is in a full analogy to the case of a magnetically  uncoupled \nFM1/NM/FM2 system [5] when the resonances coincide.  Another important point is  that diffusive regime \nof the spin transport will suppress the above described effects  due to averaging of transversal  components \nof the spin currents . \nOur study has also shown that the hybridization effect on the linewidth is of the same magnitude as \nthe non -local damping effect in the case of weak interlayer coupling , and that the hybridization fully \ndominates in the case of strongly coupled magnetic layer s. A separation of these two contributions, \nhowever, is possible due to  their different angula r behavior. In general case, contribution of the \nhybridization to the linewidth parameter will be dependent on degree of asymmetry of layers.  Thus, one \ncan expec t that, if the free and fixed layers would have the same damping , the influence of the \nhybridization would be considerably suppressed.  \n \nAcknowledgements  \nThis work was partially supported by the FCT of Portugal through the projects PEst/CTM/LA0025/2011, \nRECI/FIS -NAN/0183/2012, PTDC /CTM -NAN/112672/2009,  PTDC/FIS/120055/2010 , and grants \nSFRH/BPD/74086/2010 (A.A.T.)  and IF/00981/2013 (G.N.K.) as well as by the Euro pean FP7 project \n“Mold -Nanonet” .17 \n References  \n1 S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura, Spin Current  (OUP Oxford, 2012).  \n2 E. Y. Tsymbal and I. Zutic, Handbook of Spin Transport and Magnetism  (Taylor & Francis, \n2011).  \n3 S. Mizukami, Y. Ando, and T. Miyazaki, Japanese Journal of Applied P hysics 40, 580 (2001).  \n4 R. Urban, G. Woltersdorf, and B. Heinrich, Physical Review Letters 87, 217204 (2001).  \n5 B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. Bauer, Physical \nReview Letters 90, 187601 (2003).  \n6 K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, \nNature 455, 778 (2008).  \n7 Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004).  \n8 J. Wunderlich, B. Kaestner, J. Sinova, and T. Ju ngwirth, Physical Review Letters 94, 047204 \n(2005).  \n9 K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, \nNature materials 10, 737 (2011).  \n10 A. A. Bakun, B. P. Zakharchenya, A. A. Rogachev, M. N. Tkachuk, and V. G. Fle ǐsher, JETP \nLetters 40, 464 (1984).  \n11 E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Applied Physics Letters 88, 182509 (2006).  \n12 S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006).  \n13 L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buh rman, Physical Review Letters \n109, 096602 (2012).  \n14 M. Buhl, A. Erbe, J. Grebing, S. Wintz, J. Raabe, and J. Fassbender, Scientific reports 3, 2945 \n(2013).  \n15 V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. Schmitz, and \nS. O. Demokritov, Nature materials 11, 1028 (2012).  \n16 A. Brataas, Y. Tserkovnyak, G. Bauer, and B. Halperin, Physical Review B 66 (2002).  \n17 M. Weiler, et al., Physical Review Letters 111, 176601 (2013).  \n18 G. Woltersdorf, O. Mosendz, B. Heinrich, and C. Back, Physical Review Letters 99, 246603 \n(2007).  \n19 J.-V. Kim and C. Chappert, Journal of Magnetism and Magnetic Materials 286, 56 (2005).  \n20 X. Joyeux, T. Devolder, J. V. Kim, Y. G. de la Torre, S. Eimer, an d C. Chappert, J Appl Phys \n110, 063915 (2011).  \n21  . Salikhov,  .  brudan,  .  r ssing, S.  uschhorn, M. Ewerlin,  . Mishra,  .  adu, I.  . \nGarifullin, and H. Zabel, Applied Physics Letters 99, 092509 (2011).  \n22 R. Salikhov, R. Abrudan, F. Brüssing, K. Gr oss, C. Luo, K. Westerholt, H. Zabel, F. Radu, and I. \nA. Garifullin, Physical Review B 86, 144422 (2012).  \n23 A. A. Timopheev, N. A. Sobolev, Y. G. Pogorelov, S. A. Bunyaev, J. M. Teixeira, S. Cardoso, P. \nP. Freitas, and G. N. Kakazei, J Appl Phys 113, 17D7 13 (2013).  \n24 A. A. Timopheev, N. A. Sobolev, Y. G. Pogorelov, A. V. Talalaevskij, J. M. Teixeira, S. \nCardoso, P. P. Freitas, and G. N. Kakazei, J Appl Phys 114, 023906 (2013).  \n25 L. Nèel, Compt. Rend. 255, 1676 (1962).  \n26 M. Zwierzycki, Y. Tserkovnyak, P. Kelly, A. Brataas, and G. Bauer, Physical Review B 71, \n064420 (2005).  \n27 M. K. Marcham, et al., Physical Review B 87, 180403 (2013).  \n28 O. Mosendz, G. Woltersdorf, B. Kardasz, B. Heinrich, and C. Back, Physical Review B 79, \n224412 (2009).  \n29 H. Jiao and G. E. W. Bauer, Physical Review Letters 110, 217602 (2013).  \n \n \n "    },    {        "title": "1208.2211v2.Phase_Separation_in_Mixtures_of_Repulsive_Fermi_Gases_Driven_by_Mass_Difference.pdf",        "content": "Phase Separation in Mixtures of Repulsive Fermi Gases Driven by Mass Di\u000berence\nXiaoling Cuiy;\u0003and Tin-Lun Hoy;\u0003\nyDepartment of Physics, The Ohio State University, Columbus, OH 43210, USA\n\u0003Institute for Advanced Study, Tsinghua University, Beijing 100084, China\n(Dated: November 10, 2021)\nWe show that phase separation must occur in a mixture of fermions with repulsive interaction if\ntheir mass di\u000berence is su\u000eciently large. This phenomenon is highly dimension-dependent. Con-\nsequently, the density pro\fles of phase separated 3 dmixtures are very di\u000berent from those in 1 d.\nNoting that the ferromagnetic transition of a spin-1/2 repulsive Fermi gas is the equal mass limit\nof the phase separation in mixtures, we show from the Bethe Ansatz solution that a ferromagnetic\ntransition will take place in the scattering states when the interaction passes through the strongly\nrepulsive regime and becomes attractive.\nIn the last few years, there have been considerable in-\nterests in strongly repulsive Fermi gases. Many of these\nstudies were stimulated by the initial report of ferromag-\nnetism in the Fermi gas of6Li[1]. The possibility of itin-\nerant ferromagnetism was \frst proposed by Stoner for\nelectron gas[2]. The idea is that if Coulomb repulsion\nincreases faster than kinetic energy with increasing den-\nsity, as indicated by Hartree-Fock calculation, the sys-\ntem will turn ferromagnetic at su\u000eciently high densities\nto avoid repulsion at the expense of increasing kinetic\nenergy. However, Hartree-Fock approximation overesti-\nmates repulsion energy. So far, itinerant ferromagnetism\nhas not been found in metals.\nItinerant ferromagnetism had also been predicted for\nstrongly repulsive Fermi gas based on perturbative and\nmean \feld calculations[3, 4] prior to the MIT experiment\n[1]. However, such approaches are known to be unreliable\nin strongly interacting regime. In fact, later experiment\nhas not observed ferromagnetism in strongly interacting\n6Li Fermi gas[5]. It is hard to determine whether it is\ndue to the absence of Stoner ferromagnetism or that fer-\nromagnetism is superseded by severe atom loss. Still,\nStoner's idea of avoiding repulsion by tuning ferromag-\nnetic remains sound, and should apply to systems such\nas Fermi-Fermi mixtures, where the analog of ferromag-\nnetic transition (which leads to magnetic domains) cor-\nresponds to phase separation.\nPhase separation of Fermi-Fermi mixtures has been\nstudied in ref.[6] using mean \feld approximation and per-\nturbation methods. It is found that a6Li-40K mixture\nwill phase separate in the strongly interacting regime.\nSince mean \feld theory is know to be unreliable in the\nstrongly interacting regime, it raises the questions about\nwhether increasing repulsion can in fact cause a Fermi-\nFermi mixture to phase separate.\nIn this paper, we would like to point out that phase\nseparation in a Fermi-Fermi mixture can always be in-\nduced by increasing the mass ratio of the two fermion\nspecies, but not necessarily by increasing repulsion. The\nreason is that the kinetic energy cost for phase separation\ncan always be reduced to zero by increasing the mass ra-\ntio, thereby falling below the repulsion energy, renderingthe Stoner argument valid[7]. On the other hand, since\nthe density regime for strong interaction is dimension de-\npendent, the phenomena of phase separation changes sig-\nni\fcantly with dimensionality. Since the ferromagnetic\ntransition in spin-1/2 systems is the equal mass limit\nof the phase separation of Fermi mixtures, it is useful to\nunify these two phenomena in a global phase diagram as a\nfunction of mass ratio and interaction. In the 1 dcase, we\nshall also show from exact result that an \\upper-branch\"\nspin-1/2 Fermi gas will turn ferromagnetic as the system\npasses through the Tonks-Girardeau limit, i.e. when the\ncoupling constant jumps from strong repulsion to strong\nattraction. In the cases we consider, atom loss will not\nimpede the observation of phase separation.\n(A). A theorem on mass-di\u000berence driven phase sepa-\nration: A homogeneous Fermi-Fermi mixture with an ar-\nbitrary repulsion will phase separate for su\u000eciently large\nmass di\u000berence.\nFirst, let us introduce some de\fnitions. The energy\ndensityEhmof the ground state of a homogenous mixture\nof light and heavy fermions with masses ( mL,mH) and\ndensities (nL,nH) is\nEhm=EL+EH+ELG\u0012mL\nmH;n1=d\nLa;nH\nnL\u0013\n;(1)\nwhereEL(H)(nL(H)) =Adn(2+d)=d\nL(H)=mL(H)is the energy\ndensity of the ideal gas of the light (heavy) fermions, dis\nthe dimensionality, and Adis a constant. The last term\nU=ELGis the interaction energy in units of EL, and\nGis a dimensionless function of the variables displayed.\n\\a\" is the length scale associated with the interaction.\nIn 3d,ais the s-wave scattering length asin the pseudo-\npotential ^U= 2\u0019as=mP\ni>j\u000e(ri\u0000rj)\u0010\n@\n@rijrij\u0011\n, where\nrij=jri\u0000rjjfor two interacting atoms at riand rj,\nm\u00001=m\u00001\nL+m\u00001\nH, and we have set \u0016 h= 1. By applying\nharmonic con\fnement along the axial (with frequency\n!z) or the transverse ( !?) direction, the system can be\nreduced to a quasi 2 dor a quasi 1 dsystem. For quasi\n2dsystems,ais related to the binding energy as \u000fb=\n1=(2ma2), where\u000fb=A\n\u0019!zep\n2\u0019az=as,az=p\n1=(m!z) is\nthe con\fnement length and A\u00190:915[8]. For quasi 1 darXiv:1208.2211v2  [cond-mat.quant-gas]  21 Mar 20132\nsystems,a=\u0000a?\n2(a?\nas\u0000B) wherea?=p\n1=(m!?) and\nB\u00191:46[9]. In all dimensions, the energy satis\fes the\nadiabatic theorem, @Ehm=@\u0010=C=m> 0, whereCis the\ncontact.\u0010is\u00001=(2\u0019a), ln(k0a)=\u0019anda=4 respectively\nfor 3d, 2d and 1d systems and k0is an arbitrary momen-tum scale[10{13]. That we parametrize the interaction in\nterms of\u0010because it is proportional to the magnetic \feld\nin experiments that tunes the system across the strongly\ninteracting regime.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s49 /s50 /s51/s48/s49/s50/s51/s40/s76/s39/s44/s72/s39/s41/s38/s40/s76/s34/s44/s72/s34/s41\n/s49/s47/s40/s107\n/s70/s97\n/s115/s41/s76/s105/s45/s75/s40/s76/s38/s72/s41/s40/s76/s44/s72/s41\n/s32/s32\n/s40/s65/s41/s32/s51/s100\n/s48 /s53 /s49/s48/s48/s49/s48/s50/s48/s51/s48\n/s40/s68/s41/s32/s51/s100/s109\n/s76/s47/s109\n/s72\n/s72/s76\n/s40/s76/s44/s72/s41\n/s32\n/s32/s32\n/s40/s98/s41\n/s40/s97/s41/s76\n/s72/s40/s76/s44/s72/s41/s38/s76\n/s40/s76/s38/s72/s41/s40/s76/s44/s72/s41/s40/s67/s41/s32/s49/s100\n/s32/s32\n/s107\n/s70/s47/s40/s109/s103/s41\n/s32/s32\n/s40/s76/s44/s72/s41\n/s72/s40/s70/s41/s32/s49/s100/s32\n/s72/s76/s40/s100/s41/s40/s99/s41/s32\n/s32/s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s40/s76/s38/s72/s41/s40/s76/s44/s72/s41/s40/s66/s41/s32/s50/s100\n/s50 /s47/s40/s109/s103/s41\n/s32/s32\n/s48 /s53 /s49/s48/s48/s53/s49/s48\n/s40/s76/s44/s72/s41\n/s40/s69/s41/s32/s50/s100/s72/s76\n/s72\n/s32/s32\nFIG. 1. Figure 1A, 1B, and 1C are the phase diagrams for a 3 d, 2d, and 1dFermi-Fermi mixture with NL=NHin a volume\nV.kF= (6\u00192n)1=3and\u0019nrespectively for 3 dand 1d, withn=NL=V=NH=V. To the right of the vertical blue line, the\nmean-\feld interaction energy is less than half of total kinetic energy for a homogenous mixture (L,H), and the system is weakly\ninteracting deeper in that region. The gray dashed-dot lines indicate the case of a Li-K mixture with mL=mH= 6=40. Figure\n1D, 1E, and 1F are the phase diagrams of a 3 d, 2d, and 1dLi-K mixture in chemical potential plane for weak interactions.\n\u0016H; \u0016Lare scaled by ( mdg2)1=(2\u0000d)in 3d and 1d, and 1 =min 2d. The red dashed-dot lines in 1D and 1F represent trajectories\nfor the density pro\fles of a trapped system, corresponding to (a-d) in Fig.2, with the squares denoting the chemical potentials\nat the trap center. From Figure 1A to 1F, the black (orange) solid lines represent the 1st (2nd)-order boundaries with (without)\ndensity discontinuity. In Figure 1B, the boundary is given by the function gc(mL;mH) = 2\u0019=pmLmH. In 1E, the two solid\norange lines are the boundaries for interaction g < gc, with two slopes ( g=gc)p\nmH=mLand (gc=g)p\nmH=mLrespectively.\nWheng\u0015gc, the two boundaries merge into one (shown by dashed line) with slopep\nmH=mL.\nProof of the Theorem: Consider a system with NL\nandNHfermions in a volume V, we de\fne\nmL=mH\u0011x; NH=NL\u0011\r; (2)\nthe total energy of the homogenous mixture is\nEhm=VEL(nL) (1 +\r\u000bx+G); \u000b = 1 + 2=d: (3)\nNext, we consider the fully phase separated state. Let VH\nandVLbe the volumes of the heavy and light fermions,\nVH+VL=V. The ratio VH=VLis determined by equat-\ning the pressure Pof these two separated gases. Since\nthe pressure of an ideal gas is proportional to its energydensity,P= 2E=d, we haveEL(n0\nL) =EH(n0\nH), where\nn0\nH(L)=NH(L)=VH(L). This gives VH=VL=\rx1=\u000b.\nThe total energy of the phase separated state is EPS=\nVHEH(n0\nH) +VLEL(n0\nL) =VEL(nL)(V=VL)(2+d)=d, or\nEPS=VE(nL)\u0010\n1 +\rx1=\u000b\u0011\u000b\n: (4)\nThe phase separated state will have lower energy if Ehm\u0000\nEPS>0, or\nI(x) =G(x)\u0000h\n(1 +\rx1=\u000b)\u000b\u00001\u0000\r\u000bxi\n>0:(5)\nWhen the mass ratio is su\u000eciently small such that3\n\r1=\u000bx\u001c1, hencex\u000b<x, Eq.(5) becomes\nI(x) =G(0)\u0000\u000b\rx1=\u000b+O(x;x2=\u000b)>0; (6)\nwhereG(0)>0 is the repulsive interaction energy in the\nlimit when mH!1 [14]. Eq.(6) can always be satis-\n\fed for su\u000eciently small x, hence phase separation must\noccur for su\u000eciently large mass di\u000berence. Q.E.D .\nCorollary: Because of the adiabatic theorem, if a mix-\nture with mass ratio mL=mHphase separates at a given\ninteraction parameter \u0010, it will continue to phase sepa-\nrate at stronger interactions, i.e. at a larger \u0010.\n(B). Phase diagram: To demonstrate the e\u000bect of\nmass-imbalance on phase separation, we shall construct\nthe phase diagram as a function of interaction and mass\nratio. To obtain results with certainty, we consider a\nhomogeneous Fermi-Fermi mixture of weakly repulsion.\nIn this case, mean \feld approach is valid. The energy\ndensityEhm, the pressure P, and the chemical potential\n(\u0016L;\u0016H) for light and heavy particles are given accu-\nrately by\nEhm(nL;nH) =EL(nL) +EH(nH) +gnLnH;(7)\n\u0016L(H)(nL;nH) =@EL(H)(nL(H))\n@nL(H)+gnH(L); (8)\nP(nL;nH) =\u0016LnL+\u0016HnH\u0000E(nL;nH):(9)\nwheregis the interaction constant, g=2\u0019as\nmin 3d,\n2p\u0019\nmas\nazin quasi 2d, and2\nmas\na2\n?in quasi 1d. While we\nuse the same mean \feld approach as in ref.[6], our ideas\nare very di\u000berent. We goal is to show phase separation\nmust occur at su\u000eciently large mass ratios, even though\nthe system is weakly interacting. We therefore only draw\nconclusions in the weakly interacting regime and do not\nextend our results to strong interacting regions.\nTo derive the phase diagram, we consider a system\nwithNLlight fermions and NHheavy fermions in a vol-\numeV. The possible equilibrium con\fgurations are: ( a)\nfully phase separated state (PS), denoted as ( L&H); (b)\ncoexistence of a homogenous mixture and a single phase,\ndenoted as ( L;H)&Lor (L;H)&H; (c) coexistence of two\nhomogeneous mixtures with di\u000berent densities ( n0\nL;n0\nH)\nand (n00\nL;n00\nH), denoted as ( L0;H0)&(L00;H00); and (d) a\nsingle homogenous mixture ( L;H). To determine the\npresence of these phases, it is su\u000ecient to consider the\ngeneral case ( L0;H0)&(L00;H00), which covers all other\ncases. For example, the state ( L&H) corresponds to\nn0\nH=n00\nL= 0. The state ( L;H)&Lcorresponds to n00\nH=\n0, and the state ( L;H) corresponds to n00\nL=n00\nH= 0.\nLet (N0\nL;N0\nH) and (N00\nL;N00\nH) be particle numbers of\nthe mixtures ( L0;H0) and (L00;H00), andV0andV00be\ntheir volumes respectively. The equilibrium con\fgura-\ntion is obtained by minimizing the total energy with re-\nspect to these particle numbers and volumes, subject to\nthe constraint N0\nL+N00\nL=NL,N0\nH+N00\nH=NH; and\nV0+V00=V. The evolution of this equilibrium state as afunction of mass ratio and interaction strength yields the\nphase diagram. Figure 1A, 1B, and 1C show the phase\ndiagrams for a 3 d, 2d, and 1dmixture with NL=NHin\na volumeV. For both 1 dand 3d, there is a range of mass\nratio (for given interaction) in which the system consists\nof two di\u000berent phases in equilibrium, (( L;H)&Lfor 1d\nand (L0;H0)&(L00;H00) for 3d). This feature is absent in\n2d[15]. For all dimenson, the system is fully phase sep-\narated in the weakly interacting regime for su\u000eciently\nlarge mass di\u000berence. In this regime, atom loss will be\nstrongly suppressed[16] and will not hinder the observa-\ntion of Stoner instability.\nNote that the phase boundaries shown in Figure 1A\nto 1C are inaccurate in the strongly interacting region,\nsince they are derived from the mean \feld expressions\nEqns.(7), (8) and (9). However, the corollary in section\n(A) guarantees that the system will phase separate in\nthe strongly interacting regime over a range of mass ratio\nwider than that in the weakly interacting regime.\n(C) Ferromagnetic transition of 1dspin-1/2 Fermi gas:\nThe phase diagram for 1 dFermi-Fermi mixture is not\nonly constraint by the results in the weakly interact-\ning regime, but also by the exact Bethe Ansatz solution\nalong the line mL=mH= 1[17], which is a spin-1/2 repul-\nsive Fermi gas with interaction gP\ni>j\u000e(xi\u0000xj), where\ng=\u00004(m\u0010)\u00001. Because of the integrability of this sys-\ntem, there are two classes of eigenstates: one where all\nquasi-momenta are real, i.e., all particles are in scatter-\ning states, (denoted as class (i)), and one that contains\nat least one pair complex conjugate quasi-momenta, i.e.\nwith at least one fermion bound pair, (denoted as class\n(ii)). Repulsive Fermi gas, which falls into class (i), is\nreferred to as in the \\upper branch\"; since it is a many-\nbody eigenstate , it will not decay into class (ii)[18].\nExperimentally, one can tune the system from weak\nto strong repulsion ( \u0010= 0\u0000; g\u00001= 0+), and then to\nstrongly attraction ( \u0010= 0+; g\u00001= 0\u0000). The regime\nwhereg\u00001= 0+will be referred to as the Tonk-Girardeau\n(TG) regime. The ground state of a repulsive ( \u0010 <0 )\nspin-1/2 Fermi gas with equal spin population is a spin-\nsinglet according to the Lieb-Mattis theorem[19]. In the\nTG limit, the spatial wavefunction of the ground state is\nidentical to that of a fully spin polarized Fermi gas up to\na sign (which changes in various regions in con\fguration\nspace). As a result, its energy E(0) is given by that of a\nfully spin polarized state with huge spin degeneracy[17] {\nall spin con\fgurations including the spin con\fgurations\n(a) to (c) mentioned above are degenerate, with HandL\nnow labeling the two spin species. This means that the\ntwo phase boundaries in Fig.1C will converge to the equal\nmass point mL=mH= 1 at resonance. Crossing the TG\nlimit to the attractive side, the energies of all spin states\ncontinue to increase according to the adiabatic theorem,\nhenceE(\u0010 >0)>E(0); except for the largest spin state\nwhich remains at E(0) regardless of interaction. As a\nresult, the system will make transition to this maximum4\nspin state. In practice, such transition can be facilitated\nby the presence of small magnetic \feld gradients that\ndestroy spin conservation. It is useful to note that atom\nloss in the TG regime is vanishing small[20], and therefore\nwill not a\u000bect the observation of ferromagnetism.\n/s48 /s53 /s49/s48 /s49/s53/s48/s49/s50/s51\n/s48 /s53 /s49/s48 /s49/s53/s48/s49/s50/s51\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s49/s50/s51\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s49/s50/s51/s32/s110\n/s72/s47/s110\n/s76/s40/s48/s41\n/s32/s110\n/s76/s47/s110\n/s76/s40/s48/s41/s40/s97/s41\n/s32/s32\n/s32/s40/s98/s41\n/s32/s32\n/s40/s100/s41\n/s32/s32\n/s114/s47/s97\n/s104/s111\n/s32/s32\n/s114/s47/s97\n/s104/s111/s32/s40/s99/s41\n/s32/s32\nFIG. 2. Density pro\fles of a trapped Li(Light)-K(Heavy)\nmixture in 3d((a) and (b)) and 1d((c) and (d)), corresponding\nto the trajectories (a) to (d) in Fig.1D and 1F. The densities\n(nH;nL) have been normalized by n(0)\nL, the density of light\natoms(Li) at the trap center for non-interacting system in the\nsame setup. The position ris scaled by aho=p\n1=(mL!L),\nthe con\fnement length of light atoms. (a) and (b) are with\nthe same particle numbers ( NH;NL) = (105)(1:47;6:9) and\nthe same trapping frequency ratio !H=!L= 0:3, but with\ndi\u000berent interaction strengths a=aho= 0:055(a);0:065(b).\n(c) and (d) are with the same ( NH;NL) = (104)(1:45;0:51)\nand the same interaction \u0000aho=a= 15\u0019, but with di\u000berent\n!H=!L= 0:5(c);0:2(d).\n(D) The density pro\fle in a trap: The density pro-\n\fles of heavy and light atoms in a trap can be obtained\nfrom the equation of state nL(H)(\u0016L;\u0016H) using standard\nlocal density approximation (LDA). Since the equation\nof state depends on the nature of the equilibrium phase,\none needs to \frst determine its nature as a function of\nchemical potentials ( \u0016L;\u0016H).\nFor given (\u0016L;\u0016H), three phases are possible : the sin-\ngle component Fermi gas ( L), (H), and the homogenous\nmixture (L;H). To express the pressure of a homogenous\nmixturePhmas a function of \u0016L;\u0016H, we invert Eq.(8) to\nobtainnLandnHas a function of \u0016Land\u0016H, and then\nsubstitute them into Eq.(9). The pressure of ( L) or (H)\nisPL(H)(\u0016L;\u0016H) =Bdmd=2\nL(H)\u00161+d=2\nL(H), whereBdis a con-\nstant. The phase boundary for the full phase separationisPL(\u0016L) =PH(\u0016H), or\n\u0016H=\u0016L=\f; \f = (mL=mH)d=(d+2): (10)\nThe phase boundary between the mixture ( L;H) and\nL(or (H)) is obtained by equating Phm(\u0016L;\u0016H) =\nPL(H)(\u0016L(H)). The phase boundaries for the 3 d, 2d, and\n1dmixtures are shown in Figure 1D, 1E, and 1F respec-\ntively. Within the region of homogenous mixture, the in-\nversion of Eq.(8) may yield several solutions of densities\n(say, (n0\nL;n0\nH), (n00\nL;n00\nH)) for given chemical potentials\n(\u0016L;\u0016H). The thermodynamic state is given by the one\nwith highest pressure. In the 3 dcase, the homogeneous\nmixture is contained within the \\bubble\" in Figure 1D.\nWithin this region, the thermodynamic state is unique\nexcept on the line that is an extension of the bound-\nary Eq.(10) where two states (with densities ( n0\nL;n0\nH),\n(n00\nL;n00\nH)) have identical chemical potential and pressure.\nThis is a line of \frst order transition. Furthermore, the\ndensities of these two phases are related as n0\nL=\fn00\nH,\nn0\nH=\f\u00001nL, since Eq.(7) to (9)) are invariant under\nthis change. The density discontinuities across this line\n\u0001nL=\fnH\u0000nL, \u0001nH=\f\u00001nL\u0000nHthen has the\nratio \u0001nL=\u0001nH=\u0000\f.\nIn Fig.2a to 2d, we show the density pro\fles of the\n3dand 1dmixtures in a trap obtained by applying\nLDA to the equation of state nL(H)(r) =nL(H)(\u0016L\u0000\nVL(r);\u0016H\u0000VH(r)), whereVL(H)(r) =mL(H)!2\nL(H)r2=2\nare the harmonic potentials experienced by the light(L)\nand heavy(H) particles. Moving from the center of the\ntrap to the surface of the cloud corresponds to following\nthe trajectories indicated in Fig.1D and 1F. Fig.2a and\n2b show the density pro\fles of a 3 dmixture at di\u000berent\ninteraction strengths. The discontinuities in the densities\nobey the related mentioned above. Fig.2c and 2d show a\n1dmixture under di\u000berent trapping potentials.\nTwo features of the density pro\fles should be empha-\nsized. Firstly, the density pro\fles of a 3 dmixture di\u000ber\nsigni\fcantly from that of the 1 dmixture, (see Fig.1D\nand 1F). Phase separation takes place in the outer part\nof the atom cloud in 1 dbut in the inner part in 3 d.\nThis is because the strongly interacting regime occurs\nin the low (high) density region in 1 d(3d). Secondly,\nin Fig. 2a-2d, we note that nL(H)can increase with\nr. This is di\u000berent from the single component case,\nwheredn=dr < 0, due to the fact that dn=d\u0016 > 0\nas demanded by thermodynamic stability. In the mix-\nture case, stability against density \ructuation requires\nDet(M)>0, whereMij=@\u0016i=@nj, andi;j=L\nandH. We then have dni=dr= (M\u00001)ijd\u0016j=dr, where\nM\u00001= Det\u00001(M)\u0012AH\u0000g\n\u0000g AL\u0013\n,AL(H)=@\u0016L(H)\n@nL(H)>0.\nThatdnL(H)=drcan be positive or negative is because it\nis made up of two terms. If dnL=dr> 0, it is easily shown\nfrom stability condition ( ALAH>g2) thatdnH=dr< 0.\nThus one can have at most one species with a positive\ndensity derivative.5\nConclusion. We have shown that the Stoner instabil-\nity (phase separation) can be driven by large mass dif-\nference of Fermi-Fermi mixtures, but not necessarily by\nstrong repulsions. In all dimensions, phase separation\nwill occur for su\u000eciently large mass di\u000berence even in\nthe weak interacting regime. Furthermore, we point out\nthat the Bethe Ansatz solution implies a Stoner instabil-\nity of the 1d spin-1 =2 fermions across the TG limit, which\ninn turn allows one to constrain the phase diagram of 1d\nFermi-Fermi mixtures. In the cases we consider, atom\nloss would be suppressed and will not a\u000bect observation\nof Stoner ferromagnetism in experiments.\nXC acknowledges the support of NSFC under Grant\nNo. 11104158, and Tsinghua University Initiative Scien-\nti\fc Research Program. TLH acknowledges the support\nby NSF Grant DMR-0907366 and by DARPA under the\nArmy Research O\u000ece Grant Nos. W911NF-07-1-0464,\nW911NF0710576, by the Institute for Advanced Study\nof Tsinghua University through the Qian-Ren Program.\n[1] G.-B. Jo, Y.-R. Lee, J.-H. Choi, C. A. Christensen, T. H.\nKim, J. H. Thywissen, D. E. Pritchard and W. Ketterle,\nScience 325, 1521 (2009).\n[2] E. Stoner, Philos. Mag. 15, 1018 (1933).\n[3] R. A. Duine and A. H. MacDonald, Phys. Rev. Lett. 95,\n230403 (2005).\n[4] L. J. LeBlanc, J. H. Thywissen, A. A. Burkov and A.\nParamekanti, Phys. Rev. A 80, 013607 (2009).\n[5] C. Sanner, E. J. Su, W. Huang, A. Keshet, J. Gillen and\nW. Ketterle, Phys. Rev. Lett. 108, 240404 (2012).\n[6] C. W. vonKeyserlingk and G. J. Conduit, Phys. Rev. A\n83, 053625 (2011).\n[7] In contrast, in a lattice, even when the band mass of one\nspecies is in\fnite, the kinetic energy cost for phase sepa-ration is non-zero because the volume of heavy fermions\ncan not be compressed to zero. As a result, phase separa-\ntion can only be activated by su\u000eciently large repulsion.\nSee J. K. Freericks, E. H. Lieb and D. Ueltschi, Phys.\nRev. Lett. 88, 106401 (2002); D. Ueltschi, J. Statist.\nPhys. 116, 681 (2004) and references therein.\n[8] D.S. Petrov, M. Holzmann and G.V. Shlyapnikov, Phys.\nRev. Lett. 84, 2551 (2000).\n[9] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).\n[10] S. Tan, Ann. Phys. 323, 2971 (2008).\n[11] F. Werner and Y. Castin, Phys. Rev. A, 86,\n013626(2012); M. Valiente, N. T. Zinner and K. M\u001clmer,\nPhys. Rev. A, 84, 063626 (2011).\n[12] M. Barth and W. Zwerger, Ann. Phys. 326, 2544 (2011).\n[13] M. Valiente, N. T. Zinner and K. M\u001clmer, Phys. Rev. A,\n86, 043616 (2012).\n[14] SincementersEin the combination as=m, we have\n@E=@m < 0. Combining with Eq.(3), we have G(x)>\nG(0)\u0000\r\u000bx, which in turn implies the correction to lin-\near order in xin Eq.(6) vanishes.\n[15] For 2nd-order phase transitions, the phase boundaries\ncan be obtained by \\polaron\" method, see X. Cui and H.\nZhai, Phys. Rev. A 81, 041602(R) (2010) and P. Massig-\nnan and G. M. Bruun, Eur. Phys. J. D 65, 83 (2011).\n[16] D. S. Petrov, Phys. Rev. A 67, 010703(R) (2003).\n[17] L. Guan and S. Chen, Phys. Rev. Lett. 105, 175301\n(2010).\n[18] In quasi 1d systems, despite the lack of integrability, a\nmetastable upper branch is well-de\fned in the framework\nof high-temperature Virial expansions, see X. Cui, Phy.\nRev. A 86, 012705 (2012).\n[19] E. H. Lieb and D. Mattis, Phys. Rev. 125, 164 (1962).\n[20] In the TG limit, the atom loss of two-component fermions\nis strongly suppressed as in hard-core bosons[21, 22], as\nboth systems behave equivalent to spinless fermions.\n[21] D. M. Gangardt and G.V. Shlyapnikov, Phys. Rev. Lett.\n90, 010401 (2003).\n[22] E. Haller, M. Rabie, M. J. Mark, J. G. Danzl, R. Hart,\nK. Lauber, G. Pupillo and H.-C. N agerl, Phys. Rev. Lett.\n107, 230404 (2011)."    },    {        "title": "1010.1075v1.Ferromagnetic_Resonance_in_Spinor_Dipolar_Bose__Einstein_Condensates.pdf",        "content": "arXiv:1010.1075v1  [cond-mat.quant-gas]  6 Oct 2010Ferromagnetic Resonance in Spinor Dipolar Bose–Einstein C ondensates\nMasashi Yasunaga and Makoto Tsubota∗\nDepartment of Physics, Osaka City University, Sumiyoshi-k u, Osaka 558-8585, Japan\n(Dated: April 27, 2022)\nWe used the Gross–Pitaevskii equations to investigate ferr omagnetic resonance in spin-1 Bose–\nEinstein condensates with a magnetic dipole-dipole intera ction. By introducing the dipole interac-\ntion, we obtained equations similar tothe Kittel equations used torepresent ferromagnetic resonance\nin condensed matter physics. These equations indicated tha t the ferromagnetic resonance originated\nfrom dipolar interaction, and that the resonance frequency depended upon the shape of the conden-\nsate. Furthermore, spin currents driven by spin diffusions a re characteristic of this system.\nPACS numbers: 03.75.Mn, 03.75.Nt\nI. INTRODUCTION\nMagnetic resonance (MR) as a physical concept has\nbeen applied in various fields, enabling physical, chem-\nical, and medical experiments to obtain information on\nnuclear spin and electron spin systems. The concept has\nalso provided valuable information to help understand\nthe unknown structures of many condensed matter sys-\ntems [1].\nThe use of MR in the study of ferromagnets, e.g.\nNickel, Cobalt, and Iron, began in the 1940s. Grif-\nfiths observed that the Land´ e’s g-factor of electrons in\nferromagnets was far from the well known value, 2 [2].\nIn order to understand these anomalous results, Kittel\ntheoretically introduced a demagnetizing field into the\nequation representing the motion of the magnetization\nM= (Mx,My,Mz), obtaining an equation valid in an\nexternal magnetic field H0ˆz, withMz0=H0/Nzand de-\nmagnetizing fields [3], thereby obtaining the Kittel equa-\ntion,\ndM\ndt=γn[M×H]. (1)\nHere,γnis the nuclear gyromagnetic ratio, and H=\n(−NxMx,−NyMy,H0−NzMz) is given by the demagne-\ntizing factors Ni. By linearizing the magnetization M=\nM0+δMfromthestationarymagnetization M0=Mz0ˆz,\nKittel obtained a precession of the magnetization and a\nprecessing frequency, i.e.resonance frequency,\nω2=γ2\nn{H0+(Ny−Nz)Mz0}{H0+(Nx−Nz)Mz0},(2)\nwhich explained the anomalous g-factor. Furthermore,\nhe found that the resonance frequency depends on the\nshape of a ferromagnet because Nidepends on the shape\n[3]. Thus, ferromagnetic resonance (FMR) was estab-\nlished, and the workenabled numerousadditionalstudies\n[4].\nMR also plays an important role in quantum conden-\nsate systems. In superfluid3He, the dynamics of the spin\n∗Electronic address: tsubota@sci.osaka-cu.ac.jpvector and the d-vector are represented by the Leggett\nequation, which couples these vectors through magnetic\ndipole-dipole interactions [5]. The equation also shows\nnot only an MR typical of condensed matter, but also a\nnew MR that cannot be described using the equations of\nmotion for general paramagnets and ferromagnets. This\nMR was used to find AandBphases [6]. Parallel ring-\ning, which is an oscillation of longitudinal magnetization,\nwas also observed [7].\nSince the discovery of atomic Bose–Einstein conden-\nsates(BECs)[8,9], BECshavebeenstudiedinopticsand\natomic and condensed matter physics. We have intro-\nducedMRintoBECstorealizemagneticresonanceimag-\ning, a popular method of nondestructive testing. Spinor\nBECs are expected to be suitable for MR, since they\nhave not only internal degrees of freedom but also mag-\nnetic properties. In particular, we are interested in mag-\nnetic dipole-dipole interactions (MDDI) in spinor BECs,\nwhich have been actively studied. The interaction be-\ntween spins has a characteristic symmetry of rotation\nand spin, which is expected to result in a new quan-\ntum phase [10–12] and Einstein–de Haas effects [13]. Ex-\nperimentally, Griesmaier et al.realized spinor dipolar\ncondensates using52Cr atoms, which have a larger mag-\nnetic moment than alkali atoms [14]. The shape of the\ncondensates clearly represented the anisotropy of the in-\nteraction [15, 16]. Thus, MDDI has opened new areas of\nspinor condensate research.\nAs an introduction to MR in BECs, we numerically\nstudied spin echo in dipolar BECs with spin-1 [19]. The\nspin echo is a typical phenomenon of MR, discovered by\nHahn [17] and developed by Carr and Purcell [18]. Previ-\nously, we calculated the transition from Rabi oscillations\nto internal Josephson oscillations in spinor condensates\n[20]. In this paper, we consider MDDI in spin-1 BECs,\nexamining FMR by analyzing the Gross–Pitaevskii (GP)\nequations.\nIn section II, we derive Kittel-like equations from the\nGP equations, and analyze them. In section III, using\na single-mode approximation, we derive Kittel equations\nfrom the Kittel-like equations. The MDDI of the Kittel\nequationsisconsideredasthe originofthe demagnetizing\nfield, which is phenomenologically introduced in Eq. (1).\nIn section IV, we numerically solve the GP equations,2\nobtaining resonance frequencies that depend upon the\nshape of the condensates, and spin currents driven by\nspin diffusion which is given by the MDDI. Finally, Sec.\nV is devoted to our conclusions.\nII. FORMULATION\nIn this section, we derive the equations of motion for\nspinsfromthespin-1GPequationswithanexternalmag-\nnetic field and an MDDI [19].\ni/planckover2pi1∂ψα\n∂t=/parenleftbigg\n−/planckover2pi12\n2M∇2+V−µ+c0n/parenrightbigg\nψα\n−gµBHiFi\nαβψβ+c2FiFi\nαβψβ\n+cdd/integraldisplay\ndr′δij−3eiej\n|r−r′|3Fi(r′)Fj\nαβψβ.(3)\nHere,Vis the trapping potential, µis the chemical\npotential, and the total density n=/summationtext\niniis given\nbyni=|ψi|2. The external magnetic field is H=\n(Hx,Hy,Hz), and the components Fi\nαβof the spin ma-\ntricesˆFiare for spin-1. The interaction parameters are\nc0= (g0+2g2)/3 andc2= (g2−g0)/3 forgi= 4π/planckover2pi12ai/M\nrepresented by s-wave scattering lengths ai. The dipolar\ncoefficient is cdd=µ0g2\neµ2\nB/4π, and the unit vector is\ne= (ex,ey,ez) = (x−x′,y−y′,z−z′)/|r−r′|.\nUnder the homogeneous magnetic field H=Hˆz, the\nequations can be rewritten as,\ni/planckover2pi1∂ψ1\n∂t=/parenleftbigg\n−/planckover2pi12∇2\n2M+V−µ+c0n/parenrightbigg\nψ1−gµBHψ1\n+c2{(n1+n0−n−1)ψ1+ψ∗\n−1ψ2\n0}+D1,\n(4a)\ni/planckover2pi1∂ψ0\n∂t=/parenleftbigg\n−/planckover2pi12∇2\n2M+V−µ+c0n/parenrightbigg\nψ0\n+c2{(n1+n−1)ψ0+2ψ∗\n0ψ1ψ−1}+D0,\n(4b)\ni/planckover2pi1∂ψ−1\n∂t=/parenleftbigg\n−/planckover2pi12∇2\n2M+V−µ+c0n/parenrightbigg\nψ−1+gµBHψ−1\n+c2{(n−1+n0−n1)ψ−1+ψ∗\n1ψ2\n0}+D−1.\n(4c)\nThese dipolar terms are represented as,\nD1=/parenleftbiggψ0√\n2d−+ψ1dz/parenrightbigg\n,\nD0=/parenleftbiggψ1√\n2d++ψ−1√\n2d−/parenrightbigg\n,\nD−1=/parenleftbiggψ0√\n2d+−ψ−1dz/parenrightbigg\n,\nwith the integrations d±=dx±idyanddzgiven by,\ndi=cdd/integraldisplay\ndr′Fi(r′)\n|r−r′|3{1−3ei/summationdisplay\njej}.(5)The spin density vectors Fiare defined as,\nFx=Ψ†ˆFxΨ\n=/planckover2pi1√\n2{ψ∗\n0(ψ1+ψ−1)+ψ0(ψ∗\n1+ψ∗\n−1)},(6a)\nFy=Ψ†ˆFyΨ\n=i/planckover2pi1√\n2{ψ∗\n0(ψ1−ψ−1)−ψ0(ψ∗\n1−ψ∗\n−1)},(6b)\nFz=Ψ†ˆFzΨ=/planckover2pi1(|ψ1|2−|ψ−1|2). (6c)\nHere,Ψ= (ψ1,ψ0,ψ−1)Tis the spinor wave function.\nDifferentiating Eq. (6) with respect to time and utiliz-\ning Eq. (4), we can obtain the Kittel-like equation,\n∂F\n∂t=K+γe[F×Heff] (7)\nwith the gyromagnetic ratio γe=gµB//planckover2pi1of an electron.\nThe first term K= (Kx,Ky,Kz) becomes,\nKx=/planckover2pi1\n2Mi1√\n2{(ψ1+ψ−1)∇2ψ∗\n0−ψ∗\n0∇2(ψ1+ψ−1)\n+ψ0∇2(ψ∗\n1+ψ∗\n−1)−(ψ∗\n1+ψ∗\n−1)∇2ψ0},\nKy=/planckover2pi1\n2Mii√\n2{(ψ1−ψ−1)∇2ψ∗\n0−ψ∗\n0∇2(ψ1−ψ−1)\n−ψ0∇2(ψ∗\n1−ψ∗\n−1)+(ψ∗\n1−ψ∗\n−1)∇2ψ0},\nKz=/planckover2pi1\n2Mi(ψ1∇2ψ∗\n1−ψ∗\n1∇2ψ1\n+ψ∗\n−1∇2ψ−1−ψ−1∇2ψ∗\n−1).\nThe effective magnetic fields Heff=H+Hdd=\n(Hx\neff,Hy\neff,Hz\neff)consistoftheexternalmagneticfieldand\nthe dipolar field Hdd, given by,\nHx\neff=−cdd\ngµBdx,\nHy\neff=−cdd\ngµBdy,\nHz\neff=H−cdd\ngµBdz.\nNote that Eq. (7) does not depend on spin exchange\ninteraction, which refers to the second term with c2in\nEq. (3). Generally, the interaction affects a spin through\nthe effective magnetic fields of the other spins. However,\nexchange interaction does not appear in Heff. Therefore,\nthe isotropic exchange interaction does not affect MR in\nthese condensates.\nWe can redefine Eq. (7) as,\n∂Fk\n∂t=/planckover2pi1\n2Mi∇2Fk−∇·jk+γe[F×Heff]k,(8)\nwhere,\njx=/planckover2pi1√\n2Mi(ψ∗\n0∇(ψ1+ψ−1)+(ψ∗\n1+ψ∗\n−1)∇ψ0),\njy=/planckover2pi1√\n2M(ψ∗\n0∇(ψ1−ψ−1)−(ψ∗\n1−ψ∗\n−1)∇ψ0),\njz=/planckover2pi1\nMi(ψ∗\n1∇ψ1−ψ∗\n−1∇ψ−1).3\nThe equation of motion (8) for spins describes the prop-\nerties of spin dynamics in a ferromagnetic fluid. The\nfirst, second, and third terms of Eq. (8) represent spin\ndiffusion, spin current, and spin precession around Heff,\nrespectively.\nComparing Eq. (8) with Eq. (1), we noticed several\ndifferences. First, Eq. (8) was directly derived from\nthe GP equations, whereas Eq. (1) is a phenomenologi-\ncal equation of magnetization. The spin density vectors\nin Eq. (8) are microscopically affected by other spins\nthrough the dipolar fields in the effective magnetic fields.\nOn the other hand, the magnetization in Eq. (1) is af-\nfected by demagnetizing fields originating from macro-\nscopicallypolarizedmagnetizationin the condensed mat-\nter. Namely, Eq. (8) can describe the macroscopic de-\nmagnetizing field resulting from the microscopic dipolar\nfield. This is a very important difference between these\nequations.\nWe initially investigated the physics of the first and\nsecond terms of Eq. (8). To simplify the discussion, we\nconsidered the equation under the condition Heff=0.\nThus, we derived the continuity equations,\n∂Fi\n∂t+∇·Ji= 0, (9)\nwhereJk=jk−/planckover2pi1/(2Mi)∇Fkisaneffectivecurrentterm,\nJx=−i/planckover2pi12\n2√\n2M{ψ∗\n0∇(ψ1+ψ−1)+(ψ∗\n1+ψ∗\n−1)∇ψ0\n−ψ0∇(ψ∗\n1+ψ∗\n−1)−(ψ1+ψ−1)∇ψ∗\n0},(10a)\nJy=/planckover2pi12\n2√\n2M{ψ∗\n0∇(ψ1−ψ−1)−(ψ∗\n1−ψ∗\n−1)∇ψ0\n+ψ0∇(ψ∗\n1−ψ∗\n−1)−(ψ1−ψ−1)∇ψ∗\n0},(10b)\nJz=−i/planckover2pi12\n2M(ψ∗\n1∇ψ1−ψ1∇ψ∗\n1−ψ∗\n−1∇ψ−1+ψ−1∇ψ∗\n−1).\n(10c)\nEquation (9) can also be rewritten as,\nd\ndt/integraldisplay\nVFidV=/integraldisplay\nV∇·JidV=/integraldisplay\nSJi·ndS,\nby using the volume integral and the surface integral,\nwhose unit vector nis vertical to the surface for Stokes’\ntheorem. The equation indicates that the expectation\nvalue of the spin matrix ∝an}b∇acketle{tˆFi∝an}b∇acket∇i}ht=/integraltextdVFiin the volume V\nis conserved for the spin probability flux Jileaving and\nentering the surface.\nUnderHeff∝ne}ationslash= 0, the Kittel-like equation can be re-\nduced to the following equation,\n∂Fi\n∂t+∇·Ji= [F×Heff]i, (11)\nwhere the right side of the equation breaks the conserva-\ntion law of spin density. Therefore, the Kittel-like equa-\ntionshavetwodynamics: spin precessionswith frequency\ngiven by the effective magnetic field and spin currents\nwithout spin conservation. The spin currents of the sys-\ntem will be discussed in Sec. IVBIII. FMR UNDER SINGLE-MODE\nAPPROXIMATION\nIn order to study the basic properties of the second\nterm in Eq. (7), we introduced the single-mode approxi-\nmation,\nψi(r,t) =√\nNξi(t)φ(r)exp/parenleftbigg\n−iµt\n/planckover2pi1/parenrightbigg\n,(12)\nwhereφsatisfies the eigenvalue equation ( −/planckover2pi12∇2/2M+\nV+c0n)φ=µφwith the relation/integraltextdr|φ|2= 1. The\napproximation is effective when the shapes of the con-\ndensates are determined by the spin-independent terms,\nnamely|c0| ≫ |c2|[21]. For87Rb and23Na, the re-\nlation is satisfied. Under this approximation, the first\nterm of Eq. (7) vanishes, and we obtain the Kittel equa-\ntion for the spatially independent spin density vector\nS= (Sx,Sy,Sz),\ndS\ndt=γe[S×HSMA\neff], (13)\nwhere,\nSx=/planckover2pi1√\n2{ξ∗\n0(ξ1+ξ−1)+ξ0(ξ∗\n1+ξ∗\n−1)},\nSy=i/planckover2pi1√\n2{ξ∗\n0(ξ1−ξ−1)−ξ0(ξ∗\n1−ξ∗\n−1)},\nSz=/planckover2pi1(|ξ1|2−|ξ−1|2),\nand the effective magnetic field HSMA\neff =\n(−Nx\nddSx,−Ny\nddSy,H−Nz\nddSz) is given by\nNi\ndd=cdd\ngµBN/integraldisplay /integraldisplay\ndrdr′|φ(r)|2|φ(r′)|2\n|r−r′|3{1−3ei/summationdisplay\njej}.\n(14)\nEquation (13) also indicates that the spin vector Spre-\ncesses around HSMA\neff. The precession frequency reveals\nthe characteristic dynamics. Next, we consider a small\ndeviationδS= (δSx,δSy,δSz) around the stationary so-\nlution,S0=S0ˆzwithS0=H0/Nz\ndd, of Eq. (13), namely\nS=S0+δS. Introducing this representation into Eq.\n(13) and linearizing the equation, we derived the follow-\ning equations,\nd\ndtδSx=γe{H+(Ny\ndd−Nz\ndd)S0}δSy,\nd\ndtδSy=−γe{H+(Nx\ndd−Nz\ndd)S0}δSx,\nd\ndtδSz= 0,\nwhich give the resonance frequency,\nω2=γ2\ne{H+(Nx\ndd−Nz\ndd)S0}{H+(Ny\ndd−Nz\ndd)S0}(15)\nThespinprecesseswiththe resonancefrequency ω, which\ndepends on the dipolar terms Ni\ndd.4\nHere, we consider the single particle density distri-\nbution|φ(r)|2∝e−(x2+y2+λzz2)/a2, whereλzis the as-\npect ratio, and discuss simple situations. For the spher-\nical case of λz= 1, the integration (14) results in\nNx\ndd=Ny\ndd=Nz\ndd, givingω=γeH. The dipolar\nfields are canceled because of the isotropy, so that the\nspin precesses with Larmor frequency. For the circular\nplane (infinite cylinder) case of λz=∞(0), we obtain\nω=γe{H−(Nx\ndd−Nz\ndd)S0}forNx\ndd=Ny\ndd.\nIn this representation, it seems that the microscopic\ndipolar fields, Eq. (14), act as a macroscopic demagne-\ntizingfield tocompareEq. (2) with (15). We believethat\nthe origin of the demagnetizing field is an MDDI. If the\nabove discussion is correct, the dipolar coefficients Ni\ndd\nshoulddepend ontheshapeofthecondensates. However,\nthe single-mode approximation in spinor dipolar BECs\nis not effective in large-aspect-ratio condensates, as dis-\ncussed by Yi and Pu [22]. Therefore, we must consider\nthe spin dynamics beyond the approximation.\nIV. FMR FOR NUMERICAL CALCULATION\nA. Precession dependence on the aspect ratio λ\nIn this section, we discuss FMR by numerically calcu-\nlating the two-dimensional Eq. (3) under the condition\nof87Rb, namely c0≫ −c2>0. We begancalculatingthe\nspin precessions by applying a π/20 pulse to the ground\nstate, whose spins were polarized to the uniform mag-\nnetic field H=Hˆztrapped by V=Mω2\nx(x2+λ2y2)/2\nwithgµBH//planckover2pi1ωx= 20 and an aspect ratio λ=ωy/ωx.\nWe investigated the dynamics of ∝an}b∇acketle{tFx∝an}b∇acket∇i}htforλ= 0.5,1,\nand 1.5 with and without the MDDI. From t= 0 to\nπ/(20γeH), aπ/20 pulse was applied. Then, the spins\nwere tilted by π/20 radians from the zaxis with pre-\ncession. After turning off the pulse, the spins precessed\naround the zaxis, conserving ∝an}b∇acketle{tFz∝an}b∇acket∇i}ht. We define the nota-\ntion∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=λaand∝an}b∇acketle{tFi∝an}b∇acket∇i}htλ=λaas indicating the expectation\nvalues ofFiwith and without an MDDI in the trap with\nλ=λa.\nFirst, the typical motions of spins are shown in Fig. 1.\nInvestigatingthe time development of ∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=0.5,∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=1,\nand∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=1.5, we obtained the differences between their\nprecession frequencies, as shown in Fig. 1 (a) and (b).\nThe differences appeared at frequencies below the Lar-\nmor frequency, given by H. For 0 ≤t≥2, no devi-\nation between the precessions was observed, but devia-\ntions clearly appeared as more time elapsed. In order\nto demonstrate that the λdependence was given not by\nHbut byHdd, we show precessions for the same aspect\nratios without the MDDI in Fig. 1 (c) and (d). The\nprecession frequency did not change without the MDDI\nfor different values of λ. Therefore, the dipolar frequency\nωdd=γeHdddepends upon the shape of the condensate.\nNext, we examined the effects of the MDDI on the\nprecessions in Fig. 2. Comparing ∝an}b∇acketle{tFx∝an}b∇acket∇i}htdd\nλwith∝an}b∇acketle{tFx∝an}b∇acket∇i}htλ,!\"#$!\"#%\"\"#%\"#$\n\" % $!\"#$!\"#%\"\"#%\"#$\n&\" &% &$π/20pulse/angbracketleftFx/angbracketrightdd\nλ=0.5/¯h/angbracketleftFx/angbracketrightdd\nλ=1/¯h\n/angbracketleftFx/angbracketrightdd\nλ=1.5/¯h!\"# !$#\nωxt ωxt!\"#$!\"#%\"\"#%\"#$\n\" % $!\"#$!\"#%\"\"#%\"#$\n&\" &% &$!%# !&#\n/angbracketleftFx/angbracketrightλ=1/¯h/angbracketleftFx/angbracketrightλ=1.5/¯h\n/angbracketleftFx/angbracketrightλ=0.5/¯h\nFIG. 1: (Color online) The time development of /angbracketleftFx/angbracketrightdd\nλ, (a)\nand (b), and /angbracketleftFx/angbracketrightλ, (c) and (d). The red solid, blue dashed,\nand green dotted lines show the results of λ= 0.5, 1, and 1 .5\nrespectively. The gray zone represents the duration of a π/20\npulse.\nwe observed that the MDDI caused an effective mag-\nnetic field, because the frequency of the precession with\nthe MDDI deviated from that without the MDDI in\nFig. 2 (a) to (f). Assuming that ∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=1− ∝an}b∇acketle{tFi∝an}b∇acket∇i}htλ=1\nis represented approximately to Acosγe(H+Hdd)t−\nAcosγeHtwith an amplitude A, we extracted the dipole\nfrequency from the waveform. Since the waveform be-\ncame−2Asinωddt/2sin(ωL+ωdd/2)t, the beat consisted\nof the large frequency ωL+ωdd/2 and the small fre-\nquencyωdd/2. From Fig. 2 (h), we estimated these\nfrequencies to obtain ωdd/ωL≃6.5,9,and11×10−3for\nλ= 0.5,1,and1.5 respectively.\nFigure 3 shows the λdependence of ωdd/ωL. From the\nresults, however, we cannot safely conclude that the λ\ndependence of the frequencies is given by changing the\nshape of the condensates, since the dipolar frequencies\nmay be given by change of the density with the shape.\nFMR in condensed matters has been discussed in con-\ndensed matter of uniform density, even with changing\nshape. On the other hand, atomic BECs have tunable\ndensity and shape. Therefore, our calculations indicate\ncharacteristic of FMR in atomic cold gases.\nB. Spin current\nWe observed spin currents driven by spin diffusion,\nwhich was caused by a rdependence of the dipolar field.\nFigure 4 shows the projections of Fonto thex-yplane5\n!\"#$!\"#%\"\"#%\"#$\n&\" &% &$!\"#$!\"#%\"\"#%\"#$\n\" % $π/20pulse\nωxt ωxt!\"#\n!\"#$!\"#%\"\"#%\"#$\n&\" &% &$!$#\n!\"#$!\"#%\"\"#%\"#$\n\" % $!%#\n!\"#$!\"#%\"\"#%\"#$\n&\" &% &$!&#\n!\"#$!\"#%\"\"#%\"#$\n\" % $!'# !(#\n!\"#$!\"#%\"\"#%\"#$\n\" &\" %\" '\" $\" (\"!)# !*#\n!\"#$!\"#%\"\"#%\"#$\n\" & % ' $ ( )λ=0.5\nλ=1\nλ=1.5λ=0.5\nλ=1\nλ=1.5\nFIG. 2: (Color online) Comparing the precession with and\nwithout the MDDI. (a) and (b), (c) and (d), and (e) and (f)\nshow the precession for λ= 0.5, 1, and 1 .5, respectively. The\nsolid anddashedlines are /angbracketleftFx/angbracketrightλand/angbracketleftFx/angbracketrightdd\nλ. (g)and(h)repre-\nsent (/angbracketleftFx/angbracketrightdd\nλ=0.5−/angbracketleftFx/angbracketrightλ=0.5)//planckover2pi1(solid), ( /angbracketleftFx/angbracketrightdd\nλ=1−/angbracketleftFx/angbracketrightλ=1)//planckover2pi1\n(dot), and ( /angbracketleftFx/angbracketrightdd\nλ=1.5−/angbracketleftFx/angbracketrightλ=1.5)//planckover2pi1(dashed), respectively.\nforλ= 1.5 andωxt= 12.7. The precession with the\nMDDI lost homogeneity of the spin directions, whereas\nthe precession without the MDDI maintained this ho-\nmogeneity. This is because the precession frequency\nhas anrdependence, specifically, ω(r) =γeHeff(r) =\nγe(H+Hdd(r)).\nThe dipole interaction drives the spin diffusion, which\nis shown in Fig. 5. The figure shows Fx/|Fxy|= cosφas\na function of xaty= 0, where φis the angle between\nthe spin vector and the xaxis. In the dynamics with!\"!#!\"!#$!\"!%!\"!%$!\"!&\n!!\"$##\"$%%\"$&\nλωdd\nωL\nFIG. 3:λdependence of ωdd/ωL.\nxy\n03.637×10−3\n0!\"# !$#1.964×10−3\nFIG. 4: (Color online) Projection of Fonto the x-yplane\nforλ= 1.5 andωxt= 12.7. The figures show the results (a)\nwith MDDI and (b) without that. The vectors are nondimen-\nsionalized.\nthe dipole interaction for λ= 1.5 (a) and 1 (b), the\nspin densities lost their angular coherence, whereas the\ndynamicswithout the dipole interactionsmaintained this\ncoherence ( (c) and (d)).\nThe spin diffusion drives the spin current Jkin Eq.\n(10), which is shown in Fig. 6. In order to explain how\nthe spin current is driven by the spin diffusion, we con-\nsidered the amplitudes of the wave functions ψj=fjeiϕj\nas,\nψ1(r,t) =/radicalbig\nn(r,t)\n2(1+cosθ(r,t))eiϕ1(r,t),\n(16a)\nψ0(r,t) =/radicalbigg\nn(r,t)\n2sinθ(r,t)eiϕ0(r,t),(16b)\nψ−1(r,t) =/radicalbig\nn(r,t)\n2(1−cosθ(r,t))eiϕ−1(r,t),\n(16c)6\n!\"!#$%##$%\"\n!\"#!%# % \"#!\"#!\"!#$%##$%\"\n!\"#!%# % \"#!$#\n!%#\nx xλ=1λ=1.5\nωxt=0.128\n163240\n4848\n!\"!#$%##$%\"\n!\"#!%# % \"#λ=1\n0.128\n1632\n40\n0.12 4048 32 16\n8!&#\nλ=1.5\n!\"!#$%##$%\"\n!\"#!%# % \"#0.12 40832 16 48cosφ cosφ\nFIG. 5: Dynamics of a cross-section of Fx/|Fxy|aty= 0,\nwhere|Fxy|=/radicalbig\nF2x+F2y. From the relation Fx=|Fxy|cosφ,\nthe parameter represents cos φ. The results with the MDDI\n(a) and without it (b) are shown for λ= 1.5, and (c) and (d)\nshow results for λ= 1. The xaxis are nondimensionalized by/radicalbig\n/planckover2pi1/Mωx\nwhere the forms show the ground state of the ferromag-\nnetic state [23]. The amplitude is represented by nand\nthe angleθbetween the spin and the zaxis. We intro-\nduced this representation to demonstrate that the spin\ncurrent is derived from the spin diffusion. Of course, we\nconfirmed the validity of the ferromagnetic representa-\ntion under the pulse and magnetic field by calculating θ\ndirectly. Therefore, it can be utilized for the polarized\nspin state studied in our work. The amplitudes f±1were\nformed to represent Fz=n/planckover2pi1cosθ, andf0was deter-\nmined to satisfy the relation n=/summationtext\nj|ψj|2. For example,\n(n1,n0,n−1) = (n,0,0) led toFz=n/planckover2pi1withθ= 0, and\n(n1,n0,n−1) = (n/4,n/2,n/4) resulted in Fz= 0 with\nθ=π/2. The wave function can only express the fer-\nromagnetic states, i.e.the form cannot represent the\nantiferromagnetic state ( n1,n0,n−1) = (n/2,0,n/2) or\nthe polar state ( n1,n0,n−1) = (0,n,0). This restriction\nof the wave function is caused by the first representation\nFz=n/planckover2pi1cosθ.\nBy introducing this representation into Eqs. (6) and\n(10), we can redefine as follows,\nFx=n/planckover2pi1sinθ(cosϕrcosϕ−cosθsinϕrsinϕ),\nFy=−n/planckover2pi1sinθ(cosϕrsinϕ+cosθsinϕrcosϕ),and,\nJx=n/planckover2pi12\n4M/braceleftbigg\nsinθ(1+cosθ)cos(ϕ1−ϕ0)∇ϕ1\n+ sinθ(1−cosθ)cos(ϕ−1−ϕ0)∇ϕ−1\n−2sinθ(cosϕrcosϕ−cosθsinϕrsinϕ)∇ϕ0\n+ 2(cosϕrsinϕ+cosθsinϕrcosϕ)∇θ/bracerightbigg\n,(17a)\nJy=−n/planckover2pi12\n4M/braceleftbigg\nsinθ(1+cosθ)sin(ϕ1−ϕ0)∇ϕ1\n−sinθ(1−cosθ)sin(ϕ−1−ϕ0)∇ϕ−1\n+ 2sinθ(cosϕrsinϕ+cosθsinϕrcosϕ)∇ϕ0\n+ 2(cosϕrcosϕ−cosθsinϕrsinϕ)∇θ/bracerightbigg\n,(17b)\nJz=n/planckover2pi12\n4M{(1+cosθ)2∇ϕ1−(1−cosθ)2∇ϕ−1},\n(17c)\nwhereϕr= (ϕ1+ϕ−1−2ϕ0)/2 andϕ= (ϕ1−ϕ−1)/2 are\nrelative phases. Since the relation ϕr= 0 was satisfied in\nour calculations, we used the relation in Eqs. (17), and\nthe spin density vectorformed an azimuthal angle ϕwith\nthexaxis. Then, we derived the spin components Fx=\nn/planckover2pi1cosϕsinθ,Fy=n/planckover2pi1cosϕsinθ, andFz=n/planckover2pi1cosθ. We\ncan therefore rewrite the spin density currents,\nJx=n/planckover2pi12\n4M(4cosϕsinθ∇ϕ0\n+2cosϕsinθcosθ∇ϕ−2sinϕ∇θ),(18a)\nJy=−n/planckover2pi12\n4M(4sinϕsinθ∇ϕ0\n+2sinϕsinθcosθ∇ϕ+2cosϕ∇θ),(18b)\nJz=n/planckover2pi12\n4M{4cosθ∇ϕ0+4(1+cos2θ)∇ϕ},(18c)\nwhich are driven by the gradients of the angles, ϕandθ,\nandthephase ϕ0. IntheprecessionswithMDDI,thegra-\ndients occurred because of the dipolar fields Hdd(r). As\na result, the spin currents were clearly driven, as shown\nin Fig. 6. For ωxt= 0.12, the spin vectors were coher-\nent just after the applied π/20 pulse (Fig. 6 (a)). The\nspin densities, FxandFy, then flowed to the center of\nthe condensates from Fig. 6 (b) to (c). Then, the den-\nsities reversed, and diffused outward from Fig. 6 (d) to\n(e). This oscillation was repeated. Of course, we cannot\nobtain the spin current without the dipolar interactions,\nsince the gradients of θandϕwere not caused; the dy-\nnamics are shown in Fig. 7.\nIn order to investigate the spin fluid dynamics, we cal-\nculated the spin current Jxfor Eq. (17), as shown in\nFigs. 8 and 9. These figures represent Jxfrom the pre-\nvious calculations with λ= 1 and 1.5 respectively. De-\nspite the difference in the ratio, we observed two com-\nmon properties in these figures. The direction of the\ncurrents changed rapidly, corresponding to the large pre-\ncession frequency, and the magnitudes changed slowly7\n1.961×10−3\n0\n!\"# !$# 3.152×10−3\n0\n 0\n3.447×10−3!%#\n3.168×10−3\n0\n!&#2.113×10−3\n0\n!'# !(#3.043×10−3\n0\nωxt=0.12 ωxt=8 ωxt=16\nωxt=32 ωxt=40 ωxt=48\nFIG. 6: (Color online) Dynamics of Fprojected onto the x-\nyplane for λ= 1.5 with dipolar interaction.\nwith the small dipolar frequency, as shown in Fig. 10,\nwhich shows the time development of the xcomponent\nofJx(x= 4,y= 0). This figure indicates that the oscil-\nlation of the current direction occurred with the preces-\nsion frequency, which varied in magnitude with changing\ndipolar frequency. Eq. (11) also indicates that the spin\ndensity was not conserved because of the effective mag-\nnetic field. Therefore, the spin currents can be driven\nfrom a source and sink in the center of the condensates,\nas in Figs. 8 and 9. The two common properties were\ninsensitive to the value of λ. However, the change in spin\ndensity for λ= 1.5 exhibited quadratic pole motion in\na scissors-like mode for mass density [24], which can be\nunderstood as an oscillationbetween the spin density mi-\ngrating to the yaxis from the xaxis and back again, as\nshown in Figs. 6 (a) to (c). Therefore, the spin collec-\ntive mode was caused by spin diffusions induced by the\nMDDI. Therefore, the spin current causes the dynam-\nics of spin scissors-like mode, which was observed as a\nshrinking and expansion of the spin density in Fig. 6.\nThe shrinking and expansion were common features for\nλ= 1 and 1.5. However, the spin currents were affected\nby the symmetry of the traps, as shown in Figs. 8 and 9.\nFrom the calculations, we expected that the spin cur-\nrent would be observable when using the spinor BECs.\nRecently, spin current is focused from fields of spintron-\nics. However, it is difficult to observe the spin current in\nmetals and condensed matter. Atomic BECs, a macro-\nscopic quantum phenomenon, can show the spin current\nclearly and directly in the dynamics of the spinor den-\nsities. Therefore, we should attempt to observe various\nspin currents utilizing tunable experimental parameters,\ni.e.interaction parameters, trap frequencies, and the\nnumber of particles.\nV. CONCLUSION\nWe investigated the properties of magnetic resonance\nin spinor dipolar BECs by calculating the GP equations,\nobtainingKittel-likeequationsastheequationsofmotion!\"# !$# !%#\n!&# !'# !(#0\n1.965×10−3\n0\n1.965×10−3\n0\n1.965×10−3\n0\n1.965×10−3\n0\n1.965×10−3\n0\n1.965×10−3ωxt=0.12 ωxt=8 ωxt=16\nωxt=32 ωxt=40 ωxt=48\nFIG. 7: (Color online) Dynamics of Fprojected onto the x-\nyplane for λ= 1.5 without dipolar interaction.\nωxt=0.12 ωxt=8 ωxt=16\nωxt=32 ωxt=40 ωxt=481.95×10−4\n0\n!\"# !$# !%#\n!&# !'# !(#0\n2.69×10−4\n0\n9.8×10−5\n0\n4.6×10−5\n02.73×10−4\nFIG. 8: (Color online) Dynamics of the spin currents Jxpro-\njected onto the x−yplane for λ= 1 with dipolar interaction.\nThe vectors are nondimensionalized.\nfor the spin density vector. The equations revealed two\nproperties. One is the dynamics of the spin fluid, and\nthe other is precession under the effective magnetic field\nconsisting of the external magnetic fields and the dipolar\nfields. The magnetic resonancewith the properties of the\nspin fluid was characteristic of this system.\nIn order to extract properties from the GP equations,\nwe studied the law of conservation of spin density cur-\nrent without effective magnetic fields by first deriving\nthe continuity equations from the GP equation, obtain-\ning representations of the spin current. Second, we ana-\nlytically evaluated the precession dynamics described by\nthe Kittel equations derived from the GP equations us-\ning a single-mode approximation, where the Kittel equa-\ntions show conventional FMR. The analysis clearly in-\ndicated that the origin of the FMR in the BECs is like\nthe dipolar field, whereas the origin of the resonance in\nthe Kittel equations for condensed matter is the demag-\nnetizing field. Comparing the FMR of the BEC with\nthat of the condensed matter, we concluded that the ori-\ngin of the resonance was not the spin exchange interac-\ntion that causes magnetism in condensed matter, but the8\nωxt=0.12 ωxt=8 ωxt=16\nωxt=32 ωxt=40 ωxt=48!\"# !$# !%#\n!&# !'# !(#0\n3.59×10−4\n0\n2.76×10−4\n0\n2.13×10−4\n0\n8.5×10−5\n05.5×10−5\nFIG. 9: (Color online) Dynamics of the spin currents Jxpro-\njectedontothe x−yplanefor λ= 1.5withdipolarinteraction.\n!\"#\"\"\"$\"\"#\"\"\"$\n\" %\" $\" &\" '\" (\"!\"#\"\"\"$\"\"#\"\"\"$\n\" %\" $\" &\" '\" (\"{Jx}x(x=4,y=0)\nωxtλ=1 λ=1.5\nωxt!\"# !$#\n!\n! \" #!\n! \" #\nFIG. 10: Dynamics of the xcomponent of Jxatx= 4 and\ny= 0. The inter figures are the results for ωxt= 0 to 4.anisotropy of the MDDI. Finally, we numerically calcu-\nlated the GP equations, representing the dynamics with\nthe twocommon properties. The characteristicdynamics\nshowed that the effective magnetic field introduced spin\ndiffusion into the Larmor precession, driving the spin-\ncurrent-like scissors modes.\nThe relation between the spin current and FMR has\nnot yet been discussed for typical FMR. Therefore, it is\nimportant to study spin current in condensates. We also\nbelieve that the study of spin current will be useful for\nthe development of spintronics, because it is difficult to\ndirectly observe spin currents in condensed matter spin-\ntronics.\nVI. ACKNOWLEDGMENT\nM. Y. acknowledges the support of a Research Fellow-\nship of the Japan Society for the Promotion of Science\nfor Young Scientists (Grant No. 209928). M. T. ac-\nknowledges the support of a Grant-in Aid for Scientific\nResearch from JSPS (Grant No. 21340104).\n[1] C. P. Slichter, Principles of Magnetic Resonance , (Berlin:\nSpringer–Verlag, 1990).\n[2] J. H. E. 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(Cambridge, New York, 2008)."    },    {        "title": "1607.04535v2.Theory_of_spin_coherent_transport_through_a_defect_spin_state_in_a_metal_ferromagnet_tunnel_junction_during_ferromagnetic_resonance.pdf",        "content": "Theory of spin-coherent electrical transport through a defect spin state in a\nmetal/insulator/ferromagnet tunnel junction undergoing ferromagnetic resonance\nN. J. Harmon\u0003and M. E. Flatt ´e†\nDepartment of Physics and Astronomy and Optical Science and Technology Center, University of Iowa, Iowa City, Iowa 52242, USA\n(Dated: November 26, 2021)\nWe describe the coherent dynamics of electrical transport through a localized spin-dependent state, such as is\nassociated with a defect spin, at the interface of a ferromagnet and a non-magnetic material during ferromagnetic\nresonance. As the ferromagnet’s magnetic moment precesses, charge carriers are dynamically spin-filtered by\nthe localized state, leading to a dynamic spin accumulation on the defect. Local effective magnetic fields modify\nthe precession of a spin on the defect, which also modifies the time-integrated total charge current through\nthe defect. We thus identify a new form of current-detected spin resonance that reveals the local magnetic\nenvironment of a carrier spin located at a defect, and thus potentially the defect’s identity.\nThe emerging field of “quantum spintronics” seeks to en-\ngineer and manipulate single coherent spin systems for the\nsake of quantum-enhanced sensing/imaging technologies and\nquantum computing [1]. Defect spins in an insulating re-\ngion between a ferromagnetic metal and a nonmagnetic con-\nductor produce an array of coherent spin-dependent phenom-\nena, including defect-associated spin pumping [2–4], thermal\nspin transport [5], and small-field magnetoresistance under\nelectrical bias [6–8]. Individual spin-coherent defects have\neven been electrically detected in precisely-designed junctions\n[9, 10]. However, the potential of a coherently-precessing\nsource of spins, readily available from a ferromagnetic con-\ntact undergoing precession (such as from a spin torque oscil-\nlator) has not yet been explored; such a coherent source may\nbe able to reach a single-defect-spin regime of spin pumping\nor dynamic spin polarization.\nHere we predict observable coherent dynamics in the\ncharge and spin transport through a single defect in the junc-\ntion between a ferromagnetic material and a second, non-\nmagnetic (NM) conducting material, when the magnetism of\nthe ferromagnet (FM) precesses in time such as during fer-\nromagnetic resonance (FMR). During electrical transport the\ndefect can become dynamically spin polarized, and its spin\nmanipulated, even with negligible coupling between the de-\nfect and FM from a magnetic dipolar field or exchange in-\nteraction. This provides a single-defect-spin example of dy-\nnamic spin polarization. Analysis of the current through the\ndevice reveals the local spin character of a defect and its en-\nvironment without the need of a microwave cavity. These ef-\nfects, in the single-defect limit, would be detectable with a\nspin-polarized scanning tunneling microscope tip undergoing\nFMR, and should persist even for sequential hopping trans-\nport between the tip and the defect, as well as between the\ndefect and the second conducting contact. A slower transport\nrate between the defect and the FM provides better resolution\nof the defect’s local environment, so long as the defect spin\nstate’s coherence time is comparable to or exceeds the electri-\ncal transport rate through the junction.\nHere we focus on a defect electronic structure correspond-\ning to a single orbital state and two (oppositely-oriented)\nspin states, either unoccupied or singly occupied by a spin-\n1=2 electron. The junction is shown schematically in Fig. 1.\nNMFMUdefect energy(c)(a)\nPR(t)s(t)!0!`!d!0+!`=0yxz(b)yzx!0+!`=!d!FMFIG. 1. (a) Diagram of the energy landscape of a ferromag-\nnet/nonmagnetic (FM/NM) metal junction. The darker box specifies\nthe NM metal and the middle planes represent the two energy levels\nof the defect which are separated by an on-site Coulomb energy U.\nThe bias pushes electrons through the junction from the NM metal\n(left) to the FM (right). The vertical direction is energy whereas the\nlateral directions are spatial coordinates. (b) Schematic of the spatial\norientation of various spins: the FM’s polarization, PR(t)(green ar-\nrow) precesses around an axis wwwFM(black arrow). The spin of the\ndefect sss(t)(blue arrow) precesses in the sum of an externally ap-\nplied and a local magnetic field, at a frequency wwwd=www0+www`. In\nthis panel wwwd=0. The dynamical spin polarization of the defect fol-\nlows the FM’s polarization. (c) For wwwd6=0 the defect spin precesses\naround the static ( wFM=0) steady state spin orientation (indicated\nby the orange line).\nTransport occurs as an electron spin, of arbitrary direction,\nhops from the left contact to the previously empty defect site\nand singly occupies the level. The electron’s subsequent mo-\ntion will then be limited depending on the orientation of its\nspin relative to the majority spin polarization at the Fermi level\nin the FM; if parallel then the transport is rapid, while if an-\ntiparallel the transport is slower. Similar behavior will occur\nfor hole spin transport, with opposite bias voltage and whenarXiv:1607.04535v2  [cond-mat.mes-hall]  4 Jan 20182\nthe hole hops to a defect site that is empty (of holes, and thus\ndoubly-occupied by electrons), or for defects with different\nelectronic state ordering, so long as the transport through the\ndefect states depends on spin. For example, a ground-state\nspin-1 defect, such as a silicon carbide divacancy [11], will\nexhibit essentially the same features as our spin-1/2 system,\nbut with opposite dynamic spin polarization. We focus on the\ncase shown in Fig. 1.\nA heuristic picture helps visualize the resonance condition\nfor transport through the defect state during precession of the\nFM’s spin polarization. The spin polarization of the FM’s\nFermi-level carriers, PPPR(t)(green arrow), precesses around an\naxiswwwFM(black arrow), depicted parallel to ˆ zin Fig. 1. The\ncone angle is the angle between PPPR(t)andwwwFM. The equi-\nlibrium polarization of the FM when not undergoing FMR is\nPPPRjjˆz. The probability for a carrier at the defect to enter the\nFM depends on the relative orientation of the carrier’s spin,\nsss(t)(blue arrow), and PPPR(t). For the simplest picture consider\nthe FM to be 100% spin polarized, for which only a carrier\nwith some spin component parallel to PPPR(t)may tunnel into\nthe FM.\nThe spin on the defect site, associated with the carrier, can\nalso precess due to the influence of an applied magnetic field\nas well as a local effective field arising from hyperfine inter-\nactions, exchange interactions with neighboring sites, or other\neffects. The directions of precession vectors will be described\nusing a polar angle qrelative to wwwFMkˆzand an azimuthal an-\nglefrelative to the ˆ xaxis, with a subscript corresponding to\nthe specific precession vector. The local field is considered to\nbe independent of the applied magnetic field, and causes the\ndefect spin to precess according to the precession vector www`.\nThe applied magnetic field precesses the defect spin accord-\ning to the precession vector www0, and the total precession will\nbewwwd=www0+www`. To distinguish this precession frequency\nfrom apparent precession due to spin filtering, the precession\nfrequency wdwill be referred to as the defect spin’s Larmor\nfrequency .\nDynamic spin polarization emerges on the defect site, and\nis largest when wwwd=0, shown in Fig. 1(b). Under bias the\ndefect occupation is continually replenished, until the carrier\nspin on the defect is oriented antiparallel toPPPRand no further\ntransport occurs until the carrier spin decoheres or the FM po-\nlarization changes. This spin filtering process results in the de-\nfect spin tracking approximately antiparallel to PPPR(t), there-\nfore blocking the current through the junction. Figure 2 illus-\ntrates the details of the spin-coherent effects on charge current\nduring FMR, beginning with an unoccupied defect spin state.\nFigure 2(a) demonstrates (orange line) that sss(t)\u0001PPPR(t)!\u0000 1\nafter transient dynamics.\nFigure 1(c) shows the changing dynamics for a non-\nvanishing Larmor precession of the carrier spin on the defect,\nand for wwwdperpendicular to wwwFM. For wwwFM=0 the defect\nspin precession causes the dynamic spin polarization gener-\nated from spin filtering in transport into the FM to rotate in\nthezyplane and be oriented along the orange line, which is\ndetermined by the relative precession frequency and spin fil-\n(b)\n\u0000FMti/e\u0000RPR(t)·s(t),(a)010020030040050000.51-1-0.50ωFMti/ekR,PR(t)·s(t)\n(c)!d=0!d=1.1!FM!d=!FM✓d=⇡/2✓d=⇡/2FIG. 2. (a) Charge current [Eq. (5)] when the defect spin’s precession\nfrequency, wd, is zero. The current (black line) decreases to zero as\nthe carrier spin at the defect (orange line) becomes polarized opposite\nthat of the FM. Once the defect is completely antiparallel, no further\ncharge can occupy or leave the defect. (bc) Charge current from two\nchoices of wd, (b) non-resonant and (c) resonant, with wwwdoriented\nalong the x-axis in Fig. 1. The orange curves depict the projection\nof the carrier spin, s(t), onto the rotating polarization, PR(t), which\ndetermines the current (black lines). Parameters are fd=0, cone\nangle between PPPR(t)andwwwFMof 0:05 radians (\u001810%), gL=10wFM\nandgR=0:01wFM,PR=1, and PL=0. For clarity, each amplitude\nis enhanced by a factor of 10.\ntering rate to be\nsss(t) =\u00002gL[g2\nRPPPR+gRwwwd\u0002PPPR+(wwwd\u0001PPPR)wwwd]\n(gR(1\u0000P2\nRc(wwwd))+2gL)(g2\nR+w2\nd);(1)\nwith\nc(wwwd) =g2\nR+(wwwd\u0001ˆPR)2\ng2\nR+w2\nd; (2)\nwhere gLis the hopping rate from the left conductor to the\ndefect and gRthe hopping rate from the defect to the FM.\nForwwwFM6=0 the dynamical defect spin polarization sss(t)\nprecesses at the frequency wFMaround the orange line, as\nindicated in Fig. 1(c). Figure 2 displays the current for\nthis configuration off resonance [Fig. 2(b)] and on resonance\n[Fig. 2(c)]. When off resonance some beating occurs in the\ntransient stage until the defect spin sss(t)is syncronized with\nPPPR(t)[12]. On resonance, corresponding to wFM=jwww0+\nwww`j=wd, the amplitudes of the defect spin’s precession and\nthe current oscillations increase.3\n(b)transient0.900.951.001.051.101.151.2010-50.0010.10010\nfrequency/ωFMpower spectrum(norm.)2510200.0010.0050.0100.0500.100Pω/PωFMt-3/2integration time\u0001103ωFM-1\u0003\n0.900.951.001.051.100.00.20.40.60.81.0ω/ωFMPωFM(norm.)resonant(a)\n⇡/2.25⇡/6⇡/3integration time (103!\u00001FM)!/!FM!d/!FMP!FM(norm.)power spectrum (norm.)P!dP!FMP!d/P!FM\nnon-resonant✓d=⇡/2\nFIG. 3. (a) Power spectra for the resonant (black) and non-resonant\n(red) currents in Fig. 2, each normalized with respect to the resonant\npeak. The off-resonant spectrum shows a transient peak at wd=\n1:1wFM, in addition to a persistent peak at wd=wFMwith width\ngoverned by a damping rate G=0:005wFM. Integration times are 5\n(dotted), 10 (dashed), and 20 (solid) \u0002103w\u00001\nFM. (Inset) dependence\non the integration time of the off-resonant ratio of the power at the\nLarmor frequency to that at the FMR frequency (Pwd=PwFM). (b)\nPwFMversus wd, for several angles ( qd) between wwwdandwwwFM.PwFM\nis independent of fd. Parameters are identical to those in Fig. 2.\nThe off-resonant power spectrum (red) of the current os-\ncillations, Fig. 3(a), shows peaks at both the FMR frequency\n(wFM) and the defect spin’s precession frequency ( wd); the\npeak at wdis a transient, as shown with integration times of\n5, 10, 20\u0002103w\u00001\nFM, and in the inset. When on resonance\n(black), wFM=jwww0+www`j=wd,s(t)andPR(t)are synchro-\nnized and s(t)increases, producing larger amplitude current\noscillations. Figure 3(b) shows the dependence of the cur-\nrent’s power spectrum at the FMR frequency, PwFM, onwdfor\nseveral different orientations qd.\nWe now describe how the charge current through the junc-\ntion during FMR is calculated including the spin-coherent dy-\nnamics of the defect. The current operators involving the two\ncontacts, from the NM contact to the defect (‘left’ current),\nand from the defect to the FM (‘right’ current), are explic-\nitly constructed and combined with a coherent density matrix\ntreatment of the carrier spin dynamics. The following ansatz\ndescribes the ‘right’ current operator\nˆiR(t) =e\n2gRh\nˆPR(t)r(t)+r†(t)ˆP†\nR(t)i\n; (3)where PR(t)is the polarization operator of the FM and r(t)the\ndensity matrix of the defect’s carrier spin. The second term of\nEq. (3) ensures hermiticity. ˆPR(t) =1\n2(I+PR(t)\u0001s)describes\nan imperfect spin filter ( ˆPR(t)is not idempotent unless PR=1)\n[13]. PPPR, determined by Tr (ˆPs), precesses around wwwFMand\nis determined by\n˙ˆPR(t) =\u00001\n2i\n~[~wwwFM\u0001sss;ˆPR(t)]: (4)\nAn analytic solution for ˆPR(t)is available using an algebraic\nsolver [14]. To account for the finite line width of the FMR,\nthe power spectrum is convolved with a Lorentzian function\nof width G[15].\nˆiRrepresents the movement of charge combined with spin\ninformation encoded in the matrix elements. Charge (spin)\ncurrent is iR=TrˆiR(iiis;R=TrˆiRsss). The right charge current\nonce the defect site is filled,\ni=Tr(ˆiR) =1\n2(1+sss(t)\u0001PPPR(t))egR;with sss=Tr(rsss);(5)\nwhich illustrates the dependence of the current on the relative\nalignment of the defect spin and FM’s polarization. For gL\u001d\ngRthe defect state is predominately filled. For a spin-polarized\ncontact that is an STM tip, the tip can be moved away from the\nimpurity until gL\u001dgR. The amplitude of current oscillations,\nfor small cone angles and gL\u001dgR, scales as P2\nR.\nThe ‘left’ current (NM contact to defect) can be derived in\na similar fashion after constraining the defect to be at most\nsingly occupied. For a left conductor with a static magnetiza-\ntion,\nˆiL(t) =egL[1\u0000Trr(t)]ˆPL; (6)\nwhere ˆPL=1\n2(I+PL\u0001s)is the polarization operator of the left\nconductor. This formalism can be generalized to include dy-\nnamic magnetization of the left conductor, although here we\npresent results only for a NM, i.e.PL=0. Charge conserva-\ntion demands that the ‘left’ current be the same as the ‘right’\ncurrent for a time-independent PPPR(t)or when the current is\naveraged over a precession period of PPPR(t), soTrˆiL=TrˆiR.\nConstruction of the defect density matrix r(t)consistently\nconnects the currents and determines their sensitivity to spin\nand applied magnetic fields. The stochastic Liouville equation\nis suited well for this type of problem [16, 17], so\n˙r(t) =\u0000i\n~[H;r(t)]\u0000gRfˆPR(t);r(t)g+2gL[1\u0000Trr(t)]:(7)\nThe first term of Eq. (7) produces the coherent evolution of\nthe spin, the second term (curly braces are anti-commutators)\nthe spin-selective nature of tunneling into the FM. The last\nterm describes hopping onto the defect site from the left con-\ntact.H= (~=2)wwwd\u0001sssis the spin Hamiltonian at the defect\nsite. In typical insulators the localization length of the defect’s\nwave function is wide enough to encompass a large number of\nrandomly oriented nuclei, so a local hyperfine field www`can be4\naccurately approximated as a classical vector. The spin den-\nsity matrix is obtained from a numerical solution to Eq. (7),\nand the current from either Eq. (3) or Eq. (6).\nAlthough the resonances always occur when wFM=wd, in-\ndependent of precession axis direction, it is possible to deter-\nmine www`by measuring w0at resonance, for several different\ndirections of www0, asw0at resonance will vary with direction\nfromwFM\u0000w`towFM+w`. In Fig. 4(a) the current’s power\nspectrum at the FMR frequency, PwFM, is shown as a func-\ntion of w0for three different directions of www0, for an example\nhyperfine local field www`= (\u00000:3;0:1;0:2)wFM. This theory\napplies also to two independent defects through which paral-\nlel currents run. Fig. 4(b) displays sweeps of w0atq0=p=2\nand three different f0, similar to the single defect scenario.\nNow resonances occur at two different applied fields for each\nsweep [18].\n0.60.81.01.21.40.00.20.40.60.81.0ω0/ωFMPωFM(norm.)\n0.60.81.01.21.40.00.20.40.60.81.0ω0/ωFMpower spectrum(norm.)⇡/3⇡/2\n!0/!FMP!FM(norm.)defect 1defect 2⇡/3⇡/6(a)(b)\u00000=0\u00000=0✓0=⇡/2✓0=⇡/2\nFIG. 4. (a) Plots showing the integrated current at wFMwhen the ap-\nplied field, w0, is swept. Resonances occur when wd=jwww0+www`j=\nwFM. Here www`= (\u00000:3;0:1;0:2)wFM. (b) Two resonance features\nappear when two defects are probed. Each colored curve corresponds\nto an independent sweep of the magnetic field in the x\u0000yplane at\nan angle f0. For the two defects, www`;1= (\u00000:3;0:1;0:2)wFMand\nwww`;2= (0:4;0:1;\u00000:1)wFM. Curves in (a) and (b) are normalized to\nthe highest peak and labeled by the applied field’s azimuthal angle\nf0. Parameters are identical to those used in Fig. 3.\nwFMis fixed in Figs. 3(b) and 4 as w0varies. For a fer-\nromagnetic thin film with the easy axis of the contact in the\nfilm plane, the component of the applied magnetic field along\nthe hard axis, if sufficiently small, does not influence wwwFM\nbut does change www0andwwwd. We assume the magnetic field\ncomponent along the hard axis is varied in order to vary wwwd\nleaving wwwFMfixed.\nThe relevant timescale for differential precession of the\ncarrier spin and the FM is the timescale for hopping from\nthe defect to the FM. For typical scanning tunneling mi-\ncroscopy measurements with currents of 0 :1\u000030 nA [19],the timescale for hopping from a defect to a ferromagnetic\ntip would be 0.05–1.6 ns. For spins on the defect coherent on\nthis timescale, which is known to be the case for many exam-\nples of localized spins [20], the features described here will\nemerge. By comparing this hopping time to the precession\ntime of the carrier spin on the defect in a local magnetic field,\nthe sensitivity to local fields can be estimated to be of the order\nof\u001810 mT, characteristic of hyperfine fields for many types\nof defects. Smaller currents will improve sensitivity to www`.\nSpin-coherent evolution of a carrier spin at a defect pro-\nduces resonant features in the charge conductivity of a ferro-\nmagnet/insulator/nonmagnet junction. From this, small num-\nbers of defects, or a single defect, can be identified by matches\nbetween the ferromagnetic resonance frequency of a contact\nand the local precession of the spin(s) of the defect(s). The ap-\nproaches described here would also permit the preparation of\nspecific desired defect spin states through appropriate choices\nfor the ferromagnet’s precession frequency, leading to con-\ntrolled studies of the coupled dynamics of two coherent spins.\nThis work was supported by the U.S. Department of En-\nergy, Office of Science, Office of Basic Energy Sciences, un-\nder Award #DE-SC0016447.\n\u0003nicholas-harmon@uiowa.edu\n†michael flatte@mailaps.org\n[1] D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu, and\nJ. R. 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Tan, Juan P. Dehollain, Wee H. Lim,\nJohn J. L. Morton, David N. Jamieson, Andrew S. Dzurak, and\nAndrea Morello, “A single-atom electron spin qubit in silicon,”\nNature 489, 541–545 (2012).\n[11] William F Koehl, Bob B Buckley, F Joseph Heremans, Greg\nCalusine, and David D Awschalom, “Room temperature co-\nherent control of defect spin qubits in silicon carbide,” Nature\n479, 84 (2011).\n[12] A brief animation of the magnetization and defect spin dynam-ics is found in the online Supplementary Information.\n[13] P. S. Farago, “Electron spin polarization,” Reports Prog. Phys.\n34, 1055 (1971).\n[14] The explicit form of the matrix ˆPR(t)is found in the Supple-\nmentary Information.\n[15] A description of the convolution is found in the Supplementary\nInformation.\n[16] R. Kubo, J. Math. Phys. 4, 174 (1963).\n[17] R. Haberkorn, Mol. Phys. 32, 1491 (1976).\n[18] Further description of the resonant detection is located in the\nSupplementary Information.\n[19] C. Julian Chen, Introduction to Scanning Tunneling Microscopy\n(Oxford University Press, 2008).\n[20] P. M. Koenraad and M. E. Flatt ´e, “Single dopants in semicon-\nductors,” Nature Materials 10, 91–100 (2011)."    },    {        "title": "2007.01505v1.Light_controlled_room_temperature_ferromagnetism_in_vanadium_doped_tungsten_diselenide_semiconducting_monolayers.pdf",        "content": "1 \n Light -control led room temperature ferromagnetism in vanadium -doped tungsten \ndiselenide semiconducting monolayer s  \nValery Ortiz Jimenez1, Yen Thi Hai Pham1, Mingzu Liu2, Fu Zhang3,4, Vijaysankar Kalappattil1, \nBaleeswaraiah Muchharla1, Tatiana Eggers1, Dinh  Loc Duong6,7,   \nMauricio Terrones2,3,4,5,*, and Manh -Huong  Phan1,* \n1Department of Physics, University of South Florida, Tampa, Florida 33620, USA  \n2Department of Physics, The Pennsylvania State University, University Park, PA 16802 USA  \n3Department of Materials Science and Engineering, The Pennsylvania State University, \nUniversity Park, PA 16802, USA  \n4Center for Two Dimensional and Layered Materials, The Pennsylvania State University, \nUniversity Park, PA 16802, USA  \n5Department of Chemistry, The Pennsylv ania State University, University Park, PA 16802, USA  \n6Center for Integrated Nanostructure Physics (CINAP) Institute for Basic Science (IBS) Suwon \n16419, Republic of Korea  \n7Department of Energy Science, Sungkyunkwan  University, Suwon 16419, Republic of Korea . \n \n \n*Corresponding authors:    phanm@usf.edu , mut11@psu.edu ,   \n  2 \n Atomically thin t ransition metal dichalcogenide  (TMD) semiconductors hold enormous \npotential for modern optoelectronic device s and quantum computing applications . By \ninducing long-range ferromagnetism (FM) in these semiconductors through  the \nintroduction of  small amounts of a magnetic dopant , it is possible to extend their potential  \nin emerging spintronic  application s. Here , we demonstrate light -mediated,  room \ntemperature (RT) FM , in V-doped WS 2 (V-WS 2) monolayers . We probe this effect using the \nprinciple of magnetic LC  resonance , which employs a soft ferromagnetic  Co-based \nmicrowire coil driven near its resonance in the radio frequency (RF)  regime . The \ncombination of LC resonance  with an extraordinary  giant magneto -impedance effect , \nrenders  the coil highly sensitive to changes in the magnetic flux through its core.  We then \nplace the V-WS 2 monolayer  at the  core of the coil  where it  is excited with  a laser while  its \nchange in magnetic permeability  is measured . Notably , the magnetic permeability  of the \nmonolayer is found to depend on the laser intensity , thus confirming light control of RT \nmagnetism in this two-dimensional (2D) material . Guided  by density functional \ncalculations, we attribute this phenomenon to the presence of excess  holes in the conduction \nand valence bands, as well as carriers trapped in the magnetic doping states, which in turn \nmediate s the magnetization of the V-WS 2 monolayer.  These findings provide a unique route \nto exploit light -controlled ferromagnetism  in low powered  2D spintronic devices  capable of \noperating  at RT . \n \n  3 \n Introduction  \nDilute magnetic semiconductors (DMSs) offer an alternative path towards the realization of \ncutting -edge spintronic devices.1-6  The use of light to control magnetism in these semiconductors \nhas the added advantage of being able to control both charge and spin simultaneously, which \nsupports the demands of multifunctional (smart) sensing devices, information storage, and \nquantum comp uting technologies .7-10 So far, i t has been reported  that a carrier -mediated \nferromagnetic interaction between  the Mn ions  in p-type (In,Mn)As/GaSb  semiconductor  has \nbeen  enhanced by the illumination of light through the generation of excess holes in the \n(In,Mn)As  layer .7 Unfortunately , this effect is limited to temperatures (<  50 K), well below \nambient temperature , while  the most important  technological applications are required to operate \nat  room temperature .11-12    \nIt has recently been theoretically and experimentally shown  that the introduc tion of a  \nmagnetic transition metal atom into  semiconducting two-dimensional  (2D) TMD , such as V-\ndoped WSe 2 and V -doped WS 2, permits long-range ferromagnetic order that can be induced at \nroom temperature .13-15 Currently, t he Ruderman –Kittel –Kasuya –Yosida (RKKY) mechanism is \nbelieved to be responsible for  the long-range ferromagnetic order  in these TMD systems , where \nfree holes are the  medium that support  the interaction between V atoms.13,14 In particular , we \nhave recently demonstrated  that p-type V-doped WS 2 monolayers have strong and tunable room \ntemperature ferromagnetism .15 By replacing W , having six valence electrons , with V , having five \nvalence electrons, an electron deficiency is created in V-WS 2 that eventually becomes a p-type \ndominant  semiconductor . Unlike diamagnetic pristine WS 2 (Fig. S1(a)), ferromagnetism is \nenhanced with V doping  in monolayers of V -WS 2, which is found to be optim ized at ~2  at.% V \n(Fig. S1(b)).15 The l ong-range ferromagnetic order in this optimally magnetic 2D DMS allows us 4 \n to modify its magneto -electronic property  with external stimuli, like a magnetic field , an electric \nfield, or as we show in this letter, with light.  Photoluminescence (PL) reveal that , even after \ndoping, strong photoluminescence is still present (Fig. S1(c)). These observations lead us to \npropose that the ferromagnetism in the V-WS 2 monolayer can be mediated by illuminati on with a \nlaser of appropriate energ y, that is,  above the optical gap  (see Fig. 1). Electrons from \nphotogenerated electron -hole pairs  may be captured by the V atoms , thus creat ing an imbalance \nin the carrier population (i.e., the generat ion of  excess holes ) such that the ferro magnetism of the \nmonolayer is modifie d. While p -type (In,Mn)As/GaSb show ed light-mediated ferromagnetism at \ntemperatures  below  50 K ,7 in this manuscript we demonstrate that light control s the \nferromagnetism at ambient temperature in an atomically thin p -type V-WS 2 semiconductor. The \nlight mediated changes of the  magnetization in 2D semiconducting materials , will certainly lead \nto novel applications in spintronic devices  that have not been yet realized .  \nProbing light -induced 2D magnetism  \nProbing magnetism in atomically thin  materials is extremely challenging when compared \nto bulk systems .16,17  While techniques such as vibrating sample magnetometry ( VSM ) and \nsuperconducting quantum interference device s (SQUID ) are capable of measuring the magnetic \nmoment  of these materials ,18,19 measurement s in real time while  apply ing external  stimuli is not \neasily achievable.  Methods that require electrical contact, such as transport measurements, have \nan extra layer of difficulty since monolayer films often do not span the surface continuously and \nthe size of the electrical contacts are large compared to the surface area.20 Optical methods based \non the magneto -optical Kerr effect (MOKE) have proven very successful in thin films and  have \nhad a measure of success with 2D materials such as CrI 3.16 However, local heating from high \nlaser powers causes thermal instability , which is a significant source of noise in these 5 \n measurements.  Therefore, c ryogenic temperatures are needed to reduce thermal and mechanical \nmeasurement noise , but for room temperature, this is not an option . Despite this, optical \nmeasurements  remain  a powerful practice  that yield crucial insight into the spin dynamics of \natomically thin materials .21,22 The shortcomings of these techniques  motivate  the develop ment of  \na new approach  to measure 2D magnetization in real time as external stimul i are applied . \nIn order to probe  the light-induced  magnetization  of an atomically thin  magnetic  film \nsuch as  V-WS 2 monolayer s, we propose  a new technique utilizing our recently  developed \nmagneto -LC resonance sensor with ultrahigh magnetic field sensitivity (pT  regime) .23,24 The \nsensing element of this device is a magnetic microwire wound into a coil driven with a frequency \nof 118 MHz, which is near the coil’s LC resonance  (Fig. S2). The impedance of the coil is then \nmeasured with an impedance analyzer, from which we extract the reactance of the coil. The \nsetup is depicted in Fig. 1. The film is placed at the core of the coil and the reactance of the coil \nis measured. Theoretically, the reactance of the coil is written  as  \n𝑋coil=𝜔[L(1−𝜔2LC𝑝𝑎𝑟)−𝐶par𝑅𝑝𝑎𝑟2]\n(1−𝜔2L𝐶𝑝𝑎𝑟)2+(𝜔𝐶𝑝𝑎𝑟𝑅𝑝𝑎𝑟)2 , \nwhere 𝜔 is the angular frequency . Since the film is ferro magnetic, it will modify the relative \npermeability of the space within the coil, thus changing the magnetic flux through the coil and \nconsequently the reactance of the coil . As the microwire itself is ferro magnetic, the \nmagnetization of the film  will also lead to a change in the  effective permeability of the \nmicrowire. Therefore , the reactance of the sensor depends on this effective permeability, \n𝑋=𝑋(𝜇𝑒𝑓𝑓). Changes in the permeability of the film upon light illumination will influence  the \neffective permeability  of the coil , which can be accessed through the change in its  reactance , \n𝑋=𝑋(𝜇𝑒𝑓𝑓,𝑙𝑎𝑠𝑒𝑟  𝑜𝑛)− 𝑋(𝜇𝑒𝑓𝑓,𝑙𝑎𝑠𝑒𝑟  𝑜𝑓𝑓 ).  6 \n Results and Discussion  \n Due to the presence of defects and edge effects in WS 2 monolayers, a small magnetic \nmoment may be present in these “pristine” films . Therefore , we first  examine whether \nphotogenerated electron -hole pairs have any effect on the magnetization of the film. Figure 2(a) \nshows  the signal obtained from  a WS 2 monolayer film with and with out laser excitation . In this \nfigure,  a small change in the signal,  likely due to thermal fluctuations in the film , can be seen . \nNext, w e perform ed the measurement on the 2  at.% V-doped WS 2 monolayer (Fig. 2(b)), where \nwe observe a large change in reactance (magnetic permeability  or magnetization ) upon \nillumination with the same  laser.  The excitation wavelength is  = 650 nm ( h = 1.91 eV) with \nthe power of 5.1 mW/cm2 at the sample surface. The laser was operated for several minutes \nduring these measurements , which were carried out at room temperature . Since prolonged laser \nexposure may heat up the magnetic coil and hence change the magnetic propert ies, coil heating \neffects due to laser exposure have also been studied. As shown in Fig. S 3, the dot laser, which \ncovers  only a  small area of the coil  (0.11 cm2), has a negligible  effect on the magneti sm of the  \ncoil. These findings  indicate that the observed enhancement of the magnetization/ permeability \nfrom the experimental setup, i.e.  illuminated V -WS 2 monolayer (Fig. 2b ) within the coil,  is not \ndue to a laser/sample heating effect but originate s from  carrier -mediated ferromagnetism , similar \nto the case of a p-type (In,Mn)As/GaSb  semiconductor .7 As show n in Fig. S 4, measurements on \nthe same V -WS 2 sample were performed weeks apart to demonstrated that we can reproduce the \nchange in magnetization, and confirm th e reversibility and reproducibility of this effect.  \nMeasured magnetic hysteresis ( M-H) loops at room temperature before and after illumination \nalso confirm that the process is reversible ( Fig. S 5). It is worth mentioning that upon light \nillumination  with comparable  laser powers , the magnetization change in the illuminated 7 \n (In,Mn)As/GaSb  film ( = 685 nm, 6 mW/cm2) was only observed below 50 K, while enhance d \nmagnetization is achieved at room temperature for the illuminated V -WS 2 monolayer (  = 650 \nnm, 5.1 mW/cm2). This striking difference makes this atomically thin ferromagnetic \nsemiconductor a promising candidate for use  in light-controlled  spintronics and other \nmultifunctional nanodevices . The semiconducting nature of V -WS 2 facilitates incorporation to \ncurrent silicon -based  technology  and prov ides a platform for optoelectronic phenomena ; \ncombined  with its FM properties  we obtain a unique way to manipulate the spin states in the \nmaterial  by illuminating it with light . \nIt is of interest  to determine how  an increase in illumination area of the sample would \naffect the permeability/ magnetization,  so we illuminated the film with two 650 nm lasers , labeled \n‘dot’ and ‘target’ lasers , with 0. 11 cm2 and 0.41 cm2 coverage areas , respectively . The two lasers \nhave  different spot s izes, but the same light intensity of 4.2  mW/cm2. In Fig. 3(a,b) it is observe d \nthat increasing the coverage area increases the change in magnetization . Finally, we sought to \ndetermine the light intensity dependence of the change in magnetization. In Fig. 3(c,d) we \ndemonstrate this for both 650 nm lasers , both of which  show  a similar trend. Initially , we see a \nsharp increase in magnetization with increasing light intensity , and at higher laser intensities the \nchange in magnetization begin s to saturate. Since higher laser powers bring about a considerable \nheating effect, which may damage the coil or the sample , in this study we restricted the laser \nintensity below 6 mW/cm2 for light -induced magnetization experiments.  Photon concentration \nwas calculated as  a function of laser intensity, for each laser, using the relation E = nhν \n(where  E is the energy and ν is the frequency,  n is the photon -generated carrier concentration, \nand h is Planck's  constant),  assuming 2% absorption in the V -WS 2 layer 25 and a ssuming that  \nonly 1% of the electron s/hole s creat ed do  not immediately recombine.  We observe that by 8 \n increasing the illumination area, a smaller photon concentration , compared to the “dot” laser,  is \nrequired to achieve a change on magnetization . A smaller photon concentration is also necessary \nto achieve the saturation feature; using the “dot laser” a concentration of ~3.1 x 1012 \nphotons/cm2/s is necessary to approach saturation, while using the “target laser” saturation starts \naround ~2.7 x 1012 photons/cm2/s. This points to a long range cumulative effect in which the \nenhanced magnetic moments in the illuminated area  may couple with the moments surrounding \nit. \nTo elaborate  these findings, the band structure s of the WS 2 and V-WS 2 monolayer s were  \ninvestigated  by density functional theory  (DFT)  calculations  and the results are shown in  Fig. S6. \nIn order to establish  the change of the WS 2 band structure by  a V atom, the valence band edge \nwas used as the reference.  The V atom induces two empty doping states : one is below the \nconduction band  and the other is on the top of the valence band. These two doping bands are flat \nand localized,  implying a weak interaction with W and S atoms. The latter atom  plays  the role of \nelectron acceptor , where it  accept s thermal ly excited electrons from the valence band  and \nmanifests  the p -type characteristic of the V -WS 2 monolayer .15 In addition, there are two strong \nhybridization bands between the V and W atoms, which induce  approximately 1µ B of the \nmagnetic moment in the V-WS 2 monolayer . The energy of the ferromagnetic state is 0.24  eV, \nwhich is  lower than that of the no -spin state. Similar hybridization bands have been reported in \nmonolayers of V-doped WSe 2 13,14\n, which also shows a lowering of the energy of a FM state  with \nCurie temperature above RT . This points to V -doped WSe 2 monolayers  as another candidate for \nlight mediated magnetism.  \nGenerally,  two main  factors have been suggested to  influence  the magnetic  moment of a \nDMS  upon  light illumination :26-30  (i) the population  of the free excited  carries in the conduction 9 \n and valence bands28, and (ii) the localized  excited carriers trapped by  magnetic doping states in \nthe band gap of the host material .26 The former effect is usually dominant in lightly  doped \nsamples  (e.g. 1.1% of Mn in GaAs28), whereas the latter becomes  significant in heavily doped \nsamples  (e.g. 10% of Mn in CdSe and HgTe )26,27, in which  the dopants form a new band in side \nthe gap of the host . Due to its single layer limit and V concentration of approximately 2% , both \nfree and localized excited carriers are expected to  mediate  the magneti zation  in the V -WS 2 film \nwhen illuminated with an appropriate power laser . Figure 4(a) shows the distribution of the \nmagnetic moments under varying carrier populations. Increasing the concentration of holes \nresults in a more robust magnetic moment across the lattice , where W atoms far from the V site \nshow an enhanced magnetic moment. The evolution  of the band structure is calculated under \ndifferent carrier concentrations , as shown  in Fig. 4(b) . We find  that while the Fermi level is \nshifted deeper inside the valence band with hole injection , it is shifted toward the conduction \nband edge with electro n injection.  The evolution of the exchange energy is also calculated , the \nresult of which is  presented  in Fig. 4(c) . The exchange energy  becomes stronger with increasing \nhole concentration , so the magnetic moment must also change as the carrier concentration varies. \nIndeed,  we observe an increase in the magnetic moment with increasing hole concentration  (Fig. \n4(c)). This is consistent with our experimental findings  (Fig. 3c,d ), where a higher light intensity \nresult ed in a larger hole concentration  and consequently an increased magnetic moment . At large  \nhole concentrations the magnetic moment saturates, confirm ing the saturation feature we \nobserved experimentally ( Fig. 3c,d ). This is also consistent with another report ,31 in which hole \ninjection into a V-WSe 2 monolayer s increase the magnetic moment . We should note  that both \nelectrons and holes are populated in experiments whereas the simulation considers the separated \neffect for each type of carriers.  10 \n  In summary, we have demonstrated that magnetism can be tuned with light in V -doped \nWS 2 monolayers , by varying the light intensity or by changing the illumination area.  As the film \nis illuminate d, the absorbed photons generate electron -hole pair s and the electrons are captured \nby the electron deficient V sites,  which  generat e an imbalance in the carrier population and , \nhence , a change in magnetic moment. We have shown that the carrier concentration can be tuned \nby changing the light intensity, allowing control over the  magnetic moment of the film . Density \nfunctional c alculations confirm that the magnetic moment of the V-WS 2 monolayer can be \nenhanced by increasing the hole concentration.  All of this is achieved at room temperature which \nhas been a key obstacle to  applied  2D spintronics. These findings highlight the potential for \ncutting -edge  application s of 2D DMS and other atomically thin magnetic semiconductors . \nMethods  \nSample synthesis  and characterization:  \nThe WS 2 and 2  at.% V-doped WS 2 monolayer films were synthesized using a reliable single -step \npowder vaporization method.  STEM -EDS of the sample s were  performed by a FEI Talos F200X \nmicroscope  with a SuperX EDS detector  operating at 200 kV. Photoluminescence ( PL) spectra of \nthe samples were recorded with a  Coherent Innova 70C argon -krypton laser at 532 nm. Details of \nthe samples ’ synthesis and characterization have been reported previously .15  \nMagnetic and light -induced m easurements:  \nMagnetization  versus magnetic field  (M-H) measurements were carried out at room \ntemperature in a Physical Property Measurement System (PPMS) from Quantum Design with a \nvibrating sample magnetometer (VSM) magnetometer in fields up to 9 T.  The M-H loops of the 11 \n V-WS 2 monolayer film before and after l ight illumination (  = 650 nm, 5.1 mW/cm2) were also \nrecorded at room temperature to investigate the magnetization re laxation  of the sample.  \nLight -induced  magnetization measurements on the WS 2 and V -WS 2 monolayer films \nwere performed with  a magnetic microwire LC resonator as the sensing element  (see Fig. S 2).24 \nThe reactance (X) is monitored as the light illuminates the film  with an HP  4191A impedance \nanalyzer . Two different lasers of 650 nm wavelength  but different spot sizes were  used. One, \nwhich is referred to as the ‘dot laser ,’ has a round shaped spot and covers an area of 0.11 cm2. \nThe second, which is refer red to as the ‘target laser’ , has a target -shape d spot , in which one of \nthe lines covers the entire surface of the film with a surface area of 0.41 cm2. The target laser has \na significant heating effect on the coil  and demonstrates a  reactance change of about ∆𝑋=1 𝛺. \nTo correct for this, we subtract ed ∆𝑋 from the data obtained using this laser. To prevent \nsignificant hea ting effects, low powered lasers  were used  (< 5 mW).  \nTheoretical  calculations  and simulation:  \nThe band structure s of the pristine WS 2 and V -doped WS 2 monolayers were calculated \nusing Quantum Espresso code.32 Projector -augmented wave potentials were used with  a cut -off \nenergy of 30 Ry.33 A 8x8x1 supercell, which corresponds to 1.6% of V atoms, was used with a \n3x3x1 k -point grid. The structures of WS 2 and V -doped WS 2 were optimized until the \nconvergence of force and energy is smaller than 0.0001 Ha and 0.001 (Ha/bohr), respectively, \nwithout including spin -orbit coupling. The initial spin state was induced along z direction  and the \ntotal energy is optimized with 10-6 Ha of convergence including spin -orbit coupling . To describe \nthe strong correlation of the d electron s, the GGA+U method with U  = 3 was used for the \nvanadium atom s.  12 \n Acknowledgments  \nResearch at USF was supported by  the U.S. Department of Energy, Office of Basic Energy \nSciences, Division of Materials Sciences and Engineering under Award No. DE -FG02 -\n07ER46438 . Research at P SU was supported by the Air Force Office of Scientific Research \n(AFOSR) through grant No. FA9550 -18-1-0072  and the NSF -IUCRC Center for Atomically \nThin Multifunctional Coatings (ATOMIC) .  \nAuthor Contributions  \nV.O.J. and M.H.P. conceived  the initial concept. V.O.J. performed light -controlled \nmagneti zation  experiments and analyzed the data  with inputs from T.E. and B.M . Y.H.T.N. and \nV.K. performed magnetic measurements and analyzed the magnetic data. F.Z. and M.L. \nsynthesized the films and characterized the  structural and optical properties. L.D.D. performed  \nthe computational calculations . 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Phys. 86, 855 (2014)  \n31 Duong, D.L., Kim, S.G. & Lee, Y.H. Gate modulation of the long -range magnetic order in a \nvanadium -doped WSe 2 semiconductor. AIP Advances  10, 065220 (2020)  \n32 Giannozzi , P., et al. , QUANTUM ESPRESSO: a modular and open -source software project for \nquantum simulations of materials , J. Phys. Condens. Matter 21, 395502 (2009)  \n33 E. Kucukbenli, M. Monni, B. I. Adetunji, X. Ge, G. a Adebayo, N. Marzari, S. de Gironcoli, \nand a D.  Corso, http://Arxiv.Org/Abs/1404.3015 (2014).  16 \n Figure  caption s  \n \nFig. 1: (a) An illustration of the measurement  scheme  for light-induced magnetization of the V-\nWS 2 monolayer ; (b) a sketch showing photon absorption generating  a hole -electron pair in the \nfilm.   \nFig. 2: The c hange in reactance upon illumination with a 650 nm laser for (a) WS 2 and (b) V-\nWS 2 as measured by the microwire coil sensor . The change in reactance is proportional to the \nchange in magnetization. A negligible change in magnetization is observed in pristine WS 2 while \na significant change in magnetization was measured on the V - doped WS 2 monolayer.  \nFig. 3: The dependence of the change in reactance (∆𝑋) on illumination area is shown in (a,b) ; \n(c,d)  shows the dependence of ∆𝑋 on light intensity and photon concentration for two different \nillumination areas . All measurements shown were made using  a 650 nm light source s, labeled \n“dot” and “target lasers  with illumination area s 0.11 cm2\n and 0.41 cm2, respectively . We \nobserved a similar trend for both lasers in which increasing light intensity (and photon \nconcentration) results in an increase in magnetization, until it reaches a critical photon \nconcentration at which saturation begins. Notably, a smaller photon conc entration ( ~2.7 x 1012 \nphotons/cm2/s) is required to begin saturation using the “target” laser than with the “dot” laser  \n(~3.1 x 1012 photons/cm2/s) \nFig. 4: (a) The p roject ed magnetic moments along the c-axis in the case of an injecti on of  one \nhole and one electron ; (b) the band structures of the V -doped WS 2 monolayer with different \ndoping carrier densities ; and (c) the exchange energy and net  magnetic moments of the V-doped \nWS 2 monolayer with different carrier doping densities . Net magnetic moment increases with 17 \n increasing hole concentration, consistent with experimental results. A saturation feature is also \npresent at higher hole concentrations, confirming what we observed experimentally.  \n \n \n  18 \n Figure 1 \n \n \n \n \n \n  \n19 \n Figure 2 \n \n \n \n \n \n \n \n  \n20 \n Figure 3 \n \n \n \n \n \n \n \n \n  \n21 \n Figure 4 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n"    },    {        "title": "0809.2611v1.Stochastic_dynamics_of_magnetization_in_a_ferromagnetic_nanoparticle_out_of_equilibrium.pdf",        "content": "arXiv:0809.2611v1  [cond-mat.mes-hall]  16 Sep 2008Stochastic dynamics of magnetization in a ferromagnetic na noparticle out of\nequilibrium\nDenis M. Basko1and Maxim G. Vavilov2\n1International School of Advanced Studies (SISSA), via Beir ut 2-4, 34014 Trieste, Italy\n2Department of Physics, University of Wisconsin, Madison, W I 53706, USA\n(Dated: September 15, 2008)\nWe consider a small metallic particle (quantumdot) where fe rromagnetism arises as a consequence\nof Stoner instability. When the particle is connected to ele ctrodes, exchange of electrons between\nthe particle and the electrodes leads to a temperature- and b ias-driven Brownian motion of the\ndirection of the particle magnetization. Under certain con ditions this Brownian motion is described\nby the stochastic Landau-Lifshitz-Gilbert equation. As an example of its application, we calculate\nthe frequency-dependentmagnetic susceptibility of the pa rticle in a constant external magnetic field,\nwhich is relevant for ferromagnetic resonance measurement s.\nPACS numbers: 73.23.-b, 73.40.-c, 73.50.Fq\nI. INTRODUCTION\nThe description of fluctuations of the magnetization\nin small ferromagnetic particles pioneered by Brown1is\nbased on the Landau-Lifshitz-Gilbert (LLG) equation2,3\nwith a phenomenologically added stochastic term. This\napproach has been widely used: just a few recent appli-\ncations are a numerical study of the dynamic response\nof the magnetization to the oscillatory magnetic field,4\na numerical study of ferromagnetic resonance spectra,5\nstudy of resistance noise in spin valves,6and a study of\nthe magnetization switching and relaxation in the pres-\nence of anisotropy and a rotating magnetic field.7\nIn equilibrium the statistics of stochastic term in\nthe LLG equation can be simply written from the\nfluctuation-dissipation theorem.1However, out of equi-\nlibrium a proper microscopic derivation is required. Mi-\ncroscopic derivations of the stochastic LLG equation out\nof equilibrium, available in the literature, use the model\nof a localized spin coupled to itinerant electrons,8,9,10,11\nor deal with non-interacting electrons.12In contrast to\nthis approach, we start from a purely electronic system\nwhere the magnetization arises as a consequence of the\nStoner instability. Our derivation has certain similarity\nwith that of Ref. 13 for a bulk ferromagnet, where the di-\nrection of magnetization is fixed and cannot be changed\nglobally, so its local fluctuations are small and their de-\nscription by a gaussianaction is sufficient. This situation\nshould be contrasted to the case of a nanoparticle where\nthe direction ofthe magnetizationcan be completely ran-\ndomized by the fluctuations, so that the effective action\nfor the direction of the classicalmagnetization has a non-\ngaussian part. The bias-driven Brownian motion of the\nmagnetization with a fixed direction (due to and easy-\naxis anisotropy and ferromagnetic electrodes) has been\nalso studied in Ref. 14 using rate equations.\nWe assume that the single-electron spectrum of the\nparticle, which is also called a quantum dot in the litera-\nture, to be chaotic and described by the random-matrix\ntheory15,16. Totakeintoaccounttheelectron-electronin-teractions in the dot we use the universal Hamiltonian,17\nwith a generalized spin part, corresponding to a ferro-\nmagnetic particle. Electrons occupy the quantum states\nof the full Hamiltonian and form a net spin of the par-\nticle of order of S0≫1; throughout the paper we use\n¯h= 1. The dot is coupled to two leads, see Fig. 1, which\nwe assumed to be non-magnetic. The approach can be\neasily extended to the case of magnetic leads. The num-\nberNchof the transversechannels in the leads, which are\nwell coupled to the dot, is assumed to be large, Nch≫1.\nEquivalently, the escape rate 1 /τof electrons from the\ndot into the leads is largecompared to the single-electron\nmean level spacing δ1in the dot. This coupling to the\nleads is responsible for tunnelling processes of electrons\nbetween states in the leads and in the dot with random\nspin orientation. As a result of such tunnelling events,\nthe net spin of the particle changes. We show that this\nexchangeofelectronsgivesrisebothtotheGilbert damp-\ning and the magnetization noise in the presented model,\nand under conditions specified below, the time evolution\nof the particle spin is described by the stochastic LLG\nequation.\nWe study in detail the conditions for applicability of\nthe stochastic approach. We find that these limits are\nset by three independent criteria. First, the contact re-\nsistance should be low compared to the resistance quan-\ntum, which is equivalent to Nch≫1. If this condi-\ntion is broken, the statistics of the noise cannot be con-\nsidered gaussian. Physically, this condition means that\neach channel can be viewed as an independent source\nof noise, so the contribution of many channels results in\nthe gaussian noise by virtue of the central limit theo-\nrem ifNch≫1. Second, the system should not be too\nclose to the Stoner instability: the mean-field value of\nthe total spin S0≫√Nch. If this condition is violated,\nthe fluctuations of the absolute value of the magneti-\nzation become of the order of the magnetization itself.\nThird,S2\n0≫Teff/δ1, whereTeffis the effective tempera-\nture of the system, which is the energy scale of the elec-\ntronicdistribution function determined bya combination2\nNLNRB~\nz \nB~\nB0Drain Source\nµ+eVµFerromagnetic\nnanoparticle\nµ\nFIG. 1: (Color online). Device setup considered in this work :\na small ferromagnetic particle (quantum dot) coupled to two\nnon-magnetic leads (see text for details).\noftemperature andbiasvoltage(the Boltzmannconstant\nkB= 1throughoutthe paper). Otherwise, theseparation\nof the degrees of freedom into slow (the direction of the\nmagnetization) and fast (the electron dynamics and the\nfluctuations of the absolute value of the magnetization)\nis not possible.\nIn the present model we completely neglect the spin-\norbit interaction inside the particle, whose effect is as-\nsumedtobeweakascomparedtotheeffectoftheleads.18\nThe effects of the electron-electron interaction in the\ncharge channel (weak Coulomb blockade) are suppressed\nforNch≫1,15so we do not consider it.\nAs an application of the formalism, we consider themagnetic susceptibility in the ferromagnetic resonance\nmeasurements, which is a standardcharacteristicofmag-\nnetic samples. Recently, a progress was reported in mea-\nsurements of the magnetic susceptibility on small spatial\nscales in response to high-frequency magnetic fields.19\nMeasurements of the ferromagnetic resonance were also\nreportedfornanoparticles, connected to leadsfor asome-\nwhat different setup in Ref. 20.\nThe paper is organized as follows. In Sec. II we intro-\nduce the model for electrons in a small metallic particle\nsubject to Stoner instability. In Sec. III we analyze the\neffective bosonic action for the magnetization of the par-\nticle. In Sec. IV we obtain the equation of motion for the\nmagnetization with the stochastic Langevin term, which\nhas the form of the stochastic Landau-Lifshitz-Gilbert\nequation, and derive the associated Fokker-Planck equa-\ntion. In Sec. V we discuss the conditions for the applica-\nbility of the approach. In Sec. VI we calculate the mag-\nnetic susceptibility from the stochastic LLG equation.\nII. MODEL AND BASIC FORMALISM\nWithin the random matrix theory framework, elec-\ntrons in a closed chaotic quantum dot are described by\nthe following fermionic action:\nS[ψ,ψ∗] =/contintegraldisplay\ndt\nN/summationdisplay\nn,n′=1ψ†\nn(t)(δnn′i∂t−Hnn′)ψn′(t)−E(S(t))\n, Si≡N/summationdisplay\nn=1ψ†\nnˆσi\n2ψn. (2.1)\nHereψnis a two-component Grassmann spinor, truns along the Keldysh contour, as marked by/contintegraltext\n; ˆσx,y,zare the Pauli\nmatrices (we use the hat to indicate matrices in the spin space and us e the notation ˆ σ0for the 2 ×2 unit matrix).\nHnn′is anN×Nrandom matrix from a gaussian orthogonal ensemble, described by the pair correlators:\nHmnHm′n′=Nδ2\n1\nπ2[δmn′δnm′+δmm′δnn′]. (2.2)\nHereδ1is the mean single-particle level spacing in the dot.\nThe magnetization energy E(S) is the generalizationof the JsS2term in the universal Hamiltonian for the electron-\nelectron interaction in a chaotic quantum dot.17Since we are going to describe a ferromagnetic state with a largevalu e\nof the total spin on the dot, we must go beyond the quadratic term ; in fact, all terms should be included. E(S) can be\nviewed as the sum of all irreducible many-particle vertices in the spin c hannel, obtained after integrating out degrees\nof freedom with high energies (above Thouless energy); the corre sponding term in the action is thus local in time, and\ncan be written as the time integral of an instantaneous function E(S(t)). This functional can be decoupled using the\nHubbard-Stratonovich transformation with a real vector field h(t), which we call below the internal magnetic field:\nexp/parenleftbigg\n−i/contintegraldisplay\ndtE(S)/parenrightbigg\n=/integraldisplay\nDh(t) exp/parenleftbigg\ni/contintegraldisplay\ndt(2h·S−˜E(h))/parenrightbigg\n. (2.3)\nWe rewrite the action S[ψ,ψ∗] in the form\nS[ψ,ψ∗,h] =/contintegraldisplay\ndt\nN/summationdisplay\nn,n′=1ψ†\nn(t)/parenleftBig\nˆG−1/parenrightBig\nnn′ψn′(t)−˜E(h(t))\n, (2.4)3\nwhere the inverse Green’s function\n/parenleftBig\nˆG−1/parenrightBig\nnn′= (iˆσ0∂t+h·ˆσ)δnn′−Hnn′ˆσ0(2.5)\nis a matrix in time variables t,t′, in orbital indices nand\nn′with 1≤n,n′≤N, in spin indices, and in forward(+)\nand backward ( −) directions on the Keldysh contour.\nIntegration over fermionic fields ψn,ψ†\nnyields the purely\nbosonic action:\nS[h] =−iTr/braceleftBig\nln(−iˆG−1)/bracerightBig\n−/contintegraldisplay\ndt˜E(h(t)),(2.6)\nwhere the trace is taken over allindices of the Green’s\nfunction, listed above.\nIn the space of forward and backward directions on\nthe Keldysh contour, we perform the standard Keldysh\nrotation, introducing the retarded ( GR), advanced ( GA),\nKeldysh (GK), and zero ( GZ) components of the Green’s\nfunction:\n/parenleftbiggˆGRˆGK\nˆGZˆGA/parenrightbigg\n=1\n2/parenleftbigg\n1 1\n1−1/parenrightbigg/parenleftbiggˆG++ˆG+−\nˆG−+ˆG−−/parenrightbigg/parenleftbigg\n1 1\n−1 1/parenrightbigg\n,\n(2.7)\nas well as the classical ( hcl) and quantum ( hq) compo-\nnents of the field:\n/parenleftbigg\nhclhq\nhqhcl/parenrightbigg\n=1\n2/parenleftbigg\n1−1\n1 1/parenrightbigg/parenleftbigg\nh+0\n0−h−/parenrightbigg/parenleftbigg\n1 1\n1−1/parenrightbigg\n.\n(2.8)\nWe will also write this matrix as h=hclτcl+hqτq,\nwhereτclandτqare 2×2 matrices in the Keldysh space\ncoinciding with the unit 2 ×2 matrix and the first Pauli\nmatrix, ˆσx, respectively.\nThe saddle point of the bosonic action Eq. (2.6) is\nfound by the first order variation with respect to hcl,q(t),\nwhich gives the self-consistency equation:\nhq(t) = 0,−∂˜E(hcl(t))\n∂hcl\nj(t)=i\n2Tr\nn,σ/braceleftBig\nˆσjˆGK\nnn(t,t)/bracerightBig\n.(2.9)\nWe also note that the right-hand side of this equation is\nproportional to the total spin of electrons of the particle\nfor a given trajectory of hcl(t):\nS(t) =i\n4Tr\nn,σ/braceleftBig\nˆσjˆGK(t,t)/bracerightBig\n. (2.10)\nIn Eqs. (2.9) and (2.10), the trace is taken over orbital\nand spin indices only.\nIn the limit N→ ∞, one can obtain a closed equation\nfor the Green’s function traced over the orbital indices:21\nˆg(t,t′) =iδ1\nπN/summationdisplay\nn=1ˆGnn(t,t′). (2.11)\nThe matrix ˆ g(t,t′) satisfies the following constraint:\n/integraldisplay\nˆg(t,t′′)ˆg(t′′,t′)dt′′=τclˆσ0δ(t−t′),(2.12)where the right-hand side is just the direct product of\nunit matrices in the spin, Keldysh, and time indices.\nThe Wigner transform of ˆ gK(t,t′) is related to the spin-\ndependent distribution function ˆf(ε,t) of electrons in the\ndot:\n∞/integraldisplay\n−∞ˆgK(t+˜t/2,t−˜t/2)eiε˜td˜t= 2ˆf(ε,t).(2.13)\nIn equilibrium, ˆf(ε) = ˆσ0tanh(ε/2T).\nThe self-consistency condition (2.9) takes the form\n−∂˜E(hcl(t))\n∂hcl\ni(t)=π\n2δ1lim\nt′→tTr\nσ/braceleftbig\nˆσiˆgK(t,t′)/bracerightbig\n−2hcl\ni(t)\nδ1.\n(2.14)\nThe last term takes care of the anomaly arising from\nnon-commutativity of the limits N→ ∞andt′→t.\nIn this paper we consider the dot coupled to two leads,\nidentified as left ( L) and right ( R). The leads have\nNLandNRtransverse channels, respectively, see Fig 1.\nFor non-magnetic leads and spin-independent coupling\nbetween the leads and the particle, we can characterize\neach channel by its transmission Tnwith 0< Tn≤1\nand by the distribution function of electrons in the chan-\nnelFn(t−t′), assumed to be stationary. We consider\nthe limit of strong coupling between the leads and the\nparticle,/summationtextNch\nn=1Tn≫1.\nThe coupling to the leads gives rise to a self-energy\nterm, which should be included in the definition of the\nGreen’s function, Eq. (2.5). Without going into details\nof the derivation, presented in Ref. 21, we give the final\nform of the equation for the Green’s function traced over\nthe orbital states, Eq. (2.11):\n[∂t−ih·ˆσ,ˆg]\n=Nch/summationdisplay\nn=1Tnδ1\n2π/parenleftbigg\n−FnˆgZˆgRFn−FnˆgA−ˆgK\nˆgZ−ˆgZFn/parenrightbigg\n×/bracketleftbigg\nˆ1+Tn\n2/parenleftbigg\nˆgR−ˆ1+FnˆgZˆgRFn+FnˆgA\n0 −ˆgA−ˆ1+ ˆgZFn/parenrightbigg/bracketrightbigg−1\n.\n(2.15)\nHere the products of functions include convolution in\ntime variables. This equation is analogous to the Usadel\nequation used in the theory of dirty superconductors.22\nTo conclude this section, we discuss the dependence\n˜E(h). Deep in the ferromagnetic state, i.e.far from the\nStoner critical point, we expect the mean-field approach\nto give a good approximation for the total spin of the\ndot. Namely, the mean field acting on the electron spins,\nis given by 2 h0=dE(S)/dS≡E′(S). We then require\nthat the response of the system to this field gives the\nsame average value for the spin:\nS0=2h0\n2δ1=E′(S0)\n2δ1. (2.16)\nHere we evaluated S0from Eq. (2.10) and applied the\nself-consistency equation (2.14) to equilibrium state with4\nˆgK∝ˆσ0, when the contribution of the first term in the\nright hand side of Eq. (2.14) vanishes.\nNot expecting strong deviations of the magnitude of\nthe spin from the mean-field value, we focus on the form\nof˜E(h) when|h| ≈h0. The inverse Fourier transform of\nEq. (2.3) and angular integration for the isotropic E(S)\ngives\ne−i˜E(h)∆t= const∞/integraldisplay\n0sin2Sh∆t\n2Sh∆te−iE(S)∆tS2dS,(2.17)\nwhere ∆tis the infinitesimal time increment used in the\nconstruction of the functional integral in Eq. (2.3).\nExpanding E(S) near the mean-field value S0,\nE(S)≈E(S0)+E′(S−S0)+E′′\n2(S−S0)2,(2.18)\nperforming the integration in the stationary phase ap-\nproximation and using S0=h0/δ1=−E′/(2δ1), we ob-\ntain\n˜E(h) =−2(h−h0−E′′S0/2)2\nE′′+˜E0,(2.19)\nwhere˜E0ish−independent term. This expression for\n˜E(h) defines the action S[h], Eq. (2.6).\nThe energy E(S) does not contain the energy Eorb(S),\nassociated with the orbital motion of electrons in the\nparticle. Namely, to form a total spin Sof the par-\nticle, we have to redistribute Selectrons over orbital\nstates, which changes the orbital energy of electrons by\nEorb(S)≃δ1S2. The total energy Etot(S) of the particle\nis the sum of two terms: Etot(S) =E(S)+Eorb(S). Sim-\nilarly, we obtain the total energy of the system in terms\nof internal magnetic field\n˜Etot(h) =˜E(h)−h2\nδ1\n=−2/parenleftbigg1\nE′′+1\n2δ1/parenrightbigg\n(h−h0)2+˜E1,(2.20)\nwhere˜E1does not depend on h. We notice that the\nextremumof ˜Etot(h)correspondsto h=h0anddescribes\nthe expectation value of the internal magnetic field in an\nisolated particle. The energy cost of fluctuations of the\nmagnitude of the internal magnetic field is characterized\nby the coefficient 1 /E′′+1/2δ1.\nIII. KELDYSH ACTION\nIn this Section we analyze the action Eq. (2.6) for the\ninternal magnetic field hα. We expect that the classical\ncomponent hcl(t) of this field contains fast and small os-\ncillations of its magnitude around the mean-field value\nh0. We further expect that the orientation of hcl(t)\nchanges slowly in time, but is not restricted to smalldeviations from some specific direction. Based on this\npicture, we introduce a unit vector n(t), assumed to de-\npend slowly on time, and write\nhcl(t) = (h0+hcl\n/bardbl(t))n(t), (3.1)\nwherehcl\n/bardbl(t) is assumed to be fast and small. We expand\nthe action (2.6) to the second order in small fluctuations\nof the quantum component hq(t) and the radial classical\ncomponent hcl\n/bardbl(t):\nS[h]≈ −2π\nδ1/integraldisplay\ndtgK(t,t)hq(t)+\n+8\nE′′/integraldisplay\ndthcl\n/bardbl(t)n(t)hq(t)\n−/integraldisplay\ndtdt′ΠR\nij(t,t′)hq\ni(t)hcl\n/bardbl(t′)nj(t′)\n−/integraldisplay\ndtdt′ΠA\nij(t,t′)hcl\n/bardbl(t)ni(t)hq\nj(t′)\n−/integraldisplay\ndtdt′ΠK\nij(t,t′)hq\ni(t)hq\nj(t′).(3.2)\nThe applicability of this quadratic expansion is discussed\nin Sec. VB.\nIn Eq. (3.2) we introduced the polarization operator,\ndefined as the kernel of the quadratic part of the action\nof the fluctuating bosonic fields:\n/parenleftbigg\nΠZΠA\nΠRΠK/parenrightbigg\n≡/parenleftbigg\nΠcl,clΠcl,q\nΠq,clΠq,q/parenrightbigg\n,(3.3a)\nΠαβ\nij(t,t′) =i\n2δ2Tr/braceleftbig\nlnG−1/bracerightbig\nδhβ\nj(t′)δhα\ni(t),(3.3b)\nwhereα,β=cl,qandi,j=x,y,z. The short time\nanomaly is explicitly taken into account in the definition\nof the polarization operators, see Eq. (3.14) below.\nThefirsttermofEq.(3.2)containsthevector gKofthe\nKeldysh component of the Green function ˆ gK= ˆσ0gK\n0+\nˆσ·gK. We emphasize that the Green’s function and\nthe polarization operator in Eq. (3.2) are calculated at\nhcl\n/bardbl(t) = 0 and hq(t) = 0 for a given trajectory of the\nclassical field h0n(t).\nA. Keldysh component of the Green function\nFor the Green’s function in the classical field we have\nˆgR(t,t′) =−ˆgA(t,t′) = ˆσ0δ(t−t′),(3.4)\nwhile the Keldysh component satisfies the equation\n/bracketleftbig\n∂t−ih0n·ˆσ,ˆgK/bracketrightbig\n=Nch/summationdisplay\nn=1Tnδ1\n2π/parenleftbig\n2Fn−ˆgK/parenrightbig\n.(3.5)\nWe introduce the notation\n1\nτ=Nch/summationdisplay\nn=1Tnδ1\n2π=1\nτL+1\nτR, (3.6)5\nThen the scalar gK\n0and vector gKcomponents of ˆ gK=\nˆσ0gK\n0+ˆσ·gKsatisfy two coupled equations:\n/bracketleftbigg\n∂t+∂t′+1\nτ/bracketrightbigg\ngK\n0(t,t′) =Nch/summationdisplay\nn=1Tnδ1\n2π2Fn(t−t′)\n+ih0[n(t)−n(t′)]·gK(t,t′), (3.7a)/bracketleftbigg\n∂t+∂t′+1\nτ/bracketrightbigg\ngK(t,t′)+h0[n(t)+n(t′)]×gK(t,t′)\n=ih0[n(t)−n(t′)]gK\n0(t,t′). (3.7b)\nAs a zero approximation, we can consider the station-\nary situation: gK\n0(t,t′) =gK\n0(t−t′) andn(t) = const. In\nthis case, we have\ngK\n0(t,t′) =τ\nτL2FL(t−t′)+τ\nτR2FR(t−t′) (3.8)\nandgK= 0.\nFor an arbitrarytime dependence n(t), Eq. (3.7b) can-\nnot be solved analytically. However, if the variation of\nn(t) is slow enough, we can make a gradient expansion:\n/parenleftbigg\n∂t+1\nτ/parenrightbigg\ngK+2h0n×gK=i˜th0˙ngK\n0.(3.9)\nHere we introduced t= (t+t′)/2,˜t=t−t′,∂t+∂t′=∂t.\nThe dependence on ˜tis split off and remains unchanged,\nwhile for the dependence on tthe solution is determined\nby a linear operator L+\nn:\nL±\nn=/parenleftbigg1\nτ±∂t±2h0n×/parenrightbigg−1\n, (3.10a)\nL+\nn(ω)X(ω) =n(n·X(ω))\n−iω+1/τ+\n+1\n2/summationdisplay\n±−n×[n×X(ω)]±i[n×X(ω)]\n−i(ω±2h0)+1/τ.\n(3.10b)\nHere we assume that the direction of the internal mag-\nnetic field nchanges slowly in time, and |˙n|τ≪1.\nThus, all perturbations of gKdecay with the charac-\nteristic time τ. In particular, the solution of Eq. (3.9)\nhas the form\ngK=i˜th0L+\nn˙ngK\n0≈i˜th0τgK\n0˙n+2h0τn×˙n\n(2h0τ)2+1.(3.11)\nExpression for the first term in Eq. (3.2) can be easily\nobtained from Eq. (3.11) by taking the limit ˜t→0 and\ntaking into account that any fermionic distribution func-\ntion in the time representation has the following equal-\ntime asymptote:\ngK\n0(t,t′)≈2\niπ1\nt−t′,(t→t′).(3.12)\nWe have\ngK(t,t) =2h0τ\nπ˙n+2h0τn×˙n\n(2h0τ)2+1. (3.13)We notice that n·gK(t,t) = 0, and therefore the first\nterm in the action Eq. (3.2) is coupled only to the tan-\ngential fluctuations of hq(t)⊥n(t).\nB. Polarization operator\nWe express the polarization operator in terms of the\nunit vector n(t). The polarization operator can be rep-\nresented as the response of the Green’s functions to a\nchange in the field, as follows directly from the defini-\ntion (3.3b) and the expression (2.6) for the action:\nΠαβ\nij(t,t′) =π\n2δ1Tr4×4/braceleftbig\nταˆσiδˆg(t,t)/bracerightbig\nδhβ\nj(t′)−2\nδ1τq\nαβδijδ(t−t′).\n(3.14)\nHere the Green function δˆg(t,t) can be calculated as the\nfirst-order response of the solution of Eq. (2.15) to small\narbitrary (in all three directions) increments of δhcl(t)\nandδhq(t). The zero-order solution of Eq. (2.15) in the\nfieldhcl=h0nandhq= 0 is\nˆg(t,t) = ˆσ0/parenleftbigg\nδ(t−t′)gK\n0(t−t′)\n0−δ(t−t′)/parenrightbigg\n.(3.15)\nFirst, we calculate δˆgZ, which responds only to δhq:\n/bracketleftbigg\n∂t+∂t′−1\nτ/bracketrightbigg\nδˆgZ(t,t′)−ih0ˆσ·n(t)δˆgZ(t,t′)\n+δˆgZ(t,t′)ih0ˆσ·n(t′) = 2iˆσ·δhq(t)δ(t−t′).\n(3.16)\nSince∂t+∂t′=∂t, the solution always remains propor-\ntional toδ(t−t′):\nδˆgZ(t,t′) =−2iˆσ(L−\nnδhq)(t)δ(t−t′).(3.17)\nGivenδˆgZ, components δˆgR,Acan be found either from\nEq. (2.15), or, equivalently, using the constraint ˆ g2=ˆ1:\nˆgδˆg+δˆgˆg= 0⇒δˆgR=−ˆgKδˆgZ\n2, δˆgA=δˆgZˆgK\n2.\n(3.18)\nWe notice that both δˆgR,Arespond only to hq(t) and,\ntherefore,\nΠZ\nij(t,t′)∝Tr{ˆσi(δˆgR(t,t)+δˆgA(t,t))}\nδhcl\nj(t′)≡0.(3.19)\nThis equation ensures that the action along the Keldysh\ncontour vanishes for hq≡0.\nTo evaluate the remaining three components of the po-\nlarization operator, we can apply the variational deriva-\ntives to the sum of δˆgK(t,t) +δˆgZ(t,t) with respect to\neither classical δhcl(t′) or quantum δhq(t′) field, which\ngive ΠR\nij(t,t′) and ΠK\nij(t,t′), respectively. Then, the ad-\nvanced component ΠA\nij(t,t′) = [ΠR\nji(t′,t)]∗.6\nThe equation for δˆgK=ˆσ·δgKreads as\n/bracketleftbigg\n∂t+∂t′+1\nτ/bracketrightbigg\nδgK(t,t′)\n+h0/bracketleftbig\nn(t)×δgK(t,t′)−δgK(t,t′)×n(t′)/bracketrightbig\n=i/bracketleftbig\nδhcl(t)−δhcl(t′)/bracketrightbig\ngK\n0(t−t′)\n−2iδhq(t)δ(t−t′)−Q(t,t′),\n(3.20a)\nwhere\nQ(t,t′) =Nch/summationdisplay\nn=1Tnδ1\n2π/parenleftbigggK\n0\n2δgZgK\n0\n2+FnδgZFn/parenrightbigg\n−Nch/summationdisplay\nn=1Tn(1−Tn)δ1\n2π/parenleftbigggK\n0\n2−Fn/parenrightbigg\nδgZ/parenleftbigggK\n0\n2−Fn/parenrightbigg\n(3.20b)\nandδgZ= Tr{ˆσδˆgZ}/2 withδˆgZgiven by Eq. (3.17).\nTo calculate the retarded component ΠR\nijof the polar-\nization operator, we calculate the response of δgK(t,t′)\ntoδhqin the limit t′→t. Using the asymptotic behavior\nof the Fermi function, Eq. (3.12), we obtain:\nδgK(t,t) =/integraldisplaydω\n2π−2iω\nπL+\nn(ω)δhcl(ω)e−iωt.(3.21)Substituting this expression for δgK(t,t) to Eq. (3.14),\nwe obtain\nΠR\nij(ω) = ΠR\n/bardbl,ij(ω)+ΠR\n⊥,ij(ω),(3.22a)\nwith\nΠR\n/bardbl,ij(ω) =−2\nδ1ninj\n1−iωτ, (3.22b)\nΠR\n⊥,ij(ω) =−2\nδ1/summationdisplay\n±δij−ninj±ieijknk\n2(3.22c)\n×(1±2ih0τ)\n1−i(ω∓2h0)τ.\nHere we represented the polarization operator ΠR\nij(ω) as\na sum of the radial, ΠR\n/bardbl,ij(ω), and tangential, ΠR\n⊥,ij(ω),\nterms. We note that the action Eq. (3.2) contains only\ntheradialcomponentoftheretardedandadvancedpolar-\nization operators because we do not perform expansion\nin terms of the tangential fluctuations of the classical\ncomponent of the field hcl(t).\nIn response to δhq, both corrections δgK(t,t′) and\nδgZ(t,t′) contain terms ∝δ(t−t′), However, their sum\nδgK(t,t′)+δgZ(t,t′) remains finite in the limit t→t′:\nδgK(t,t′)+δgZ(t,t′) =−2i/integraldisplay\ndt′′/integraldisplay\ndt1dt2L+\nn(¯t−t1)Q(t1−t2+˜t/2;t2−t1+˜t/2)L−\nn(t2−t′′)δhq(t′′),(3.23a)\nQ(τ1;τ2) =2\nτδ(τ1)δ(τ2)−Nch/summationdisplay\nn=1Tnδ1\n2π/bracketleftbigggK\n0(τ1)\n2gK\n0(τ2)\n2+Fn(τ1)Fn(τ2)/bracketrightbigg\n(3.23b)\n−Nch/summationdisplay\nn=1Tn(1−Tn)δ1\n2π/bracketleftbigggK\n0(τ1)\n2−Fn(τ1)/bracketrightbigg/bracketleftbigggK\n0(τ2)\n2−Fn(τ2)/bracketrightbigg\nwith¯t= (t+t′)/2 and˜t=t−t′.\nFrom Eq. (3.23) we obtain the following expression for\nthe Keldysh component of the polarization operator:\nΠK\nij(ω) = ΠK\n/bardbl,ij(ω)+ΠK\n⊥,ij(ω), (3.24a)\nΠK\n/bardbl,ij(ω) =−ininj\nω2+1/τ2R(ω), (3.24b)\nΠK\n⊥,ij(ω) =−i\n2/summationdisplay\n±δij−ninj±ieijknk\n(ω∓2h0)2+1/τ2R(ω).(3.24c)\nHere function R(ω) coincides with the noise power of\nelectric current through a metallic particle in the approx-imation of non-interacting electrons\nR(ω) =Nch/summationdisplay\nn=1/integraldisplaydε\n8πTn\n×/braceleftBig/bracketleftbig\n8−gK\n0(ε)gK\n0(ε+ω)−4Fn(ε)Fn(ε+ω)/bracketrightbig\n+(1−Tn)/bracketleftbig\ngK\n0(ε)−2Fn(ε)/bracketrightbig/bracketleftbig\ngK\n0(ε+ω)−2Fn(ε+ω)/bracketrightbig/bracerightBig\n.\n(3.25)\nIn principle, electron-electron interaction in the charge\nchannel can be taken into account. The interaction mod-\nifies the expression Eq. (3.25) for R(ω) to the higher\norder23inτδ1≪1 and we neglect this correction here.\nIn this paper we consider a particle connected to\nelectron leads at temperature Twith the applied bias\nV. In this case, FL,R(ε) = tanh(ε−µL,R)/(2T) with7\nµL−µR=V, and the integration over εgives\n2πτR(ω) = 4ωcothω\n2T+ΞΥT(V,ω),(3.26)\nwhere\nΥT(V,ω)≡/summationdisplay\n±2(ω±V)cothω±V\n2T−4ωcothω\n2T\n(3.27)\nand Ξ is the ”Fano factor” for a dot\nΞ =τ2\nτLτR+τ3δ1\n2πτ2\nR/summationdisplay\nn∈LTn(1−Tn)\n+τ3δ1\n2πτ2\nL/summationdisplay\nn∈RTn(1−Tn).(3.28)\nAt|V| ≫Tthe function Υ T(V,ω) has two scales of ω:\n(i)Tsmears the non-analyticity at ω→0, but the value\nof ΥT(V,ω) deviates from Υ T(V,0) at|ω| ∼ |V|. Thus,\nthe typical time scale above which one can approximate\nΠK(ω) by a constant is at least ω≪max{T,|V|}. In the\nlimitω→0 we have\nΠK\nij(ω= 0) =−i8τTeff\nδ1/parenleftbigg\nninj+δij−ninj\n(2h0τ)2+1/parenrightbigg\n.(3.29)\nThe effective temperature Teffis given by\nTeff≡T+Ξ/parenleftbiggV\n2cothV\n2T−T/parenrightbigg\n.(3.30)\nC. Final form of the action\nWe can rewrite the action for magnetization field h=\n{hcl;hq}withhclin the form of Eq. (3.1) as a sum of\nthe radial and tangential terms:\nS[h] =S/bardbl[hcl\n/bardbl,hq\n/bardbl]+S⊥[n(t),hq\n⊥]. (3.31)\nThe radial term in the action has the form\nS/bardbl[hcl\n/bardbl,hq\n/bardbl] = (D−1\n/bardbl)αβ(t,t′)hα\n/bardbl(t)hβ\n/bardbl(t′),(3.32)\nwhere the inverse function of the internal magnetic field\npropagator is given by\n(D−1\n/bardbl)αβ(t,t′) =4\nE′′/parenleftbigg\n0 1\n1 0/parenrightbigg\nδ(t−t′)\n−/parenleftBigg\n0 ΠR\n/bardbl(t,t′)\nΠA\n/bardbl(t,t′) ΠK\n/bardbl(t,t′)/parenrightBigg(3.33)\nand Παβ\n/bardbl(t,t′) =ninjΠαβ\n/bardbl,ij(t,t′). From this equation we\nfind\nDR\n/bardbl(ω) =Dq,cl\n/bardbl(ω) =E′′\n4−iω+1/τ\n−iω+(δ1+E′′/2)/(τδ1),\n(3.34)andDA\n/bardbl(ω) = [DR\n/bardbl(ω)]∗. The Keldysh component is\nDK\n/bardbl(ω) =Dq,q\n/bardbl(ω) =DR\n/bardbl(ω)ΠK\n/bardbl(ω)DA\n/bardbl(ω).(3.35)\nThe tangential term in the action is\nS⊥[n(t),hq\n⊥] =−4h0τ\nδ1/integraldisplay\ndt(˙n+2h0τn×˙n)hq\n⊥\n(2h0τ)2+1\n−4\nδ1/integraldisplay\ndt[n(t)×[n(t)×B]]·hq\n⊥(t)\n−/integraldisplay\ndtdt′hq\n⊥,i(t)ΠK\n⊥,ij(t−t′)hq\n⊥,j(t′).\n(3.36)\nHere we recovered the external magnetic field B(t). The\npolarization operator ΠK\n⊥,ijis given by Eq. (3.24c).\nIV. LANGEVIN EQUATION\nA. Langevin equation for the direction of the\ninternal magnetic field\nIn this section we consider evolution of the direction\nvectorn, described by the tangential terms in the action,\nEq. (3.36). We neglect fluctuations of the magnitude\nof the internal magnetic field, h/bardbl, the conditions when\nthese fluctuations can be neglected are listed in the next\nsection.\nWe decouple the quadratic in hq\n⊥component of the\nactionin Eq. (3.36) by introducingan auxiliaryfield w(t)\nwith the probability distribution\nP[w(t)]∝exp/braceleftbigg4i\nδ2\n1/integraldisplay\ndtdt′(ΠK\n⊥)−1\nij(t,t′)wi(t)wj(t′)/bracerightbigg\n,\n(4.1)\nand the correlation function\n∝an}bracketle{twi(t)wj(t′)∝an}bracketri}ht=δ2\n1\n8iΠK\n⊥,ij(t,t′).(4.2)\nThe field w(t) plays the role of the gaussian random\nLangevin force. Integration of the tangential part of\nthe action, Eq. (3.36), over hq\n⊥produces a functional\nδ-function, whose argument determines the equation of\nmotion:\n˙n+2τh0[n×˙n]\n4τ2h2\n0+1−1\nτh0(w−n×[n×B]) = 0.(4.3)\nThe above equation can be resolved with respect to ˙n:\n˙n=−2[n×(w+B)]−1\nh0τ[n×[n×(w+B)]].(4.4)\nThis equation is the Langevin equation for the direc-\ntionn(t) of the internal magnetic field in the presence\nof the external magnetic field B(t) and the Langevin\nstochastic forces w(t).8\nB. The Fokker-Plank equation\nNext, we follow the standard procedure of derivation\nof the Fokker-Plank equation for the distribution P(n)\nof the probability for the internal magnetic field to point\nin the direction n. The probability distribution satisfies\nthe continuity equation:\n∂P\n∂t+∂Ji\n∂ni= 0, (4.5)\nwhere the probability current is defined as\nJ=−/parenleftbigg\n2n×B+1\nh0τn×[n×B]/parenrightbigg\nP\n+1\n2/angbracketleftBig\nξ/parenleftbigg\nξ·∂P\n∂n/parenrightbigg/angbracketrightBig (4.6)\nand the stochastic velocity ξis introduced in terms of\nthe field was\nξ=−2[n×w]−1\nh0τ[n×[n×w]].(4.7)\nThe derivative ∂/∂nis understood as the differentiation\nwithrespecttolocalEuclideancoordinatesinthetangent\nspace. Performing averaging over fluctuations of win\nEq. (4.6), we obtain\n∂P\n∂t=∂\n∂n/braceleftbigg(2h0τ)[n×B]+[n×[n×B]]\nh0τP/bracerightbigg\n+1\nT0∂2P\n∂n2,(4.8)\nwhere the time constant T0is defined as\nT0=2(h0τ)2\nτTeffδ1. (4.9)\nBelow we use the polar coordinates for the\ndirection of the internal magnetic field, n=\n{sinθcosϕ,sinθsinϕ,cosθ}. In this case the Fokker-\nPlank equation can be rewritten in the form\n∂P\n∂t=1\nsinθ∂\n∂ϕ/bracketleftbigg\nFϕP+1\nT01\nsinθ∂P\n∂ϕ/bracketrightbigg\n+1\nsinθ∂\n∂θ/bracketleftbigg\nsinθFθP+sinθ\nT0∂P\n∂θ/bracketrightbigg\n,(4.10)\nwhere\nFϕ=Bxsinϕ−Bycosϕ\nh0τ\n+ 2cosθ(Bxcosϕ+Bysinϕ)−2sinθBz,(4.11)\nFθ= 2(Bxsinϕ−Bycosϕ)\n−cosθ\nh0τ(Bxcosϕ+Bysinϕ)+sinθ\nh0τBz.(4.12)It should be supplemented by the normalization condi-\ntion:\n2π/integraldisplay\n0dϕπ/integraldisplay\n0sinθdθP(ϕ,θ) = 1, (4.13)\nwhich is preserved if the boundary conditions at θ= 0,π\nare imposed:\nlim\nθ→0,πsinθ2π/integraldisplay\n0dϕ∂P\n∂θ= 0. (4.14)\nBelow we apply the Fokker Plank equation for calcu-\nlations of the magnetization of a particle\nM=/integraldisplaydΩn\n4πnP(n) (4.15)\nV. APPLICABILITY OF THE APPROACH\nIn this section we discuss the conditions of validity\nof the stochastic LLG equation, see Eq. (4.10), for the\nmodel of ferromagnetic metallic particle connected to\nleads at finite bias. We briefly listed these conditions\nin the Introduction. Here we present their more detailed\nquantitative analysis.\nA. Fluctuations of the radial component of the\ninternal magnetic field\nWe represented the classical component of the internal\nmagnetic field hclin terms of a slowly varying direction\nn(t) and fast oscillations hcl\n/bardblof its magnitude around\nthe average value h0. Now, we evaluate the amplitude of\noscillations of the radial component hcl\n/bardblof the field, using\nthe radial term in the action, see Eqs. (3.31) and (3.32).\nThe typicalfrequencies fortime evolutionofsmallfluc-\ntuations of the internal magnetic field in the radial direc-\ntion are of order of\nω∼δ1+E′′/2\nδ11\nτ(5.1)\nas one can conclude from the explicit form of the prop-\nagatorDR\n/bardbl(ω), Eq. (3.34), of these fluctuations. This\nscale has the meaning of the inverse RC-time in the\nspin channel. Deep in the ferromagnetic state (i. e., far\nfrom the Stoner critical point E′′+2δ1= 0) we estimate\nδ1+E′′/2∼δ1(which is equivalent to E′′∼h0/S0),\nso this spin-channel RC-time is of the same order as the\nescape time τ. This estimate for the frequency range is\nconsistent with the simple picture, which describes the\nevolution of the internal magnetic field of the grain as\na response to a changing value of the total spin of the\nparticle due to random processes of electron exchange9\nbetween the dot and the leads. The electron exchange\nhappens with the characteristic rate 1 /τ.\nThe correlation function ∝an}bracketle{thcl\n/bardbl(t)hcl\n/bardbl(t′)∝an}bracketri}htcan be evalu-\nated by performing the Gaussian integration with the\nquadratic action in hcl\n/bardblandhq\n/bardbl. Using Eq. (3.35), we ob-\ntain the equal-time correlation function\n∝an}bracketle{t(hcl\n/bardbl)2∝an}bracketri}ht=i\n2/integraldisplaydω\n2πDK\n/bardbl(ω)\n=(E′′)2\n32τδ1/integraldisplaydω\n2π2πR(ω)\nω2+[1+E′′/(2δ1)]2/τ2.(5.2)\nThis equation gives the value of fluctuations of the radial\ncomponent of the internal magnetic field of the particle.\nThese fluctuations survive even in the limit T= 0 and\nV= 0, when R(ω) = 2|ω|/πτ. We have the following\nestimate\n∝an}bracketle{t(hcl\n/bardbl)2∝an}bracketri}ht=(E′′)2\n16πτδ1lnETτ\n1+E′′/(2δ1),(5.3)\nthe upper cutoff ETis the Thouless energy, ET=vF/L\nfor a ballistic dot with diameter Land electron Fermi\nvelocityvF.\nThe separation of the internal magnetic field into the\nradial and tangential components is justified, provided\nthat the fluctuations/radicalBig\n∝an}bracketle{t(hcl\n/bardbl)2∝an}bracketri}htof the radial component\nare much smaller than the average value of the field h0,\ni.e.∝an}bracketle{t(hcl\n/bardbl)2∝an}bracketri}ht ≪h2\n0. Using the estimate Eq. (5.3), we\nobtain the necessary requirement for the applicability of\nequations for the slow evolution of the vector of the in-\nternal magnetic field of a particle:\nS0≫/radicalbigg\n1\nτδ1ln(ETτ), (5.4)\nwhereS0is the spin of a particle in equilibrium and\nwe again used the estimate E′′∼h0/S0. Condition of\nEq. (5.4) requires that the system is not close to the\nStoner instability.\nB. Applicability of the gaussian approximation\nLet us discuss the applicability of the gaussian approx-\nimation for the action in hcl\n/bardblandhq. The coefficients in\nfront of terms hq(t)h/bardbl(t1)...hcl\n/bardbl(tn) are obtained by tak-\ning thenth variational derivative of δgK(t,t)+δgZ(t,t),\nor, equivalently, byiteratingthe Usadelequation ntimes.\nSince the typical frequencies of h/bardblareω∼1/τ, the left-\nhand side of the equation is ∼δ(n+1)gK/τ, while the\nright-hand side is hcl\n/bardblδ(n)gK. Since the only time scale\nhereisτ, allthe coefficientsofthe expansionofthe action\ninhcl\n/bardbl(ω) atω∼1/τare of the same order:\nSn+1∼τn−1\nE′′/integraldisplaydω1...dωn\n(2π)n×\n×hcl\n/bardbl(ω1)...hcl\n/bardbl(ωn)hq(−ω1−...−ωn).(5.5)At the same time, the typical value of hcl\n/bardbl(ω∼1/τ),\nas determined by the gaussian part of the action, was\nestimated in the previous subsection to be of the order of/radicalBig\nτDK\n/bardbl(ω∼τ)∼√τδ1≪1, so the higher-order terms\nare indeed not important.\nFor the quantum component of the field the quadratic\nand quartic terms in the action are estimated as\nSn+1∼Teffτn\nδ1/integraldisplaydω1...dωn\n(2π)n×\n×hq(ω1)...hq(ωn)hq(−ω1−...−ωn).(5.6)\nIfTeff≫1/τ, then the typicalfrequencyscaleis ω∼1/τ,\nsothe quadraticterm gives hq(ω∼1/τ)∼/radicalbig\nδ1/Teff, and\nSn∼(δ1/Teff)n/2−1∼[τδ1/(τTeff)]n/2−1. IfTeff≪1/τ,\nat the typical scale ω∼Teffwe obtainhq(ω∼Teff)∼/radicalbig\nδ1/(T2\neffτ), so again Sn∼(τδ1)n/2−1≪1 forn>2.\nPhysically, the parameter 1 /(τδ1) =Nch(orTeff/δ1,\nif it is larger) can be identified with the number of the\nindependent sources of the noise acting on the magne-\ntization field. Thus, the smallness of the non-gaussian\npart of the action is nothing but the manifestation of the\ncentral limit theorem.\nC. Applicability of the Fokker-Plank equation\nFrom the above analysis we found that evolution of\nthe direction of the internal magnetic field in time is\ndescribed by a characteristic time T0, introduced in\nEq. (4.9). From the analysis of the fluctuations of the\nmagnitude of the internal magnetic field, see Eq. (5.1),\nwe obtain the following condition when the separation\ninto slow and fast variables is legitimate. The criterium\ncan be formulated as T0≫τ, which can be presented as\nTeff\nδ1≪/parenleftbiggh0\nδ1/parenrightbigg2\n=S2\n0. (5.7)\nVI. MAGNETIC SUSCEPTIBILITY OF\nMETALLIC PARTICLES OUT OF EQUILIBRIUM\nThe LLG equation derived in this paper for a ferro-\nmagnetic particle with finite bias between the leads can\nbe applied to a number of experimental setups. More-\nover, the derivation of the equation can be generalized\nto spin-anisotropic contacts with leads or Hamiltonian of\nelectron states in the particle. In this paper we apply\nthe stochastic equation for spin distribution function to\nthe analysis of the magnetic susceptibility at finite fre-\nquency. The susceptibility is the basic characteristic of\nmagnetic systems, it can often be measured directly, and\ndetermines other measurable quantities.\nBelow, we calculate the susceptibility of an ensemble\nof particles placed in constant magnetic field of an ar-\nbitrary strength and oscillating weak magnetic field, see10\nFig. 1. We consider the oscillating magnetic field with\nits components in directions parallel and perpendicular\nto the constant magnetic field.\nA. Solution at zero noise power\nAtTeff= 0 when w(t) = 0, and at fixed direction\nof the field, B(t) =ezB(t), equation of motion (4.4) is\neasily integrated for an arbitrary time dependence B(t):\nϕ=ϕ0+t/integraldisplay\n02B(t′)dt′, (6.1)\ntanθ\n2= tanθ0\n2exp\n−t/integraldisplay\n0B(t′)\nh0τdt′\n.(6.2)\nHere the direction of magnetic field correspondsto θ= 0.\nB. Constant magnetic field\nAt finiteTeffin constant magnetic field B0the Fokker-\nPlank equation has a simple solution\nP0(θ) =b\nsinhbebcosθ\n4π, (6.3)\nwhere the strength of constant magnetic field is written\nin terms of the dimensionless parameter\nb≡(2h0τ)B0\nτδ1Teff. (6.4)\nSubstituting this probability function to Eq. (4.15), we\nobtain the classical Langevin expression for the magne-\ntization of a particle in a magnetic field\nMz= cothb−1\nb, Mx=My= 0.(6.5)\nThis expression for the magnetization coincides with the\nmagnetization of a metallic particle in thermal equilib-\nrium, provided that the temperature is replaced by the\neffective temperature Teffdefined by Eq. (3.30).\nThe differential dc susceptibility is equal to\nχdc\n/bardbl=dMz(b)\ndb=1\nb2−1\nsinh2b. (6.6)\nC. Longitudinal susceptibility\nWe now consider the response of the magnetization to\nweakoscillations ˜Bz(t)oftheexternalmagneticfieldwith\nfrequencyωin direction parallel to the fixed magnetic\nfieldB0. We write the oscillatory component of the field\nin terms of the dimensionless field strength:\nb/bardble−iωt+b∗\n/bardbleiωt=2h0˜Bz(t)\nδ1Teff. (6.7)The linear correction to the probability distribution can\nbe cast in the form\nP(θ,t) =/bracketleftBig\n1+b/bardblu/bardbl(θ)e−iωt+b∗\n/bardblu∗\n/bardbl(θ)eiωt/bracketrightBig\nP0(θ),\n(6.8)\nwithP0(θ) defined by Eq. (6.3). The magnetic ac sus-\nceptibility can be evaluated from Eq. (6.8) as\nχ/bardbl(ω,b) = 2ππ/integraldisplay\n0u/bardbl(θ)P0(θ)cosθsinθdθ. (6.9)\nTheequationfor u/bardbl(θ) isobtainedfromEq.(4.10)with\nBz=B0+˜Bz(t):\n∂2u/bardbl\n∂θ2+cosθ−bsin2θ\nsinθ∂u/bardbl\n∂θ+iΩu/bardbl=bsin2θ−2cosθ,\n(6.10)\nwhere we introduced the dimensionless frequency\nΩ =ωT0, (6.11)\nand the time constant T0is defined in Eq. (4.9).\nNote the symmetry of Eq. (6.10) with respect to the\nsimultaneous change b→ −bandθ→π−θ. Also, the\nnormalization condition for the probability function re-\nquires that\nπ/integraldisplay\n0u/bardbl(θ)P0(θ)sinθdθ= 0. (6.12)\nThe latter holds if the boundary conditions Eq. (4.14)\nare satisfied, which in the case of axial symmetry can be\nwritten as\nlim\nθ→0,π/braceleftbigg\nsinθ∂u/bardbl(θ)\n∂θ/bracerightbigg\n= 0. (6.13)\nThedifferentialequation(6.10)withtheboundarycon-\ndition Eq. (6.13) can be solved numerically and then the\nsusceptibility is evaluated according to Eq. (6.9). The\nresult is shown in Figs. 2 and 3, where the susceptibility\nis shown as a function of frequency ωor magnetic field b,\nrespectively. We also consider various asymptotes for the\nacsusceptibility, obtainedfromthe solutionofEq.(6.10).\nAt zero constant magnetic field, b= 0, we find the\nexact solution of Eq. (6.10) explicitly:\nu/bardbl(θ) =cosθ\n1−iΩ/2. (6.14)\nThis solution allows us to calculate the ac susceptibility\nin the form\nχ/bardbl(Ω,b= 0) =1\n31\n1−iΩ/2. (6.15)\nForb≫1 only cosθ∼1/bmatter, and we can find a\nspecific solution of the inhomogeneous equation:\nu/bardbl(θ) =1−b(1−cosθ)\nb−iΩ/2, b≫1. (6.16)11\n0.000.100.200.30\n02468100.000.040.080.120.16 PSfrag replacementsReχ/bardbl(Ω,b) Imχ/bardbl(Ω,b)\nΩb= 0.1\nb= 0.5\nb= 1\nb= 2\nb= 5\nFIG. 2: (Color online). Plot of the real and imaginary parts\nof the susceptibility χ/bardbl(Ω,b) as a function of the dimension-\nless frequency Ω = ωT0. The oscillatory field at frequency ω\nis parallel to the constant magnetic field with strength b. The\nreal part of the susceptibility decreases monotonically fr om\nits dc value, Eq. (6.6), as frequency increases, while the im ag-\ninary part increases linearly at small Ω ≪1, see Eq. (6.20),\nand decreases at higher frequencies.\nThe requirement of regularity at the opposite end can\nbe replaced by the probability normalization condition,\nEq. (6.12), which is satisfied by this solution. Substitut-\ning this solution to Eq. (6.9), we obtain the strong field\nasymptote for the ac susceptibility\nχ/bardbl(Ω,b) =1\nb(b−iΩ/2). (6.17)\nFor Ω≫1 and Ω ≫b, we can neglect the derivatives\nin Eq. (6.10) and find the solution in the form\nu/bardbl(θ)≈bsin2θ−2cosθ\niΩ, (6.18)\nThis solution u/bardbl(θ) also satisfies Eq. (6.12). For the sus-\nceptibility, Eq. (6.9), we obtain\nχ/bardbl(Ω→ ∞,b) =2i\nΩ/parenleftbiggcothb\nb−1\nb2/parenrightbigg\n.(6.19)\nFinally, the low frequency limit can be also analyzed\nanalytically. The real part of the susceptibility coincides\nwith the differential susceptibility in dc magnetic field,\nEq. (6.6), for the imaginary part to the first order in\nfrequency we obtain, see Appendix,\nImχ/bardbl(Ω,b) = Ωf/bardbl(b). (6.20)\nThe function f/bardbl(b) has a complicated analyticalform and\nis not presented here, but its plot is shown in Fig. 4.\nIn all considered four limiting cases, the asymptotic\napproximations hold regardless the order in which the\nlimits are taken. Indeed, the asymptote of the expression\nfor the susceptibility in the zero field, Eq. (6.15), has the\nasymptote at Ω → ∞consistent with Eq. (6.19) at b= 0.0.000.100.200.30\n02468100.000.040.080.120.16 PSfrag replacementsReχ/bardbl(Ω,b) Imχ/bardbl(Ω,b)\nbΩ = 0 .1\nΩ = 0 .5\nΩ = 1\nΩ = 2\nΩ = 5\nFIG. 3: (Color online). Plot of the real and imaginary parts\nof theacsusceptibility χ/bardbl(Ω,b) at several values of the dimen-\nsionless frequency Ω of the oscillating magnetic field along the\nconstant magnetic field with strength b. In general, magnetic\nfield suppresses both real and imaginary parts of the suscep-\ntibility.\nSimilarly, the high frequency limit of Eq. (6.17) coincides\nwith the limit b→ ∞of Eq. (6.19). Both limits of weak\nand strong magnetic field of the imaginary part of the\nsusceptibilityatlowfrequencies, Eq.(6.20), coincidewith\nthe imaginarypartof χ/bardbl(Ω,b), calculatedfromEq.(6.15)\nand Eq. (6.19), respectively.\nIn general, we make a conjecture that the ac suscepti-\nbility is given by the following expression:\nχ/bardbl(Ω,b) =/summationdisplay\nnχ/bardbl\nn(b)\n1−iΩ/Γ/bardbl\nn(b),(6.21)\nwhere functions χ/bardbl\nn(b) and Γ/bardbl\nn(b) are real and describe\nthe degeneracy points of the homogeneous differential\nequation Eq. (6.10) with real iΩ. This expansion is re-\nlated to the expansion of time-dependent Fokker-Plank\nequations in the spherical harmonics, analyzed in Ref. 1.\nIn particular, χ/bardbl\nn>1(b→0) =O(b) and Γ/bardbl\nn(b→0) =\nn(n+1)+O(b).\nFor practical purposes, we found from a numerical\nanalysis that even the single-pole approximation,\nχapp\n/bardbl(ω,b) =/parenleftbigg1\nb2−1\nsinh2b/parenrightbigg/bracketleftbig\n1−iωT/bardbl(b)/bracketrightbig−1,(6.22)\ngives a very good estimate of the susceptibility for all\nvalues ofωandb. The analysis shows that the suscep-\ntibility, Eq. (6.9), obtained from a numerical solution of\nEq. (6.10), is within a few per cent of the estimate given\nby Eq. (6.22). The characteristic time constant, T/bardbl(b),\nas a function of magnetic field bis chosen from the high\nfrequency asymptote Eq. (6.19):\nT/bardbl(b) =T0\n2sinhbsinh2b−b2\nbcoshb−sinhb.(6.23)\nTo evaluate the accuracy of the above approximation,\nEq. (6.22), we consider the opposite limit of low frequen-12\n012345670.00.050.100.15\nPSfrag replacementsf/bardbl(b), fapp\n/bardbl(b)\nbf/bardbl(b)\nfapp\n/bardbl(b)\nΩ = 5\nFIG. 4: (Color online). Dependence on magnetic field bof\nthe slope of the imaginary part of the linear in frequency\nsusceptibility χ/bardbl(Ω,b) at low frequencies Ω ≪1, calculated\naccording toEq. (6.20). For comparison, we also plot functi on\nfapp\n/bardbl(b), see Eq. (6.24).\ncies, Ω≪1, and compare the exact result for the imagi-\nnary part of the susceptibility, Eq. (6.20), with\nImχapp\n/bardbl(ω,b) =ωT0fapp\n/bardbl(b),\nfapp\n/bardbl(b) =1\n2b2sinh3b(sinh2b−b2)2\nbcoshb−sinhb.(6.24)\nFor visual comparison of functions f/bardbl(b) andfapp\n/bardbl(b), we\nplot both functions in Fig. 4, where these curves are\nnearly indistinguishable. The difference between these\ntwo curves vanishes at b→0 andb→ ∞, and has a\nmaximal difference at b≈2, which constitutes only tiny\nfraction off/bardbl(b).\nD. Transverse susceptibility\nNext, we consider the response of the magnetization\nto weak oscillations ˜B⊥(t) of the external magnetic field\nwith frequency ωin direction perpendicular to the fixed\nmagnetic field B0. We write the oscillatory component\nof the field in the form:\n˜B⊥(t) =δ1Teff\n2h0/bracketleftbig\nb⊥(ex+iey)e−iωt+b∗\n⊥(ex−iey)eiωt/bracketrightbig\n.\n(6.25)\nThis field represents a circular polarization of an ac\nmagnetic field in the ( x,y) plane, perpendicular to\nthe fixed magnetic field in the z-direction: B=\n{B⊥cosωt;B⊥sinωt;B0}. We look for the linear cor-\nrection to the probability distribution in the form\nP(ϕ,θ,t) =P0(θ)\n×/bracketleftbig\n1+b⊥u⊥(θ)eiϕ−iωt+b∗\n⊥u∗\n⊥(θ)e−iϕ+iωt/bracketrightbig\n.(6.26)\nThe equation for u⊥(θ) is obtained from the Fokker-\nPlank equation Eq. (4.10), linearized in the parame-terb⊥:\n∂2u⊥\n∂θ2+cosθ−bsin2θ\nsinθ∂u⊥\n∂θ+/parenleftbigg\niΩ⊥−1\nsin2θ/parenrightbigg\nu⊥\n=−sinθ(2+2ih0τb+bcosθ).\n(6.27)\nHere the dimensionless frequency is a difference between\nthe drive frequency ωand the precession frequency in\nexternal field B0:\nΩ⊥= (ω−2B0)T0= Ω−2(h0τ)b, (6.28)\nwhereT0is defined in Eq. (4.9) and the right equality is\nwritten in terms of dimensionless variables Ω, Eq. (6.11),\nandb, Eq. (6.4). Equation (6.27) is symmetric with re-\nspect to the simultaneous change θ→π−θ,b→ −b,\ni→ −i, Ω⊥→ −Ω⊥(“parity”). The function P(ϕ,θ,t)\nis single-valued at the poles θ= 0 andθ=π, only if\nu⊥(θ= 0) = 0, u⊥(θ=π) = 0.(6.29)\nThe latter equations establish the boundary conditions\nfor the differential equation (6.27). We also note that\nthe normalization condition is satisfied for any function\nu⊥(θ).\nWe define the susceptibility in response to the ac mag-\nnetic field, Eq. (6.25), as\nχ⊥(Ω,b) = 2ππ/integraldisplay\n0u⊥(θ)P0(θ)sin2θdθ. (6.30)\nThis expression for the susceptibility can be used to cal-\nculate the magnetization of a particle\nM(t) =/integraldisplay\nn(ϕ,θ)P(ϕ,θ,t)sinθdθdϕ (6.31)\ntothe lowestorderin the acmagnetic field. Inparticular,\nMx(t) = Re(χ⊥b⊥e−iωt), My(t) = Im(χ⊥b⊥e−iωt).\n(6.32)\nSolving numerically the differential equation (6.27)\nwith the corresponding boundary conditions, Eq. (6.29),\nwe obtain the transverse susceptibility, Eq. (6.30), shown\nin Figs. 5 and 6. Below we analyze several limiting cases.\nIn zero fixed magnetic field, b= 0, we have the exact\nsolution of Eq. (6.27):\nu⊥(θ) =sinθ\n1−iΩ⊥/2. (6.33)\nThis solution corresponds to the solution in the longitu-\ndinal case, rotated by 90◦, cf. Eq. (6.14).\nAtω= 0 and Ω ⊥=−2bh0τ, the solution of Eq. (6.27)\nhas a simple form u/bardbl(θ) = sinθand corresponds to a tilt\nof the external field. The susceptibility due to such tilt\nis\nχ⊥(Ω = 0,b) =2\nb2(bcothb−1).(6.34)13\nt\n−0.20.00.20.40.6\n−5 0 5 10 15−0.20.00.20.40.6PSfrag replacementsReχ⊥(Ω, b) Imχ⊥(Ω, b)\nΩbb= 0.4;h0τ= 5\nb= 1;h0τ= 2\nb= 2.5;h0τ= 0.8\nΩ = 2; h0τ= 2\nΩ = 2; h0τ= 0.5\nΩ = 0 .5;h0τ= 2\nΩ = 0 .5;h0τ= 0.5\nΩ = 2\nΩ = 5\nFIG. 5: (Color online). Plot of the real and imaginary parts\nof the transverse susceptibility χ⊥(Ω,b) as a function of the\ndimensionless frequency Ω. Negative frequency correspond s\nto the opposite sense of the circular polarization of the ac\nmagnetic field in a plane, perpendicular to the constant mag-\nnetic field with strength b. The parameters of the three shown\ncurves are chosen so that h0τb= 2.\nIn strong fixed magnetic field, b≫1, we need to con-\nsider small angles θ∼1/√\nb, therefore, we can approxi-\nmate cosθ≈1 in Eq. (6.27) and obtain:\nu⊥(θ) =b+2+2ih0τb\nb+2−iΩ⊥sinθ. (6.35)\nThe susceptibility in the limit b≫1 is given by\nχ⊥(Ω,b) =1+2ih0τ\nb(b(1+2ih0τ)−iΩ). (6.36)\nAt Ω⊥≫1,bwe can disregard the terms in Eq. (6.27)\nwith derivatives. Moreover, the contribution to the sus-\nceptibility, Eq. (6.30), from the vicinity of θ= 0 and\nθ=πis suppressed as sin2θ. This observation allows us\nto write the solution in the form\nu⊥(θ) = sinθ2+2ih0τb+bcosθ\n−iΩ⊥, (6.37)\nConsequently, we obtain the following high frequency,\nΩ⊥≫1, asymptote for the susceptibility:\nχ⊥(Ω,b) =i\nΩ−2h0τb/bracketleftbiggbcothb−1\nb2(2ih0τb−1)+1/bracketrightbigg\n.\n(6.38)\nWe can use the approximate expression for the suscep-\ntibility in response to the transverse oscillating magnetic\nfield\nχapp\n⊥(ω) =bcothb−1\nb2[1+2iB0T⊥(b)]\n×[1+i(2B0−ω)T⊥(b)]−10.00.10.20.30.4\n−10 −5 0 5 100.00.10.20.3PSfrag replacementsReχ⊥(Ω, b) Imχ⊥(Ω, b)\nΩ\nbΩ = 2; h0τ= 2\nΩ = 2; h0τ= 0.5Ω = 0 .5;h0τ= 2\nΩ = 0 .5;h0τ= 0.5\nΩ = 2\nΩ = 5\nFIG. 6: (Color online). Plot of the real and imaginary parts\nof the transverse susceptibility χ⊥(Ω,b) as a function of the\nstrength bof a constant magnetic field, shown for two values\nof frequency Ω and two values of the “damping factor” h0τ.\nNegative values of bcorresponds to the opposite sense of the\ncircular polarization of the ac magnetic field in a plane, per -\npendicular to the constant magnetic field. The real part of\nthe susceptibility exhibits a strong non-monotonic behavi or\nat weak magnetic fields.\nThecorrespondingcharacteristictimeconstant T⊥(b)can\nbe found for any bfrom the asymptotic behavior of\nχ⊥(Ω,b) at Ω ⊥≫1,b:\nT⊥(b) =T0bcothb−1\nb2+1−bcothb. (6.39)\nVII. CONCLUSIONS\nWe have studied the slow dynamics of magnetization\nin a small metallic particle (quantum dot), where the fer-\nromagnetism has arisen as a consequence of Stoner insta-\nbility. Theparticleisconnectedto non-magneticelectron\nreservoirs. A finite bias is applied between the reservoirs,\nthus bringing the whole electron system away from equi-\nlibrium. The exchange of electrons between the reser-\nvoirs and the particle results in the Gilbert damping3of\nthe magnetization dynamics and in a temperature- and\nbias-driven Brownian motion of the direction of the par-\nticle magnetization. Analysis of magnetization dynam-\nics and transport properties of ferromagnetic nanoparti-\ncles is commonly performed4,5,6,7,11within the stochas-\ntic Landau-Lifshitz-Gilbert (LLG) equation2,3, which is\nan analogue of the Langevin equation written for a unit\nthree-dimensional vector.\nWe derived the stochastic LLG equation from a mi-\ncroscopicstarting point and established conditions under\nwhich the description ofthe magnetizationofaferromag-\nnetic metallic particle by this equation is applicable. We\nconcluded that the applicability of the LLG equation for14\na ferromagneticparticle is set by three independent crite-\nria. (1)Thecontactresistanceshouldbelowcomparedto\nthe resistance quantum, which is equivalent to Nch≫1.\nOtherwise the noise cannot be consideredgaussian. Each\nchannel can be viewed as an independent source of noise\nand only the contribution of many channels results in\nthe gaussian noise by virtue of the central limit theorem\nforNch≫1. (2) The system should not be too close\nto the Stoner instability: the mean-field value of the to-\ntal spinS2\n0≫Nch. Otherwise, the fluctuations of the\nabsolute value of the magnetization become of the or-\nder of the magnetization itself. (3) S2\n0≫Teff/δ1, where\nTeff≃max{T,|eV|}is the effective temperature of the\nsystem, which is the energy scale of the electronic dis-\ntribution function. Otherwise, the separation into slow\n(the direction of the magnetization) and fast (the elec-\ntron dynamics and the magnitude of the magnetization)\ndegrees of freedom is not possible.\nUnder the above conditions, the dynamics of the mag-\nnetization is described in terms of the stochastic LLG\nequation with the power of Langevin forces determined\nby the effective temperature of the system. The effective\ntemperature is the characteristic energy scale of the elec-\ntronic distribution function in the particle determined by\na combination of the temperature and the bias voltage.\nIn fact, for a considered here system with non-magnetic\ncontacts between non-magnetic reservoirs and a ferro-\nmagnetic particle the power of the Langevin forces is\nproportional to the low-frequency noise of total charge\ncurrent through the particle. We further reduced the\nstochastic LLG equation to the Fokker-Planck equation\nfor a unit vector, corresponding to the direction of the\nmagnetization of the particle. The Fokker-Plank equa-\ntion can be used to describe time evolution of the distri-\nbution of the direction of magnetization in the presence\nof time-dependent magnetic fields and voltage bias.\nAs an example of application of the Fokker-Plank\nequation for the magnetization, we have calculated the\nfrequency-dependent magnetic susceptibility of the par-\nticle in a constant external magnetic field (i. e., linear\nresponse of the magnetization to a small periodic mod-\nulation of the field, relevant for ferromagnetic resonance\nmeasurements). We have not been able to obtain an ex-\nplicit analytical expression for the susceptibility at ar-\nbitrary value of the applied external field and frequency;\nhowever, analysisofdifferent limiting caseshas lead us to\na simple analytical expression which gives a good agree-\nment with the numerical solution of the Fokker-Planck\nequation.\nAcknowledgements\nWe acknowledge discussions with I. L. Aleiner, G.\nCatelani, A. Kamenev and E. Tosatti. M.G.V. is grate-\nful to the International Centre for Theoretical Physics(Trieste, Italy) for hospitality.\nAPPENDIX A: LONGITUDINAL\nSUSCEPTIBILITY AT LOW FREQUENCIES\nWe find the linearin frequency Ω ≪1 correctionto the\ndc susceptibility. For this purpose, we look for a solution\nto Eq. (6.10) in the form\nu/bardbl(θ) =u(0)\n/bardbl(θ)+u(1)\n/bardbl(θ), (A1)\nwhereu(0)\n/bardbl(θ) is the solution of Eq. (6.10) at Ω = 0 and\nu(1)\n/bardbl(θ)∝Ω. We choose\nu(0)\n/bardbl(θ) =1\nb−cothb+cosθ, (A2)\nsincethis formof u(0)\n/bardbl(θ) preservesthe normalizationcon-\ndition (6.12). This function can be found directly as a\nsolution of Eq. (6.10) with Ω = 0 or as a variational\nderivative of function P0(θ), defined in Eq. (6.3), with\nrespect tob.\nThe linear in Ω correction u(1)\n/bardbl(θ) is the solution to the\ndifferential equation\n∂2u(1)\n/bardbl(θ)\n∂θ2+cosθ−bsin2θ\nsinθ∂u(1)\n/bardbl(θ)\n∂θ=−iΩu(0)\n/bardbl(θ).(A3)\nFrom this equation, we can easily find\n∂u(1)\n/bardbl(θ)\n∂θ=−iΩ\nbsinθ/bracketleftbigg\ncothb−cosθ−e−bcosθ\nsinhb/bracketrightbigg\n.(A4)\nWe notice that the solution to the latter equation will\nautomatically satisfy the boundary conditions, given by\nEq. (6.13). Integrating Eq. (A4) once again, we obtain\nthe following expression for function u(1)\n/bardbl(θ):\nu(1)\n/bardbl(θ) =C(b)\n−iΩ\nb/integraldisplayθ\n0/bracketleftBigg\ncothb−cosθ′−e−bcosθ′\nsinhb/bracketrightBigg\ndθ′\nsinθ′.(A5)\nHere the integration constant C(b) has to be chosen to\nsatisfy the normalization condition, Eq. (6.12), which re-\nsults in complicated expression for the final form of the\nfunctionu(1)\n/bardbl(θ).\nTo obtain function f/bardbl(b), introduced in Eq. (6.20), we\nhave to perform the final integration\nf/bardbl(b) =2π\nΩ/integraldisplayπ\n0u(1)\n/bardbl(θ)P0(θ)sinθcosθdθ. (A6)\nThe result of integration is shown in Fig. 4.15\n1W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).\n2L. Landau and E. Lifshitz, Phys. Z. Sowietunion 8, 153\n(1935).\n3T. Gilbert, Phys. Rev. 100, 1243 (1955).\n4J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Phys. Rev. B 58,\n14937 (1998).\n5K. D. Usadel, Phys. Rev. B 73, 212405 (2006).\n6J. Foros, A. Brataas, G. E. W. Bauer, and Y. Tserkovnyak,\nPhys. Rev. B 75, 092405 (2007).\n7S. I. Denisov, K. Sakmann, P. Talkner, and H¨ anggi, Phys.\nRev. B75, 184432 (2007).\n8A. Rebei and M. Simoniato, Phys. Rev. B 71, 174415\n(2005).\n9H. Katsura, A. V. Balatsky, Z. Nussinov, and N. Nagaosa,\nPhys. Rev. B 73, 212501 (2006).\n10A. S. N´ u˜ nez and R. A. Duine, Phys. Rev. B 77, 054401\n(2008).\n11A. L. Chudnovskiy, S. Swiebodzinski, and A. Kamenev,\nPhys. Rev. Lett. 101, 066601 (2008).\n12J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,Phys. Rev. Lett. 95, 016601 (2005).\n13R.A.Duine, A.S.N´ u˜ nez, J.Sinova, andA.H.MacDonald,\nPhys. Rev. B 75, 214420 (2007).\n14X. Waintal and P. W. Brouwer, Phys. Rev. Lett. 91,\n247201 (2003).\n15I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys.\nReports 358, 309 (2002).\n16C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).\n17I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, Phys.\nRev. B62, 14886 (2000).\n18Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n19S. Tamaru et al., J. Appl. Phys. 91, 8034 (2002).\n20J. C. Sankey et al., Phys. Rev. Lett. 96, 227601 (2006).\n21Y. Ahmadian, G. Catelani, and I. L. Aleiner, Phys. Rev.\nB72, 245315 (2005).\n22K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n23G. Catelani and M. G. Vavilov, Phys. Rev. B 76, 201303\n(2007)."    },    {        "title": "0711.0405v2.Spin_torque_driven_ferromagnetic_resonance_of_Co_Ni_synthetic_layers_in_spin_valves.pdf",        "content": "arXiv:0711.0405v2  [cond-mat.mes-hall]  7 Jan 2008Spin-torque driven ferromagnetic resonance\nof Co/Ni synthetic layers in spin valves\nW. Chen, J-M. L. Beaujour, G. de Loubens, A. D. Kent\nDepartment of Physics, New York University, New York, NY 100 03\nJ. Z. Sun\nIBM T. J. Watson Research Center, Yorktown Heights, NY 10598\n(Dated: November 30th, 2007)\nSpin-torque driven ferromagnetic resonance (ST-FMR) is us ed to study thin Co/Ni synthetic\nlayers with perpendicular anisotropy confined in spin-valv e based nanojunctions. Field swept ST-\nFMR measurements were conducted with a magnetic field applie d perpendicular tothe layer surface.\nThe resonance lines were measured under low amplitude rf exc itation, from 1 to 20 GHz. These\nresults are compared with those obtained using conventiona l rf field driven FMR on extended films\nwith the same Co/Ni layer structure. The layers confined in sp in valves have a lower resonance field,\na narrower resonance linewidth and approximately the same l inewidth vsfrequency slope, implying\nthe same damping parameter. The critical current for magnet ic excitations is determined from\nmeasurements of the resonance linewidth vsdc current and is in accord with the one determined\nfrom I-V measurements.\nSpin-transfer torque has been theoretically predicted\nand experimentally demonstrated to drive magnetic ex-\ncitationsinnanostructuredspinvalvesandmagnetictun-\nneljunctions[1, 2, 3, 4, 5]. Withanrfcurrent, spintrans-\nfer can be used to study ferromagnetic resonance [6, 7].\nThis technique, known as spin-torque driven ferromag-\nnetic resonance (ST-FMR), enables quantitative studies\nofthe magneticpropertiesofthin layersin aspin-transfer\ndevice. Specifically, the layer magnetic anisotropy and\ndamping can be determined [8], which are important pa-\nrameters that need to be optimized in spin-torque-based\nmemory and rf oscillator applications.\nSpin-transfer memory devices will likely include mag-\nnetic layers with perpendicular magnetic anisotropy that\ncounteracts their shape-induced easy-plane anisotropy.\nThis will allow efficient use of spin current for magnetic\nreversal with a reduced switching threshold [9] and a\nfaster switching process [10]. Recent work by Mangin et\nal.[11] has demonstrated improvements of spin-torque\nefficiency in a spin valve that has perpendicularly mag-\nnetized Co/Ni synthetic layers. For further optimization\nof perpendicular anisotropy materials, it is important to\nhave quantitative measurements of their anisotropy field\nand damping in ananostructured device, asboth ofthese\nparameters directly affect the threshold current for spin-\ntransfer induced switching.\nIn this Letter, we present ST-FMR studies of bilayer\nnanopillars, where the thin (free) layer is composed of a\nCo/Ni synthetic layer and the thick (fixed) layer is pure\nCo. The magnetic anisotropy and damping of the Co/Ni\nhave been determined by ST-FMR. We compare these\nresults with those obtained from extended films with the\nsameCo/Nilayerstackmeasuredusingtraditionalrffield\ndriven FMR.\nPillar junctions with submicron lateral dimensions\n(Fig. 1(a)) were patterned on a silicon wafer using a\nnanostencil process [12]. Junctions were deposited us-\ning metal evaporation with the layer structure /bardbl1.5 nm4.454.504.554.604.65\n-2 -1 0 1 21.531.541.551.56dV/dI [Ohm](d)\ndV/dI [Ohm]\n-15 -10 -5 0 5 10 152468\nMagnetic Field [kOe]\ndc Current Bias [mA](c)\n(b)\nMagnetic Field [kG](a)\nV\nH\n2o\nFIG. 1: (a): Sample layer structure and ST-FMR circuit.\n(b): Zero current in-plane MR hysteresis loop for a 50 ×150\nnm2spin valve junction with t=0.4. (c): dV/dIvsI of the\nsame junction with a perpendicular magnetic field of 9.5 kOe.\n(d): Contour plot of dV/dIas a function of both dc current\nand perpendicular magnetic field. Data points: critical cur -\nrents determined from ST-FMR at three different fields and\nfrequencies (see text).\nCr|100 nm Cu |20 nm Pt |10 nm Cu |[tnm Co|2tnm\nNi]×1.2/t|10 nm Cu |12 nm Co |200 nm Cu /bardbl. We var-\nied the Co thickness tfrom 0.1 to 0.4, tuning the magni-\ntude of the Co/Ni composite layer’snet anisotropy, while\nkeeping the total magnetic moment and thickness of the\nfree layer constant. For ST-FMR measurements, an rf\ncurrent generated by a high frequency source is added to\na dc current using a bias-T (the dashed-line box in Fig.\n1(a)). Positivedccurrentsaredefined suchthatelectrons\nflow from the free layer to the fixed layer.\nThe magnetoresistance (MR) was measured with a\nmagnetic field applied in the film plane using a 4-point2\ngeometry. A typical MR hysteresis loop of a 50 ×150nm2\njunction with t=0.4 is shown in Fig. 1(b). The magne-\ntoresistance MR= ( RAP−RP)/RPis≃2.3±0.3 % for\nall junctions, independent of t, within the range inves-\ntigated. Here RAP(RP) represents the static junction\nresistance when the free layer and fixed layer magneti-\nzations are antiparallel (parallel). Current-voltage mea-\nsurements were conducted with a magnetic field applied\nnearlyperpendicular to the sample surface (The field\nwas applied 2◦from the film normal to produce a small\nin-plane field along the easy axis of the junction. This\nwas done to suppress vortex states in the magnetic lay-\ners.) Measurements were conducted in a 2-point geome-\ntry where lead resistances are included. Fig. 1(c) shows\ndV/dIvsI of the same junction in a 9.5 kOe applied\nfield. A peak without hysteresis is observed at 9.1 mA,\nwhich we interpret asthe critical current Icfor excitation\nof the free layer [13]. A contour plot of 2-point dV/dIas\nthe function of both current and perpendicular magnetic\nfield is shown in Fig. 1(d). The peak in dV/dIis seen as\nthe bright color at high field and current.\nAt resonance, the rf current and spin valve resistance\noscillate at the same frequency resulting in a dc voltage\n(V=< I(t)R(t)>) [6, 7]. This voltage can be expressed\nasV=1\n4(RAP−RP)Irfsinβsinθ. Hereβis the an-\ngle between the free and fixed layers before applying the\nrf current and θis the precession angle. Irfrepresents\nthe rf current amplitude. This is a simplified formula\nthat assumes small angle precession and a sinusoidal an-\ngular dependence of junction resistance between parallel\nand antiparallel states. With a perpendicular magnetic\nfield greater than the free layer’s easy-plane anisotropy\nfield, the free layer magnetization is normal to the sur-\nface, while the fixed layer, which has a larger easy-plane\nanisotropy field, is still mainly magnetized in the film\nplane. This non-collinear arrangement of the layer mag-\nnetizations ( β/lessorsimilarπ/2) enhances the ST-FMR signal. To\nfurtherincreasethesignal(typicallyinthesub- µVrange)\nto noise ratio, we modulate the rf current on and off at\n800 Hz and use a lock-in amplifier to detect the voltage\nat this frequency.\nST-FMR measurements were conducted with the cir-\ncuit shown in Fig. 1(a). Resonance lines under low am-\nplitude rf current at zero dc current and different rf fre-\nquencies fare plotted in Fig. 2(a) versus perpendicular\nmagneticfield. Differentfrequencies(3 ∼20GHzin1GHz\nsteps) are plotted with each adjacent curve offset by 0.4\nµV. The voltage signals are shown on the left vertical\naxis. From the peak height VpeakandIrf, we estimate\nthe precession angle to be ∼4◦. We verified that this\nset of data was taken in a linear response regime with\nVpeak/I2\nrfindependent of Irf. These data are typical of\nall junctions with t=0.4. However, much broader reso-\nnance peaks and multiple peaks were found on samples\nwitht=0.1, 0.2 and 0.3. This is likely associated with\nthe excitation of higher order spin wave modes, but is\nnot presently understood. Therefore, data analysis and\ndiscussion mainly focus on samples with t=0.4.2 4 6 80246\n0 10 20 300.00.20.40.60.81.0\n510152020 GHz\n(b)Vlock-in [PV]\nMagnetic Field [kOe](a)\n3 GHz\nFrequency [GHz]\nLinewidth [kOe]\nFrequency [GHz]\nFIG. 2: (a): Lock-in voltage signal as a function of applied\nperpendicular magnetic field at different rf frequencies fro m\n3 up to 20 GHz in 1 GHz steps. /trianglesolid:Hresof a 50×150 nm2,\nt=0.4 Co/Ni synthetic free layer in a spin valve; black dashed\nline: corresponding linear fit; gray dashed line: a linear fit\nofHresvsfof an extended film with the same Co/Ni layer\nstack. (b): ∆ Hvsffor the spin valve junction ( /trianglesolid) and the\nextended film ( /squaresolid), together with their corresponding linear\nfits.\nWe also measured resonance lines on an extended film\nwith the same Co/Ni synthetic layer stack sandwiched\nbetween 10 nm Cu on each side. These measurements\nwere conducted with a traditional rf field driven FMR\nusing a flip-chip method [14]. Broader resonance peaks\nwere not found in extended films with t=0.1, 0.2, and\n0.3.\nThe resonance field Hresof the Co/Ni element in the\nspin valve increases linearly with fabove 4 GHz, as\nshown in Fig. 2(a) ( /trianglesolidsymbols). At lower frequencies,\nthe free layer magnetization tilts into the plane, lead-\ning to a lower resonance field. A linear fit of Hresvsf\nof the extended film is also plotted with a gray dashed\nline (to the right of the /trianglesolidsymbols) in Fig. 2(a). A\nlinear relationship between fandHresin extended mag-\nnetic films is expected when the magnetization is normal\nto the film surfaceh\nµBf=g(Hres−4πMeff) [15]. Here\ngis the Land´ e gfactor and the easy-plane anisotropy\nis 4πMeff= 4πMs−HP, where MsandHPrepre-\nsent the saturation magnetization and the perpendicular\nanisotropyfield. Alinearfitofeachdataset(dashedlines\nin Fig. 2(a)) gives g=2.17 and 4 πMeff= 2.58 kOe for\nthe extended film, and a slightly larger slope (2.28) and\na smaller field-axis intercept (1.92 kOe) for the Co/Ni\nelement confined in the spin valve. This consistency be-\ntween data sets confirms that the main peak of the ST-\nFMR signal is associated with the Co/Ni synthetic free\nlayer rather than the other magnetic layers. The differ-\nences are associated with the static dipolar fields from\nother magnetic layers and finite size effects on the spin\nwavemodes, which is discussed in detail in a forthcoming3\n6.0 6.5 7.0 7.5 8.0-0.50-0.250.000.250.500.75\n-4 -2 0 2 4 6 80.00.10.20.30.40.50.6(b)\nMagnetic Field [kG]Vlock-in [PV]\n-4 mA6 mA (a)\nCurrent Bias [mA]\n'H [kOe]\nFIG. 3: (a): ST-FMR signal as a function of applied field at\ndifferent dc currents. The rf frequency was set at 18 GHz,\nand the rf amplitudes were 595, 595, 470, 470, 315 and 315\nµA respectively for each dc current from -4 to 6 mA in 2 mA\nsteps. Each adjacent curve is offset by 0 .20µV. Solid lines\nare Lorentzian fits of each data set. (b): ∆ H(full width at\nhalf maximum) vsdc current.\npublication [16].\nHere we focus on the resonance linewidth ∆ H. ∆Hvs\nfat zero dc bias is plotted in Fig. 2(b). ∆ Hof both\nthe Co/Ni layer in spin valve ( /trianglesolid) and the same-stack\nextended film ( /squaresolid) increases linearly with f. Linear fits\nareshownassolidlinesinFig. 2(b), andgiveanintercept\nand slope:\n∆H= ∆H0+2αh\ngµBf (1)\nwherehis the Planck Constant and µBis the Bohr Mag-\nneton. The first term ∆ H0describes the inhomogeneous\nbroadening, and the second term is related to the damp-\ningα[17]. ∆Hvsfof the spin valve and that of the ex-\ntendedfilmhaveasimilarslope,implyingasimilardamp-\ning parameter ( α=0.036±0.002for the extended film and\n0.033±0.003 for the magnetic layer in the spin valve).\nHowever, the intercepts are quite different: ∆ H0=24±15\nOe in the spin valve, which is much lower than that of\nthe extended film, 284 ±30 Oe.\nWhen adc currentbias isapplied to a spin-value, there\nis an additional spin transfer torque that modifies the\nfree layer’s effective damping, αeff=α(1−I\nIc), where\nIis the dc current. Thus αeffdecreases with increasing\npositive current up to a critical current Ic, that defines\nthe threshold for magnetic excitation of the free layer.\nThe critical current in the Slonczewski model [1] is given\nbyIc=2e\n/planckover2pi1PαMsV\ncosβ(H−4πMeff), where Pis the spin po-larization factor and Vis the volume of the magnetic el-\nement. So with a dc bias, ∆ H-∆H0=2αhf\ngµB(1−I\nIc), and\ntherefore at fixed frequency ∆ H-∆H0depends linearly\non current and goes to zero at the critical current. We\nplot resonance lines of the spin valve with f= 18 GHz\nat different dc currents from -4 to 6 mA in 2 mA steps in\nFig. 3(a). ∆ Hvsdc current bias is shown in Fig. 3(b).\nThe intercept of ∆ HvsI is 7.8 mA. Inclusion of ∆ H0\ndecreases the intercept by no more than 0.2 mA, because\n∆H0is small compared to the linewidth at the dc cur-\nrents studied. Critical currents determined for f=10, 14\nand 18 GHz are plotted in Fig. 1(d), and agree well with\nthose obtained from the I-V measurements. Further, Ic\nis quantitatively consistent with the Slonczewski model\ntaking a spin polarization factor P ∼0.3.\nThe frequency independent term ∆ H0originates from\nfilm inhomogeneities: roughness, polycrystalline struc-\nture, as well as defects. The scale of the inhomogeneities\nis likely the film grain size, 5 ∼10 nm. In a simple model,\nfluctuations in Hresfrom grain to grain result in an in-\nhomogeneously broadened resonance line [18]. However,\nit is likely that the exchange coupling between grains is\nimportant to a detailed understanding of the linewidth\n[19].\nThe free layer in the nanostructured device contains\nat most a few hundred grains, therefore one expects less\ninhomogeneity than that in extended film. More im-\nportantly, the lateral magnetic confinement results in\nstrongly varying internal field in the plane of the nanos-\ntructure that lifts the degeneracy between different spin\nwave modes. Numerical and analytical calculations of\nnormal modes in the Co/Ni rectangular element are pre-\nsentedandcomparedwithourST-FMRdatainRef. [16].\nThe separation between them is more than the inhomo-\ngeneous broadening ∆ H0in the extended film, therefore\nwe expect that the linewidth measured on an individ-\nual mode is close to its intrinsic value (the term propor-\ntional to fin Eq. 1) [20]. The remaining inhomogeneous\nbroadeninginthenanostructuremaybeattributedtothe\nquasi-degeneracy subsisting between some very closely\nspaced modes resulting from film inhomogeneities.\nInsummary, ST-FMRhasbeen usedto studythe mag-\nneticpropertiesofCo/Nisyntheticlayerswithperpendic-\nular anisotropy in spin valves. The ST-FMR resonance\nlines were compared with those of traditional FMR on\nsame-stackextended film. The damping ofthe ST-device\nfree layer is essentially the same as that of an unpat-\nterned film and the critical currents determined from the\nST-FMR homogeneous linewidth are in agreement with\nthose of quasistatic I-V measurements.\nThis research is supported by NSF-DMR-0706322 and\nan NYU-Research Challenge Fund award.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).[2] L. Berger, Phys. Rev. B 54, 9353 (1996).4\n[3] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[4] J. Z. Sun, J. Magn. Magn. Mater. 202, 157 (1999).\n[5] Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. Valet,\nAppl. Phys. Lett. 84, 3118 (2004).\n[6] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub-\nota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira,\nN. Watanabe, and S. Yuasa, Nature 438, 339 (2005).\n[7] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N.\nKrivorotov, R.A. Buhrman, andD. C. Ralph, Phys. Rev.\nLett.96, 227601 (2006).\n[8] G. D. Fuchs, J. C. Sankey, V. S. Pribiag, L. Qian,\nP. M. Braganca, A. G. F. Garcia, E. M. Ryan, Z.-P. Li,\nO.Ozatay, D.C. Ralph, andR.A. Buhrman, Appl.Phys.\nLett.91, 062507 (2007).\n[9] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[10] A. D. Kent, B. ¨Ozyilmaz, and E. del Barco, Appl. Phys.\nLett.84, 3897 (2004).\n[11] S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey,\nB. D. Terris, and E. E. Fullerton, Nature Materials 5,\n210 (2006).\n[12] J. Z. Sun, Appl. Phys. Lett. 81, 2202 (2002).[13] B.¨Ozyilmaz, A. D. Kent, D. Monsma, J. Z. Sun, M. J.\nRooks, and R. H. Koch, Phys. Rev. Lett. 91, 067203\n(2003).\n[14] J.-M. Beaujour, W. Chen, K. Krycka, C.-C. Kao, J. Z.\nSun, and A. D. Kent, Eur. Phys. J. B 59, 475 (2007).\n[15] C. Kittel, Introduction to Solid State Physics (John Wi-\nley & Sons, Inc., New York, 1996).\n[16] W. Chen, G. de Loubens, J.-M. L. Beaujour, A. D.\nKent, and J. Z. Sun, to be published in J. Appl.\nPhys. as MMM’07 Conference Proceeding, preprinted at\narXiv:0712.0404 (2007).\n[17] D. L. Mills and S. M. Rezende, Spin Dynamics in\nConfined Magnetic Structures II (Springer, Heidelberg,\n2002).\n[18] H. Hurdequint, J. Magn. Magn. Mater. 242-245 , 521\n(2002).\n[19] R.D.McMichael, D.J.Twisselmann, andA.Kunz,Phys.\nRev. Lett. 90, 227601 (2003).\n[20] G. de Loubens, V. V. Naletov, O. Klein, J. B. Youssef,\nF. Boust, and N. Vukadinovic, Phys. Rev. Lett. 98,\n127601 (2007)."    },    {        "title": "1804.06320v1.Ultrasensitive_multi_mode_ESR_probed_ferromagnetic_two_level_system_of__Mn__4____impurity_ion_in_the_insulated__MnO_6__complex_of__SrLaAlO_4__at__20_mK_.pdf",        "content": "Ultrasensitive multi-mode ESR probed ferromagnetic two-level system of\nMn4+impurity ion in the insulated MnO 6complex of SrLaAlO 4at20mK\nM. A. Hosain1,a)\nARC Centre of Excellence for Engineered Quantum Systems, School of Physics, University of Western Australia,\n35 Stirling Highway, Crawley WA 6009, Australia.\nUltrasnsitive multi-mode electron spin resonance spectroscopy in the SrLaAlO 4dielectric resonator at 20 mK\nreveals ferromagnetic states of Mn4+impurity ion. The formation of ferromagnetic states in the MnO 6\ncomplex implies to oxygen de\fciency of this multi-valence Mn4+ion. Experiment results supports that\nan intricate electronic hybridization in MnO 6structural instability is related to Pseudo Jahn-Teller e\u000bect.\nMeasured dipolar hyper\fne structure parameter of nucleus is Pk=\u00003:7\u000210\u00004cm\u00001. Mean inverse third\npower of the electron distance is hr\u00003\nqi= 3:325a:u: assuming nuclear electric quadruple moment Q=\n+0:33(1)barn. In such a state, giant g-factor is observed due to magneto (ferromagnetic) impedance taking in\nto account a two-level system on the adiabatic-potential-energy-surface. The spins exhibited parity is opposite\nin the interaction of highest-occupied-molecular-orbital and lowest-unoccupied-molecular-orbital coupling.\nA. Introduction:\nUltrasensitive electron spin resonance (ESR) with high\nprecision in dielectric crystal multi-mode resonators as-\nsimilating as hybrid-quantum system is a road-map for\nthe development of quantum technologies1{4. This exper-\nimental study is on impurity paramagnetic ion's unpaired\nelectron spin states using electron spin resonance (ESR)\nspectroscopy in a suitable dielectric crystal resonator\nexciting microwave whispering gallery (WG) modes5{7.\nWG multi-mode ESR spectrum works as a direct probe\nproviding information of electronic and magnetic states\nof paramagnetic impurity ions8. In this process, Mn4+\nion has been detected in the dielectric single crystal\nSrLaAlO 4(SLA ), and analysed taking into account an\nintricate electronic hybridization due to its extra charge\ninMnO 6complex at 20 millikelvin ( mK). Thisd\u0000p\nmetal-ligand orbital hybridization is mediated in MnO 6\nstructural instability, and plays a vital role in the mech-\nanism of spontaneous polarization and/or magnetization\nforming two-level system on the adiabatic potential en-\nergy surface (APES)9. Naturally, paramagnetic ion's\nthree phenomena ferro-electricity, magnetization, and\nspin-crossover are observed as coexisted10. Ohkoshi et\nal.11demonstrated unpaired electrons as a spin-crossover\nmagnet in the mechanism of light induced phase tran-\nsition. Such a transition process can be used to mon-\nitor magnetization saturation ( Ms), Curie temperature\n(Tc), coercive (magnetic) \feld ( Hc) and/or the mag-\nnetic pole11. Optical spectroscopy and X-ray di\u000braction\n(XRD) results are available providing localization infor-\nmation ofMn4+ion in this type of crystal12,13.\nMany studies have been devoted towards better un-\nderstanding of the main mechanisms governing electron\ndelocalization and electron intervalence absorptions of\ntransition metals in metal-ligand complexes14,15. These\ntransition metal based crystals like LaMnO 3exhibits in-\nteresting magnetic behaviours, such as, mono-metallic\na)Electronic mail: akhter361@yahoo.co.ukcomplexes in the crystal exhibiting single ion mag-\nnetic (SIM) behaviour14,16, and two or more metal sites\nof varying oxidation numbers ( known as mixed va-\nlence (MV) sites16,17) exhibits single molecular mag-\nnet (SMM) behaviour14,17. This is essentially important\ndue to the fact that these metal-ligand complex struc-\ntures possess several potential applications in quantum\ntechnology14,15. However these characteristics due to\ntheir intrinsic magnetic properties, SMMs are distinctly\ndetected only at liquid helium temperatures14,18. Among\nthe MV metal complexes, dual-exchange (DE)14,16is gen-\nerated due to the presence of an itinerant electron in two\ndi\u000berent valence sites in the crystal. As instance, elec-\ntron exchanges between two neighbouring sites Mn3+\nandMn4+in theMn\u0000O\u0000Mn chain ofLaMnO 3\ncrystal19. The MV metal ion complexes exhibit coupling\nof electron movement with the structural distortion, and\nsubsequently a\u000bects the degree of localization of the extra\nelectron14,20,21. The oxidation variation of MV sites of\nthe metal-ligand complex produces distortions as Jahn-\nTeller e\u000bect (JTE) within trigonal plane of lower symme-\ntry, and can be con\frmed from ESR spectrum15,19,20.\nIntriguingly, we examine an insulated octahedral\nmono-metallic complex MnO 6inSrLaAlO 4whereAl3+\nion is substituted by Mn4+ion12,17. In this case the\nmanganese ion shows multi-valence behaviour instead of\nMV behaviour. Interesting magnetic behaviours of this\ntype of transition metal complex are observed in a linear\ncombination of atomic orbitals. Neither the DE mecha-\nnism which is a type of magnetic exchange (whether ma-\nterials are ferromagnetic or antiferromagnetic) that may\narise between MV ions in the Mn\u0000O\u0000Mn link, nor\nthe super-exchange (SE) (or Kramers-Anderson super-\nexchange) which is a strong (usually) antiferromagnetic\ncoupling between two next-to-nearest neighbour cations,\nis possible for an insulated MnO 6individual unit in\nSrLaAlO 4. We will justify this multi-valance man-\nganese ion in a metal-ligand charge transfer (oxidation\nvariation) produced spontaneous magnetization with the\nmeasured spin-Hamiltonian parameters along with site\nsymmetry22{25.arXiv:1804.06320v1  [cond-mat.other]  17 Apr 20182\nThe magnetic Mn4+ion inMnO 6structure has a cer-\ntain spin parity in the formed molecular orbitals10,26.\nHere is our description for an appropriate realization\nof detected Mn4+ion's spin quantum state, which\nis dealt with ferromagnetism empirically rationalized\nto intricate electronic hybridization in MnO 6com-\nplex referring to PJTE27. Formation of two-level\nsystem is taken into account in the two minima on\nAPES in the mechanism of highest-occupied-molecular-\norbital (HOMO ) and lowest-unoccupied-molecular-\norbital (LUMO ) coupling.10,26{29.\nB. ESR spectroscopy experiment using WG modes:\nField con\fnement of the WG mode in SrLaAlO 4crys-\ntal allows loss mechanisms to be minimized to achieve a\nhigh Q-factor at 20 mK30{33, which is required for an\nultrasensitive ESR spectroscopy. Using X-band to Ku-\nband frequency WG multi-mode ESR at this tempera-\nture, high precision is achieved in the measurements of\nthe spin-Hamiltonian parameters34,35. Di\u000berent process\nare devoted in measuring sensitivity of di\u000berent type of\nresonator of wide range of frequency with varieties of\nprobing system36{40. Benmessai et al.6described a con-\ncentration level measurement process of Fe3+impurity\nion exciting WG modes at millikelvin temperatures in\nsapphire. Anders et al.39described a single-chip electron\nspin resonance detector operating at 27 GHz .\nFor such a spectroscopy, a cylindrical light yellow\nSrLaAlO 4single crystal of height 9 :04mm and diam-\neter 17:18mm was inserted centrally in an oxygen-free\ncylindrical copper cavity. The crystal loaded cavity was\ncooled in a dilution refrigerator (DR) to less than 20 mK.\nPractically microwave-power and other terms are kept\nconstant then the required minimum number of impurity\nion follows the proportionality23Nmin/1\n!QLfor detec-\ntion of ESR transition spectrum. The required minimum\nspin number is estimated generally as:\nNmin=\u00103KBVsTs\ng2e\f2\u0016\u000eS(S+ 1)\u0011\u0010\u0001!\n!\u0011\u00101\n\u0011QL\u0011\u0010Pn\nP\u00111\n2(1)\nWhereVsis the mode volume, Tsis the sample temper-\nature,Sis the electron e\u000bective spin, geis the electron\ng-factor,\fis the Bohr electron magneton, \u0016\u000eis the mag-\nnetic permeability of free space, !is the resonance fre-\nquency,\u0011is the \flling factor, Pnis the noise power at\nthe detector, Pis the microwave input power, and \u0001 !\nis the width of aggregated spin frequency at resonance\nwhich is depended on the shape-function f(!) normal-\nized asR1\n0f(!)@!= 1 for a wide range of Larmor pre-\ncession (!L) of magnetic dipoles22. Signi\fcant output\n(transmission) occurs only at resonance in a very narrow\nfrequency width \u0001 !in the region !\u0019!Lat ESR. In\nthis experiment, for Mn4+ion \u0001!'40kHz (red band\nin the Fig. 1). The variation of Q-factor was small due\nto a little dielectric variation among the selected modes,\nand observed loaded Q-factor QLwas always more than50,000 at 20 mK. Assuming, \u0001 !in the order of the\nline-width of all the selected WG modes, the minimum\nnumber of detectible ions settingPn\nP\u00191 (see Eq.1) may\nbe as low as 0 :1ppblevel of concentration.\nFifteen WG modes with high-azimuthal-mode-number\nwith a frequency range of 7 GHz to 18GHz , and thus\nelectromagnetic energy \flling factors of order unity were\nmonitored. The static magnetic \feld between \u00000:2Tto\n1Twas varied through the use of computer control in a\nstep of sweep 4\u000210\u00004T. Each WG mode was scanned\nfor a period of \fve seconds at each step of magnetic \feld\nsweep. This slow sweep of magnetic \feld was applied\nunder control of an in-house MATLAB program to avoid\nheating above 20 mK, with the microwave input power\nof\u000060dBm .\nTo avoid the addition of thermal noise from room tem-\nperature, a 10 dBmicrowave attenuator was used at 4 K\nstage and another one at 1 Kstage of the DR. Also, a\n20dBattenuator was added at 20 mK stage of the DR.\nThese cold stage attenuation plus the use of a low noise\ntemperature cryogenic ampli\fer after the resonator en-\nsures good enough signal to noise ratio ( SNR ). From this\nmulti-mode ESR characteristics with hyper\fne structure\nplotting as a map ( Fig:2), we were able to identify the\ntypes of paramagnetic impurities present in the crystal.\nC. Results and Discussion\nUsing the prescribed technique of experiment, the\nmonitored ESR spectrum is mapped as in Fig-2. The\nisoelectronic Cr3+ion ofMn4+ion41has a nuclear spin\nI=7\n2, but manganese has nuclear spin I=5\n2which is re-\nsponsible for hyper\fne structure of 6-lines ( Fig: 1;2 and\n3). The observed ESR spectrum assures the presence\nofMn4+ion in theSrLaAlO 4crystal lattice. Optical\nspectroscopy study of Zhydachevskii et al.12shows that\nmanganese ion is present exclusively in doped SrLaAlO 4\ncrystal tetragonal lattice in the form of Mn4+ion occupy-\ning six fold coordinated Al3+sites. Hence the presence\nofMn4+ion in lower valence Al3+site is enhanced13.\nAlso, the higher valence state of Mn4+ion resulting\nin stronger Coulomb interaction between Mn4+ion and\nO2\u0000ion12,13. The fact that in the4T2gtriplet,Mn4+ion\nhas energy level overlapping depending on local charge\ndensity12,13. This overlapping in the perovskite crys-\ntal SLA structure display variety of magnetic proper-\nties as a linear combinations of the atomic Hartree-\nFock orbitals14,15,42,43in the molecular orbitals (MO) of\nMnO 6complex15,16.\nSome anisotropy of g-factors and hyper\fne line space\nbroadening is observed, opposite to the direction of the\nincrease of applied DC magnetic \feld ( Fig: 2 and 3).\nThe geometrical anisotropy terms of the single MnO 6\nstructure is an important case where local order, as es-\ntablished by local interactions, cannot be freely propa-\ngated throughout space. The system can lift degeneracy\nresulting charge or spin ordering of manganese ion44,45.3\n0DJQHWLF\u0003)LHOG\u0003\u000b7\f5HVRQDQFH\u0003)UHTXHQF\\\u0003\u000b*+]\f\u0013\u0011\u0013\u001a \u0013\u0011\u0013\u001b \u0013\u0011\u0013\u001c \u0013\u0011\u0014\u0014\u0013\u0011\u0019\u0016\u00142\u0014\u0013\u0011\u0019\u0016\u00144\n\u0013\u0011\u0014\u0014\u0013\u0011\u0016\u0013\u0011\u0017\u0013\u0011\u0018\u0013\u0011\u0019\u0013\u0011\u001a\n+\\SHUILQH\u00033HUWXUEDWLRQ\nFIG. 1. Transmission spectrum colour density plot of Mn4+\nion nuclear spin perturbation with WGH 5;1;1mode of reso-\nnance frequency 10 :6313 GHz .\nMagnetic Field (T)0.050.060.070.080.090.10.110.120.130.140.15Frequency (Hz)\u00011010\n0.80.911.11.21.31.41.51.6gII=14.871 GHz\n12.264 GHz10.631 GHz9.072 GHz7.613\n224.5 G249 G\n556 G531.5 G\n7.4467.5307.6887.7457.789\n+5/2+1/2+3/2-3/2-1/2-5/2\nFIG. 2. g-factor map of Mn4+ion ESR spectroscopy show-\ning hyper\fne structure broadening. Six lines of nuclear mag-\nnetic quantum numbers +5\n2;+3\n2;+1\n2;\u00001\n2;\u00003\n2and\u00005\n2shows\ndi\u000berent g-factors.\nCrystal distortion relates to Jahn-Teller distortion46, and\nmetal-ligand charge transfer with orbital ordering plays\nan essential role in stabilizing ferromagnetic states10,28,45.\nThe measured parallel g-factors decreases in the or-\nder of nuclear magnetic quantum number +5\n2;+3\n2;+\n1\n2;\u00001\n2;\u00003\n2;\u00005\n2with the increase of magnetic\n\feld (Fig: 2 and 3). Measured parallel g-factors are\ngkMn= 7:789;7:745;7:688;7:613;7:530 and 7:446 ac-\ncording to the order +5\n2to\u00005\n2. Similarly, hyper\fne line\nspacings are AkMn=\u0000209:8\u000210\u00004cm\u00001;\u0000196:4\u0002\n10\u00004cm\u00001;\u0000182:7\u000210\u00004cm\u00001;\u0000169:1\u000210\u00004cm\u00001;\u0000\n159:7\u000210\u00004cm\u00001;\u0000153:7\u000210\u00004cm\u00001according to\nthe order of nuclear magnetic quantum number +5\n2to\n\u00005\n2at 10:6313GHz (WGH 5;1;1) (Fig. 2 and 3).\nThe crystal \feld created large gap between t2gand\ne2gstabilizing 4+ oxidation stage of manganese ion47.\nIt is a worthy remark that, in principle, any decrease\ninMnO 6symmetry results in at least partial lifting of\nthe orbital degeneracy, no matter how small the dis-\nplacements are. Also, it has been observed by ESR in\n0.060.070.080.110.120.13−4.25−4.2−4.15−4.1−4.05−4\u0011\u0013−3.95−3.9\nMagnetic Field (T)7UDQVPLVVLRQ\u00036 \u0015\u0014\u0003\u000bG%\f\n0.1\u0013 0.\u0013\u001c\u000e\u0018\u0012\u0015\u000e\u0016\u0012\u0015\u000e\u0014\u0012\u0015\u0010\u0014\u0012\u0015\u0010\u0016\u0012\u0015\u0010\u0018\u0012\u0015\n−4.\u0016+\\SHUILQH\u00030XOWLSOHW\u0003%URDGHQLQJFIG. 3. ESR spectrum of WGH 5;1;1mode of resonance\nfrequency 10 :6313 GHz . Spectrum of hyper\fne line shows\naverage spacing 44 :16G(\u0000178:5\u000210\u00004cm\u00001) ofMn4+ion\nat 20 mK.\n.\nPbTiO 3that valence state of doped manganese in Ti3+\nsite changes from Mn2+toMn4+with increase of its\nconcentration21. This is usually accompanied by a dis-\ntortion of crystal structure, typically through an interac-\ntion with the lattice48. Likewise any orbital degeneracy\nlifting in the crystallographic sites due to structural dis-\ntortion is bound to entail di\u000berences in the total electron\ncharges leading to a non-integer oxidation state. Plausi-\nbly, elongated MnO 6octahedron due to tetragonal dis-\ntortion inSrLaAlO 4may implies on the modulation of\ncharge density and variation of oxidation in the covalency\nstate of the Mn4+ion extra charges in the substituted\nAl3+ion sites20,47. The tetragonal elongation (along c-\naxis) is 0:236\u0017A in theAl3+site oxygen octahedron of\ntwo bonds Al\u0000O2 of length Rk= 2:121\u0017A and four\ncoplanar bonds Al\u0000O1 of length R?= 1:885\u0017A between\naluminium and oxygen49,50. As an instance, it may be\nmentioned that about 100 cm\u00001energy change in aver-\nage can be caused by 0 :01\u0017A o\u000b-center displacement of\nthe impurity ion (due to structural distortion)51. This\ncrystal may have a little rhombic distortion at 20 mK\ntemperature and is not identi\fed as a ferroelectric crys-\ntal.\nObserved giant g-factor indicates high magnetic mo-\nment of electron in the Mn4+site at 20mK. ESR spec-\ntrum reveals this magnetization state directly as a spin-\nHamiltonian parameter rationalizing to intricate elec-\ntronic hybridization. In this hybridization, paramagnetic\nion's three phenomena ferro-electricity, magnetization,\nand spin-crossover are observed as coexisted10. Empiri-\ncally, giant g-factor due to ferromagnetic two-level system\nwhich is formed in two potential minima on the APES be-\ntween metal Mnand ligandOdue to HOMO and LUMO\ncoupling (Fig.4). The octahedral central manganese ion\nshifting with respect to oxygen Q\u000b= (Qx; Qy; Qz)\nin normal coordinates creates MnO 6structural insta-4\nbility under a condition that the curvature of resultant\nspring constant K= (@2E\n@Q2\u000b)\u000e(deviating from cubic sym-\nmetry) negative10,52.E=h \u000ejHj \u000eithe energy at high-\nsymmetry (cubic), the ground state wave function is  \u000e\nand H is the metal-ligand interaction Hamiltonian.\nWe consider that K=K\u000e+Kv. Where10;\nK\u000e=\u001c\n \u000e\f\f\f\f\u0012@2H\n@Q2\u000b\u0013\n\u000e\f\f\f\f \u000e\u001d\n(2)\nis the ground state diagonal matrix element. It de-\nscribes the \fxed (rigid) nucleus high symmetry electron\ndensity distribution j \u000ej2re\recting sti\u000bness of the lat-\ntice as a long-range (whole crystal) feature. Whereas, the\ntermKvis always negative due to the Born-Oppenheimer\nground state wave function and does not include long-\nrange inter-cell interaction. This o\u000b-diagonal matrix ele-\nments are described in terms of second order perturbation\nas:27,52\nKv=\u00002X\nnjh \u000ej(@H\n@Q\u000b)\u000ej nij2\nEn\u0000E\u000e(3)\nInstability arises in the structure under the condition\nK\u000e+Kv<0 in strong enough PJTE lower-symmetry\ndue to manganese ions additional covalency with oxy-\ngen. We can consider that K\u000eandKvcancel one another\napproximately52. This allows us to focus on electronic\ninteraction Hamiltonian for valence electrons only. In\ncase ofMnO 6elongated (along z-axis) octahedron form-\ning molecular orbitals as a linear combination of single\nelectron Hartree-Fock wave function, the structure has\nattained the PJTE state of vibronic coupling at 20 mK.\nTheHOMO ist1uenergy level of oxygen p\u0019function and\nLUMO ist2genergy level of Mn4+iond\u0019function in\nhybridization52. Referring the wave functions  \u000eand n\nto HOMO and LUMO respectively, the vibronic coupling\nconstant of PJTE for Mn4+ion additional covalency can\nbe given as:52;10,27\nF=\u001c\n2pz(O)\f\f\f\f\u0012@H\n@Qx\u0013\n\u000e\f\f\f\f3dyz(Mn)\u001d\n(4)\nThis perturbation forms a two-level system in the pro-\n\fle of APES (Fig.4).\nThe local character of the negative Kvcontribution\nto the curvature indicates that the instability producing\nPJTE is essentially of local origin, and the long range\n(whole crystal) interaction of K\u000eis important in realiza-\ntion of instability condition jKvj>K\u000e. This means that\nthe PJTE of manganese ion center can be taken into ac-\ncount as two-level system27approximately reducing the\ndenominator of the equation-3. Evidence of this insta-\nbility reveals by the increased energy (or frequency) of\nspin transition in ESR due to higher magnetization of\nthe ionMn4+creating giant g-factor as observed in the\nexperiment. In such a two-level state, higher frequency\nWG mode ESR transmissions should be noisy, and ob-\nserved same results as shown in the \fgure-5a,b,c,d. The\nQ 2∆ 𝑄\n𝑄 − 𝑄 𝐸 (𝑄)  \nΙ ӀӀ \n𝑂ଶ- 𝑂ଶ- \n𝑂ଶ- \n𝑂ଶ- 𝑂ଶ- 𝑂ଶ- \n𝑀𝑛ସା \n(a) (b) FIG. 4. - (a) Two-level system with two minima on APES at\n+Q\u000eand\u0000Q\u000edue to PJTE atF2\nK\u000e>\u0001 state (\u0001 is the energy\ngap) of distortion in normal mode Q=Q\u000b(Qx; Qy; Qz). At\nJTE stage the pro\fles are joining along dotted crossing lines:\none from left to up right another from right to up left in a\nsingle distortion mode. (b) Elongated MnO 6octahedral four-\nfold axes shape.\nd3electron con\fguration of Mn4+ion e\u000bective spin state\nS=3\n2in thet2gorbital triplet is realised in the high\ncrystal \feld of SrLaAlO 4. In contrast, according to the\nobserved ESR spectrum of giant g-factor and low \fne\nstructure term, neither high-spin (HS) state S=5\n2nor\nlow-spin (LS) state S=1\n2ofd5electron of Mn2+ion is\nto be considerable in this substantially elongated MnO 6\noctahedral structure at 20 mK temperature. There-\nfore, considering the typical molecular-orbital in energy\nscheme ofd3electron spin con\fguration, the HOMO is\n(t1u#)3(t1u\")3(t2g\")3with the ground state term4A1g\nand the LUMO is ( t1u#)2(t1u\")3(t2g\")3(t2g#)1with\nthe lowest ungerade term4T1uof odd parity10,52. This\nground and excited states of opposite parity mediates the\ntwo-level vibronic coupling in the PJTE. An important\nfeature is that it takes place as a magnetic dipolar e\u000bect\nin theMnO 6complex.\nThe measured high parallel g-factor gkMnofMn4+\nion ESR in SrLaAlO 4may compare with Suchocki et\nal.53observed enhanced zeeman e\u000bect with an e\u000bective g-\nfactor in the range about 6 to 8 for Mn4+ion in gadolin-\nium gallium garnet (GGG) at 25 mK. It includes the de-\ngree of localization of the extra electron of Mn4+ion, and\nindicates that spin transition has magnetic impedance\ndue to inherent 'frustration' coupling.\nThe excess positive charge of Mn4+increases its\nCoulomb force on the O2\u0000surrounded site, and cause\nof orbital energy reduction. Meddey et al.54claimed\nevidence of oxygen-vacancy induced ferromagnetic or-\nder ofMn4+ion in the Mn doped SrTiO 3single crys-\ntal. Oxygen vacancy or de\fciency ( \u000e= 0:15) in the\nsample is formed as SrTi 1\u0000xMnxO3\u0000\u000e, and its ferro-\nmagnetic phase shows a clear hysteresis loop at low\ntemperature54,55. Hence, the excess positive charge\nofMn4+in theMnO 6elongated octahedral sites of\nSrLaAlO 4substituting Al3+having oxygen de\fciency\ncreates order-disorder state in ferromagnetic phase and\ninteracts with external applied magnetic \feld of ESR5\n\u000bD\f \u0003\u000bE\f\n\u000bF\f \u000bG\f\nFIG. 5. Noisy hyper\fne perturbation of Mn4+ion ESR spec-\ntrum due to magnetic impedance of mode frequency with the\nincrease of frequency 9 :0719 GHz to 14 :8718 GHz arises in\ncascade due to coupling with ferromagnetic two-level system\nof higher local magnetization respectively. Also, slope of the\nelectron transmission among consecutive nuclear spin pertur-\nbation is lowering with the increase of frequency. Figure-(a)\nshows clear hyper\fne structure and ESR transmission spec-\ntrum, and sti\u000b slope of electron transmission spectrum among\nconsecutive nuclear spin hyper\fne coupling.\nspectroscopy. Typically, another example of oxygen va-\ncancy mechanism for the reason of observed giant g-factor\nis as Gorni et al56reported in their X-band parallel mode\nESR experiment that the g-factor of manganese is 8.1 in-\ncluding 6:6mThyper\fne structure line spacing at 5 K,\ncentered at 86 mT. More clearly, the resonant frequency\nwith applied external DC magnetic \feld Bparallel to\nthe crystal axis may be given by the Kittel formula57\n!=\rp\nB(B+\u00160M). WhereMis the magnetization\nof the ferromagnet and \ris the gyromagnetic ratio. As\na result, determination of g-factor depends on the rela-\ntive spin and orbital moments of a material which can be\nevaluated by use of the well-known relation57\u0016F\n\u0016S=g\u00002\n2.\nWhere,\u0016Fis the moment of spin in ferromagnetic state,\nand\u0016S=\u0016Bis the free spin moment.\nHence, we observed a giant g-factor as ESR is ob-\nserved at higher resonance frequency with applied DC\nmagnetic \feld B in addition with local magnetization\nM. In such a local environment with intricate electronic\nhybridization, measured nuclear hyper\fne parameter is\nPk' \u00003:7\u000210\u00004cm\u00001as an impact of nuclear sec-\nond order perturbation (calculated using spin Hamilto-\nnian). Using the value of manganese nuclear quadru-\nple moment58Q= +0:33(1)barn, the measured mean\ninverse third power of the electron distance is hr\u00003\nqi=\n3:325a:u:at 20mK. According to the theory, this term\nis approximately 10 \u000020% higher in values for un\flled\nd\u0000shell unpaired electrons. Ionic radius of Mn4+ion in\noctahedral structure roh= 0:67\u0017A is in good agreement\nwith these measurements59.\nAlthough, Prodi et al presented that with temper-\nature, mixed-valence manganites with the ABO 3per-ovskite structure display variation of properties with the\nrelative concentration of Mn3+andMn4+in the octa-\nhedral corner-sharing network42. Instead of this mixed-\nvalance sites, we may consider this case in the insulated\nsingleMnO 6structure taking into account a non-integer\noxidation state of Mn4+ion reducing from the 4+ oxi-\ndation state. The spin still reasonably same as it was in\ncase ofd3electron con\fguration in orbital triplet with\nhigh energy gap between egandt2g. Although, spin-\ncrossover mediates in orbital ordering resulting to struc-\ntural change60;61. Without of spin-crossover, the orbital\ntriplet splitting is viable for d3con\fguration taking into\naccount only the excited (lifted) jxziandjyzistates of\nt2gorbital triplet in the elongation along z-axis.\nIt is observed that the hyper\fne perturbation be-\ncomes noisy with the increase of resonance frequency\n(Fig:5a;b;c;d ). Apparently, the magnetic impedance\ndue to ferromagnetic order induces the electron spin\ntransitions. Increase of magnetic reminiscence in the\nferromagnetic hysteresis loop can deplete the distinc-\ntion of spin interactions. Both the ferroelectric order\nand ferromagnetic order has hysteresis loop in the above\nmentioned metastable two-level system of order-disorder\nphase. Although, the experimental results reveals the\nimpact of magnetic impedance but not able to identify\nthe accurate scale of ferromagnetic two-level system of\norder-disorder.\nRegarding this two-level system, one electron Hamil-\ntonian can be presented as H=H\u000e+HPwith the part\nof PJTE perturbation HP. In the second quantization\nformation due to Mn4+ion andO2\u0000ion (HOMO and\nLUMO) electronic coupling, it is described with raising\noperatoray\ni(O\u0016) and lowering operator aj(O\u0017) as52-\nHP=X\n\u000bX\n\u0016;\u0017X\ni;j\u001c\ni;O\u0016\f\f\f\f@H\n@Q\u000b\f\f\f\fj;O\u0017\u001d\nQ\u000bay\ni(O\u0016)aj(O\u0017) (5)\nWhere;\u0016,\u0017denotes di\u000berent atoms and i, j denotes\ndi\u000berent orbitals of coupling. Also, O\u0016andO\u0017denotes\noctahedral symmetry of the orbitals.\nD. Conclusion:\nThe elongated MnO 6octahedral structure become un-\nstable in lower symmetry raising metastable two-level\nsystem on the APES at 20 mK. WG multi-mode ESR\nprobes this instability directly revealing the Mn4+ion's\nvalence electron states in situ. Structural anisotropy\nand oxygen de\fciency has the vital role in formation of\nmetastable two-level system in electronic hybridisation\nstate which has been explained in terms of vibronic the-\nory of HOMO and LUMO coupling. Hyper \fne line width\nbroadening measurement and covalent e\u000bect is impor-\ntant for microscopic state analysis at millikelvin temper-\natures revealing nuclear dipolar hyper\fne parameter ( Pk)\nand mean inverse third power of the electron distance\nhr\u00003\nqi. Measured value of Pkis negative as manganese6\nnuclear electric quadruple moment is positive. 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Reson. 46, 85 (2015)."    },    {        "title": "1910.08148v1.Phonon_Transport_Controlled_by_Ferromagnetic_Resonance.pdf",        "content": "Phonon Transport Controlled by Ferromagnetic Resonance \n \nChenbo Zhao1,2, Yi Li1, Zhizhi Zhang1, Michael V ogel1, John E. Pearson1, Jianbo Wang2, Wei \nZhang3,1, Valentine Novosad1, Qingfang Liu2*, and Axel Hoffmann1,4* \n \n1 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA \n2 Key Laboratory of Magnetism and Magnetic Ma terials of the Ministry of Education, \nLanzhou University, Lanzhou 730000, People’s Republic of China \n3 Department of Physics, Oakland University, Rochester, MI 48309, USA \n4 Department of Materials Science and En gineering, University of Illinois at \nUrbana-Champaign, Illinois, IL 61801, USA \n \nABSTRACT \nThe resonant coupling of phonons and magnons is important for the \ninterconversion of phonon ic and spin degrees of freedom. We studied the \nphonon transmission in LiNbO 3 manipulated by the dyna mic magnetization in \na Ni thin film. It was observed that the phonons could be absorbed strongly \nthrough resonant magnon-phonon coupli ng, which was reali zed by optimizing \nthe interfacial coupling between Ni and LiNbO 3. The line shapes of phonon \ntransmission were furt her investigated consid ering the ma gnon-phonon \ninterconversion in the elastically driven  ferromagnetic resonance process. The \nresults promote unique routes for pho non manipulation and detection in the \npresence of magnetization dynamics. \n  INTRODUCTION \nElastically driven ferromagnetic re sonance (FMR) is at the core of \ncombining straintronics and spintronics [1 -5], which is drawing much attention \ndue to both interesting fundamental p hysics and potential applications. This \nincludes among others, el astically driven spin pum ping [6,7], phonon driven \ninverse Edelstein effect [8] and field-fr ee magnetization switching [9]. Recently, \nseveral studies on magnons-p honons interconversions have emerged [10-14]. In \nparticular, theoretical mode ls were developed in orde r to provide a microscopic \nunderstanding of how magnon -phonon interaction infl uences the damping and \ntransport of magnons [13,15]. The mechanisms for the coupling between \nphonons and magnons in fe rromagnetic materials include, magnetostriction \n[1-9,16] and spin-rotation coupling [14,17,18]. The latter mechanism is a \nmanifestation of the Einstein–de Haas and Barnett effects corresponding to the \ntransfer of the angular momentum betw een spin and mechanical degrees of \nfreedom [19-21]. In other words, the transfer between magnon- and \nphonon-angular momentum can be mani pulated by magnetization dynamics \n[15,19,21-24]. These earlier research wo rks shed light on the opportunity for \nphonon manipulation  [25] and detection using magnetization dynamics. \nHowever, a remaining challenge is to accurately measure the changes of the \nphonon systems due to the ma gnon-phonon interaction [4]. Therefore, we focus \nhere on how the propagation of phonons  can be modulated via ferromagnetic \nresonance. \nBased on surface acoustic wave  (SAW) magneto-transmission \nmeasurements, the interaction of SAWs and ferromagnetic th in films has been \nstudied experimentally by several groups [1,2,5,8,26]. However, important \naspects of the interaction mechanism stil l remain to be resolved. An important \nparameter to characterize the interaction mechanis m between the SAWs and \nferromagnetic thin films is the linewidth of the transmission power. In general, \nthe linewidth of the transmission power obtained from phonon driven FMR is \nmuch larger than that in cavity FMR [1,2 ,26]. In order to fit the broadening of \nthe linewidth of the transmission powe r obtained from phonon driven FMR, the \nposteriori damping constant used is se veral times larger than the damping \nvalues obtained from cav ity FMR experiments [1,2]. At the same time, the \nsignificant broadening of the transmission  power line shape contains important \ninformation about the magnon  and phonon coupling [4], e.g., angular \nmomentum transformation, during phonon  driven FMR. Therefore, the phonon \ntransport properties in the presence of  phonon driven FMR also provide an \neffective approach for exploring the an gular momentum transformation [11,27] \nvia the coupled phon ons and magnons.  \nIn the present work, we have invest igated the dependence of the phonon \ntransport on magnetization dynamics in Ni/LiNbO 3 hybrid heterostructures. We observe the simultaneous existence of di ps and peaks in the transmission power \nof the surface acoustic waves (SAW) devi ce, which we interpret by considering \nboth phonon attenuation and gene ration [28,29]. By  fitting the theory [28,29] to \nthe phonon transmission power, the value of the FMR field, resonance \nlinewidth and the ex change stiffness parameter of  the magnon system can be \nestimated. The obtained resonant linewidth of the SAW driven FMR is very \nsimilar to those from waveguide FMR experiment. This suggests that the \nphonon driven FMR linewidth broadeni ng is mainly due to magnon-phonon \ninteraction, instead of the nonuniform excitation fields induced by SAW. Our \nfindings thus generate new opportunities  for phonon control and detection of \nmagnetization dynamics. \n \nEXPERIMENT AND SET UP \nFigure 1(a) shows a schematic il lustration of the magnon-phonon \ninteraction in our SAW driven FMR measur ement. In these devices, we excite, \nat microwave frequencies, the SAW using interdigitated transducers (IDT) on a 127.86\no Y-X cut LiNbO 3 substrate with the thickness of 500 μm. The SAW then \ndrives subsequently FMR in the Ni fi lm. The optical images of the Ni/LiNbO 3 \nhybrid device with a 50-nm thick Ni rectangular film  is shown in Fig. 1(b) and \n(c). The IDTs of 50-nm thick Al were fabricated by using maskless \nphotolithography and electron-beam evaporation. The finger width and the \nspacing of the IDT were both 3 μm, launching SAWs with a wavelength ߣௌௐ \nof 12 μm. The Ni film was deposited via DC  sputtering using 10 W power, and \nan Ar pressure of 3 ×10-3 Torr. The length of the delay line L was set at L = 438 \nμm, and the Ni film was deposited in the center of delay line. In order to \nobserved phonon driven FM R, a DC magnetic field H is applied in a direction \nwith an angle θ with respect to the SAW wavevector ݇ௌௐ set to θൌ60º and \nθൌ90º, respectively. During ph onon driven FMR measurements H is varied \nfrom -1.1 to 1.1 kOe. As shown in Fig. 1(d), the reflection parameters of the \ndevice exhibit one peak at f0= 325 MHz, measured using a vector network \nanalyzer (VNA). The velocity of th is SAW mode can be determined as \nݒௌௐ ൌ 3912 m/s, according to the foll owing relation between the velocity \nݒௌௐ and frequency  f0 of SAW: ݒௌௐ ൌߣ ௌௐ݂. The measured ݒௌௐ is \ntypical for the velocity of a Rayleigh SAW [30].  \nTo separate the SAW signals and the electromagnetic wave, the time-gating \nfunction of VNA was used [1,26]. Fi gure 1(e) shows the transmission \nparameters of SAW signal for the device with gating on (centered at 145 ns and \nspanning over 45 ns) [1,26]. The SAW odd harmonics from 3th to 13th have \nbeen launched and the highest harmonics is located at 4.23 GHz. Because the \n9th and 13th harmonics are weak, this work focuses on the 3th, 5th, 7th and 11th \nharmonics to investigate th e phonon transport during  the coupling of magnon and phonon relevant to the Rayleigh SAW. Some of th e device with Ni films \nwere annealed at 400 oC for 30 minutes in vacuum, and the as-grown and \nannealed devices both were measured to study the phonon transport properties \n[see Fig. 1(a)]. \n \n \n \nFigure 1 (a) shows schematic illustration of the magnon-phonon interaction during SAW \ndriven FMR; (b) and (c) show optical images of the Ni/LiNbO3 hybrid device with 50-nm \nthick Ni rectangular film; (d) shows the re flection parameters for the device with H = 1000 \nOe applied at θ = 60º; (e) shows the transmission parameters of SAW signal with gating on \n(centered at 145 ns and spanning over 45 ns) H applied at θ = 60º. \n \nRESULTS AND DISSCUSSION \nThe SAW is generated and detected by electromagnetic wave using a pair \nof spatially separate d IDTs on the LiNbO 3 crystal. Coupling with the magnon \nsystem (Ni films) is achieved during the SAW propagation in between the two \nIDTs. Phonon-driven FMR can be characterized by the transmission parameter \nS21 using the VNA with time gating  function[1,26]. To enhance the \nsignal-to-noise ratio, the transmission parameter S 21 of SAW was converted \ninto transmission power P. Figure 2(a-f) show the colormap of magnetic field \nH dependence ( H is applied at θ = 60º) of the normali zed transmission power P \nof the SAW signal: (a-c) for an as-grown device at 5th, 7th and 11th harmonics, \nrespectively; (d-f) for an annealed device at 5th, 7th and 11th harmonics, \nrespectively. The color codes represent in tensity of transmission power of SAW, \nwith red indicating maximum transmissi on. The dotted green lines represent \nthe Kittel formula, whic h is based on waveguide FMR measurements for an \nextended Ni film grown on LiNbO 3. The colormap data sh ows that the intensity \nof SAW exhibits a minimum near zero fi eld for the as-grown device, but an \nobvious attenuation near the FMR field is observed for the annealed devices. \nThis shows that the annealing treatm ent improves the resonant coupling \nbetween the LiNbO 3 and the ferromagnetic Ni ma terial. The enhancement of \nthe resonant magnon-phonon coupling can be attri buted to improving the \ninterface of the magnon -phonon system [31] via the thermal annealing \ntreatment. This magnon-ph onon coupling, as indi cated by the dispersion \ncrossing, is enhanced due to the strong abso rption of phonons at f = 2.24 GHz \nand 3.56 GHz for the annealed devi ce. The attenuation of the SAW \ntransmission power near zero field [Fig s. 2(a-c)] and the FMR field [Figs. \n2(d-f)] can be attributed to the magne tization switching [1,32] and the FMR \nabsorption [1-5], respectively.  \nThe plots of the magnetic field de pendence of the SAW signal peak \nposition are shown in Fig. 3 and illu strate the phonon transport properties. \nFigure 3(a) and (c) show the magnetic field ( H) dependence of the normalized \ntransmission power for th e as-grown device at θൌ 60º and 90º, respectively. \nWhen H is applied at θൌ 60º, the transmission power shows large dips near \nthe zero field region due to the magn etization switching, which has been \nreported before [1,32]. Notably, peaks for transmission power near the FMR \nfield are also observed [Fig. 3(a)].  These peaks indicate that the SAW \ntransmission intensity increase s near the FMR field. When H is applied at θൌ \n90º, only the magnetization switching dips  remain [Fig. 3(c)]. This shows that \nfor θ = 60º ferromagnetic resonance is excited. However, there is only the \nuniform mode that particip ates in the coupling for θ = 90º. \nWe find that the annealing treatment for the device can markedly tune the \nphonon transport. Figure 3(b) and (d) show the H dependence of the \ntransmission power for the annealed device at θൌ 60º and 90º, respectively. \nThe dips at the FMR field and the peaks above the FMR field in the \ntransmission power spectra  are both observed at θൌ 60º [Fig. 3(b)] \nsimultaneously, but they are not present when θൌ 90º [Fig. 3(d)]. This \nsuggests that both attenu ation and amplification of  the phonon transport is \npossible due to the magnon- phonon interaction. The H position of the peaks \n[Fig. 3(a)] and dips [Fig. 3( b)] for as-grown and anneal ed devices are plotted in Figs. 3(e) and (f), respectiv ely, as well as the expected behavior based on the \nKittel formula. The H positions are in good agreem ent with the Kittel formula, \nwhich indicate that the phonons transport is mani pulated by magnetization \ndynamics during FMR.  \n \n \nFigure 2: Colormap of magnetic field dependen ce of the normalized transmission power of \nSAW signal for the as-grown and the annealed devices at different SAW harmonics with H \napplied at θ = 60º: (a-c) for the as-grown device at the 5th, 7th and 11th harmonic; (d-f) for the \nannealed device at the 5th, 7th and 11th harmonic. The dotted lines represent the Kittel formula, \nwhich come from waveguide FMR experiment for the full Ni film. The color codes represent \nintensity of transmission power of SAW, with red indicating maximal transmission. \n \nTo model the phonons transport, we take into consideration both phonon \nattenuation and generation  [28,29] manipulated by  the magnetic dynamics, \nduring the SAW-driven FMR. Angular momentum interconversion between \nmagnon and phonons is relate d to the Einstein–de Haas  effect [14,18] and the \nBarnett effect [21]. Due to the ma gnon-phonon interconve rsion during FMR, \nphonons will redistribute the angular momentum and energy between magnons \nand phonons [15,17], and thus macroscopic mechanical rotation can be \nstrengthened, leading to an enhancement of the transmission  of the SAW. The \nchange of the SAW transmitted power ( E) related to the phonon attenuation and \ngeneration during FMR can be rewritten as [28,29]: \nܧൌܧ ௩\nమమ\nరరቚቚ\n∆ுమ(1ଶ)ିభ\nమሾ1 ()ߚ ଶሿିభ\nమexp ିആ\nమ\nଵା(ାఉ )మ൨ቂ 1\nexp ቀିఎబு\nଶቁቃ(1), \nwhere, \nൌுିு \n∆ு (2),  \nߟൌଶగ మమ\nరర௩ெೞ∆ுఓ బ (3),  \nߚൌଶఘ(ଶగ)మ\nరరெೞ∆ுఓ బ (4), \nand the  H dependence for the line shape of the transmission power is \ndetermined by  the parameters p, η and β. Equation (1) corresponds to the \nconventional microwave transmission through a Ni film [28,29], but employs \ninstead of an externally applied real microwave magnetic field the effective ac \nmagnetic field ݄ୄ due to the magnon-phonon in teraction defined in Eq. (5) \nbelow. The parameters in Eqs. (1-4) are: ݒ௧ = 3912 m/s is the Rayleigh \nacoustic wave velocity, ܿ = 3×108 m/s is the speed of light, B 2 = 8.7 ×106 N/m2 \nis the magnetoelastic parameter of Ni, C 44 = 1.22 ×1011 N/m2 is the elastic \nmodulus of Ni, ∆ܪ is the linewidth of the FMR, ܮ = 12 μm is characteristic \nlength related to the SAW wavelength, H is the external magnetic field, ܪ is \nthe FMR field, ܯ௦ = 4.7 ×105 A/m is the saturation magnetization, ܣ is the \nexchange stiffness parameter, ߩ = 8900 kg/m3 is the density of Ni, ݂ the \nfrequency of the SAW and ߤ is the permeability of vacuum. The last term in formula (1) is the phonon scatte ring due to the viscosity and \nthermo-conductivity [28,33,34]. \nThe field angle dependence ( H at θൌ 60º and 90º) for the line shape of \nthe transmission power [Figs. 3(a) and (b); Figs. 3(c) and (d)] exhibits the fingerprint for the Rayleigh SAW driven FMR [1,5], which is relevant to the \ncomponents ห݄\nୄห of the effective ac magnetic field ݄ perpendicular to \nthe magnetization [1,2,5]: ห݄\nୄหൌห ݄ ௫ߠ݊݅ݏ െ ݄ ௬ߠ݊݅ݏ െ ݄݅ ௭หൌ2 (ܾଵെܾ ଶ)ߝ௫௫cos ߠ sin ߠ \n4ܾߝ௭௫cos ( ߠ5) , \nwhere b1 and b2 are the longitudinal-type magnetoelastic coupling constants, b6 \nis the shear-type magnetoelastic coupling constant, ߝ௫௫and ߝ௭௫ are pure \nRayleigh SAW strain components. As can  be seen from Eq. (5), the value of \nห݄ୄห is zero at θൌ90º. Thus, the SAW cannot excite the FMR in that \nconfiguration. Therefore, both the dips and peaks due to the phonon attenuation \nand generation observed for θൌ60º [Figs. 3(a) and (b)] are related to the FMR \nand vanish at θൌ90º [Figs. 3(c) and (d)]. Unlike the asymmetry of the dips for \nopposite H orientations in previous results [1 ,5,8], the dips are nearly symmetry \nin Fig. 3(b). The asymmetry has b een found being pr oportional to the \nmagnetoelastic coupling constant  (ܾଵെܾ ଶ) [5]. Therefore, when the values of \nܾଵ and ܾଶ are close to each other, the asy mmetry becomes less apparent. This \nalso indicates that the shear- type magnetoelastic coupling ( b6) is dominant over \nthe longitudinal-type magnetoelastic couplings ( b1 and b2) for our device. \n \n \n  \n  \n \n  \n \n \n  \n  \n  \n \nFigure 3: (a-d) Magnetic field H dependence of the transmissi on power [the symbols are \nexperimental data, and the soli d lines are fit to theory using Eq. (1)]: (a) as-grown at θൌ60º; \n(b) annealed device at θൌ60º; (c) as-grown at θൌ90º; (d) annealed device at θൌ90º. (e) \nshows the H position of the transmission power peaks at angle θൌ60º for the as-grown \nsample; (f) shows the H position of transmission power dips at angle θൌ60º for the annealed \ndevice. The Kittel formula is determined from waveguide FMR experiments for full Ni film. \n \n  \n \n \n \nFigure 4: (a-c) Theoretical calculation results of the magnetic field H dependence of the \ntransmission power at different frequenc ies using Eq. (1): (a) With different ߟ at f=3.56 GHz; \n(b) and (c) with values of ߟ=3.1 and ߟ = 0.3 at different frequencies. The transmission \npower are relative values and the cures are shifte d with respective to each other. (d) Linear \nfitting of ߟ and ∆ܪ using ∆ߟܪ= ݂݇(k is a constant), where ߟ and ∆ܪ are obtained from \ntheory fits to the line shape of the tr ansmission power at different frequency f for the annealed \ndevice using Eq. (1), as shown in Fig. 3(b). \n \nThe experimental results for the anneal ed device are show n in Fig. 3(b). \nThe line shape of the transmission power can be fitted using Eq. (1), and the fit \nparameters ߟ ,ߚ and ∆ ܪare shown in the table in  Fig. 4(d). Since only ∆ܪ \nand ߟ change with f, the expression of ߟ can be simplified as ∆ߟܪ=( ݂݇k is a \nconstant ), resulting in a good fit for ߟ and ∆ܪ ,as shown in Fig. 4(d). This \nsuggests that the model describes the phonon transp ort in these SAW devices \nvery well. By putting the fitted value of ߚ into Eq. (4), the exchange stiffness \nparameter of Ni film is estimated as A = 0.54 ×10-6 ergs/cm ( f= 3.56 GHz) and \n0.64×10-6 ergs/cm ( f= 2.24 GHz). These obtained values for A are slightly \nsmaller than A = (0.76 ±0.03) ×10-6 ergs/cm obtained from elastic small-angle \nneutron scattering for nanocrystalline Ni  film [35]. At a lower frequency, f = \n0.97 GHz, the anomalous value of η and β obtained from theory fits can be \nattributed to the nonuniform magnetizatio n distribution, whic h also results in \nlarge inhomogeneities of th e magnetization precession.  \nHowever, for the transmission power of  the as-grown device [Fig. 3(a)], \nthe experimental data cannot be fitt ed using Eq. (1) due to the strong \nattenuation given by the magnetiza tion switching. According to the \nmaterials-specific parameters fo r Ni and the FMR parameters at f = 3.56 GHz \ndetermined from an independent  waveguide-FMR experiments, ߟ is \ncalculated to be ߟ = 3.1 using Eq. (3). This calculated value of ߟ is consistent \nwith fits of the data to  Eq. (1). We neglect the phonon scattering due to the \nviscosity and thermo-conductivity and set ߟ=0 [28]. The theoretical \ncalculations using Eq. (1), i.e., the H dependence of the transmission power \nwith different values of ߟ at f = 3.56 GHz, are shown in Fig. 4(a). The \ntransmission power is shown using rela tive values and the cures are shifted \nwith respective to each other. The ca lculations show that the dips at H = ܪ \nare increasingly obvious with the increase of ߟ ,which means the phonons are \nstrongly attenuated at the ferromagnetic  resonance field. The peaks related to \nthe phonon gene ration at ܪൌܪ ∆ ܪ ඥ2ߟ െ 1  (ߟ1 / 2 ) become more \nobvious with the decrease of ߟ .When ߟ1 / 2 , only the peaks at H = ܪ are \nobserved. Figures 4(b) and (c) show  theoretical calculations of the H \ndependence of the transmissi on power with  values of ߟ = 3.1 and ߟ = 0.3 at \ndifferent frequencies, respectively. It ca n be seen that the line shape of the \ntransmission power has a good agreem ent with the annealed [Fig. 3(b), \nߟ1 / 2 ] and as-grown device [Fig. 3(a), ߟ1 / 2 , except the dips near zero \nfield due to magnetization switchin g], respectively. The changes in ߟ related \nto magnetoelastic coupling [Fig. 3(a)  and (b)], can be  attributed to \nimprovement the interface of the magnon -phonon [31] system  by the annealing \ntreatment. \nFurthermore, the linewidth ∆ܪ of the transmission power can be \nobtained to be ∆ܪ = 325 Oe and 500 Oe at f = 2.24 GHz and 3.56 GHz from the transmission power dips plotted in Fig. 3(b), which are approximately 1.7 \ntimes larger than the values ∆ܪ = 185.3 Oe and 290.6  Oe obtained from \nwaveguide FMR experiments. These values are consistent to the theory fitting \nresults ( ∆ܪ = 186 Oe and 292 Oe) using Eq. (1 ). Therefore, compared to the \nFMR measurements, any broadening effect coming from nonuniform excitation \nfields induced by SAW [1,2 ,26] can be ignored. The in crease of linewidth can \nmainly be attributed to the magnon -phonon interaction, e.g., angular \nmomentum transformation [15,23]. These results show that the coupling \nbetween elastic and magnetic degrees of freedom open additional channels for information interconversion between phononic and magn onic components. \n \nCONCLUSION \nIn conclusion, phonon transport prop erties during th e phonon driven \nferromagnetic resonance has been inves tigated. The ferromagnetic resonance is \ndriven acoustically, since no external rf magnetic field is applied to the \nferromagnet. Rather, a purely internal rf magnetic field arises due to \nmagnetoelastic coupling between the surf ace acoustic wave elastic strain field \nand the ferromagnet. Annealing of the sample results in increase interfacial \nmagnon-phonon coupling, and thus en ables tuning of the line shape of \ntransmission power. Considering both the phonon attenuation and generation \nsimultaneously during the phonon driven FMR, the p honon transport properties \nand line shape of transmis sion power can be ex plained in a quantitative fashion. \nBy analyzing the pho non transmission power, th e magnetization dynamics of \nthe magnon system can be detected. We also demonstrate that the broadening \neffect of the transmission power line shape during phonon driven FMR can be \nmainly attributed to the interaction of magnons and phonons, instead of the \nnonuniform excitation fields i nduced by surface acoustic wave. \n \nACKNOWLEDGEMENTS \nThis work was performed at the Argonne National Laboratory and supported by the \nDepartment of Energy, Office of Science, Materials Science and Engineering Division. \nThe use of the Centre for Nanoscale Materi als was supported by the US. Department \nof Energy (DOE), Office of Sciences, Basic Energy Sciences (BES), under Contract No. DE-AC02-06CH11357. 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Bauer,3,2Wolfgang Belzig,4,2and Yaroslav Tserkovnyak5,2\n1Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Centre for Advanced Study at the Norwegian Academy of Scienc e and Letters, Drammensveien 78, NO-0271 Oslo, Norway\n3Kavli Institute of NanoScience, Delft University of Techno logy, 2628 CJ Delft, The Netherlands\n4University of Konstanz, Department of Physics, D-78457 Kon stanz, Germany\n5Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n(Dated: December 17, 2007)\nThe injection of pure spin current into superconductors by t he dynamics of a ferromagnetic\ncontact is studied theoretically. Taking into account supp ression of the order parameter at the\ninterfaces (inverse proximity effect) and the energy-depen dence of spin-flip scattering, we determine\nthe temperature-dependent ferromagnetic resonance linew idth broadening. Our results agree with\nrecent experiments in Nb |permalloy bilayers [C. Bell et al., arXiv:cond-mat/0702461].\nPACS numbers: 74.25.Fy, 74.78.Na, 85.75.-d,72.25.-b\nCooperpairsin conventionalsuperconductorsarespin-\nsinglet states and therefore cannot carry a spin cur-\nrent. Some aspects of the resilience of the supercon-\nducting state against spin-current injection have been\nexperimentally demonstrated in hybrid ferromagnet-\nsuperconductor spin valves [1], switches [2], and π-\njunctions [3]. In these experiments, the spin current\nflow in the superconducting state can only be inferred\nvia charge current measurements. This complicates the\nunderstanding of the spin current flow in superconduc-\ntors.\nInjection of a pure spin current into a superconduc-\ntor has recently been demonstrated by Bell et al.[4] in\nferromagnet |superconductor structures under ferromag-\nnetic resonance (FMR) conditions, in which the precess-\ning magnetization acts as a “spin pump” [5]. The spin\nangular momentum lost by the ferromagnet can be ob-\nserved directly in terms of an increased broadeningof the\nFMR spectrum. In this Letter we demonstrate theoreti-\ncally that the spin transport thus measured as a function\nof temperature and device/material parameters offers di-\nrect insight into spin-flip relaxation and the inverse prox-\nimity effect in superconductors. Our theory agrees well\nwith the recent experimental results [4], and we provide\nsuggestions and predictions for future experiments.\nThe theoretical challenge of spin-pumping into su-\nperconductors as compared to normal conductors is\nthe strong energy dependence of quasiparticle trans-\nport properties around the superconducting energy gap\n[6]. Also, the energy dependent spin-flip scattering rates\ncaused by spin-orbit coupling or magnetic impurities dif-\nfer. Experimentsthat directly probe spin transport, such\nasRef. 4, thereforeprovideuniqueinformationaboutthe\nspin-flip scattering mechanism. A complicating factor is\nthe inverse proximity effect [7] that suppresses the super-\nconducting order parameter close to a metallic interface\nwith ferromagnets like Ni, Co, and Fe. The resulting\nspatial dependence of the superconducting gap requires\nsolution of the full transport equations in the entire su-perconducting layer. The spin currents measured at such\ninterfaces therefore serve as probes of superconducting\ncorrelations in magnetic heterostructures, and the tem-\nperature dependence of the FMR linewidth near and be-\nlow the critical temperature can provide a wealth of in-\nformation about spin-flip processes and superconducting\nproximity physics, with potential implications for differ-\nent areas of mesoscopic physics.\nR S Fm∆(x)\nxg⊥g\nFIG. 1: The ferromagnet |superconductor |spin reservoir\n(F|S|R) structure. Precession of magnetization m(t) pumps\nspins into S, which can diffuse and dissipate in R. The F |S in-\nterface spin-mixingconductancefor spins polarized trans verse\ntothemagnetization directionis g⊥andtheS |Rinterface con-\nductance is g. The superimposed superconducting gap ∆( x)\nis suppressed close to the interfaces (inverse proximity eff ect).\nIn the following we develop a theory\nof energy-dependent spin pumping at a\nferromagnet |superconductor interface and the re-\nsulting spectral spin current flow in the superconductor.\nWe consider a diffusive metallic heterostructure con-\nsisting of a superconducting layer (S) of thickness L\nthat is sandwiched by a ferromagnet (F) of thickness\ndand a spin reservoir (“spin sink”) (R), see Fig. 1.\nThe slowly precessing magnetization m(t) emits a spin\ncurrent that is transversely polarized with respect to\nthe instantaneous magnetization direction [5]. The spin\ncurrent that flows through S is immediately dissipated\nupon reaching R. R thus increases the sensitivity of the\nexperiments to the spin transport properties of S. R\nrepresents either a cap of an efficient spin-flip scattering2\nmaterial such as Pt or a large reservoirof a high mobility\nmetal [5]. We assume sufficient thermal anchoring so\nthat heating from absorbed FMR microwave radiation\ncan be disregarded.\nThe magnetizationdynamics is determined by the gen-\neralized Landau-Lifshitz-Gilbert equation,\ndm\ndt=−γm×Heff+G0\nγMSm×dm\ndt+γ\nMSVIs.(1)\nHereγis the gyromagnetic ratio, Heffis the effective\nmagnetic field, MSis the saturation magnetization, and\nVis the volume of the ferromagnet. The intrinsic dis-\nsipation in the bulk ferromagnet is parametrized by the\nGilbert damping constant G0.Isis the total spin (i.e.\nangular momentum) current generated by the precessing\nferromagnet. This loss of angular momentum is equiv-\nalent to an interface contribution to the magnetization\ndampingandisobservableintermsoftheenhancedFMR\nlinewidth broadening. Our task is to evaluate the effect\nof superconducting correlations on Is. The results can\nbe summarized in terms of an effective resistor model for\nthe spin transport. We find an energy-dependent spin\ntransport resistance of S in series with the spin-mixing\nresistance r⊥= 1/g⊥of the F|S interface in the normal\nstate and the conventional resistance r= 1/gof the S|R\ninterface.\nTo illustrate the physics we first sketch the results for\nm(t) rotating in the xy-plane and in the absence of spin-\nflip scattering (the derivation for the general situation\nwill be outlined subsequently). The magnetization then\nemits a time-independent spin current that is polarized\nalong the z-axis [5]. The superconducting condensate\nconsists of spin-singlet Cooper pairs. A spin current can\ntherefore only be carried in S by excited quasiparticles.\nSincethelow-energydensityofquasiparticlestatesissup-\npressed by superconducting correlations, the spin trans-\nport resistivity is enhanced when S is in the supercon-\nducting state, resulting in reduced spin injection from\nthe ferromagnet. The energy-dependent spin resistance\nis governed by a spectral Ohm’s law,\nR⊥\neff(E) =r⊥\nN(0,E)+/integraldisplayL\n0dx′ρL(x′,E)\nA+r\nN(L,E),(2)\nwhereρL= 1/(hN0DL) is the effective resistivity of the\nsuperconductor for spin transport in units of e2/h,N0\nis the density of states at the Fermi level in the normal\nstate,DL(x,E) andN(x,E) are the effective spin diffu-\nsion coefficient and the normalized density of state at po-\nsitionxand energy E, respectively [6]. At zero tempera-\nture, therelevantquasiparticleenergy Eisdeterminedby\nthe FMR frequency which is typically fFMR∼10GHz.\nFor BCS superconductors hfFMR/∆0≈0.3 K/Tcwhere\n∆0is the bulk zero-temperature energy gap and Tcthe\ncritical temperature of the superconductor. For small-\nangle precession, the effective “rotation” frequency canbe introduced as f∼φfFMR, where φis the angle of\nprecession. Thus the relevant energy scale for FMR-\ngenerated excitations is in practice expected to be much\nsmaller than hfFMR, and the characteristic energy of\npumped electrons is set by the temperature, see Eq. (3)\nbelow. At the F |S interface N(x= 0,E)≈1 due to\nthe inverse proximity effect (see below). R⊥\neffdepends on\ntemperature through the local gap ∆( x,T) which deter-\nminesN(x,E) andρL(x,E). The spin current loss of the\nferromagnet is consistent with the Gilbert phenomenol-\nogy in terms of an increased damping parameter G. It is\ndetermined by the spin angular momentum escape rate\nthrough S and reads\nG=G0+(gLµB)2\n2πℏ1\nd/integraldisplay\ndE−dfFD(E)/dE\nAR⊥\neff(E),(3)\nwheregLis theg-factor,µBis the Bohr magneton, Ais\nthe sample crosssection area, and fFDis the Fermi-Dirac\ndistribution function.\nAt temperatures T≪Tc, ∆(x) asa function ofthe dis-\ntancefromthe F |Sinterfaceapproachesthe bulkvalueon\nthe scale of the bulk superconducting coherence length\nξ0=/radicalbig\nℏD/2πkBTc. Since the relevant spin resistivity\nρL(x,E) and thus R⊥\neffare very large for E <∆,ξ0\nsets the penetration length scale for spin current into the\nsuperconductor. At low temperatures and L > ξ 0the\nGilbert damping (3) will therefore be weakly enhanced.\nOn the other hand, at T/lessorsimilarTcthe gap is suppressed\nthroughout S and transport channels at energies E/greaterorsimilar∆\nbecome accessible. R⊥\neffand the Gilbert damping then\napproach the normal state values.\nSpin-flip scatteringin S dissipates spin currentemitted\nfrom F, and enhances Gby suppressing the back-flow of\nspins into the ferromagnet. The spin-flip length in the\nnormal state is given by lsf=√Dτsf, where Dis the\nnormal state diffusion coefficient. We take spin-flips into\naccountthat arecaused by magneticimpurities aswell as\nspin-orbit coupling at impurities in terms of the spin-flip\nrate 1/τsf= 1/τm+ 1/τso[6]. The spin-orbit coupling\nrespects the symmetry of singlet Cooper pairs, whereas\nthe pair-breaking scattering by magnetic impurities sup-\npresses superconductivity and reduces Tc. BelowTc, the\nspin-flip rates in S depend on energy. For E <∆ spin-\nflip rates both due to spin-orbit coupling and magnetic\nimpurities are suppressed. For T≪TcandL > ξ0, the\nGilbert damping will therefore be weakly enhanced. On\nthe other hand, for E >∆ the spin-flip rate due to mag-\nnetic impurities is enhanced whereas the spin-flip rate\ndue to spin-orbit coupling is similar to that in the normal\nstate. Wethereforepredictanon-monotonictemperature\ndependence of the Gilbert damping close to the critical\ntemperaturewhenspin-flip isdominatedbymagneticim-\npurities. Experimental data indicate that lsf> ξ0for\ntypical S. lsf= 48 nm and ξ0= 13 nm has been reported\nfor Nb [1] (which is used in Ref. 4) whereas lsf= 1.1µm\nandξ0= 124 nm for Al [8, 9]. When L≤ξ0spin-flip in S3\nis therefore inefficient since L≤ξ0< lsfin these materi-\nals. We are then allowed to disregard spin-flip scattering\n[5]. On the other hand, when L≫lsfthe spin current\nneverreachesRsothat Gisgovernedexclusivelyby spin-\nflip in S for all temperatures. In the interesting regime\nwherelsf≈L, the full theoretical treatment sketched in\nthe following has to be invoked in order to compute the\ncompeting effects that determine G.\nThe total spin current leaving the ferromagnet in the\nF|S|R heterostructurecanbeexpressedasanenergyinte-\ngral over the balance of the spectral pumping and back-\nflow currents Is=/integraltext\ndE(iinj\ns−iback\ns). The spin current\ninjected into S by the precessing magnetization is [5, 10]:\niinj\ns(E) =ℏN(0,E)\n4πfFD(E−hf/2)−fFD(E+hf/2)\nhf\n×/parenleftbigg\ng⊥\nrm×dm\ndt+g⊥\nidm\ndt/parenrightbigg\n, (4)\nwherefis the instantaneous rotation frequency. Here,\ng⊥\nrandg⊥\niare the real and imaginary parts of spin-\nmixing conductance. For metallic interfaces, g⊥\nr≫g⊥\ni\n[11]. We therefore disregard the “effective field” g⊥\niin\n(4), although it contributes to the interface boundary\nconditions discussed below. The magnetization damp-\ning that follows from (4) is frequency dependent beyond\nthe Gilbert phenomenology. We have checked numeri-\ncally that the f-dependent terms contribute weakly to\nthe damping even when hf/lessorsimilar∆0for the parameters\nstudied. We therefore restrict attention to the linear\nresponse regime in which the Fermi-Dirac functions in\n(4) can be expanded to first order in hf. This leads to\nfrequency-independent enhanced Gilbert damping in (1).\nThe spectral back-flow of spin current into F induced by\nthe spin accumulation on the S side is\niback\ns(E) =−N(0,E)\n4πg⊥\nrhTS(0,E).(5)\nThe nonequilibrium spin distribution function hTS(x,E)\ncan be computed by Keldysh transport theory [6].\nIn the S bulk, the total spin current Is(x) =\nℏAN0/integraltext∞\n−∞dEDL(E,x)∂xhTS(x,E)/2 follows from the\ndiffusion equation\n/parenleftbigg\nN∂t+∂xDL∂x−αm\nTSTS\nτm−αso\nTSTS\nτso/parenrightbigg\nhTS= 0.(6)\nDiffusion through S is taken to be instantaneous on\nthe scale of the FMR frequency as long as f < D/L2\nand/orf≪1/τsfso that hTSin (6) becomes time-\nindependent. αm(so)\nTSTS= [Re cosh θ]2+(−)[Re sinh θ]2are\nenergy-dependent renormalization factors for the spin-\nflip rates due to magnetic impurities (spin-orbit cou-\npling), and the energy dependent spin diffusion coeffi-\ncientDL/D=αso\nTSTS. The spectral properties of the\nsuperconductor parametrized by θ(x,E) are determinedby the Usadel equation for the retarded Green function\nˆGR= ˆτ3coshθ+iˆτ2sinhθ,\nℏD\n2∂2θ\n∂x2=i∆cosh(θ)−iEsinh(θ)+3\n8ℏ\nτmsinh(2θ),(7)\nto be solved with the BCS gap equation ∆ =\n(N0λ/2)/integraltextED\n0dEtanh(E/2kBT)Re sinh( θ) [6]. Here, ED\nis the Debye cut-off energy and λthe interaction param-\neter.\nThe boundary condition for the diffusion equation (6)\nisconservationofspincurrentattheinterfaces. At x= 0,\nℏAN0DL∂xhTS/2 =iinj\ns−iback\ns. We use boundary con-\nditions derived in Ref. 12 for (7) at the S |R interface.\nAt the F |S interface we impose complete suppression of\nsuperconducting correlations, θ(x= 0,E) = 0 for the\nfollowing reasons. The large exchange energy in transi-\ntion metal ferromagnets completely suppress supercon-\nducting correlations, so that the F adjacent to S is a\nsource of incoherent particles. Additionally, spin de-\npendent interface scattering at the S side [13] induces\nan effective pair-breaking exchange field, which we es-\ntimate as Beff=ℏg⊥\ni/e2gLµBN0Aξ0[14]. Here, N0Aξ0\nis the number of states at the Fermi energy within ξ0\nfrom the interface. With g⊥\ni≈0.05gSh, where gShis\nthe Sharvin conductance [11], and approximating N0\nby the free-electron value, µBBeffis comparable to ∆ 0,\ne.g,µBBeff(Nb)∼0.56meV, µBBeff(Al)∼69µeV. The\nbulk F exchange splitting and the induced Beffby spin-\ndependent interface scattering leads to a vanishing gap\n(andθ) at the F |S interface [15, 16].\nThe spin diffusion equation (6) can be solved analyti-\ncally in the absence of spin-flip, proving (2). We now use\nthe full machinery sketched above to make contact with\nexperimental results for a F |S device (without R) similar\nto sample C in Ref. 4. Numerically computing Isinclud-\ning spin-flip caused by magnetic impurities [17], we ob-\ntaintheenhancedGilbertdamping Gfrom(1). Intheex-\nperiment, F is a permalloylayerwith d= 2nm, and gL=\n2.1. S is Nb with L= 70 nm, bulk critical temperature\nTc0= 8.91 K,lsf= 48 nm, and D= 5.41 cm2s−1[1, 18].\nFor the interface conductances we use Ar= 3 fΩm2[19].\nWe find G−G0= 0.777×108s−1atTc/2 = 3.6 K and\n1.19×108s−1in the normal state. When the inhomo-\ngeneous linewidth broadening is small, the width of the\nFMR spectra are proportional to Gand the experimen-\ntal data gives [ G(T > T c)−G(T=Tc/2)]/G(T > T c)≈\n21 %. Using G0= 0.7×108s−1[5] we obtain 22 %. The\nmeasured reduction of the Gilbert damping upon cooling\nthe sample from above TctoTc/2 agrees quantitatively\nwith our calculation.\nWe can make additional predictions for the Gilbert\ndamping in F |S|R systems, focusing on Al as S since its\nspin-flip length is much larger than that of Nb, and as\na weak coupling superconductor is better described by\nBCS theory. The Al material parameters are Tc0= 1.26\nK,lsf= 1.1µm, andD= 160 cm2s−1. In the left panel4\nof Fig. 2 we show the temperature dependence of G−G0\nfor three different thicknesses Lwhen spin-flip is induced\nexclusively by either magnetic disorder or spin-orbit cou-\npling to impurities. In contrast to spin-orbit scatterers,\nmagnetic impurities reduce Tcdue to the pair-breaking\nterm in (7). For L > l sfandT≪Tc, as well as for\nT > T c, the results do not depend on the nature of\nthe spin-flip scattering. In general, we observe that Tc\nstrongly depends on Ldue to the inverse proximity ef-\nfect. Wealsonotethatthedifferenceindampingbetween\nthe normal state and the superconducting state is small\nwhenL∼ξ0since only a small gap develops.\nThe experiments of Ref. 4 probed the regimes L≪ξ0\nas well as L≫ξ0. We also present results for arbitrary\nL/ξ0. In the normal state, Gdecreases with increasing\nLdue to increasing bulk spin transport resistance, which\nlimits relaxation in R, until Lreaches the value of lsf\nwhereRbecomesirrelevant(inset Fig.2). When T≪Tc,\non the other hand, the relevant length scale for spin pen-\netration into S is ξ0. This explains the more rapid decay\nofG−G0asa function of Lin the superconducting state.\nWhenL > ξ0, the spin-current absorption is completely\ndetermined by the inverse proximity effect: Spin dissipa-\ntion in R by transport through S is suppressed by the\nsuperconducting gap, and, furthermore, spin relaxation\ndeep in S is suppressed by the superconductivity. How-\never, the inverse proximity effect enhances the density of\nstates at low energy as well as spin-flip scattering rates\nclose to the F |S interface.\nWhenL < lsf, the results depend strongly on the S |R\ncontact described by g. In the right panel of Fig. 2, we\nshow the temperature dependence of G−G0forL=\n900 nm in an F |S system (no R or g= 0). At T > T c,\nthe damping is much smaller in the F |S system (the right\npanel) than in the F |S|R system with the same L(the\nmiddle pair of curves in the left panel). Tcis also higher\nsince there is no inverse proximity effect at x=L. At\nvery low temperatures, T≪Tc,G−G0saturates at the\nsame value for the F |S system as the F |S|R system with\nthe larger thickness, L= 1300 nm. For such thick S,\nTcis unaffected by R and spins cannot diffuse through S\nand dissipate in R, so that the resulting damping is the\nsame as in the F |S system. We also see from the right\npanel of Fig. 2 that when T/lessorsimilarTcthe enhanced Gilbert\ndamping can be somewhat larger than above Tcwhen\nspin-flip is induced by magnetic impurities, because the\ninduced spin accumulation of quasiparticles with energy\nkBT/greaterorsimilar∆ experiences an enhanced spin-flip rate through\nαm\nTSTS. In the F |S|R system, this effect is overwhelmed\nby the spin accumulation drain in R.\nIn conclusion, our theory quantitatively repro-\nduces the measured FMR linewidth broadening in\nferromagnet |superconductor structures. We make addi-\ntional predictions for varying system sizes and temper-\natures, and the nature and strength of spin-flip scatter-\ning. We hope to stimulate more experiments that should 0.2 0.4 0.6\n 0.3 0.6 0.9T [K]F-S-Rτsf=τmτsf=τso\n0.20.40.6\n 0.3 0.6 0.9F-S0.20.40.6\n0.81.21.6\nL [µm]T=Tc/2T>Tc\nFIG. 2: Calculated G−G0[108s−1] (same ordinate in all\nplots). Red solid (green dashed) lines for system where τsf=\nτm(τsf=τso). Left panel: F |S|R system with L[nm] from\ntop to bottom: 600, 900, 1300. Right panel: F |S system (no\nR) with L= 900 nm. Inset: Ldependence [ µm] ofG−G0\nforT > T c(green dashed line) and T≪Tc(red solid line).\nreveal information about the strong inverse proximity ef-\nfectandenergydependence ofspinflip scatteringinthese\nsystems.\nWe would like to thank C. Bell and J. Aarts for discus-\nsions. This work has been supported by NanoNed, the\nEC Contracts NMP-505587-1 ”SFINX” and IST-033749\n”DynaMax”, the DFG through SFB 513 and the Lan-\ndesstiftung Baden-W¨ urttemberg.\n[1] J. Y. Gu et al., Phys. Rev. B 66, 140507(R) (2002).\n[2] J. Y. Gu et al., Phys. Rev. Lett. 89, 267001 (2002); A.\nPotenza and C. H. Marrows, Phys. Rev. B 71, 180503(R)\n(2005); I. C. Moraru, W. P. Pratt Jr., and N. O. Birge,\nPhys. Rev. B 74, 220507 (2006); A. Yu. Rusanov, S.\nHabraken, and J. Aarts, Phys. Rev. B 73, 060505(R)\n(2006).\n[3] V. V. Ryazanov et al., Phys. Rev. Lett. 86, 2427 (2001);\nV. V.Ryazanov, V.A. Oboznov, A.V. Veretennikov,and\nA. Yu. Rusanov, Phys. Rev. B 65, 020501(R) (2001); T.\nKontoset al., Phys. Rev. Lett. 89, 137007 (2002); H.\nSellier, C. Baraduc, F. Lefloch, and R. Calemczuk, Phys.\nRev. B68, 054531 (2003); A. Bauer et al., Phys. Rev.\nLett.92, 217001 (2004); V. A. Oboznov et al., Phys.\nRev. Lett. 96, 197003 (2006). M. Weides et al., Phys.\nRev. Lett. 97, 247001 (2006).\n[4] C. Bell, S. Milikisyants, M. Huber, and J. Aarts (2007),\ncond-mat/0702461, accepted for publication by Phys.\nRev. Lett.\n[5] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and\nB. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005); Y.\nTserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002); Y. Tserkovnyak, A.\nBrataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403\n(2002).\n[6] J. P. Morten, A. Brataas, and W. Belzig, Phys. Rev. B\n70, 212508 (2004); J. P. Morten, A. Brataas, and W.\nBelzig, Phys. Rev. B 72, 014510 (2005).5\n[7] M. A. Sillanpaa, T. T. Heikkila, R. K. Lindell, and P. J.\nHakonen, Europhys. Lett. 56, 590 (2001).\n[8] F. J. Jedema, M. S. Nijboer, A. T. Filip, and B. J. van\nWees, Phys. Rev. B 67, 085319 (2003).\n[9] G. R. Boogaard, A. H. Verbruggen, W. Belzig, and T. M.\nKlapwijk, Phys. Rev. B 69, 220503(R) (2004).\n[10] M. B¨ uttiker, H. Thomas, and A. Prˆ etre, Z. Phys. B 94,\n133 (1994).\n[11] A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep.\n427, 157 (2006).\n[12] M. Y. Kupriyanov and V. F. Lukichev, Sov. Phys. JETP\n67, 1163 (1988).[13] D. Huertas-Hernando and Y. Nazarov, Eur. Phys. J. B\n44(2005).\n[14] G.E.W.Bauer et al.,Phys.Rev.Lett. 92, 126601(2004).\n[15] G. Sarma, J. Phys. Chem. Solids 24, 1029 (1963).\n[16] W. Belzig, A. Brataas, Y. V. Nazarov, and G. E. W.\nBauer, Phys. Rev. B 62, 9726 (2000).\n[17] N. Poli et al.(2007), arXiv:cond-mat/0707.2879.\n[18] R. T. W. Koperdraad and A. Lodder, Phys. Rev. B 51,\n9026 (1995).\n[19] J. Bass and W. P. Pratt Jr., J. Magn. Magn. Mater. 200,\n274 (1999)."    },    {        "title": "2007.02850v1.Transverse_and_Longitudinal_Spin_Torque_Ferromagnetic_Resonance_for_Improved_Measurements_of_Spin_Orbit_Torques.pdf",        "content": "Transverse and Longitudinal Spin-Torque Ferromagnetic Resonance for Improved\nMeasurements of Spin-Orbit Torques\nSaba Karimeddiny,1,\u0003Joseph A. Mittelstaedt,1,\u0003Robert A. Buhrman,1and Daniel C. Ralph1, 2\n1Cornell University, Ithaca, NY 14850, USA\n2Kavli Institute at Cornell, Ithaca, NY 14853, USA\n(Dated: July 7, 2020)\nSpin-torque ferromagnetic resonance (ST-FMR) is a common method used to measure spin-orbit\ntorques (SOTs) in heavy metal/ferromagnet bilayer structures. In the course of a measurement,\nother resonant processes such as spin pumping (SP) and heating can cause spin current or heat\n\rows between the layers, inducing additional resonant voltage signals via the inverse spin Hall\ne\u000bect (ISHE) and Nernst e\u000bects (NE). In the standard ST-FMR geometry, these extra artifacts\nexhibit a dependence on the angle of an in-plane magnetic \feld that is identical to the recti\fcation\nsignal from the SOTs. We show experimentally that the recti\fcation and artifact voltages can be\nquanti\fed separately by measuring the ST-FMR signal transverse to the applied current (i.e., in a\nHall geometry) in addition to the usual longitudinal geometry. We \fnd that in Pt (6 nm)/CoFeB\nsamples the contribution from the artifacts is small compared to the SOT recti\fcation signal for\nCoFeB layers thinner than 6 nm, but can be signi\fcant for thicker magnetic layers. We observe a\nsign change in the artifact voltage as a function of CoFeB thickness that we suggest may be due to\na competition between a resonant heating e\u000bect and the SP/ISHE contribution.\nI. INTRODUCTION\nCurrent-induced spin-orbit torques (SOTs) have the\npotential to provide improved e\u000eciency in the control of\nmagnetic memory and logic devices, enabling new tech-\nnologies that are fast, non-volatile, high-density, and of\nin\fnite endurance [1{3]. The metrology of SOT mate-\nrials and devices is critical to these developments. Sev-\neral di\u000berent techniques have been developed to quan-\ntify spin-orbit torques, including spin-torque ferromag-\nnetic resonance (ST-FMR) [4{6], second-harmonic (low-\nfrequency) Hall measurements [7{9], optical measure-\nments of current-induced magnetic de\rection [10, 11],\ndetermination of the threshold currents for switching of\nnanoscale magnets with in-plane anisotropy [5, 12], mea-\nsurements of spin Hall magnetoresistance [13, 14], and\nmeasurements of current-induced domain wall motion\nwithin perpendicular magnetic \flms [15, 16]. However,\ndi\u000berent techniques sometimes produce inconsistent re-\nsults [17, 18] and can even give internal discrepancies.\nFor example, independent second harmonic Hall stud-\nies on layers with in-plane and out-of-plane magnetic\nanisotropy [19, 20] have measured discrepant (and some-\ntimes unphysical) results for the damping-like torque ef-\n\fciency\u0018DL, and ST-FMR and second-harmonic Hall\nmeasurements on samples with in-plane anisotropy can\ndi\u000ber by tens of percent. Therefore, there is a continuing\nneed to examine possible artifacts a\u000becting the di\u000berent\nmeasurement approaches and to improve their accuracy.\nHere we consider one of the most popular techniques\nto measure SOTs, ST-FMR. A known artifact in ST-\nFMR is that the measured signals can include contribu-\ntions from spin pumping (SP) together with the inverse\nspin Hall e\u000bect (ISHE) [21{24]. In addition, there can\n\u0003These authors contributed equallybe thermoelectric contributions resulting from resonant\nheating that gives rise to a longitudinal spin Seebeck ef-\nfect (LSSE) together with the ISHE [25, 26], or Nernst\ne\u000bects (NE) [27{30]. In the standard ST-FMR measure-\nment con\fguration, these artifact signals are challenging\nto disentangle from the primary spin-torque diode (rec-\nti\fcation) signal because they all have identical depen-\ndences on the angle of a magnetic \feld applied within\nthe device plane [23, 29].\nPrevious studies attempting to separate artifact volt-\nages from the ST-FMR signal have largely been focused\non SP/ISHE contributions [31{33]. One previous study\nhas attempted to separate SP/ISHE by using the exter-\nnal \feld to tilt the magnetization partly out of plane [32],\nbut this con\fguration can be tricky to implement and in-\nterpret due to the large demagnetization \felds of typical\ndevices and the possibility of spatially non-uniform mag-\nnetization states. We demonstrate a straightforward al-\nternative approach to separately quantify both the spin-\norbit torque and the spin-pumping/resonant-heating ar-\ntifact signals using only in-plane magnetic \felds, by mea-\nsuring the ST-FMR signal transverse to the applied cur-\nrent (i.e., in a Hall geometry) in addition to the usual\nlongitudinal geometry.\nII. BACKGROUND\nIn conventional ST-FMR, a microwave current is in-\njected along a rectangular sample of a heavy metal\n(HM)/ferromagnet (FM) bilayer to induce FMR through\ncurrent-induced torques acting on the magnetization.\nWithin a simple macrospin model, the Landau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation captures the re-arXiv:2007.02850v1  [cond-mat.mes-hall]  6 Jul 20202\nsulting dynamics of the magnetic moment:\n_^m=\r^m\u0002dF\nd^m+\u000b^m\u0002_^m\n+\u001c0\nDL^m\u0002(^\u001b\u0002^m) +\u001c0\nFL^\u001b\u0002^m(1)\nwhere ^mis the normalized magnetic moment of the FM,\nFis the free energy density of the FM, \r= 2\u0016B=~is the\ngyromagnetic ratio with \u0016Bthe Bohr magneton, and \u000b\nis the Gilbert damping parameter. The \fnal two terms\nrepresent the current-induced damping-like and \feld-like\ntorques, with prefactors\n\u001c0\nDL=\u0018DL\u0016BJe\neMStFM(2)\n\u001c0\nFL=\u0018FL\u0016BJe\neMStFM: (3)\nHere\u0018DLand\u0018FLare dimensionless spin-torque e\u000ecien-\ncies that one might wish to measure for a given material\nsystem.Jeis the charge current density in the HM, eis\nthe magnitude of the electron charge, MSis the satura-\ntion magnetization of the FM, tFMis the thickness of the\nferromagnetic layer, and ^ \u001bdenotes the polarization of\nthe spin current incident on the ferromagnet. For a non-\nmagnetic heavy metal with an ordinary high-symmetry\ncrystal structure, ^ \u001bis required by symmetry to be in-\nplane and perpendicular to the applied current so that,\nfor an in-plane magnetization, the damping-like torque\npoints in the sample plane and the \feld-like torque points\nout of plane; we will assume this to be the case through-\nout this paper.\nThe magnetic resonance can be detected via a recti\fed\nlongitudinal DC voltage (oriented along the length of the\nwire parallel to the current) caused by the mixing of the\nmicrowave current with resistance oscillations produced\nby the precessing magnet via the anisotropic magnetore-\nsistance (AMR) or spin Hall magnetoresistance (SMR)\n[34, 35]. The resonance peak shape as a function of mag-\nnetic \feld magnitude at a constant \feld angle for this\nrecti\fed signal is the sum of symmetric and antisym-\nmetric Lorentzian functions. For a magnetic layer with\nin-plane anisotropy and and in-plane magnetic \feld, the\nsymmetric component arises from \u001c0\nDLand the antisym-\nmetric component from the combination of the current-\ninduced Oersted \feld and \u001c0\nFL. Once the microwave cur-\nrent is calibrated, the measurement allows determina-\ntions of both \u0018DLand\u0018FL, assuming there are no other\nartifacts contaminating the signal.\nWhen the FM layer is resonantly excited, a pure spin\ncurrent resulting from SP or LSSE can also \row from the\nFM layer into the HM layer and produce a measurable\nvoltage through the ISHE of the HM [21{25, 36{39]. Fur-\nthermore, an out-of-plane temperature gradient within\nthe heterostructure due to resonant heating can pro-\nduce a thermoelectric voltage from ordinary or anoma-\nlous Nernst e\u000bects [26, 30]. In all of these processes,\nthe result is a DC voltage perpendicular to the mag-\nnetization axis with a symmetric Lorentzian lineshape[23, 40, 41]. Consequently, if these artifact signals are\nsu\u000eciently large, they can contaminate ST-FMR mea-\nsurements of \u001c0\nDL. The signals from spin-torque rec-\nti\fcation and the spin-pumping/resonant-heating arti-\nfacts all have the same dependence on the angle of an\nin-plane magnetic \feld: /sin(2\u001e) cos(\u001e), with\u001emea-\nsured relative to the positive applied current direction\n[23, 29, 31, 33], making artifact e\u000bects di\u000ecult to disen-\ntangle.\nIn this work, we demonstrate that if one performs a\nST-FMR experiment as a function of the angle of an\nin-plane magnetic \feld by measuring the resonant DC\nvoltage transverse to the current (i.e., in a Hall geome-\ntry) the recti\fed spin-torque contribution and the spin\npumping/resonant heating can be distinguished. We are\naware of previous works that have performed ST-FMR in\nthe transverse geometry [33, 42], but these studies did\nnot illustrate how to separate the recti\fed spin-torque\ncontribution from the artifact signals. A closely-related\nidea was used previously in experiments which studied\nSP/ISHE signals from magnetic precession excited us-\ning oscillating magnetic \felds, in order to separate out\nunwanted (in that context) recti\fcation signals [36, 43].\nHarder et al. have published a review mapping out the\n\feld-angle dependence expected for resonance experi-\nments in both longitudinal and transverse geometries for\ndi\u000berent orientations of excitation [44].\nIII. THEORY\nWe consider a thin-\flm macrospin magnet with\nin-plane anisotropy subject to an external in-plane\nmagnetic \feld oriented at an angle \u001ewith respect to the\npositive current direction, that aligns the equilibrium\ndirection of the magnetization (see Fig. 1). We de\fne\nthe ^yaxis to be parallel to the equilibrium direction\nof the magnetization and ^ zto be perpendicular to the\nsample plane so that ^ x= ^y\u0002^zis in-plane. We will\nalso use capital letters to indicate a separate coordinate\nsystem \fxed with respect to the sample, where ^Xis\nalong the current direction, ^Z= ^z, and ^Y=^Z\u0002^X.\nSpherical polar coordinates \u0012;\u001efor the magnetization\norientation are de\fned relative to the X;Y;Z axes.\nA microwave current IRFRe\u0002\ne\u0000i!t\u0003\nis applied, produc-\ning alternating torques with amplitudes \u001cx=\u001c0\nDLcos(\u001e)\nand\u001cz=\u001c0\nzcos(\u001e) = (\u001c0\nFL+\u001c0\nOe) cos(\u001e) in the ^xand ^z\ndirections. With these de\fnitions, \u001c0\nOetakes a positive\nvalue by Ampere's Law and \u001c0\nDLis positive for the spin\nHall e\u000bect of Pt. Linearization and solution of the LLGS\nequation (see Supplmentary Information [45]) allows us\nto calculate the oscillatory components of the magnetic\nmoment, in complex notation,\nmx=\u0000!2\u001cz+i!\u001cx\n\u0000\r(B\u0000B0)!++i\u000b!!+\nmz=!1\u001cx+i!\u001cz\n\u0000\r(B\u0000B0)!++i\u000b!!+:(4)3\n(a)\n (b)\nFigure 1: (a)Optical image of our Hall ST-FMR\ndevice, showing the geometry of the contact pads. This\nparticular device featured a Pt(6)/CoFeB(6) bilayer\nmeasuring 20\u000280\u0016m2(in the center, dark blue). The\nscale bar is 100 \u0016m.(b)Zoomed-in optical image of\nthe bilayer and contacts with our coordinate\nde\fnitions. The XYZ (capital) coordinates are \fxed\nrelative to the device geometry while xyz(lowercase)\ncoordinates are relative to the equilibrium orientation\nof the magnetization. The scale bar is 20 \u0016m.\nHereB0is the resonance \feld, Bis the applied external\n\feld,!1=\rB0,!2=\r(B0+\u00160Me\u000b), and!+=!1+\n!2;Me\u000bis the in-plane saturation magnetization ( MS)\nminus any out-of-plane anisotropy. Note that by our\nde\fnition of coordinate axes, during the precession mx=\n\u0000d\u001eandmz=\u0000d\u0012.\nAssuming that the anisotropic magnetoresistance has\nthe formRXX=R0+RAMRm2\nX, the spin-torque mixing\nvoltage in conventional ST-FMR can be written\nVmix\nXX=IRF\n2RAMRRe [mx] sin 2\u001e; (5)\nor\nVmix\nXX=IRFRAMR\n2\u000b!+sin(2\u001e) cos(\u001e)\n\u0002\u0010\nS(B)\u001c0\nDL+A(B)!2\n!\u001c0\nz\u0011 (6)\nwhere we have de\fned the symmetric Lorentzian S(B) =\n\u00012=[(B\u0000B0)2+ \u00012], the antisymmetric Lorentzian\nA(B) = (B\u0000B0)\u0001=[(B\u0000B0)2+ \u00012] and the half-width\nat half-maximum linewidth \u0001 = \u000b!=\r . HereRAMR in-\ncludes contributions from both the anisotropic magne-\ntoresistance in the magnet and the spin Hall magnetore-\nsistance in the Pt layer, as these produce identical contri-\nbutions to the ST-FMR signals for our sample geometry\n(see Supplementary Information [45]).\nWe can compute the transverse spin-torque mixing\nvoltage within the same framework. We assume that the\nHall resistance has the symmetry RXY=RPHEmXmY+\nRAHEmZ, whereRPHEis the scale of the planar Hall ef-\nfect andRAHE is the scale of the anomalous Hall e\u000bect,\nin which case [9]\nVmix\nXY=IRF\n2(\u0000RPHEcos 2\u001eRe [mx] +RAHERe [mz]):\n(7)Using the results from Eq. (4),\nVmix\nXY=\u0000IRFRPHE\n2\u000b!+cos (2\u001e) cos(\u001e)\n\u0002\u0010\nS(B)\u001c0\nDL+A(B)!2\n!\u001c0\nz\u0011\n+IRFRAHE\n2\u000b!+cos(\u001e)\n\u0002\u0010\nS(B)\u001c0\nz\u0000A(B)!1\n!\u001c0\nDL\u0011\n:(8)\nThe artifact signals due to spin pumping and resonant\nheating can also contribute to both the longitudinal and\ntransverse ST-FMR voltages [31{33]. All of the artifacts\nwe consider, SP/ISHE, LSSE/ISHE, and NE, produce\nresonant DC electric \felds that are in-plane and perpen-\ndicular to the magnetization axis, and proportional to\nthe square of the precession amplitude (with the preces-\nsion amplitude/cos\u001e). Because these signals depend\nonly on the precession amplitude and not phase, they\nhave symmetric lineshapes. Taking the components in\nthe longitudinal and transverse directions, the artifact\nvoltages are therefore\nVart=E0\nartS(B) cos2\u001e\u001aLsin\u001e longitudinal\nWcos\u001etransverse(9)\nwhereE0\nart=E0\nSP+E0\nLSSE+E0\nNEis the total electric \feld\ngenerated by all artifact signals. The artifact voltages\nfor the longitudinal and transverse measurements di\u000ber\nonly by geometric factors and angular symmetry: Lis\nthe device length (parallel to the current \row) and Wis\nthe transverse device width.\nThe electric \feld due to the spin pumping/inverse spin\nHall e\u000bect can be calculated by the method of ref. [21, 23]\n(see Supplementary Information [45])\nE0\nSP=e\u0012SHg\"#\ne\u000b\n2\u0019P\ni\u001biti\u0015sdtanh\u0012tHM\n2\u0015sd\u0013\n\u0002\n\"\n(\u001c0\nDL)2!1+ (\u001c0\nz)2!2\n\u000b2(!+)2#\n:(10)\nHere\u0012SHis the spin Hall ratio in the HM (related to the\ndamping-like spin torque e\u000eciency by \u0012SH=\u0018DL=Tint,\nwhereTintis an interfacial spin transmission factor), g\"#\ne\u000b\nis the real part of the e\u000bective spin mixing conductance,\n\u001bi(ti) the charge conductivity (thickness) of layer i, and\n\u0015sdthe spin di\u000busion length of the HM.\nIf one assumes that the artifacts due to resonant\nheating by the current-induced torques are proportional\nto the energy absorbed by the magnetic layer during\nresonant excitation, the peak DC electric \feld due to\nLSSE/ISHE and NE can be calculated similarly [26, 28]\n(see Supplementary Information [45])\nE0\nLSSE +E0\nNE=CMstFM\u000b!+\n2\rP\ni\u001biti\"\n(\u001c0\nDL)2!1+ (\u001c0\nz)2!2\n\u000b2(!+)2#\n:\n(11)4\nHereCis a material-dependent prefactor. Due to the\nfactor oftFM\u000b!+in the numerator, the resonant heating\ncontributions scale di\u000berently than the SP/ISHE as a\nfunction of FM thickness, damping, and measurement\nfrequency.\nAdding the recti\fcation and artifact contributions\n[and using that cos2\u001esin\u001e= (sin 2\u001ecos\u001e)=2 and\ncos3\u001e= (cos\u001e+ cos 2\u001ecos\u001e)=2], the amplitudes of the\nsymmetric and antisymmetric components of the total\nlongitudinal and transverse ST-FMR signals have the\nangular dependence\nSXX(\u001e) =SAMR/art\nXX sin 2\u001ecos\u001e\nAXX(\u001e) =AAMR\nXX sin 2\u001ecos\u001e\nSXY(\u001e) =SPHE/art\nXY cos 2\u001ecos\u001e+SAHE/art\nXY cos\u001e\nAXY(\u001e) =APHE\nXYcos 2\u001ecos\u001e+AAHE\nXYcos\u001e(12)\nwith the amplitude coe\u000ecients\nSAMR/art\nXX =IRF\n2\u000b!+RAMR\u001c0\nDL\u0000L\n2E0\nart\n\u0011SAMR\nXX +Vart\nAAMR\nXX =IRF\n2\u000b!+RAMR!2\n!\u001c0\nz\nSPHE/art\nXY =\u0000IRF\n2\u000b!+RPHE\u001c0\nDL\u0000W\n2E0\nart\nAPHE\nXY=\u0000IRF\n2\u000b!+RPHE!2\n!\u001c0\nz\nSAHE/art\nXY =IRF\n2\u000b!+RAHE\u001c0\nz\u0000W\n2E0\nart\nAAHE\nXY=\u0000IRF\n2\u000b!+RAHE!1\n!\u001c0\nDL:(13)\nOne can see that all of the SXXandSXYrecti\fca-\ntion signals are contaminated by artifact voltages. If\none measures just SXXandAXXfor in-plane mag-\nnetic \felds (as in conventional ST-FMR) there is no\nway to distinguish \u001c0\nDLfrom the artifact contributions.\nHowever,\u001c0\nDLappears by itself, without any artifact\ncontamination, in the coe\u000ecient AAHE\nXY. One way to\nachieve a measurement of \u001c0\nDL, free of these artifacts,\nis therefore to directly use the expression for AAHE\nXY\nin Eq. (13) along with careful calibration of IRF,\u000b,\nandRAHE. The out-of-plane torque \u001c0\nzcan similarly\nbe determined from AAMR\nXX orAPHE\nXY. Alternatively,\nthe expressions in Eq. (13) also allow E0\nartand the\ntorque e\u000eciencies \u0018DLand\u0018FLto be measured with-\nout calibrating IRF,\u000b, and the the magnetoresistance\nscales by taking appropriate ratios to cancel prefac-\ntors. We can do so using measurements of either the\nset of parameters fSAMR/art\nXX;AAMR\nXX;SAHE/art\nXY;AAHE\nXYgor\nfSPHE/art\nXY;APHE\nXY;SAHE/art\nXY;AAHE\nXYg. We do not expect\nthat the equations involving RAMR andRPHEare physi-\ncally independent because anisotropic magnetoresistance\nand the planar Hall e\u000bect originate from the same mi-\ncroscopic mechanism. Therefore if the assumptions of\nour model are correct these two strategies for taking ra-\ntios to cancel prefactors must agree modulo experimentalnoise. We will perform both calculations, and test their\nagreement as a consistency check.\nFirst, using that on resonance !=p!1!2we calculate\nthe ratio\u0011\u0011(\u001c0\nDL=\u001c0\nz)p\n!1=!2employing the pair of\nparameters SandAassociated with each of the AMR,\nPHE, and AHE:\n\u0011=\u0000AAHE\nXY\nSAHE/art\nXY +W(Eart=2)=\n8\n>>>><\n>>>>:SPHE/art\nXY +W(Eart=2)\nAPHE\nXY\nSAMR/art\nXX +L(Eart=2)\nAAMR\nXX(14a)\n(14b)\nUsing the measured amplitude coe\u000ecients, one can solve\nseparately for Eartusing either Eq. (14a) or (14b), and\ncheck consistency.\nIt still remains to determine \u001c0\nDLand to separate the\ntwo contributions to \u001c0\nz=\u001c0\nFL+\u001c0\nOe. We choose to do this\nusing a method from ref. [17], in a way that determines\nboth the of the spin-torque e\u000eciencies \u0018DLand\u0018FLat\nthe same time without requiring a separate calibration\nofIRF. We perform measurements for a series of samples\nwith di\u000berent thicknesses of the ferromagnetic layer and\ndetermine\u0011= (\u001c0\nDL=\u001c0\nz)p\n!1=!2for each sample from\nany of the expressions in Eqs. (14a,14b), after solving\nforEart. We then de\fne\n\u0018FMR\u0011\u0011e\u00160MstHMtFM\n~r\n1 +\u00160Me\u000b\nB0(15)\nso that using Equations (2) & (3), and that by Ampere's\nLaw\u001c0\nOe=\r\u00160JetHM=2 one has\n1\n\u0018FMR=1\n\u0018DL\u0012\n1 +~\ne\u0018FL\n\u00160MstFMtHM\u0013\n: (16)\nPerforming a linear \ft of 1 =\u0018FMR vs. 1=tFMthen can be\nused to determine 1 =\u0018DL(from the intercept) and \u0018FL\n(from the slope).\nIV. MEASUREMENTS\nWe used DC-magnetron sputtering\nto grow multiayers with the structure\nsubstrate/Ta(1)/Pt(6)/ferromagnet( tFM)/Al(1) (where\nnumbers in parentheses are thicknesses in nm), using\nthree di\u000berent ferromagnets (FMs): Co 40Fe40B20\n(CoFeB), permalloy (Ni 81Fe19= Py) and Co 90Fe10\n(CoFe). Each of the three FMs is expected to have\ndi\u000berent AMR, PHE, and AHE values, and therefore dif-\nferent strengths of recti\fed spin-torque signals relative\nto the artifacts. In particular, CoFeB has weak planar\nmagnetoresistances (AMR and PHE), and has been\nargued previously to exhibit a signi\fcant contribution\nfrom SP/ISHE in ST-FMR [31, 32]. The CoFeB devices5\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFigure 2: ST-FMR measurements of a Pt(6 nm)/CoFeB(6 nm) sample for a measurement frequency f= 8 GHz.\n(a)Longitudinal resonant signals for \feld sweeps with two di\u000berent \feld angles. (b) & (c) Symmetric ( SXX) and\nantisymmetric ( AXX) Lorentzian \ft components for the longitudinal resonant signal as a function of the external\n\feld angle. (d)Transverse resonant signals for \feld sweeps with two di\u000berent \feld angles. (e) & (f) Symmetric\n(SXY) and antisymmetric ( AXY) Lorentzian \ft components for the transverse resonant signal as a function of the\nexternal \feld angle. The orange \ft line in (b) & (c) is a \ft to sin 2 \u001ecos\u001e(AMR); the light and dark blue \ft lines\nin (e,f) are \fts to cos 2 \u001ecos\u001e(PHE) and cos \u001e(AHE), respectively, and their sum (orange) \fts the data.\nwere grown with tFM=f2;3;4;6;8;10gin separate\ndepositions. The Py and Co 90Fe10devices were grown\nwith single relatively-large thicknesses to give measur-\nable artifact signals: tPy= 8 nm and tCoFe= 6 nm. All\ndevices were grown on high-resistivity ( >2\u0002104\n-cm),\nthermally-oxidized silicon wafers to prevent RF current\nleakage or capacitive coupling. The Ta was used as a\nseed layer and has negligible contribution to the SOTs\nwe measure due to the low conductivity of Ta relative\nto Pt (\u001aPt= 20.4\u0016\ncm,\u001aCoFeB = 110\u0016\ncm). The Al\ncap layer protects the layers below it, and is oxidized\nupon exposure to atmosphere.\nThe as-deposited samples were patterned using pho-\ntolithography and Ar ion-milling to de\fne rectangular\nbars ranging in size from 20 \u000240\u0016m to 40\u000280\u0016m with\nvarious aspect ratios. The transverse leads and contact\npads were then made using a second photolithography\nstep, deposited by sputtering Ti(3 nm)/Pt(75 nm) and\nformed by lift-o\u000b so that the side channels extended a\nfew microns on top of the main bar (see Fig. 1). We were\ncareful that the magnetic layer did not extend beyond\nthe de\fned rectangle into the transverse leads. In earlydevices, we etched full Hall-bar shapes within the \frst\nlayer of lithography so that the transverse leads included\nsome of the same magnetic layer as the main channel.\nFor those early devices, we found that the resulting anal-\nyses of spin-orbit torques produced anomalous results,\nvarying with the dimensions of the leads and the con-\ntact separation. This could possibly be due to spatial\nnon-uniformities in the magnetic orientation and preces-\nsion, as was speculated in [42]. Ultimately, the magnetic\nbilayer was left to be simply rectangular to promote uni-\nform precession modes, and this removed the anomalous\ngeometry dependence.\nFor the ST-FMR measurements, we connected the de-\nvices to an amplitude-modulated (\\AM\" with fAM\u0019\n1700 Hz) microwave source through the AC port of a bias\ntee and to a lock-in ampli\fer through the DC port, which\ndetected the longitudinal signal. Another lock-in ampli-\n\fer measured the DC voltage across the Hall leads of the\ndevice. Both lock-in ampli\fers referenced the same AM\nsignal, and we collected ST-FMR data in both the lon-\ngitudinal and transverse directions simultaneously. An\nin-plane applied magnetic \feld was applied at varying6\nangles\u001eusing a projected-\feld magnet. We used \fxed\nmicrowave frequencies in the range 7-12 GHz, applied\n20 dBm of microwave power, and all measurements were\nperformed at room temperature. In Figs. 2(a) and 2(d)\nwe show examples of the detected resonant signals from\nthe parallel ( XX) and transverse ( XY) lock-ins for the\nPt(6)/CoFeB(6) sample.\nBoth the longitudinal and transverse resonances are\nwell-\ft to a sum of symmetric and antisymmetric\nLorentzian peaks, with varying relative weights. For\neach sample we performed \feld-swept measurements at\na variety of angles \u001e, extracting the symmetric and\nantisymmetric components of the resonances for both\nthe longitudinal and transverse signals. The results for\na Pt(6)/CoFeB(6) sample are shown in Fig. 2(b,c,e,f),\nalong with \fts to Eq. (12). Analogous results for\nPt(6)/Py(8) and Pt(6)/CoFe(6) samples are shown in\nthe Supplementary Information [45].\nWe \fnd excellent agreement with the expected angu-\nlar dependences for SXX,AXX, andAXY. ForSXYthe\ndominant contributions to the angular dependence are,\nas expected the cos 2 \u001ecos\u001eand cos\u001eterms, but in ad-\ndition, we detect a small component approximately pro-\nportional to sin 2 \u001e. This additional contribution is less\nthan 10% of the larger terms in SXYfor all thicknesses\nof CoFeB, small enough that it is not included in the \ft\nshown in Fig. 2(e). It is more signi\fcant in the CoFe and\nPy samples that we measured, though still smaller than\nthe cos 2\u001ecos\u001eand cos\u001eamplitudes in SXY(see Sup-\nplementary Information [45]). A sin 2 \u001econtribution can\nonly arise from a breaking of mirror symmetry relative to\nthe sample's ^Y-^Zplane (see Supplementary Information\n[45]). This symmetry is broken in our samples by the dif-\nferent contact geometries on the two ends of the sample\nwire (see Fig. 1(a)). The form of the sin 2 \u001esignal can\nbe explained as due resonant heating that produces an\nin-plane thermal gradient in the longitudinal direction of\nthe sample (due e.g.to di\u000berences in heat sinking at the\ntwo ends) that is transduced to a tranverse voltage with\nthe symmetry of the planar Hall e\u000bect ( /mXmY). We\nhave checked that the signal is not due to a sample tilt\nor to a non-resonant DC current that might arise from\nrecti\fcation of the applied microwave signal at the sam-\nple contacts. All of the other Fourier components that\nare the main subject of our analysis maintain the ^Y-^Z-\nplane mirror symmetry, and so they cannot be altered\nat \frst order by a process that breaks this symmetry.\nBeing a separate Fourier component, the sin 2 \u001econtri-\nbution also does not a\u000bect the \fts to Eq. (12) to de-\ntermine the six amplitude coe\u000ecients SAMR/art\nXX ,AAMR\nXX,\nSPHE/art\nXY ,APHE\nXY,SAHE/art\nXY , andAAHE\nXY. Using these coef-\n\fcients, we calculate Eartby solving Eqs. (14a) or (14b).\nThere is a potential ambiguity in which roots of Eqs.\n(14a) and (14b) to select when applying the quadratic\nformula. In our measurements, one root would give un-\nphysical results, e.g. a sign change of \u0018DL. An important\ncheck of our method (and a check that the sin 2 \u001eterm in\nSXYdoes not contaminate the analysis) is that these two\nindependent methods for determining E0\nart(Eqs. (14a)and (14b)) give consistent results. We show below that\nthis is indeed the case.\n(a)\n(b)\nFigure 3: (a)The uncorrected measured value of SAMR\nXX\nvs.tFM, together with the value corrected by removing\nthe artifact voltage. (b)The inverse \u0018FMR vs. inverse\ntFM. The y-intercept of the line is 1 =\u0018DLand the slope\nis proportional to \u0018FLas in Eq. (16). The two \ft lines\nare color-matched \fts to the data points from the\nAHE/PHE and AHE/AMR corrections.\nFigure 3(a) shows the total amplitude of the longitu-\ndinal symmetric ST-FMR component (labeled as \\Mea-\nsured\"), and the corrected value SAMR\nXX from which Vart\nhas been subtracted. For CoFeB layer thicknesses 6 nm\nand below, the magnitude of Vartis much less than the\nmagnitude of SAMR\nXX, so that the artifacts have little ef-\nfect on ST-FMR measurements of the spin-orbit torques.7\nFigure 4:\u0018FMR for various device stacks. The gray\n(left) bars show values without correction for the\nartifacts, and the orange and blue (center, right) bars\nshow values corrected using the determination of the\nartifact voltages using Eqs. (14a) and (14b),\nrespectively.\nHowever, with increasing CoFeB thickness the magni-\ntude ofSAMR\nXX decreases and Vartgrows, so we \fnd ex-\nperimentally that for the CoFeB layers thicker than 6 nm\nthe artifact voltage becomes a signi\fcant fraction of the\ntotal signal. In this regime, VartandSAMR\nXX contribute\ntoSXX(\u001e) with opposite signs [46], with the consequence\nthat if the artifact contributions are neglected in the con-\nventional ST-FMR analysis, the result is an underesti-\nmate of the strength of \u001c0\nDL. In this respect our results\ncon\rict with some conclusions [31, 32] that neglecting\nthe SP/ISHE contribution produces an overestimate of\n\u001c0\nDL.\nAnalysis of the dependence of 1 =\u0018FMR as a function\nof 1=tFMallows a determination of the underlying spin-\ntorque e\u000eciencies \u0018DLand\u0018FLusing Eq. (16). The re-\nsults for the CoFeB series of samples is shown in Fig.\n3(b). If one does not correct for the contribution of\nthe artifacts, the calculated values of 1 =\u0018FMR depart up-\nward from the expected linear dependence for tFM&6\nnm. Similar results have been reported previously in\n[17] where the non-linearity was speculated to be from\nSP/ISHE, and the spin-torque e\u000eciencies were deter-\nmined by \ftting only to the thinner FM stacks. After we\ncorrect for the artifact contribution, we \fnd good agree-\nment with the expected linear dependence over the full\nthickness range. From the linear \ft, we determine \u0018DL=\n0.090(6) and \u0018FL= -0.020(2).\nFor the Pt(6 nm)/Py(8 nm) and Pt(6 nm)/CoFe(6\nnm) samples we \fnd the same con\fguration of signs as\nfor the thicker Pt/CoFeB samples: Vartpartially cancels\nSAMR\nXX so that the true mixing signal is larger than the\nmeasured amplitude of SXX(\u001e). The results of the calcu-\nlation of\u0018FMR according to Eq. (15) are shown in Fig. 4\nfor \fve selected samples, both without and with the cor-\nrection for artifacts. In determining \u0018FMR we use values\nforMsdetermined by room temperature vibrating sam-\nple magnetometry (VSM) and values for \u00160Me\u000bdeter-\nmined by \fts of the ST-FMR resonant \felds as a function\nof frequency. These values are: for CoFeB Ms= 9:8\u0002105A/m,\u00160Me\u000b= 0.6 { 1.4 T (depending on thickness); for\nPyMs= 7:5\u0002105A/m,\u00160Me\u000b= 1:01 T; and for\nCoFeMs= 9:1\u0002105A/m,\u00160Me\u000b= 1:66 T. If a mag-\nnetic dead layer was observed in VSM, the dead layer\nthickness was subtracted from tFM. In all cases shown\nin Fig. 4, we \fnd that correcting for the artifact con-\ntribution increases our estimates for the values of \u0018FMR.\nThe value of \u0018FMR is smaller for the Pt/Py sample than\nfor Pt/CoFeB or Pt/CoFe primarily because \u0018FLis both\nsmall and has a positive sign for Pt/Py [47, 48].\nThe dependence of the artifact voltage, Vart, on the\nferromagnetic layer thickness is shown in Fig. 5 for the\nlongitudinal ST-FMR component of the Pt/CoFeB se-\nries of samples. The data are compared to an estimate\nof the SP/ISHE contribution from Eq. (10), using the pa-\nrameters (appropriate for the resistivity of our Pt layers,\n\u001aPt= 20:4\u0016\ncm):\u0012SH= 0:32 [17, 49], g\"#\ne\u000b= 8:26\u00021018\nm\u00002[49], and\u0015sd= 3:7 nm [50]. The other quantities in\nEq. (10) were measured for our samples, including the\nvariation as a function of CoFeB thickness. The com-\nparison therefore includes no adjustable \ftting parame-\nters, but given that there is considerable disagreement\nin the literature about the values of the parameters \u0012SH,\ng\"#\ne\u000b, and\u0015sd, one should still be careful about drawing\nquantitative conclusions. The comparison indicates to\nus that for the samples with tFM\u00153 nm the SP/ISHE\ntheory predicts the correct sign and can roughly capture\nthe overall magnitude and thickness-dependence of the\nmeasured artifact signal. However, the measured artifact\nvoltage for tFM= 2 nm has the opposite sign, inconsis-\ntent with the SP/ISHE. We are con\fdent that the mea-\nsured sign change is real, because we have measured and\nperformed the analysis on \fve Pt(6 nm)/CoFeB(2 nm)\ndevices with varied geometries, with consistent results.\nGiven that the SP/ISHE cannot explain the sign\nchange in the artifact voltage for our tFM= 2 nm sam-\nples, we suggest that resonant heating e\u000bects might be\ncomparable to the SP/ISHE in our Pt(6 nm)/CoFeB\nsamples, with su\u000ecient strength to reverse the overall\nsign of the artifact voltage for our thinnest samples.\nThis suggestion di\u000bers from previous studies on Pt/YIG\nsamples, for which frequency-dependent measurements\ndemonstrated that SP/ISHE signals dominate over res-\nonant heating artifacts [41, 51]. However, the relative\nstrength of the heating e\u000bects and SP/ISHE should scale\nproportional to the damping \u000b(compare Eqs. (10) and\n(11)), so that the heating e\u000bects should be more signi\f-\ncant in higher-damping ferromagnetic metals compared\nto lower-damping YIG. We calculate that the resonant\nheating due to the excitation of magnetic precession for\nour 2 nm samples is \u00182:5\u0002104Wm\u00002(Supplementary\nInformation [45]), only about a factor of 5 less than the\nOhmic heating per unit area in the CoFeB, \u00181:2\u0002105\nWm\u00002. We suggest that this is su\u000ecient to measurably\nalter the thermal gradients within the sample at reso-\nnance and induce resonant signals from the LSSE and/or8\nFigure 5: The artifact voltage as a function of the FM\nthickness in Pt(6 nm)/CoFeB samples. The two types\nof data points re\rect the two correction equations\n((14a) and (14b)). The line is the estimated SP/ISHE\ncontribution, determined using the parameters\ndescribed in the text, with no adjustable parameters.\nNernst e\u000bects. Due to an increase in the damping coef-\n\fcient\u000bwith decreasing magnetic thickness, the ratio\nof the resonant heating to Ohmic heating is signi\fcantly\ngreater for the 2 nm CoFeB samples than for the thicker\nmagnetic layers (see Supplmentary Information [45]).\nAs noted in the introduction, past experiments have\nshown a discrepancy between measurements of \u0018DLus-\ning low frequency second harmonic Hall and ST-FMR\ntechniques. To see if our correction for the artifact volt-\nages in ST-FMR alleviates the discrepancy between the\ntwo techniques, we carried out low frequency second har-\nmonic Hall measurements on the same Pt/CoFeB bilay-\ners [45]. We found that the low frequency second har-\nmonic measurements of \u0018DLwere still approximately 60%\nlarger than what we measured by ST-FMR, even after\ncorrecting ST-FMR for spin pumping and resonant heat-\ning. This persisting quantitative di\u000berence suggests that\nthe assumptions used in analyzing one or both of these\nexperiments are missing an important bit of physics. Our\nanalysis indicates that this missing physics is not simply\nthe neglect of spin pumping or a simple heating-induced\nvoltage in the ST-FMR results, and therefore more work\nmust be done to understand the source of the disagree-\nment.\nV. CONCLUSION\nIn conclusion, we have demonstrated that the recti-\n\fcation signal used to measure the strength of spin-\norbit torques in spin-torque ferromagnetic resonance\n(ST-FMR) can be separated from artifact voltages that\nmay arise due to spin pumping and resonant heating byperforming ST-FMR in the transverse (Hall) con\fgura-\ntion as well as the usual longitudinal con\fguration. For\nPt(6 nm)/CoFeB( tFM) samples, the artifact voltages are\nsmall compared to the recti\fcation signal for tFM<6\nnm, but they can become a signi\fcant part of the mea-\nsured signal for thicker magnetic layers. The sign and\noverall magnitude of the measured artifact voltage for\nthese thicker layers are consistent with expectations for\nthe SP/ISHE e\u000bect signal. However, the sign of the arti-\nfact voltage is reversed for our thinnest magnetic layers,\nwithtFM= 2 nm. This sign reversal cannot be explained\nby the SP/ISHE, so we suggest that it may be caused\nby a resonant heating e\u000bect.\nVI. ACKNOWLEDGEMENTS\nThis research was supported in part by task 2776.047\nin ASCENT, one of six centers in JUMP, a Semi-\nconductor Research Corporation program sponsored by\nDARPA, and in part by the National Science Founda-\ntion (DMR-1708499). 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Rev.\nLett.123, 057203 (2019).\n[50] M.-H. Nguyen, D. C. Ralph, and R. A. Buhrman, Phys.\nRev. Lett. 116, 126601 (2016).\n[51] R. Iguchi, K. Ando, R. Takahashi, T. An, E. Saitoh, and\nT. Sato, Japanese Journal of Applied Physics 51, 103004\n(2012).Supplementary Information:\nTransverse and Longitudinal Spin-Torque Ferromagnetic Resonance for\nImproved Measurements of Spin-Orbit Torques\nSaba Karimeddiny,1,∗Joseph A. Mittelstaedt,1,∗Robert A. Buhrman,1and Daniel C. Ralph1, 2\n1Cornell University, Ithaca, NY 14850, USA\n2Kavli Institute at Cornell, Ithaca, NY 14853, USA\n(Dated: July 3, 2020)\nCONTENTS\nI. Definitions of coordinate axes 2\nII. Solving the LLGS equation for the oscillatory components of the magnetization 2\nIII. Magnetoresistances and Rectification in the Longitudinal Measurement Geometry 5\nIV. Spin-pumping contribution 5\nV. Energy absorption during magnetic resonance 7\nVI. Dependence of the artifact voltage on RF power 8\nVII. Characterization by vibrating sample magnetometry 9\nVIII. Longitudinal and transverse ST-FMR data for other ferromagnets 9\nIX. Low-frequency second harmonic Hall measurements 11\nReferences 15\n∗These two authors contributed equally\n111I. DEFINITIONS OF COORDINATE AXES\nIt will be convenient to define two coordinate systems. Capital letters ( ˆX,ˆY,ˆZ) will denote\na coordinate system fixed with respect to the device structure. ˆXis the direction of microwave\ncurrent flow, ˆZis perpendicular to the sample plane, and ˆY=ˆZ׈X. Lower-case letters (ˆ x,ˆy,ˆz)\nwill denote a coordinate system defined relative to the precession axis of the magnetization. ˆ yis\nalong the precession axis, ˆ z=ˆZwithz= 0 corresponding the the HM/FM interface, and ˆ x= ˆy׈z.\nSpherical polar coordinates will also be used to specify the magnetization direction, with the polar\nangleθmeasured with respect to ˆZand the azimuthal angle φmeasured with respect to ˆX.\nII. SOLVING THE LLGS EQUATION FOR THE OSCILLATORY COMPONENTS OF\nTHE MAGNETIZATION\nWe wish to compute the dynamics of the magnetization in a heavy metal (HM)/ferromagnet (FM)\nbilayer in response to a microwave current applied within the sample plane. Since by our convention\nthe precession axis lies along ˆ y, in response to the microwave current the oscillatory components\nof the magnetization are mxandmz. We begin with Landau-Lifshitz-Gilbert-Slonczewski (LLGS)\nequation [1, 2].\n˙ˆm=αˆm×˙ˆm+/vector τneq+/vector τeq (1)\nwhere ˆmis the magnetization orientation, αis the Gilbert damping parameter, and /vector τneqis the\ncurrent-induced non-equilibrium spin-orbit torque (SOT). /vector τeq=−γ/parenleftBig\nˆm×−d/tildewideF\ndˆm/parenrightBig\nis the equilibrium\ntorque (γ= 2µB//planckover2pi1is the gyromagnetic ratio with µBthe Bohr magneton), which can be found\nfrom the magnetic free energy density, /tildewideF. We may write the magnetic free energy density as\n/tildewideF=F/Ms=−/vectorB·ˆm+1\n2µ0Msˆm·← →N·ˆm−K⊥\nMs/parenleftbig\nˆm·ˆn/parenrightbig2−K/bardbl\nMs/parenleftbig\nˆm·ˆu/parenrightbig2(2)\nwhereMsis the saturation magnetization, /vectorBis the applied magnetic field,← →Nis the demagnetization\ntensor,K⊥(K/bardbl) is the strength of the out-of-plane (within-plane) anisotropy, ˆ nis the film normal\nand ˆuis the in-plane anisotropy direction. Our polycrystalline samples have negligible anisotropy\nwithin the plane so we neglect the final term, and we use the thin-film approximation that the\ndemagnetization tensor has only one nonzero element NZZ= 1. Our equilibrium torque is therefore\n/vector τeq=γ\nmymzµ0Meff+mzB\n−mxmzµ0Meff\n−mxB\n. (3)\n212Hereµ0Meff=µ0Ms−2K⊥/Ms. We are working in coordinates where my≈1 andmx,mz/lessmuch1 so\nto leading order\n/vector τeq=\nmzω2\n0\n−mxω1\n, (4)\nwhere we have defined ω1=γBandω2=γ(B+µ0Meff).\nThe damping term is of the form, to leading order\nαˆm×˙ˆm=α\nmy˙mz−mz˙my\nmz˙mx−mx˙mz\nmx˙my−my˙mx\n≈α\n˙mz\n0\n−˙mx\n. (5)\nThe amplitudes of the oscillatory non-equilibrium torques have the form\n/vector τneq=\nτx\n0\nτz\n. (6)\nWe will assume that the torques are in-phase with the applied current and arise from a spin current\nwith the symmetry required for the spin Hall effect in polycrystalline materials:\nτx∼( ˆm׈σ׈m) =−ˆm×(ˆY׈m)\nτz∼(ˆσ׈m) =−ˆY׈m,(7)\nwhere for Pt the spin Hall effect orients the spin moment ˆ σ=−ˆY for current flowing along + ˆX.\nWe can write this more explicitly:\nτx=τ0\nDLcos(φ)\nτz=τ0\nzcos(φ) = (τ0\nFL+τ0\nOe) cos(φ).(8)\nHere we have defined\nτ0\nDL=ξDLµBJe\neMStFM\nτ0\nFL=ξFLµBJe\neMStFM\nτ0\nOe=µ0γJetHM\n2,(9)\nwhereξDLis the damping-like spin-orbit torque efficiency, ξFLis the field-like spin-orbit torque\nefficiency,Jeis the charge current density in the heavy metal, eis the electron charge, tFMis the\n313thickness of the ferromagnet layer, and tHMis the thickness of the heavy metal. The damping-like\ntorque efficiency ξDLwill have a value reduced from the intrinsic spin Hall ratio within the heavy\nmetal (θSH) by the spin-transparency factor of the HM/FM interface.\nAssembling Equations (4)-(6), the linearized LLGS equation we aim to solve takes the form\n\n˙mx\n0\n˙mz\n=\nα˙mz+mzω2+τx\n0\n−α˙mx−mxω1+τz\n. (10)\nWe assume that we will have an oscillatory solution and so use the ansatz mx(z)(t) =mx(z)e−iωt.\nWith this substitution, the LLGS equation becomes\n\n−iωmx\n0\n−iωmz\n=\n−mz(iωα−ω2) +τx\n0\nmx(iωα−ω1) +τz\n, (11)\nwith the solution\nmx=−ω2τz+iωτx\n(ω2−ω2\n0) +iωαω+(12)\nmz=ω1τx+iωτz\n(ω2−ω2\n0) +iωαω+. (13)\nWe have defined ω2\n0=ω1ω2andω+=ω1+ω2, and have made an assumption that αis small so\nthatτx+ατz≈τxandτz+ατx≈τz.\nIn ST-FMR measurements, we usually perform field sweeps instead of frequency sweeps. To\nconvert our expression, we need to expand about the resonant field:\nω2\n0≈ω2\n0|B0+ (B−B0)d(ω2\n0)\ndB/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nB0, (14)\nwhere the derivative is\nd(ω2\n0)\ndB/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nB0=ω2,B0dω1\ndB/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nB0+ω1,B0dω2\ndB/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nB0=γω+\nB0. (15)\nTheB0subscript indicates that those frequencies are evaluated at the resonant field, so that ω2\n0|B0=\nω2.\nω2−ω2\n0≈−γ(B−B0)ω+\nB0(16)\nand hence\nmx=−ω2τz+iωτx\n−γ(B−B0)ω++iωαω+(17)\nmz=ω1τx+iωτz\n−γ(B−B0)ω++iωαω+. (18)\n414Near resonance, we can evaluate these expressions using ω1=γB0,ω2=γ(B0+µ0Meff), and\nω+=γ(2B0+µ0Meff), as in the main text.\nIII. MAGNETORESISTANCES AND RECTIFICATION IN THE LONGITUDINAL\nMEASUREMENT GEOMETRY\nThe total magnetoresistance for a longitudinal measurement can be written in spherical coordi-\nnates as\nRXX=R0+RAMRsin2θcos2φ−RSMRsin2θsin2φ. (19)\nwhereR0is a constant offset, RAMR is the scale of the anisotropic magnetoresistance, and RSMR\nis the scale of the spin Hall magnetoresistance [3]. We consider small angle precession such that\nθ=θ0+ ∆θandφ=φ0+ ∆φwith ∆θ,∆φ/lessmuch1 and expand to get\nRXX=R0+RAMR/parenleftbig\nsin2θ0cos2φ0+ ∆θsin 2θ0cos2φ0−∆φsin2θ0sin 2φ0/parenrightbig\n−RSMR/parenleftbig\nsin2θ0sin2φ0+ ∆θsin 2θ0sin2φ0+ ∆φsin2θ0sin 2φ0/parenrightbig\n.(20)\nThe only pieces of Eq. (20) that are current-rectifiable (able to produce a mixing voltage with the\nrf current) are the terms linear in ∆ θand ∆φ. For an in-plane magnet we have θ0=π/2, and\ntherefore the mixing voltage becomes\nVmix\nXX=IRF\n2(RAMR+RSMR) (−∆φsin 2φ0) =IRF\n2(RAMR+RSMR) (Re[mx] sin 2φ0). (21)\nOnly the in-plane deflections of the magnet are rectified to produce a mixing voltage, and the\nAMR and SMR contributions simply add. For simplicity of notation in the main text, we therefore\nincorporate both the AMR and SMR contributions in one magnetoresistance amplitude RAMR.\nIV. SPIN-PUMPING CONTRIBUTION\nThe precessing magnetization at FMR causes the ferromagnetic layer to inject a spin current\ninto the heavy metal layer; this can create a voltage through the ISHE. The time-averaged spin\ncurrent in the heavy-metal layer can be written as [4–6]\n← →jSP\nˆσ(z) = ˆσ⊗/vectorjSP(z) =/planckover2pi1\n4πg↑↓\neffsinh [(tHM+z)/λsd]\nsinh [tHM/λsd]/angbracketleftBig\nˆm×˙ˆm/angbracketrightBig\n⊗(−ˆz) (22)\nwhere/vectorjSP∝−ˆzis the direction of the spin current flow, ˆ σ∝/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nˆm×˙ˆm/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle/bardbl−ˆmis the polarization\nof the pumped spin current (where the negative sign is to account for enhanced Gilbert Damping\n515due to Spin pumping [7, 8]), g↑↓\neffis the effective spin mixing conductance at the interface, and λsd\nis the spin diffusion length of the HM. The resultant voltage is\nVSP=−RtotI=−Rtot/integraldisplay\nΣHM/vectorjHM\ne·d/vectorA (23)\nwhereRtotis the total device resistance (that will differ for the longitudinal and transverse cases),\nΣHMis the cross-section of the heavy-metal layer, /vectorjHM\ne= (2e//planckover2pi1)θSH/vectorjSP׈σis the charge current\narising from the ISHE [6] and d/vectorAis a differential surface area normal, which points along the vector\nconnecting the leads that we are measuring across. The negative sign in Eq. (23) is due to the fact\nthat we are measuring the electric field that arises from the open circuit condition of the device [6].\nSimplifying the integrals we have (for the longitudinal geometry)\nVSP=−Rtot/integraldisplay\nΣHM/vectorjHM\ne·d/vectorA (24)\n=−Rtot/integraldisplayW/2\n−W/2/integraldisplay−tHM\n0/vectorjHM\ne·d/vectorA (25)\n=−RtotWsinφ/integraldisplay−tHM\n0/vextendsingle/vextendsingle/vextendsingle/vectorjHM\ne/vextendsingle/vextendsingle/vextendsingledz (26)\nwhere W is the width of the Hall bar (dimension along ˆY). Note that in the transverse measurement\nWsinφ→LcosφwhereLis the device bar length. The only part of /vectorjHM\nethat depends on the\nthickness is\n/integraldisplay−tHM\n0sinh((tHM+z)/λsd)\nsinh(tHM/λsd)dz=λsdtanh/parenleftbiggtHM\n2λsd/parenrightbigg\n. (27)\nAt this point, we have (for the longitudinal geometry)\nVSP=−2e\n/planckover2pi1θSHRtotWsinφ/planckover2pi1\n4πg↑↓\neffλsd/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nˆm×˙ˆm/angbracketrightBig/vextendsingle/vextendsingle/vextendsingletanh/parenleftbiggtHM\n2λsd/parenrightbigg\n. (28)\nWe now only need to calculate/angbracketleftBig\nˆm×˙ˆm/angbracketrightBig\n, but we already have the oscillatory magnetization com-\nponents from Sec. II. We can write/angbracketleftBig\nˆm×˙ˆm/angbracketrightBig\n=ωIm [mxm∗\nz] (−ˆm), so therefore\nωIm [mxm∗\nz] =ω2\n(γω+)2(B−B0)2+ (ωαω+)2/bracketleftbig\nω1τ2\nx+ω2τ2\nz/bracketrightbig\n(29)\n=ω1τ2\nx+ω2τ2\nz\n(αω+)2S(B), (30)\nwhereS(B) = ∆2/[(B−B0)2+ ∆2] is a symmetric Lorentzian and ∆ ≡ωα/γ . The voltage in the\ndevice resulting from the pumped spin can then be written as (for the longitudinal geometry)\nVSP=−2e\n/planckover2pi1θSHRtotWsinφ/planckover2pi1\n4πg↑↓\neffλsd/bracketleftbiggω1τ2\nx+ω2τ2\nz\n(αω+)2S(B)/bracketrightbigg\ntanh/parenleftbiggtHM\n2λsd/parenrightbigg\n. (31)\n616Putting all of this together, using that ω1=γB0,ω2=γ(B0+µ0Meff), andω+=γ(2B0+µ0Meff),\nnoting that τxandτzhave the angular dependence specified in Eq. (8), and that for the transverse\ncase one has Wsinφ→Lcosφ, the spin-pumping voltages in the longitudinal and transverse\ndirections become\nVSP=−eB0RtotθSH\n2πα2γ(2B0+µ0Meff)2g↑↓\neffλsdtanh/parenleftbiggtHM\n2λsd/parenrightbigg/bracketleftbigg\n(τ0\nSH)2+/parenleftbigg\n1 +µ0Meff\nB0/parenrightbigg\n(τ0\nz)2/bracketrightbigg\n×S(B) cos2φ\n\nWsinφ,longitudinal\nLcosφ, transverse(32)\n=−eB0θSH\n2πα2γ(2B0+µ0Meff)21/summationtext\nitiσig↑↓\neffλsdtanh/parenleftbiggtHM\n2λsd/parenrightbigg/bracketleftbigg\n(τ0\nSH)2+/parenleftbigg\n1 +µ0Meff\nB0/parenrightbigg\n(τ0\nz)2/bracketrightbigg\n×S(B) cos2φ\n\nLsinφ, longitudinal\nWcosφ,transverse.(33)\nIn the final equation we have expressed the values of Rtotfor the longitudinal and transverse cases\nin terms of the conductivities of the i= HM and FM layers added as parallel conductors.\nV. ENERGY ABSORPTION DURING MAGNETIC RESONANCE\nWe begin with the magnetic free energy per unit area (see section II) assuming no in-plane\nanisotropy\nF/A =−/vectorB·M+1\n2µ0tFMMsMeffm2\nz. (34)\nWe assume the external field saturates the magnetization in the y-direction\nF/A =−BmyMstFM+1\n2µ0tFMMsMeffm2\nz (35)\nand using|m|= 1,\nF/A =−BMstFM+MstFM\n2/bracketleftbig\nBm2\nx+ (B+µ0Meff)m2\nz/bracketrightbig\n. (36)\nTaking a time derivative, we have\n∂tF/A =MstFM/bracketleftbigg\nBmxdmx\ndt+ (B+µ0Meff)mzdmz\ndt/bracketrightbigg\n. (37)\nTo calculate the energy absorbed from the current-induced torques, we set dmx/dt=τxand\ndmz/dt=τz. Averaging over one precession cycle, the power per unit area absorbed by the magnet\n717is\n/angbracketleft∂tF/A/angbracketright=MstFM\n2{BτxRe[mx] + (B+µ0Meff)τzRe[mz]} (38)\n=MstFMαω+\n2γ/bracketleftbiggω1τ2\nx+ω2τ2\nz\n(αω+)2/bracketrightbigg\nS(B). (39)\nThe in-plane torque τxcontains contributions from both the antidamping spin-orbit torque and the\nout-of-plane component of the Oersted field, but when averaged over the width of the sample the\nantidamping spin-orbit torque gives the larger contribution. Using Eq. (39), with parameter values\ndetermined as described in the main text, we have calculated the power absorbed per unit area\nwithin the magnetic layer of the Pt/CoFeB samples as a function of the magnetic layer thickness.\nThis is plotted as a fraction of the Ohmic dissipation in the magnetic layer in supplementary Fig.\n1. The relative amount of heating for the thinnest samples is greater primarily because of increased\nmagnetic damping for the thinnest samples.\nFIG. 1: The ratio of resonant power absorbed to the Ohmic dissipation in the ferromagnetic layer\nas a function of the ferromagnetic layer thickness, for the Pt(6 nm)/CoFeB( tFM) series of samples.\nVI. DEPENDENCE OF THE ARTIFACT VOLTAGE ON RF POWER\nBoth of the artifact effects discussed in the previous two sections (spin pumping and heating)\ndepend quadratically on the spin torque excitations of the ferromagnet and should therefore depend\nlinearly on the applied RF power. In supplementary Fig. 2 below we show the artifact voltage from\nthe longitudinal ST-FMR signal of a Pt(6)/CoFeB(10) heterostructure. Only one set of points is\nshown as the average of the AHE/PHE and AHE/AMR correction methods.\n818FIG. 2: The logarithm of the artifact voltage vs. the applied RF power in dBm.\nThe artifact voltage is indeed linearly proportional to the applied power in the regime that we have\nmeasured. We note that all measurements in the main text at were performed at 20 dBm.\nVII. CHARACTERIZATION BY VIBRATING SAMPLE MAGNETOMETRY\nIn this section we show the results of room temperature vibrating sample magnetometry (VSM),\nwhich we use to determine the saturation magnetization, Ms, and the magnetic dead layer thick-\nness,tdfor each set of ferromagnetic layers. We measure VSM hysteresis loops (not shown) for\neach thickness of FM that was grown, extract the saturation magnetic moment per unit area for\neach, and plot the results as in supplementary Fig. 3. We determine Msof the FM from the slope\nof data and tdfrom thex-intercept.\nFor the Co 40Fe40B20sample series, we find Ms= 1.233(31) T and td= 0.056(25) nm. The FM\nthicknesses used in Fig. 3 and Fig. 5 in the main text are adjusted for the dead layer thickness; i.e.,\ntFM=tnominal\nFM−td.\nVIII. LONGITUDINAL AND TRANSVERSE ST-FMR DATA FOR OTHER FERRO-\nMAGNETS\nThe results of longitudinal and transverse ST-FMR measurements with different ferromagnet\nmaterials are shown in supplementary figures 4 and 5: for a Pt(6 nm)/Co 90Fe10(6 nm) sample in\nsupplementary Fig. 4 and a Pt(6 nm)/Ni 81Fe19(8 nm) sample in supplementary Fig. 5. The results\n919FIG. 3: Saturation Magnetization per unit area vs. the nominal thickness for sputtered\nCo40Fe40B20layers, all on 6 nm of Pt.\nare similar to the CoFeB samples discussed in the main text, except that to obtain good fits for the\nangular dependence of SXYcomponent requires an additional term approximately proportional to\nsin 2φ:\nSXY=SPHE\nXYcos 2φcosφ+SAHE\nXYcosφ+S2φ\nXYsin 2φ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.\nThe sin 2φterm could also include additional angular dependence proportional to even powers of\ncosφor sinφ; such variations are difficult to distinguish in the fits. No extra term is needed for\nthe fits toAXYor the longitudinal signals SXXandAXX. A contribution SXY∝sin 2φreflects a\ndifference of overall signal magnitudes between the magnetic field angles φand−φ, which (because\nmagnetic field is a pseudovector) can occur only if there is a breaking of mirror symmetry relative\nto the sample’s ˆY-ˆZplane. We speculate that the breaking of symmetry is caused by the different\ncontact geometries at the two end of our sample wire, which might cause a longitudinal thermal\ngradient during resonant heating, and an associated transverse voltage signal with the symmetry of\nthe planar Nernst effect ( ∝mXmY). As a separate Fourier component, whether or not the sin 2 φ\nis included in the fits does not affect the extraction of parameters analyzed in the main text. Table\n1 shows how the size of the sin 2 φcomponent varies for the different types of magnetic layers we\nhave studied.\n1020(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 4: Results of longitudinal and transverse ST-FMR measurements at room temperature for a\nPt(6 nm)/Co 90Fe10(6 nm) sample. (a) Resonance lineshapes for longitudinal ST-FMR. (b,c) Angle\ndependences of the (b) symmetric and (c) antisymmetric resonance components for longitudinal\nST-FMR. (d) Resonance lineshapes for transverse ST-FMR. (e,f) Angle dependences of the (e)\nsymmetric and (f) antisymmetric resonance components for transverse ST-FMR. The lines in\n(b,c,e,f) show fits to the angular components described in the main text as well as the sums of the\nfit components.\nIX. LOW-FREQUENCY SECOND HARMONIC HALL MEASUREMENTS\nIt is widely known (but not explained clearly in the literature) that resonant ST-FMR measure-\nments and non-resonant second-harmonic Hall measurements of spin-orbit torques can differ even\nfor identical layer structures, with results from the low-frequency second harmonic Hall measure-\nments resulting in spin Hall torque efficiencies larger by tens of percent. Since our correction to\nST-FMR for the presence of artifact voltages tends to increase the measured spin Hall torque effi-\nciency, we performed second harmonic Hall measurements on our samples to see if the discrepancy\n1121(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 5: Results of longitudinal and transverse ST-FMR measurements at room temperature for a\nPt(6 nm)/Ni 81Fe19(8 nm) sample. (a) Resonance lineshapes for longitudinal ST-FMR. (b,c) Angle\ndependences of the (b) symmetric and (c) antisymmetric resonance components for longitudinal\nST-FMR. (d) Resonance lineshapes for transverse ST-FMR. (e,f) Angle dependences of the (e)\nsymmetric and (f) antisymmetric resonance components for transverse ST-FMR. The lines in\n(b,c,e,f) show fits to the angular components described in the main text as well as the sums of the\nfit components.\nseen in previous measurements could be explained.\nWe carried out these measurements on standard Hall bar shaped devices patterned on the same\nsamples we used for our ST-FMR devices. We performed these measurements on samples with 2\nand 8 nm thick CoFeB since these samples represent the extremes of the artifact voltage’s effect\non ST-FMR measurements. We employed the standard methodology for investigating in-plane\nmagnetized samples [9] which involves measuring the first and second harmonic Hall response as\na function of in-plane external field angle. The fitting functions we use for the first and second\n1222CoFeB Avg. CoFe(6) Py(8)\nS2φ\nXY/SXY(%) 7 14.6 15.8\nRAMR/RXX(%) 0.03 1 0.83\nRAHE/RXX(%) 0.2 0.2 0.06\nRAMR/RAHE 0.15 5.0 14\nTABLE I: Comparison of the relative magnitude of the sin 2 φcomponent measured in SXYfor\ndifferent samples, together with a comparison of the size of the anisotropic magnetoresistance and\nanomalous Hall resistance relative to the overall sample resistance.\nharmonics are\nVω=IRPHE\n2sin 2φ (40)\nV2ω=IRPHEτ0\nz\nγcos 2φcosφ\nB−/parenleftbiggIRAHE\n2τ0\nx\nγ1\nB+µ0Meff+VANE/parenrightbigg\ncosφ (41)\nwhereIis the current in the bar and VANEis the thermal voltage due to the anomalous Nernst\neffect. The angular dependent data and fits are shown in supplementary Figure 6(a,b) for 8 nm\nthick CoFeB with a 2000 G applied external field.\nThe dampinglike torque can be obtained from the magnetic-field dependence of the amplitude of\nthe cosφpart of the second harmonic voltage, respectively (supplementary Fig. 6(c). The amplitude\nof the cosφcontribution follows the expected linear trend well. From this, we obtain a dampinglike\ntorque efficiency of ξDL= 0.147±0.003 for the 2 nm CoFeB film and 0 .145±0.008 for the 8 nm\nfilm, both roughly 60% higher than for the corrected ST-FMR measurements. The discrepancy in\nthe dampinglike torque efficiency between the two measurement techniques remains, indicating that\nthe artifact voltages that we correct for in this paper cannot explain the difference.\n1323(a)\n (b)\n (c)\nFIG. 6: (a)The angular dependence of the first harmonic Hall voltage which allows extraction of\nthe planar Hall resistance. (b)The angular dependence of the second harmonic Hall voltage,\nshowing the decomposition into cos φ(light blue) and cos 2 φcosφ(dark blue) components which\nrelate to the dampinglike and fieldlike torques, respectively. (c)Field dependence of the cos φ\ncomponent amplitude of the second harmonic Hall voltage. The slope of this line relates to the\ndampinglike torque and the intercept to the anomalous Nernst voltage. All data shown here are\nfrom measurements on the sample with 8 nm of CoFeB.\n1424[1] J. Slonczewski, Journal of Magnetism and Magnetic Materials 159, L1 (1996).\n[2] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M. Hu, Phys. Rev. B 84, 054423 (2011).\n[3] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 87, 184421 (2013).\n[4] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n[5] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002).\n[6] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann,\nPhys. Rev. B 82, 214403 (2010).\n[7] S. Mizukami, Y. Ando, and T. Miyazaki, Japanese Journal of Applied Physics 40, 580 (2001).\n[8] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and P. J. Kelly, arXiv:1108.0385 (2011), arXiv:1108.0385\n[cond-mat.mes-hall].\n[9] C. O. Avci, K. Garello, M. Gabureac, A. Ghosh, A. Fuhrer, S. F. Alvarado, and P. Gambardella, Phys.\nRev. B 90, 224427 (2014).\n1525"    },    {        "title": "2006.16061v4.Ultrafast_optically_induced_ferromagnetic_state_in_an_elemental_antiferromagnet.pdf",        "content": "Ultrafast optically induced ferromagnetic state in an elemental antiferromagnet\nE. Golias,1,\u0003I. Kumberg,1I. Gelen,1S. Thakur,1J. G ordes,1R. Hosseinifar,1Q. Guillet,1\nJ. K. Dewhurst,2S. Sharma,3C. Sch u\u0019ler-Langeheine,4N. Pontius,4and W. Kuch1\n1Institut f ur Experimentalphysik, Freie Universit at Berlin, Arnimallee 14, 14195 Berlin, Germany\n2Max-Planck-Institut f ur Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany\n3Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Strasse 2A, 12489 Berlin, Germany\n4Helmholtz-Zentrum Berlin f ur Materialien und Energie,\nAlbert-Einstein Stra\u0019e 15, 12489 Berlin, Germany\nWe present evidence for an ultrafast optically induced ferromagnetic alignment of antiferromag-\nnetic Mn in Co/Mn multilayers. We observe the transient ferromagnetic signal at the arrival of the\npump pulse at the Mn L 3resonance using x-ray magnetic circular dichroism in re\rectivity. The\ntimescale of the e\u000bect is comparable to the duration of the excitation and occurs before the mag-\nnetization in Co is quenched. Theoretical calculations point to the imbalanced population of Mn\nunoccupied states caused by the Co interface for the emergence of this transient ferromagnetic state.\nControlling magnetic order at high speeds requires the\nultrafast manipulation of the spin degree of freedom, a\ncentral goal of spintronics [1]. Progress in lasers rendered\nultrashort optical pulses as the most promising route to-\nwards the ultrafast control of magnetization [2]. The im-\nportance for technological applications and the scienti\fc\ninterest for the physical processes underlying ultrafast\ndemagnetization focused a lot of research on ultrafast op-\ntical quenching of magnetic order in itinerant ferromag-\nnetic (FM) materials after a non-adiabatic excitation at\ntimescales comparable or even shorter than the exchange\ninteraction [3{13].\nOn the contrary, reports on itinerant antiferromagnets\nare scarce because the absence of a macroscopic magnetic\nmoment makes these systems di\u000ecult to study. Lately,\nThielemann-K uhn et al. [14] showed that manipulation\nof antiferromagnetic (AFM) order is considerably faster\nthan FM, a pivotal \fnding given the modern perspective\nfor the cooperative utilization of FM and AFM compo-\nnents in future (opto-) spintronics [15, 16]. In this con-\ntext, materials that can switch between AFM and FM\norder on ultrafast timescales could o\u000ber unprecedented\nopportunities. Such a transition has been observed in the\ntime domain for the \frst time in FeRh [17, 18] on sub-\nps timescale after excitation with fs laser pulses. Later,\nRadu et al. [19] reported on the formation of a transient\nFM state at the ps timescale during the magnetization\nreversal of the ferrimagnetic material GdFeCo.\nA critical question is: how fast can we induce such\na transient FM state in an AFM? In FeRh, the laser\npulse heats the electronic system with the concomitant\nmodi\fcation of the exchange \feld that couples the spins\nantiferromagnetically. Afterwards, the FM state emerges\nbecause of a Rh-mediated strong FM exchange interac-\ntion of Fe atoms [17]. In the case of GdFeCo, the com-\npetition between thermal and exchange energy can tran-\nsiently drive the two sublattices to a FM alignment [20].\nEventually, the lowest temporal limit for thermally acti-\nvated processes, common in itinerant systems, is set by\nthe timescale of the exchange interaction ( /100 fs) [21].Nevertheless, as the electronic response to an electric \feld\nis virtually instantaneous, optical excitations might allow\nfor the control of magnetic order at timescales shorter\nthan the exchange interaction [22, 23].\nA mechanism that enables the all-optical manipulation\nof magnetic order on sub-exchange timescales is the op-\ntically induced intersite spin transfer (OISTR) [24]. It is\nof pure optical origin, as spin-selective transfer is taking\nplace between neighboring atoms driven by the oscillat-\ning electric \feld of light. The process is universal, i.e., it\ndoes not depend on the material, and allows control of\nmagnetic order only with the structure of the excitation\npulse. After its theoretical prediction [24], experiments\nusing time-resolved magnetic circular dichroism with ex-\ntreme ultraviolet photons in Ni/Pt multilayers con\frmed\nthe presence of OISTR in the ultrafast demagnetization\nof the FM Ni layer [25], opening the way for the magnetic\ncontrol on attosecond timescales, an order of magnitude\nfaster than the exchange interaction. Shortly afterwards,\nother experimental studies concluded the existence of\nOISTR at the Co/Cu(001) interface [26] using second-\nharmonic generation and by tracing the demagnetization\nin CoPt [27] and FeNi alloys [28] using extreme ultravi-\nolet magnetic circular dichroism and transverse Kerr ef-\nfect, respectively. Nevertheless, the most intriguing pre-\ndiction of OISTR [24] is yet to be observed: an ultrafast\noptically-driven transient FM state in an AFM material.\nIn this Letter, we report the observation of such a tran-\nsient FM state in the AFM Mn in a Co/Mn multilayer,\nafter an ultrashort laser excitation. Our sample consists\nof repetitions of 3 monolayers (MLs) of Mn and 3 MLs\nof Co, in which, under static conditions, the magnetiza-\ntions of the FM Co layers are FM aligned and the net\nMn magnetization of the AFM Mn layers is close to zero.\nWe unambiguously observe a transient FM state in the\nAFM Mn by magnetic circular dichroism in time-resolved\nresonant magnetic x-ray re\rection (RMXR), which is es-\ntimated to last as long as the pump pulse duration. Our\nexperimental observations are in agreement with ab-initio\ncalculations and identify the OISTR e\u000bect as the under-arXiv:2006.16061v4  [cond-mat.mes-hall]  12 Feb 20212\nΔtθ2θsamplemagnetic fielddetector(a)(b)(c)\nDistance from surface (nm)Laser absorption (% nm  )-1Mn\nMn\nMn\nCo\nCo\nCu(001)\n00123456\n2468106xCapHIRx-ray\nAFMFM\nFIG. 1. (a) Schematic representation of our experimental geometry. The intensity of the re\rected x-rays is probed by a\nphotodiode in a \u0002 \u00002\u0002 specular geometry with incidence angle \u0012= 8.75o. Oscillating red and spiraling blue arrows indicate\nthe linearly-polarized infrared laser and the circularly-polarized soft-x-ray pulses, respectively. (b) The underlying principle of\nthe OISTR mechanism: the electric \feld of light can, above a threshold value, transfer electrons coherently between atoms. In\nan AFM material, majority states from the \frst atom are transferred to the minority of the second (and vice-versa). When\nan asymmetry is present, such as the interface between Mn and Co, a charge \flling imbalance is introduced and a transient\nFM alignment emerges. (c) Calculation of the di\u000berential laser power absorption on each layer of the Co/Mn multilayer. Red\nand blue lines correspond to the absorption from Co and Mn layers, respectively. The gold line corresponds to the absorption\nfrom the Cu(001) substrate, here, only a small part close to the interface with the multilayer is shown. The absorption in\nthe substrate slowly becomes virtually zero within 40 nm from the interface. On the upper part, a schematic of our sample is\ndisplayed. Red oscillating arrows represent the incoming and re\rected laser light, while red, blue, and gold blocks represent\nthe Co, Mn layers and the Cu(001) substrate, respectively.\nlying mechanism for the emergence of this transient FM\nstate due to the imbalanced population of unoccupied\nminority states in Mn layers caused by the contribution\nfrom the AFM-coupled interfacial Co.\nOur sample was grown in an ultra-high vacuum cham-\nber with a base pressure of 1 \u000210\u00009mbar, on a Cu(001)\nsubstrate held at room temperature using e-beam evap-\noration from a Co rod (99.998% purity) and Mn \rakes\n(99.99% purity) in a Ta crucible. We deposited six repe-\ntitions of 3 MLs of Co and Mn and on top 14 MLs of Co\nas a capping layer to prevent the oxidation of the under-\nlying multilayers by residual gas molecules in the ultra-\nhigh vacuum. During deposition, the thickness was de-\ntermined by the intensity oscillations of di\u000braction spots\nin medium-energy electron di\u000braction while the sample\ncleanliness was veri\fed by Auger electron spectroscopy.\nAfter growth, the sample was stored in a vacuum suit-\ncase with a base pressure better than 2 \u000210\u000010mbar until\nits in-vacuum transfer to the magnetic characterization\nchamber.\nWe characterized the sample at the FemtoSpeX slicing\nfacility [29] at the BESSY II synchrotron of Helmholtz-\nZentrum Berlin. Static and dynamic RMXR measure-\nments have been conducted using a magnetic \feld of 0.2\nT with alternating direction between parallel and anti-\nparallel orientation relative to the x-ray propagation di-\nrection and with a \fxed x-ray light helicity (see Fig.\n1(a)). The time-resolved RMXR measurements have\nMagnetic contrast (%)(b)(a)MnCo6040201.51.00.50.0-0.5Time delay (ps)-10-50510FIG. 2. Time-resolved magnetic contrast from the L 3edge of\n(a) Mn and (b) Co. Blue bars in (a) and red shaded regions in\n(b) correspond to the statistical errors for the measurements\nbased on Poisson statistics. The zero time delay is de\fned\nhere at the maximum of the laser pump pulse. The red os-\ncillating and the dashed lines represent the pump pulses and\ntheir full width at half maximum, respectively, while the light-\nred shaded area indicates the experimental time resolution.\nbeen performed by exciting the sample with linearly-\npolarized 60-fs laser pulses of 800 nm wavelength and\nincident \ruence F = 12 mJ/cm2, nearly parallel to the\nx-ray incidence. The magnetic signal was probed with x-3\nray pulses of 100 fs duration, reaching the sample with a\n6 kHz repetition rate, while the pump laser was operated\nat 3 kHz in order to detect in succession re\rected x-rays\nfrom the sample with and without laser excitation. The\ndynamic magnetic signals have been obtained from the\ndi\u000berence of the re\rected signal with and without laser\nexcitation at the L 3edge of Co and Mn. The total time\nresolution of our experiment was 120 fs and during all\nmeasurements the sample was kept at room temperature.\nBecause of the low intensity of the fs x-ray pulses, our\nexperimental error was determined by photon-counting\nstatistics. Additional characterization of the static mag-\nnetic and structural properties of our sample has been\nperformed at the VEKMAG end-station at BESSY II af-\nter the dynamic measurements at FemtoSpeX (see sup-\nplementary information [30]).\nA schematic representation of our sample can be seen\nin Fig. 1(c). The dominant laminar character of our\nCo/Mn multilayers has been con\frmed by the analysis\nof x-ray resonant re\rectometry oscillations measured at\nspecular geometry. Our sample has a periodicity accord-\ning to the nominal deposition pro\fle with slight inter-\nfacial di\u000busion. We have calculated the intensity of the\npump pulse's electric \feld as a function of the distance\nfrom the sample's surface (see Fig. 1(c)). Our calcula-\ntions show that 56% of the incoming infrared light is re-\n\rected while \u001926.6% is absorbed by the Co layers (13.9%\nis the share of the cap layer), 13.3% by the Mn layers and\n4.1% by the Cu(001) substrate. We estimate that 0.23\nand 0.26 photons are absorbed per pulse per Mn and\nCo atom in our sample, respectively (see supplementary\ninformation [30]).\nFigure 2 displays the time-resolved magnetic signal\nmeasured at the L 3resonances of Mn and Co. In static\nconditions, Co layers do not experience AFM interlayer\ncoupling while the applied magnetic \feld (0.2 T) is\nenough to achieve full magnetic saturation. After the\nexcitation pulse, in Fig. 2(b), we detect a strong demag-\nnetization of Co. Fitting the demagnetization curve to\nan exponential decay function [30] results in a demagne-\ntization time constant of 155 \u000629 fs, in agreement with\nstudies on Co/Pt [31] and Co/Pd [32] multilayers.\nMost importantly, in Fig 2(a) we observe virtually no\nmagnetic contrast in Mn at negative time delays in accor-\ndance with the AFM nature of Mn thin \flms. Statically,\nthe sample shows a small magnetic dichroism ( \u00191.8%)\nantiparallel to the Co magnetization [30], which is be-\nlow the experimental error in the time-resolved RMXR\nmeasurement of Mn. One would expect a higher un-\ncompensated magnetic moment in a perfectly smooth 3-\nML Mn \flm with collinear layer-wise AFM alignment\nequal to one third of the value the \flm would have if\nit was FM aligned. However, the aforementioned condi-\ntion is relaxed due to imperfections and intermixing at\nthe Mn/Co interface. Moreover, atomic-scale or surface\nroughness can lead to frustration of the exchange inter-action at step edges, leading to canted moments and a\ndeviation from a layer-wise parallel spin structure, result-\ning in the nearly vanishing static net magnetization we\nobserve in the Mn layers. At the arrival of the excitation\npulse, we detect an onset of the magnetic signal of Mn\nthat peaks at 8.2%. The data points around the peak\nshow a Gaussian trend, consistent with the convolution\nof our laser-pump and x-ray-probe pulse, with more than\n3000 times higher statistical likelihood compared to the\naverage baseline. Right after the pump pulse, the Mn\nsignal returns to its initial ground state value. The max-\nimum lifetime of the transient FM state in Mn is equal\nor lower than the time resolution of our experiment.\nWe attribute the observed transient FM order in Mn\nto the OISTR e\u000bect. We surmise that the FM state in\nMn lives roughly as long as the pump pulse is present\n(\u001960 fs), given the reported observations of the same\ne\u000bect in Ni/Pt [25] and theoretical considerations [24].\nThe estimated lifetime of the transient FM state is con-\nsistent with the timescale needed for this excited state to\nlose coherence due to spin-orbit coupling in an itinerant\nmagnet [23]. We have to stress that the Co magnetic\nmoment sets the preferential orientation of the transient\nFM magnetization of Mn. We note that, much later, at\n\u00191 ps, we see a negative magnetic signal that we can\nnot explain with either OISTR or electronic spin cur-\nrents. We tentatively assign it to coherent phonons in\nthe multilayer, which change the interatomic magnetic\ncoupling and transiently lead to this signal, for example,\nby reducing the frustration of Mn magnetic moments at\nstep edges.\nThe transient FM alignment in Mn emerges at a de-\nlay time when Co has not yet considerably demagnetized\n[33], suggesting the optical nature of the e\u000bect. Another\nmechanism that might play a role in our experiment is\nsuperdi\u000busive transport [34]. However, as the transient\nFM alignment of Mn layers occurs synchronously with\nthe pump pulse arrival and in the meantime the magnetic\nsignal reduction in Co is small, superdi\u000busive transport\nlikely does not play a signi\fcant role at this early time\nperiod.\nIn order to identify the processes underlying our ex-\nperimental observations we employed ab-initio time-\ndependent density functional theory (TD-DFT) calcula-\ntions. We utilized a stack of 2 ML Mn on top of 3 ML\nCo with an impinging pump pulse with 20 fs full width\nat half maximum (FWHM) and 19 mJ/cm2of incident\nlaser pump \ruence, as the only input parameters of the\ncalculation. Our model calculations are based on a fully\nnon-collinear version of the Elk code [35, 36], where elec-\ntron dynamics after laser excitation is treated by taking\ninto account relativistic e\u000bects. Our theoretical approach\nconsiders spin and charge currents including superdi\u000bu-\nsive currents [34, 37], spin-orbit induced \rips, electron-\nelectron scattering and charge- and spin-density waves\nwith unit vectors larger than the size of a unit cell. Dur-4\n m(t) (μB)-2.0-1.5-1.0-0.50.00.51.0\n706050403020100 Time (fs)706050403020100 Time (fs)Mn2Mn1Co3Co2Co1\nMn2Mn1Mn2Mn1Co3Co2Co1Co3Co2Co11.00.0-0.50.51.00.0-0.50.51.00.0-0.50.51.00.0-0.50.5-8-404Differential PDOS(a)(e)(f)(g)(h)Mn2Mn11.61.41.21.00.80.64.54.03.53.02.52.0(b)\n(d)(c)CoMn m(t) (μB)\n2.01.00.0-1.0-2.02.01.00.0-1.0-2.0  Transient PDOS\n Transient PDOS\n-8-6-4-2024 E-EF (eV) E-EF (eV)-8-6-4-2024 E-EF (eV)  DOS t = 14.5 fs t = 24.2 fs DOS t = 14.5 fs t = 24.2 fs\nFIG. 3. TD-DFT calculations for a \fve-atomic-layer stack comprised of 3 ML Co and 2 ML Mn. (a) Time-dependent magnetic\nmoment of each layer. Note that the stacking order of the layers in the simulation corresponds to the atomic order of the\nlegend and time zero is de\fned here at the beginning of the excitation pulse, as shown in panel (b). (b) Time-dependent total\nmagnetic moment of all Mn (blue line) and all Co (red line) atoms in the stack. The gray oscillating line corresponds to the\ntemporal pro\fle of the vector potential of the pump pulse. (c), (d) Partial DOS (PDOS) of the Mn atoms at the \frst (Mn1),\nsecond (Mn2) Mn layer, respectively, in states/eV/spin. Black dashed lines represent the full (i.e. occupied and unoccupied)\npartial DOS of the ground state. Solid blue and red lines represent the transiently occupied/populated part of the PDOS at t =\n14.5 fs and t = 24.2 fs, respectively. The zero of the energy scale corresponds to the Fermi energy of the ground state. Up- and\ndown-pointing arrows mark the PDOS for the spin-up (positive values) and spin-down states (negative values), respectively.\n(e)-(h) Di\u000berential partial density of occupied states between t = 24.2 fs and t = 0 fs (equal to the occupied ground state\nPDOS) for (e) spin up in Co layers, (f) spin down in Co layers, (g) spin up in Mn layers and (h) spin down in Mn layers,\nrespectively.\ning these simulations nuclei were kept \fxed, as the atomic\nHellmann-Feynman forces are very small during the ex-\ncitation, when our system is in a highly non-equilibrium\nstate, justifying the use of the Born-Oppenheimer ap-\nproximation.\nWe choose to compare our sample with a system with\n2 ML Mn on top of 3 ML Co to minimize the total start-\ning magnetization from Mn layers. The simulation of a\nlayered system with zero Mn magnetization as the one\nstudied experimentally would require a large supercell,\nmaking the ab-initio approach unfeasible. The main con-\nclusions from the calculations do not change, since the\nparity of Mn layers does not play a role in the emergence\nof the FM state, as OISTR is mainly an e\u000bect between\nnearest neighbors and decays fast with distance [24]. Fi-\nnally, the laser excitation was selected shorter for con-\nvenience, as the timescale of the AFM-to-FM transition\ndepends only on the FWHM of the excitation pulse [24].\nAs shown in Ref. 25, longer and weaker pulses resultin the same physics but with higher computational cost.\nTherefore, our current approach and conclusions are also\nvalid on the timescale of our experimental excitation.\nOur \frst-principles calculations can qualitatively ex-\nplain our experimental observations. In Figs. 3(a), (b),\nwe clearly show the transition from AFM to FM align-\nment of the Mn layers after the arrival of the pump pulse.\nThe onset of the FM state starts right before our pump\npulse reaches its half maximum and peaks simultaneously\nwith its vector potential, while Co shows a slower de-\nmagnetization in agreement with our experimental ob-\nservations (see Fig. 2). The underlying mechanism for\nthe transient FM alignment is revealed in Fig. 3 (c)-\n(h), where the unoccupied minority spin density of states\n(DOS) acts as a sink for excited majority spin electrons\nfrom the neighboring Mn layer. The spin swapping be-\ntween Mn neighbors, facilitated by their AFM coupling,\nas well as the higher unoccupied state \flling of the atoms\nat the interface (Mn1) from the AFM-coupled reservoir5\nof Co majority electrons drive the transient FM state in\nMn [38].\nIn summary, we presented compelling evidence of a\ntransient FM state of AFM Mn in Co/Mn multilayers\ndue to the OISTR e\u000bect. The transition is driven by the\nelectric \feld of the pump pulse in a fs timescale, much\nfaster than the FM-order quenching in Co, while the in-\nduced macroscopic magnetic moment of Mn aligns with\nthe adjacent ferromagnet. Our calculations show that\nthe transient FM state originates from the imbalance of\nintersite transfer of electrons in Mn atoms due to the\nasymmetry introduced by the Co interface. The lifetime\nof the e\u000bect is comparable to the pump-pulse duration in\nagreement with theoretical predictions. 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Data Nucl.\nData Tables 54, 181 (1993)."    },    {        "title": "2102.10598v1.Direct_and_inverse_spin_orbit_torques_in_antiferromagnetic_and_ferromagnetic_FeRh_W_001_.pdf",        "content": "Direct and inverse spin-orbit torques in antiferromagnetic and ferromagnetic\nFeRh/W(001)\nFrank Freimuth1;2,\u0003Stefan Bl ugel1, and Yuriy Mokrousov1;2\n1Peter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich and JARA, 52425 J ulich, Germany and\n2Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany\nWe use ab-initio calculations to investigate spin-orbit torques (SOTs) in FeRh(001) deposited on\nW(100). Since FeRh undergoes a ferromagnetic-antiferromagnetic phase transition close to room\ntemperature, we consider both phases of FeRh. In the antiferromagnetic case we \fnd that the\ne\u000bective magnetic \feld of the even torque is staggered and therefore ideal to induce magnetization\ndynamics or to switch the antiferromagnet (AFM). At the antiferromagnetic resonance the inverse\nSOT induces a current density, which can be determined from the SOT. In the ferromagnetic case our\ncalculations predict both even and odd components of the SOT, which can also be used to describe\nthe ac and dc currents induced at the ferromagnetic resonance. For comparison we compute the\nSOTs in the c(2 \u00022) AFM state of Fe/W(001).\nI. INTRODUCTION\nSwitching of the N\u0013 eel vector in antiferromagnets\n(AFMs) by the spin-orbit torque (SOT) is promising\nto achieve higher writing rates on the terahertz scale\nin magnetic memory devices (See Refs. [1{3] for recent\nreviews). Additionally, AFM magnetic memory allows\nhigher data density than its ferromagnet (FM) counter-\npart. Moreover, it has been proposed that SOT triggers\nself-sustained THz oscillations in AFMs [4]. N\u0013 eel vector\nswitching through SOT has been demonstrated experi-\nmentally \frst in the bulk antiferromagnets (AFMs) CuM-\nnAs [5] and Mn 2Au [6]. Planar Hall e\u000bect and magne-\ntoresistance may be used to read the state of the AFM bit\nin the memory device and they have also been used to ob-\ntain an indirect proof of AFM switching. Direct evidence\nof AFM switching is available through x-ray magnetic\nlinear dichroism-photoemission electron microscopy [6].\nExperimentally, it has been shown that the N\u0013 eel or-\nder can be switched by the SOT also in bilayers com-\nposed of a heavy metal layer (HM) and an AFM layer,\nsuch as Pt/Fe 2O3[7]. However, it has been pointed out\nthat measurements of planar Hall e\u000bect and magnetore-\nsistance in HM/AFM bilayers do not always provide clear\nevidence of AFM switching [8, 9]. Most SOT switching\nexperiments on HM/AFM bilayers considered insulating\nAFMs, such as NiO. In CoO/Pt and Fe 2O3/Pt bilayers a\nthermomagnetoelastic mechanism rather than the SOT\nhas been identi\fed as dominating mechanism for current-\ninduced switching [8, 10].\nIn this paper we study the SOT in FeRh/W(001) bi-\nlayers. This choice is motivated by three key assets of\nthis system: (i) The use of W in magnetic bilayers leads\nto large SOTs [11, 12]. (ii) Epitaxial layers of FeRh(001)\ncan be deposited on W(001) single crystals [13]. (iii)\nFeRh has gained considerable interest in the spintronics\ncommunity. Since it exhibits an AFM-FM phase transi-\ntion close to room temperature, femtosecond laser-pulseshave been used to study this phase transition at sub-\npicosecond timescales [14]. The AFM-FM phase tran-\nsition in FeRh has also been used to tune the damp-\ning dynamically [15]. Lateral spin-pumping between FM\nand AFM domains has been found to be crucial for the\ndamping enhancement [16]. When FeRh/Pt is excited\nby femtosecond laser pulses superdi\u000busive spin currents\nare generated in FeRh and converted into a charge cur-\nrent by the inverse spin Hall e\u000bect, which leads to a\nmeasurable THz signal [17{19]. This THz signal varies\nstrongly as FeRh goes through the AFM-FM phase tran-\nsition because it is suppressed in the AFM state of FeRh.\nSimilarly, the Hall e\u000bect changes signi\fcantly across the\nAFM-FM phase transition [20].\nPhenomenology suggests that the antiferromagnetic\nresonance is accompanied by spin-pumping and that con-\nversely spin current from the spin Hall e\u000bect injected into\nan AFM layer exerts staggered e\u000bective magnetic \felds\non it, which are e\u000ecient to induce magnetization dynam-\nics in the AFM [21{23]. However, these concepts have\nnot yet been investigated by ab-initio methods. There-\nfore, in this paper we investigate direct and inverse spin-\norbit torques in FeRh/W(001) based on \frst-principles\ndensity-functional theory calculations. The direct SOT\ndescribes the e\u000bective magnetic \felds acting on the mag-\nnetic moments when an electric current is applied, which\nmay excite magnetization dynamics or switch the AFM\nor FM magnetization. The inverse SOT [24, 25] describes\nthe electric current induced by magnetization dynamics\nand is therefore related to vertical spin pumping, which\nhas been investigated experimentally in the similar sys-\ntem of FeRh/Pt [16]. The inverse SOT is also related\nto the helicity-dependent component of the THz signal\nthat follows excitation by a fs laser-pulse, and which was\nmeasured experimentally in FeRh/Pt as well [18, 19]. For\ncomparison we compute the SOTs also in an Fe mono-\nlayer on W(001), which exhibits a c(2 \u00022) AFM con\fg-\nuration [26] and is therefore compensated like FeRh.\nThis paper is structured as follows. In Sec. II we ex-arXiv:2102.10598v1  [cond-mat.mtrl-sci]  21 Feb 20212\nplain \frst our computational formalism for the direct\nSOT (Sec. II A), followed by the formalism for the in-\nverse SOT (Sec. II B). In Sec. III we discuss our ab-initio\nresults in the following order: In Sec. III A we specify the\ncomputational details, and in Sec. III B and Sec. III C\nwe present our results on the even and odd direct SOT,\nrespectively, which we obtained in the AFM phase of\nFeRh/W(001). We discuss the inverse SOT in the AFM\nphase of FeRh/W(001) in Sec. III D. Next, Sec. III E is\ndevoted to the direct and inverse SOT in the FM phase\nof FeRh/W(001). Finally, Sec. III F treats the SOT in\nFe/W. This paper ends with a summary in Sec. IV.\nII. FORMALISM\nA. Direct SOT\nIn order to compute the SOT we use the formalism\ndescribed in Ref. [12]. The torque T\u0016on atom\u0016can be\nexpressed in terms of the torkance tensor t\u000b\f;\u0016 as\nT\u0016=X\n\u000b^e\u000bt\u000b\f;\u0016E\f (1)\nwhereE\fis the\f-th component of the applied electric\n\feld and ^e\u000bis a unit vector pointing into the \u000b-th Carte-\nsian direction. We separate the torkance into even and\nodd parts with respect to inversion of the magnetization\ndirection. Since we study antiferromagnets in this work\nwe consider the atom-resolved torkances t\u000b\f;\u0016, where the\nindex\u0016selects the atom. The even torkance is given by\nteven\n\u000b\f;\u0016=e~\n2\u0019NX\nkn6=mIm\u0002\nh knjT\u000b;\u0016j kmih kmjv\fj kni\u0003(\n\u0000(Ekm\u0000Ekn)\u0002\n(EF\u0000Ekn)2+ \u00002\u0003\u0002\n(EF\u0000Ekm)2+ \u00002\u0003+\n+2\u0000\u0002\nEkn\u0000Ekm\u0003\u0002\n(EF\u0000Ekm)2+ \u00002\u0003+\n+2\n\u0002\nEkn\u0000Ekm\u00032Im logEkm\u0000EF\u0000i\u0000\nEkn\u0000EF\u0000i\u0000)\n(2)\nand the odd torkance is given by\ntodd\n\u000b\f;\u0016=e~\n\u0019NX\nknm\u00002Re\u0002\nh knjT\u000b;\u0016j kmih kmjv\fj kni\u0003\n\u0002\n(EF\u0000Ekn)2+ \u00002\u0003\u0002\n(EF\u0000Ekm)2+ \u00002\u0003;\n(3)\nwhereNis the number of k-points used to sample the\nBrillouin zone,T\u000b;\u0016is the\u000b-th cartesian component of\nthe torque operator of atom \u0016,v\fis the\f-th cartesian\ncomponent of the velocity operator, \u0000 is the quasiparticle\nbroadening, and  knandEkndenote the Bloch function\nfor bandnatk-point kand the corresponding band en-\nergy, respectively.Experimental works on the SOT typically discuss the\ne\u000bective magnetic \feld that one would have to apply in\norder to generate a torque of the same size as the SOT.\nFor a given torque T\u0016this e\u000bective magnetic \feld may\nbe computed from\nBe\u000b\n\u0016=T\u0016\u0002^M\u0016\nm\u0016; (4)\nwherem\u0016is the magnetic moment of atom \u0016and ^M\u0016\nis its direction. In order to switch an antiferromagnet,\nthe e\u000bective magnetic \feld Be\u000b\n\u0016needs to be staggered,\ni.e., its sign needs to be opposite between antiparallel\nmagnetic moments.\nB. Inverse SOT\nWhile the direct SOT is the generation of a torque on\nthe magnetization when an electric \feld is applied, the\ninverse SOT is the induction of a current density jby\nmagnetization dynamics [24, 25]. When this current den-\nsity is expressed in terms of the atom-resolved torkance,\nEq. (2) and Eq. (3), a summation over the atomic site\nindex\u0016is required:\nj\u001f\n\u000b(t) =1\nVX\n\f;\u0016t\u001f\n\f\u000b;\u0016(\u0000^M\u0016(t)) \n^M\u0016(t)\u0002d^M\u0016(t)\ndt!\n\f;\n(5)\nwhere the superscript \u001fis set to 'even' and 'odd' to ad-\ndress the even inverse SOT and the odd inverse SOT,\nrespectively. When the total inverse SOT is meant, \u001fis\nleft blank.\nIn antiferromagnets with two sublattices we introduce\nthe two vectors\n~L(t) =1\n2h\n^M\"(t)\u0000^M#(t)i\n(6)\nand\n~M(t) =1\n2h\n^M\"(t) + ^M#(t)i\n; (7)\nwhere ^M\"(t) and ^M#(t) are the magnetization directions\non the two sublattices, which we denote by \"and#, re-\nspectively. These vectors satisfy ~L\u0001~L+~M\u0001~M= 1.\nSimilarly, we de\fne the two torkances\n~t\u001f\n\f\u000b=1\n2h\nt\u001f\n\f\u000b;\"(\u0000^M\") +t\u001f\n\f\u000b;#(\u0000^M#)i\n(8)\nand\n\u0016t\u001f\n\f\u000b=1\n2h\nt\u001f\n\f\u000b;\"(\u0000^M\")\u0000t\u001f\n\f\u000b;#(\u0000^M#)i\n: (9)\nWith these de\fnitions the pumped charge current density3\nin a 2-sublattice AFM can be written as\nj\u001f\n\u000b(t) =2\nVX\n\f~t\u001f\n\f\u000b \n~L\u0002d~L\ndt+~M\u0002d~M\ndt!\n\f\n+2\nVX\n\f\u0016t\u001f\n\f\u000b \n~L\u0002d~M\ndt+~M\u0002d~L\ndt!\n\f:(10)\nAt the antiferromagnetic resonance the two magneti-\nzation directions ^M\"and ^M#precess with slightly dif-\nferent cone angles, which results in a non-zero vector\n~M[27]. However, usually j~Mj\u001cj ~Ljis satis\fed, which\ncan be used to replace Eq. (10) by approximated expres-\nsions. For a layerwisely compensated layered AFM in an\nAFM/HM bilayer with two magnetic sublattices we can\napproximate\n~teven\n\f\u000b;`=1\n2h\nteven\n\f\u000b;`\"(\u0000^M`\") +teven\n\f\u000b;`#(\u0000^M`#)i\n\u0019teven\n\f\u000b;`\"(~L`);\n(11)\nwhereteven\n\f\u000b;`\"(\u0000^M`\") andteven\n\f\u000b;`#(\u0000^M`#) are the two\ntorkances of the two sublattices in layer `of the AFM\nand\n~L`(t) =1\n2h\n^M`\"(t)\u0000^M`#(t)i\n(12)\nis the generalization of Eq. (6) to a layerwisely compen-\nsated layered AFM. Therefore, in this case the even com-\nponent of the pumped current density can be approxi-\nmated as\njeven\n\u000b(t)\u00192\nVX\n\f;`~teven\n\f\u000b;`\" \n~L\u0002d~L\ndt!\n\f; (13)\nwhere we further approximated ~L`=~L. Similarly, we\napproximately obtain\njodd\n\u000b(t)\u00190 (14)\nbecause\ntodd\n\f\u000b;`\"(^M`\") =\u0000todd\n\f\u000b;`#(^M`#) (15)\nwhen ^M`\"=\u0000^M`#for a layerwisely compensated lay-\nered AFM in an AFM/HM bilayer.\nHowever, this vanishing jodd\n\u000b(t) is a special case\nand the odd inverse SOT does not always vanish in\nAFMs. Consider for example Mn 2Au [6, 28] and CuM-\nnAs [5]. In these bulk AFMs the torkance tensor satis\fes\ntodd\n\f\u000b;\"(^M\") =todd\n\f\u000b;#(^M#) when ^M\"=\u0000^M#. Thus, we\nmay approximate\n~todd\n\f\u000b=1\n2h\ntodd\n\f\u000b;\"(\u0000^M\") +todd\n\f\u000b;#(\u0000^M#)i\n\u0019\u0000todd\n\f\u000b;\"(~L)\n(16)\nand\njodd\n\u000b(t)\u0019\u00002\nVX\n\f;`~todd\n\f\u000b;\" \n~L\u0002d~L\ndt!\n\f: (17)\nz←−−−\ny−−−→⊙x\nFe3←−Fe2−→Fe4−→\nFe1←−\nFe6−→Fe7←−Fe5←−\nFe8−→\nRh Rh W W W W W Rh RhFIG. 1. One unit cell of the FeRh/W/FeRh \flm. The unit\ncell is repeated periodically along the xandyaxes. Red\narrows illustrate the directions of the magnetic moments on\nthe Fe sites.\nIII. RESULTS\nA. Computational details\nIt has been shown in experiments that ultrathin epi-\ntaxial layers of FeRh(001) can be deposited on W(100)\nsingle-crystals [13]. Since the mismatch between the\nFeRh (CsCl structure) bulk lattice constant of 2.99 \u0017A\nand the one of bcc W of aW= 3:165\u0017A is 5%, pseudo-\nmorphic growth of the FeRh(001) layer leads to tetrago-\nnal distortion.\nWe use the \flm mode [29] in the ab-initio program\nFLEUR [30] in order to compute the electronic structure\nof FeRh on W. In this mode the unit cell is repeated\nperiodically only in the in-plane directions and the re-\nsulting \flm structure is embedded into vacuum. Since\nsystems with inversion symmetry take less computational\ne\u000bort in the \flm mode we consider the centrosymmetric\nsystem FeRh/W/FeRh, where 11 layers of W(001) are\nsandwiched between two layers of FeRh on both sides.\nIn Fig. 1 we show the corresponding unit cell.\nIn order to compute the antiferromagnetic state we\nneed a magnetic unit cell with in-plane area twice as large\nas the one of the crystal unit cell. Therefore, we set the\nin-plane lattice constant top\n2aW= 8:459a0, wherea0\nis Bohr's radius. In our calculations, the inter-layer dis-\ntance is 0:5aW= 2:99a0in W and 2 :67a0in FeRh. The\ndistance between the W layer and the Fe layer at the in-\nterfaces is 2.83 a0. We chose mu\u000en-tin radii of 2 :37a0in\nFe and Rh and of 2 :57a0in W and performed the calcu-\nlations with the generalized gradient approximation [31]\nto density-functional theory. Spin-orbit coupling is in-\ncluded in the calculations. The magnetic moments are\n2.63\u0016Bin Fe1 and Fe2, while they are 3.21 \u0016Bin Fe3 and\nFe4.\nThe computational parameters used in our calculations\nof a monolayer of Fe on W(001) (we refer to this sys-\ntem simply by Fe/W(001) in the following) are given in\nRef. [26]. Also in this case we compute the inversion sym-\nmetric system Fe/W/Fe in order to reduce the numerical4\ne\u000bort.\nIn order to evaluate the SOT according to Eq. (2) and\nEq. (3) we make use of Wannier interpolation [32] for\ncomputational speed-up. For this purpose we disentangle\n18 maximally localized Wannier functions per transition\nmetal atom, where we employ our interface [33] between\nFLEUR and the Wannier90 program [34].\nB. Even torkance\nWe show the atom-resolved even torkance in Fig. 2.\nWhile the magnetic moments in Fe1 and Fe2 are aligned\nantiferromagnetically, their torkances agree: teven\nyx;Fe1=\nteven\nyx;Fe2. Similary, the torkances agree on atoms Fe3 and\nFe4, i.e.,teven\nyx;Fe3=teven\nyx;Fe4. This property of layerwisely\ncompensated layered AFMs is the basis for Eq. (11).\nAdditionally, the four torkances on atoms Fe1 through\nFe4 all have the same sign. Consequently, the e\u000bective\nmagnetic \feld, Eq. (4) is staggered, i.e., it has opposite\nsign on Fe1 through Fe4 between magnetic moments that\npoint in opposite directions. Such a staggered e\u000bective\nmagnetic \feld is precisely what is necessary to switch the\nantiferromagnetic layer composed of Fe1 through Fe4.\nFig. 2 also shows that the torques on Fe5 and Fe6 are\nequal but opposite to the torques on Fe1 and Fe2. Simi-\nlarly, the torques on Fe7 and Fe8 are equal but opposite\nto the torques on Fe3 and Fe4. This follows from the\nfact that the space inversion operation maps Fe1 on Fe5,\nFe4 on Fe8, Fe2 on Fe6, and Fe3 on Fe7. We only show\ntheyx-component of the torkance, because the xxand\nyycomponents are zero. The xy-component may be ob-\ntained from teven\nxy;\u0016=\u0000teven\nyx;\u0016.\nIn the limit \u0000!0 we \fnd the torkances teven\nyx;Fe1=\n\u00000:61ea0andteven\nyx;Fe4=\u00000:51ea0. At high quasipar-\nticle broadening \u0000 = 100 meV the torkance on Fe4 is\nsigni\fcantly reduced, namely teven\nyx;Fe4=\u00000:12ea0, while\nthe torkance on Fe1 is still of similar magnitude, namely\nteven\nyx;Fe1=\u00000:56ea0. In Ref. [12] we have shown that the\neven torkance is described by a scattering-independent\nmixed Berry curvature in the limit \u0000 !0. This predicts\nthe even torkance to be \u0000-independent at small \u0000. Indeed\nat small \u0000, e.g. \u0000 <10 meV, both torkances are roughly\nconstant.\nIn Ref. [12] we determined the even torkance to be\nteven\nyx=\u00000:83ea0in Mn(1)/W(9) and teven\nyx=\u00000:56ea0in\nMn(1)/W(15) in the limit \u0000 !0, while at \u0000 = 100 meV\nwe found the torkance to be teven\nyx=\u00000:47ea0in both\nMn/W(001) systems. Since the in-plane unit cell area of\nFeRh/W/FeRh is twice as large as the one of Mn/W(001)\nwe need to compute teven;tot\nyx =teven\nyx;Fe1+teven\nyx;Fe4in order to\nperform a meaningful comparison of torkances between\nFeRh/W/FeRh and W/Mn(001), i.e., we have to consider\nhalf of the total torque on Fe1 through Fe4. This sum is\nalso shown in Fig. 2. We \fnd teven;tot\nyx =\u00001:12ea0in the\nlimit \u0000!0 andteven;tot\nyx =\u00000:68ea0at \u0000 = 100 meV.\n1 10 100\nBroadening Γ [meV]-1-0.500.5Torkance tyx,µeven [ea0]Fe1, Fe2\nFe3, Fe4\nFe5, Fe6\nFe7, Fe8\nFe1+Fe4FIG. 2. Atom-resolved even torkances vs. quasiparticle\nbroadening \u0000 in the AFM phase of FeRh/W/FeRh. The\natomic site labels Fe1 through Fe8 are explained in Fig. 1.\nThe sum of the torkances on Fe1 and Fe4 is also shown\n('Fe1+Fe4'). The product of elementary positive charge e\nand Bohr radius a0used as unit of torkance amounts to\nea0= 8:478\u000110\u000030Cm.\nThus, the torkances in FeRh/W/FeRh are larger than in\nW/Mn. However, since the sign of the torkances agrees\nbetween the two systems, and since the magnitudes are\nsimilar, we assume that the even SOT is generated by\nthe same mechanism in both systems. In Ref. [12] we\nhave shown that the even SOT in W/Mn arises from\nspin currents and may be attributed to the spin Hall\ne\u000bect from W. Therefore, we attribute the even torque\nin FeRh/W/FeRh also to spin currents from the SHE of\nW.\nThe quasiparticle broadening \u0000 also determines the ef-\nfective spin-di\u000busion length. Consequently, we assume\nthatteven\nyx;Fe4decays stronger with increasing \u0000 than teven\nyx;Fe1\nbecause the Fe4 is further away from W than Fe1 and\ntherefore a larger fraction of spin current is lost for high\n\u0000 before the spin current reaches Fe4.\nC. Odd torkance\nIn Fig. 3 we show the odd torkance. We only show the\ncomponent todd\nxx;\u0016, because due to symmetry the xyand\nyxcomponents are zero and todd\nyy;\u0016=todd\nxx;\u0016. For atoms re-\nlated by space inversion, e.g. Fe1 and Fe5, the torques are\nagain equal but opposite. The magnetic moments on Fe1\nand Fe2 are antiparallel, but the odd torkances at these\nsites are opposite as well. Consequently, the e\u000bective\n\feld is not staggered for atoms Fe1 and Fe2. Similarly,\nthe magnetic moments on Fe3 and Fe4 are antiparallel\nand their torkances are opposite such that the e\u000bective\n\feld is not staggered for these two atoms either. This5\n10 100\nBroadening Γ [meV]-2-1012Torkance txx,µodd [ea0]Fe1, Fe6\nFe2, Fe5\nFe3, Fe8\nFe4, Fe7\nFe1-Fe4\nFIG. 3. Atom-resolved odd torkances vs. quasiparticle broad-\nening \u0000 in the AFM phase of FeRh/W/FeRh. The di\u000berence\nof the torkances on Fe1 and Fe4 is also shown ('Fe1-Fe4'),\nbecause it can be used to obtain the total odd torkance ap-\nproximately in the FM phase (discussed in Sec. III E).\nis a property of layerwisely compensated layered AFMs,\nwhich we expressed also in Eq. (15).\nOn the other hand, Fe1 and Fe4 are antiparallel, their\nodd torkances are not staggered, but their e\u000bective \felds\nare staggered. Similarly, Fe2 and Fe3 are antiparallel,\ntheir odd torkances are not staggered, but their e\u000bective\n\felds are staggered. In order to induce magnetization\ndynamics in an AFM, the e\u000bective \feld should ideally be\nstaggered consistently in the AFM. This criterion is not\nsatis\fed by the odd torkance in this system. The total\ntorkance on Fe1, Fe2, Fe3, and Fe4 is zero for the odd\ntorque, in contrast to the even torque discussed in the\nprevious section.\nAt small broadening \u0000 the sign of todd\nxx;Fe1is di\u000berent\nto the one of todd\nxxin Mn/W(001), which we attribute\nto di\u000berent interfacial spin-orbit interactions in the two\nsystems. These di\u000berences may be described by opposite\nsigns of the e\u000bective Rashba parameter in the two cases.\nD. Inverse SOT\nPrevious works on spin-pumping in AFM/HM bilay-\ners focused on the dc spin current pumped at the an-\ntiferromagnetic resonance [21{23]. This dc component\nis observed when the staggered magnetization is paral-\nlel to the bilayer interface. Here, we consider a di\u000berent\ngeometry (see Fig. 1) with staggered magnetization per-\npendicular to the bilayer interface. In this geometry only\nac spin currents can be pumped. In FM/HM bilayers it\nhas been pointed out that the pumped ac spin current is\nlarger in magnitude than its dc counterpart [35]. AC spin\ncurrents can even be measured directly [36]. Moreover,only the phase-sensitive measurement of the ac inverse\nSOT allows us to access both its even and odd compo-\nnents [24, 25] in FM/HM bilayers.\nSimilarly, we expect the ac inverse SOT to induce\nlarger voltages than its dc counterpart in the present\nAFM/HM bilayer. However, in the present case the\nodd part is not easy to access even in a phase-sensitive\nmeasurement of the ac inverse SOT, because it is ap-\nproximately zero according to Eq. (14). This vanishing\nodd inverse SOT corresponds to the vanishing total odd\ntorkance discussed in the previous Sec. III C.\nAssuming that ~Lprecesses around the zaxis according\nto\n~L(t) = [sin(\u0012) cos(!t);sin(\u0012) sin(!t);cos(\u0012)]T(18)\nat the antiferromagnetic resonance, we obtain from\nEq. (13) the current densities\njeven\nx\u0019\u0000!\nVteven\nyxsin(2\u0012) sin(!t) (19)\nand\njeven\ny\u0019!\nVteven\nyxsin(2\u0012) cos(!t); (20)\nwhere\nteven\nyx=teven\nyx;Fe1+teven\nyx;Fe3=teven\nyx;Fe1+teven\nyx;Fe4=teven;tot\nyx (21)\nshould be used to describe the current induced in\nFeRh/W(001), i.e., Fe5, Fe6, Fe7, and Fe8 have to be\nskipped in this summation, because in the centrosymmet-\nric FeRh/W(001)/FeRh system the inverse SOT current\ndensity is zero if both the upper (Fe1-Fe4) and the lower\n(Fe5-Fe8) AFMs perform the same precession Eq. (18).\nE. Ferromagnetic FeRh\nWhen we \rip all magnetic moments in the system, the\neven torkance on a given atom does not change, while the\nodd torkance on a given atom changes sign. This holds\nexactly, because the torkances in Eq. (2) and Eq. (3) are\neven and odd, respectively, with respect to inversion of\nmagnetization, i.e., with respect to \ripping all magnetic\nmoments. When we \rip only the magnetic moment of\natom\u0016but keep all other magnetic moments unchanged,\nthe even torque on atom \u0016stays approximately the same,\nwhile the odd torque on atom \u0016changes sign and stays\napproximately the same in magnitude. These relations\nhold only approximately, because by \ripping only a sin-\ngle magnetic moment we obtain a new system that is not\nrelated by any symmetry operation to the original sys-\ntem. We may use these approximate relations in order\nto describe the ferromagnetic system. Thus, the even\ntorkances shown in Fig. 2 for the AFM case apply ap-\nproximately also to the FM case, i.e., the even torkances6\nare approximately constant as FeRh passes through the\nAFM-FM phase transition. The torkance may therefore\nbe used to describe the SOT in the FM case or to compute\nthe voltage induced by the inverse SOT at the ferromag-\nnetic resonance.\nIn order to obtain approximately the odd torkances\nfor the FM case, i.e., the case in which the magnetic\nmoments of Fe2, Fe4, Fe6, and Fe8 are \ripped relative to\nwhat is shown in Fig. 1, we only need to \rip the signs\nof the odd torkances of those atoms. The resulting total\ntorkance on Fe1 and Fe4 is also shown in Fig. 3 (label\n'Fe1-Fe4'). It is the di\u000berence between the odd torkance\nof Fe1 and the one of Fe4, because Fig. 3 discusses the\ntorkances in the AFM phase and therefore we need to\nmultiply the torkance of Fe4 by -1 if we use it to describe\nthe FM case. Consequently, the sum of the odd torkances\non Fe1 and Fe4 in the FM phase is approximated by the\ndi\u000berence between the odd torkances on Fe1 and Fe4 in\nthe AFM phase.\nIn FeRh/Pt the spin pumping and inverse dc SOT\nhave been investigated experimentally across the AFM-\nFM phase transition [16]. Similarly, our results can be\nused to determine the ac and dc inverse SOT in the FM\nphase of FeRh/W. In the case of FMR-driven magneti-\nzation precession around the zaxis according to\n^M(t) = [sin(\u0012) cos(!t);sin(\u0012) sin(!t);cos(\u0012)]T(22)\nwe obtain the current densities [24]\njeven\nx\u0019\u0000!\nV\u0002\nteven\nyx;Fe1+teven\nyx;Fe4\u0003\nsin(2\u0012) sin(!t) (23)\nand\njodd\nx\u00192!\nV\u0002\ntodd\nxx;Fe1\u0000todd\nxx;Fe4\u0003\nsin(\u0012) cos(!t): (24)\nHere,\u0002\ntodd\nxx;Fe1\u0000todd\nxx;Fe4\u0003\nrefers to the di\u000berence of\ntorkances shown in Fig. 3 (label 'Fe-Fe4').\nIn Ref. [18, 19] a helicity-dependent THz signal was\nmeasured in FeRh/Pt after illumination with a fs laser-\npulse, which can be explained by the model described\nin Ref. [37]. Similarly, the odd torkance in the FM\nphase obtained from our calculations may also be used\nto predict a helicity-dependent THz signal in the similar\nsystem FeRh/W. For this purpose one may neglect the\nanisotropy of the odd torkance and apply our result for\nmagnetization along zto the case with in-plane magne-\ntization.\nF. Fe/W(001)/Fe\nIn our calculation there are two Fe monolayers in the\nc(2\u00022) AFM state that sandwich the W(001) layer. Each\nFe monolayer is described by an in-plane unit cell con-\ntaining two Fe atoms, which we label as follows: The top\n10 100\nBroadening Γ [meV]-1-0.500.51Torkance tyx,µeven [ea0]Fe1, Fe2\nFe3, Fe4 FIG. 4. Atom-resolved even torkances vs. quasiparticle\nbroadening \u0000 in the AFM state of Fe/W(001)/Fe. Fe1 and\nFe2 are in the top layer ( z>0) while Fe3 and Fe4 are in the\nbottom layer ( z<0).\nlayer (z>0) in Fe/W(001)/Fe is composed of the atoms\nFe1 and Fe2, while the bottom layer ( z < 0) is com-\nposed of the atoms Fe3 and Fe4. In Fig. 4 we show the\neven torkance in the AFM con\fguration. At high quasi-\nparticle broadening \u0000 = 100 meV we \fnd the torkance\nteven\nyx;Fe1=\u00000:31ea0, which is slightly smaller than the\nvalue found in Mn/W(001) of teven\nyx =\u00000:47ea0[12].\nAt \u0000 = 25 meV we \fnd teven\nyx;Fe1=\u00000:68ea0, which is\nlarger than teven\nyx\u0019\u00000:3ea0found in Fe/Mn(110) stud-\nied in Ref. [38]. However, it is smaller than teven;tot\nyx\nin FeRh/W/FeRh. The agreement in sign of the even\ntorque between Mn/W(001), Fe/W(001), Fe/W(110) and\nFeRh/W(001)/FeRh suggests that it is dominated by the\nabsorption of spin current from the spin Hall e\u000bect of\nW irrespective of W orientation (i.e., both in (001) and\n(110)) and for di\u000berent FMs and AFMs, namely for Mn,\nFe, and FeRh. However, this behaviour cannot be gen-\neralized to all magnets. For example, it has been shown\nthat in Ni/W(110) the even torque arises from the or-\nbital torque and that it is opposite in sign compared to\nFe/W(110) [38].\nIn Fig. 5 we show the odd torkance in the AFM con\fg-\nuration. At high quasiparticle broadening \u0000 = 100 meV\nwe \fnd the torkance todd\nxx;Fe1= 0:15ea0, which is oppo-\nsite in sign and larger in magnitude when compared with\nMn/W, where we found todd\nxx=\u00000:082ea0[12]. However,\nthe sign agrees with the one of todd\nxx;Fe1in FeRh/W/FeRh.\nThe sign agrees also with the one of Fe/W(110) studied\nin Ref. [38].\nThe inverse SOT in Fe/W may be obtained in the same\nway as discussed in Sec. III D.7\n10 100\nBroadening Γ [meV]-6-4-20246Torkance txx,µodd [ea0]Fe1, Fe4\nFe2, Fe3\nFIG. 5. Atom-resolved odd torkances vs. quasiparticle broad-\nening \u0000 in the AFM state of Fe/W(001)/Fe. Fe1 and Fe2 are\nin the top layer ( z>0) while Fe3 and Fe4 are in the bottom\nlayer (z<0).\nIV. SUMMARY\nWe use ab-initio calculations in order to study the SOT\nin FeRh/W(001) bilayers. Both the AFM and the FM\nphase of FeRh are interesting for spintronics applications.\nIn the AFM phase the even SOT leads to a staggered ef-\nfective \feld, which couples favourably to the staggered\nmagnetization. In contrast, the e\u000bect of the odd torque\nin the AFM phase is negligible. We derive expressions\nfor the inverse SOT in AFMs, i.e., formulas that express\nthe current-density induced by magnetization dynamics\nin terms of the torkance tensors. We discuss the modi-\n\fcations of the torkance tensor as the system goes from\nthe AFM state to the FM state. In the FM phase both\neven and odd SOT are signi\fcant, and both of them con-\ntribute to the ac inverse SOT at the ferromagnetic reso-\nnance. For comparison we also compute the SOTs in the\nc(2\u00022) AFM state of Fe/W(001), where we \fnd smaller\neven torkances.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from Leibniz Collab-\norative Excellence project OptiSPIN \u0000Optical Control\nof Nanoscale Spin Textures, and funding under SPP 2137\n\\Skyrmionics\" of the DFG. We gratefully acknowledge\n\fnancial support from the European Research Coun-\ncil (ERC) under the European Union's Horizon 2020\nresearch and innovation program (Grant No. 856538,\nproject \"3D MAGiC\"), from the DARPA TEE program\nthrough grant MIPR (No. HR0011831554) from DOI,\nand ITN Network COMRAD. 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Zhang1,∗\n1Department of Physics, Indiana University, Bloomington, I ndiana 47405\n2Department of Physics, Ohio State University, Columbus, Oh io 43210\nAbstract\nPyrochlore iridates have recently attracted growing inter est in condensed matter physics because\nof their potential for realizing new topological states. In order to achieve such quantum states, it is\nessential to understand the magnetic properties of these co mpounds, as their electronic structures\nare strongly coupled with their magnetic ground states. In t his work, we report a systematic study\nof the magnetic properties of pyrochlore Y 2Ir2O7and its hole-doped compounds by performing\nmagnetic, electron spin resonance (ESR), electrical trans port and x-ray photoelectron spectroscopy\n(XPS) measurements. We demonstrate the existence of weak fe rromagnetism on top of a large\nantiferromagnetic background in the undoped compound. Hol e-doping by calcium was found to\nenhance both the ferromagnetism and the electrical conduct ivity. The XPS characterization shows\nthe coexistence of Ir4+and Ir5+in the undoped compound, and the amount of Ir5+increases with\nCa-doping, which highlights the possible origins of the wea k ferromagnetism associated with the\nformation of Ir5+. We also observe a vertical shift in the M-Hcurves after field cooling, which\nmay arise from a strong coupling between the ferromagnetic p hase and the antiferromagnetic\nbackground.\nPACS numbers: 75.47.Lx, 72.20.-i, 76.30.-v, 79.60.-i\n1I. INTRODUCTION\n5d transition metal oxides (TMOs) provide a fascinating system to s tudy the interplay\nand competition between spin-orbit coupling (SOC) and electron cor relation1–6that have\ncomparable energy scales.1,7Among all the 5d TMOs, the pyrochlore iridates A 2Ir2O7(A-\n227, A = Y, Bi, or lanthanide element) have attracted particular inte rests8–29because of the\nrecent theoretical predictions of novel topological phases.2–4Indeed, a number of topologi-\ncally non-trivial states, such as topological Mott insulators,2,3,30Weyl semimetals,4,31axion\ninsulators4,32and topological crystalline insulators,33have been predicted to exist in certain\ncorrelation regimes that can potentially be accessed by tuning the A -site elements.6,34\nTo realize novel topological states inpyrochlore iridates, it is essen tial to understand their\nmagnetic properties as their electronic structures are strongly c oupled with the magnetic\nground states.4,14,31In the pyrochlore lattice, the magnetic Ir4+ions form a corner-sharing\ntetrahedralnetwork, andthecompetitionbetween thethreema jormagneticinteractions(i.e.\nthe Heisenberg-type antiferromagnetic coupling, the Dzyaloshins kii-Moriya interaction, and\nthe single-ion anisotropy)4,35,36gives rise to a variety of magnetic configurations. When the\nA3+ion is magnetic, the possible f-dexchange interaction14between the Ir4+and A3+leads\nto even more complex magnetic structures.9,17,23,25,37,38Therefore, to clarify the fundamental\nmagnetic properties associated with the iridium tetrahedral netwo rk, it is desired to choose\nthe compounds with non-magnetic A-site ions, e.g. Eu3+,18,19Bi3+,26–29and Y3+.20,21,39,40\nThe Y 2Ir2O7(i.e. Y-227) is of particular interest because it is predicted to be a We yl semi-\nmetal when its magnetic ground sate is the peculiar all-in/all-out antif erromagnetic (AFM)\nphase.4Neutron scattering measurements do not show clear evidence of lo ng-range magnetic\norder,20,21which may be due to the low scattering intensity arising from the large neutron\nabsorption cross-section of iridium. Muon spin rotation and relaxat ion experiments yield\na well-defined muon spin precession frequency, indicating a long-ran ge magnetic order at\nlow temperatures.21Further analysis of the spontaneous muon spin precession freque ncy in\ncombination with the probabilistic and ab-initio modeling techniques suggest that the all-\nin/all-out AFM is indeed the magnetic ground state.40Despite these advances, the precise\nnature of the magnetism in this compound is far from being fully under stood. For example,\na small hysteresis loop was observed in the magnetization versus ma gnetic field ( M-H)\ncurve taken at low temperatures,20which cannot be explained by the perfect all-in/all-out\n2antiferromagnetic scenario.\nIn this work, we have performed systematic magnetization measur ements, electron spin\nresonance (ESR) studies, transport characterizations and x-r ay photoelectron spectroscopy\nmeasurements (XPS) to probe the magnetic properties of undope d and Ca-doped Y 2Ir2O7\ncompounds. We demonstrate the existence of weak ferromagnet ism (FM) in the undoped\nsample, and the enhancement of both FM and electrical conductivit y by Ca-doping. We\nalso provide evidence of strong exchange coupling between the FM a nd the large AFM\nbackground. The XPS studies show mixed valence states of Ir, i.e. 0 , 4+ and 5+, in both\nthe undoped and doped samples.\nII. EXPERIMENTAL METHODS\nThe (Y 1−xCax)2Ir2O7samples were prepared by conventional solid state reaction. Mix-\ntures of high purity Y 2O3(99.99%), CaCO 3(99.997%) and IrO 2(99.99%) in stoichiometric\nratios were heated in air at temperatures between 900◦C and 1050◦C for about 6 days with\nseveral intermediate grindings. The structure and phase purity w ere characterized by X-\nray powder diffraction using a PANalytical EMPYREAN diffractometer (Cu Kαradiation).\nThe samples have a nearly pure phase of cubic Fd-3m(227) except for some minor impurity\nphases of the source materials (see Fig. S1 in Supplemental Materia l),41which is consistent\nwith what has been reported by other groups.20,21These impurity phases are either para-\nmagnetic or diamagnetic, and hence have negligible contribution to th e weak ferromagnetic\nproperties thatwill bediscussed below. Themagnetization measure ments were performedin\na Quantum Design Magnetic Property Measurement System. The ES R spectra were taken\nininaBRUKER EMX plus spectrometer withXbandmicrowave (frequen cyν=9.40GHz),\nfollowing either zero-field cooling (ZFC) or field cooling (FC). The resis tivity was measured\nusing a Linear Research LR-700 AC Resistance Bridge. The XPS meas urement was carried\nout in a PHI VersaProbe II Scanning XPS Microprobe and the fitting t o the spectra was\ndone using a standard software package CasaXPS provided by Cas a Software Ltd.\n38\n6\n4\n2M/H (10-3emu/Oe mol Ir)\n300 200 100 0\nT (K)x=0.10\nx=0.05\nx=0.01\nx=0    (a)\n(b)-12-8θ (102K)\n0.100.050.00\nx\n-0.10.00.1M (emu/g)\n-60-40-200204060\nH (kOe) x=0\n x=0.01\n x=0.05\n x=0.105K  ZFC\n102103104HC (Oe)\n0.100.050.00\nx20\n10\n0Mr (10-3emu/g)\n4\n0\n-4M (10-3emu/g)\n-101\nH (kOe)x=0.08\n7M/H\n(10-4emu/Oe mol Ir)         \n200180160\nT (K)x=0\nTM2TM1TM2\nFIG. 1. (a) Temperature dependence of ZFC and FC susceptibil ities of (Y 1−xCax)2Ir2O7samples\nmeasured in a magnetic field H= 1 kOe. The cooling field HFC= 1 kOe. Upper inset: magnetic\nsusceptibilities of the undoped sample near 190 K; Lower ins et: the Weiss constant as a function\nof doping concentration x. (b) Magnetic hysteresis loops ( M-H) taken at 5 K after ZFC. Upper\ninset: the coercivity HCand remanent Mrversus doping concentration x; Lower inset: low field\nM-Hloop of undoped sample at 5 K.\nIII. RESULTS AND DISCUSSION\nThemagneticsusceptibilitiesof(Y 1−xCax)2Ir2O7(x=0,0.01,0.05and0.10)samplestaken\nafter ZFC and FC are plotted as a function of temperature in Fig. 1(a). For the undoped\nsample (x=0), a clear magnetic transition is observed at TM1∼158 K, below which there\nis a large hysteresis difference between the ZFC and FC magnetic sus ceptibilities. This\nresult is consistent with earlier studies,20,21,37,39,42confirming our sample quality. We also\nnoticed that there is a very small but visible kink around 190 K (upper inset of Fig. 1(a)),\n4indicating a possible transition. A more pronounced transition was ob served by Disseler et\nal.21With Ca-doping, the FC magnetization is enhanced and the transition atTM2∼190 K\nbecomes more prominent. For higher doping level of x=0.05 and 0.10, the transition at TM2\ndominates over the one at TM1. By fitting the magnetic susceptibility data above TM2, we\nobtain the Weiss constant θas shown in the lower inset of Fig. 1(a). All the Weiss constants\nare negative with unusually large absolute values, suggesting a stro ng AFM interaction in\nall the samples. With Ca-doping, the constant increases, which indic ates the weakening of\nAFM interaction.\nWhile the AFM interaction and the small magnitude of magnetization ( ∼10−4µB/Ir)\nare consistent with the all-in/all-out configuration, the M-Hdata taken at 5 K show hys-\nteresis loops (Fig. 1(b)), which suggests the existence of ferromagnetic component on top of\nthe AFM background. We plot the magnetic coercivity HCand the remnant magnetization\nMras a function of doping concentration in the upper inset of Fig. 1(b). As the magnetic\nhysteresis loops of the x=0.05 and 0.10 samples are not saturated and there is a large AFM\nbackground for all the samples, their HCandMrvalues may be underestimated. Never-\ntheless, these two parameters show clear increase with increasing the doping concentration,\nsuggesting the enhancement of ferromagnetism by doping.\nSince the magnetic hysteresis loop of the undoped sample is small (low er inset of Fig.\n1(b)), we further performed ESR measurements to confirm the we ak-ferromagnetism. Fig.\n2(a) and (b) show the ESR spectra (i.e. first derivative of absorptio n with respect to field)\nof the undoped sample taken at variable temperatures after the s ample was cooled down\nin zero field and in a field of 1 kOe, respectively. To obtain the resonan t fieldBr, we\nfit the absorption spectra using the Gaussian function plus a polyno mial function which\nrepresents the background (see Fig. S2 in Supplemental Material) .41The resonant field\nwas found to be ∼3050 G at 300 K where the sample is in a paramagnetic state, and it\nalmost remains the same above 100 K (Fig. 2(a) and (b)). When the temperature is below\n100 K, it shifts towards lower values, which suggests the appearan ce of an effective local\nmagnetic field that is created by the ferromagnetically ordered mom ents. The resonant\nfield decreases until 30 K, after which the ZFC and FC curves behav e differently. The\nZFC resonant field has a minimum value at 30 K, and then increases with the decrease of\ntemperature. In contrast, the FC resonant field decreases mon otonically with decreasing\ntemperature towards 2 K. We note that recently Liu et al. also repo rted a small shift of\n5900\n600\n300\n0Local Field Bloc (G)\n300 200 100 0\nT (K)30 K FC\n ZFCInt. (arb. unit)\n6000400020000\nB (G)FC 300 K\n280\n260\n240\n220\n200\n180\n160\n140\n120\n100\n80\n60\n50\n40\n30\n20\n10\n2Int. (arb. unit)\n6000400020000\nB (G)ZFC 300 K\n280\n260\n240\n220\n200\n180\n160\n140\n120\n100\n80\n60\n50\n40\n30\n20\n10\n2(a) (b)\n(c)\nFIG. 2. ESR spectra of undoped Y 2Ir2O7at different temperatures after (a) ZFC and (b) FC\n(HFC= 1 kOe). The spectra are shifted vertically for clarity. The red dashed lines indicate the\nlocation of resonant fields. (c) Temperature dependence of e ffective local field Bloc.\nresonant field in their ESR measurements,43but their data were taken under ZFC with less\nclear temperature dependence.\nWe calculated the effective local field Bloc(T) =Br(PM) -Br(T), where Br(PM) = 3050\nG is the resonant field at 300 K (i.e. in the paramagnetic regime) and Br(T) represents the\nresonant field at a temperature T. Fig.2(c) shows the temperature dependence of local\nfield. Upon warming, the ZFC local field increases from ∼210 G at 2 K to ∼520 G at 30 K.\nIt then decreases with further increase of temperature andrea ches zero above 100 K. The FC\nlocal field is ∼870 G at 2 K, and it decreases monotonically with increasing temperat ure.\nThe different temperature dependence between ZFC and FC should be attributed to the\nfreezing of ferromagnetic domains at low temperatures. In brief, the FM domains are frozen\n6in nearly random directions after the sample was cooled down in zero fi eld, while they are\naligned along the field direction after FC. As a result, the effective loc al field is lower in the\nZFC measurement than in the FC. We also note that although the non -zero local field is\nan indication of ferromagnetism, weak or short-range ordered FM may not produce a large\nenoughlocalfieldthatcanbedetectedbyESR,44,45Therefore, wecannotdrawtheconclusion\nthat the ferromagnetic phase only exists below 100 K. In fact, as w e will discuss later the\nferromagnetic phase persists up to TM2∼190 K. In addition to the FM component, we also\nobserved a hyperfine resonance at a higher field ( ∼3350 G) (see Fig. S3 in Supplemental\nMaterial),41which suggests the existence of diluted and isolated paramagnetic m oments\nbesides the strong AFM and weak FM phases. This paramagnetic pha se is consistent with\nthe sharp increase of magnetic susceptibility upon cooling at low temp eratures (Fig. 1(a)).\nTheESRspectra takenontheCa-dopedsamplesdo notshow avisible resonance response\nnear 3050 G. We note that ESR mainly detects the local environment of the localized\nunpaired electrons. The absence of resonance could be due to the decrease and/or the\ndelocalization of unpaired electrons arising from Ca-doping: first, t he substitution of Y3+\nby Ca2+is expected to change Ir4+to Ir5+that does not have the unpaired electron; second,\nthis doping could lead to the delocalization of the unpaired electron of Ir4+, as evidenced by\nthe significant decrease of electrical resistivity (Fig. 5).\nWe further performed M-Hmeasurements after the samples were cooled down in a mag-\nnetic field to study the possible exchange coupling between the ferr omagnetism and the\nantiferromagnetic background. Fig. 3(a) shows three M-Hcurves that were taken after the\nundoped sample was cooled down to 10 K in 1 kOe, 0 and -1 kOe, respec tively. The mag-\nnetic field was swept between -2 T and 2 T, which is high enough to satu rate the hysteresis\nloop. While the ZFC M-Hcurve passes through the origin, the two FC curves show clear\nshift along the vertical axis. In particular, a positive cooling field lead s to a positive shift\nwhile a negative cooling field gives rise to a negative shift. We noticed th at a similar vertical\nshift was observed in the Sm 2Ir2O7compound,37but its underlying mechanism was unclear.\nBased on the above analysis regarding the coexistence of FM and AF M phases, we propose\nthat the strong exchange coupling at the interface between thes e two separated phases is\na possible origin of the shift. As demonstrated in some FM-AFM couple d systems46,47in-\ncluding the phase-separated manganites,48the magnetic moments in the shell of the FM\nphase (i.e. at the interface between FM and AFM domains) are stron gly pinned by the\n720\n0\n-20M (10-3emu/g)\n-20 0 20\nH (kOe) 80 K\n 100 K\n 120 K\n 140 K\n 160 K\n 180 K\n 200 K\n 220 Kx=0.0\nFC(a) (b)\n(c)40\n0\n-40M (10-3emu/g)\n-20 0 20\nH (kOe) HFC=1kOe\n ZFC\n HFC=-1kOex=0.0\n10 K\n100\n50\n0\n-50\n-100M (10-3emu/g)  80 K\n 100 K\n 120 K\n 140 K\n 160 K\n 180 K\n 190 K\n 230 Kx=0.01\nFC 10\n0M (10-3 emu/g)\n-2 0 2\nH (kOe)\n100\n50\n0\n-50\n-100M (10-3emu/g)\n-60-40-200204060\nH (kOe) 120 K\n 140 K\n 160 K\n 180 K\n 200 K\n 230 Kx=0.10\nFC(d)\n-12-8M (10-2emu/g)\n-60-40\nH (kOe) 140 K\nFIG. 3. (a) M-Hcurves of undoped Y 2Ir2O7taken at 10 K after ZFC and FC ( HFC= 1 kOe,\n-1 kOe). (b) M-Hcurves of undoped sample taken at variable temperatures nea rTM1andTM2.\nM-Hcurves of (c) x=0.01 and (d) x=0.10 samples taken after FC ( HFC= 1 kOe). The inset of\n(c) shows the M-Hcurves at low field; the inset of (d) shows a typical M-Hcurve at high field.\nexchange coupling from the AFM phase. The magnetic moments in the FM domains are\naligned with the cooling field, which gives rise to a net moment when the m agnetic field is\nremoved. When the magnetic field is swept from 2 T to -2 T, the negat ive field is able to\nalign the moment in the bulk FM, but not strong enough to rotate the moments in the shell\nof the FM that is pinned by the AFM phase. These uncompensated mo ments hence give\nrise to a vertical shift of the hysteresis loop. We note that in conve ntional exchange bias\nsystems47the magnetic hysteresis loop is shifted horizontally. The vertical sh ift observed\nhere (similar to the observation in manganites48) suggests that the FM and AFM phases\n8are strongly coupled, resulting in a strong pinning of interfacial mom ents which cannot be\nrotated/aligned by the sweeping field.\n15\n10\n5\n0Mh (10-3emu/g) x=0.0\n x=0.01\n x=0.10\n8\n6\n4\n2\n0Msh (10-3emu/g)\n200 160 120 80\nT (K) x=0.0\n x=0.01\n x=0.10(a)\n(b)\nTM1TM2\nFIG. 4. (a) The hysteresis Mhand (b) the shifted magnetization Mshas a function of temperature\nfor thex=0.0, 0.01 and 0.10 samples.\nWe carried out more FC M-Hmeasurements at various temperatures near TM1∼158 K\nandTM2∼190 K for the x=0, 0.01 and 0.10 samples. At low temperatures, the magnetic\nfield is swept between -7 T and 7 T for the doped samples to ensure th at their magnetic\nhysteresis loops are saturated (inset of Fig. 3(d)). As shown in Fig. 3(b), (c) and (d), all\nthe hysteresis loops show vertical shift below TM1. We plot the temperature dependence of\nthe hysteresis Mhand shift Mshin Fig.4(a) and (b), respectively. Here the hysteresis Mh\nis defined as one half of the difference between the two magnetizatio n values at zero field,\nand the shift Mshis defined as the average of these two values. It is clear that the Msh\nreaches zero above TM1∼158 K while the Mhpersists up to about TM2∼190 K. These\nresults suggest that the weak ferromagnetism exists up to TM2, while the coupling between\nFM and AFM occurs below TM1. This is consistent with the recent muon spin resonance\n9measurement which suggests that the all-in/all-out AFM ordering is f ormed below ∼150\nK in the undoped Y-227.21At each temperature, Mhincreases with doping concentration,\nconsistent with the enhancement of ferromagnetism. On the othe r hand, the Mshwhich\nis related to the pinned magnetic moments decreases with doping. Th is suggests that the\ninterfacial coupling between the FM and AFM becomes weaker. We no te that recent µSR\nmeasurement on the undoped sample does not seem to show signatu re of FM phase,21which\ncould be due to the small faction of FM component. Indeed a rough e stimate based on the\nmagnetic hysteresis loop suggests that the volume fraction of the FM phase is less than 0.1%\nfor the undoped sample, which may not give rise to a detectable signa l inµSR.\nWe further performed electrical transport studies to gain insight into the nature of the\nobserved weak-ferromagnetism. Fig. 5shows the temperature dependence of electrical\nresistivity for all four samples. The undoped sample shows an insulat ing behavior and the\nρ-Tcurve has a kink at the magnetic transition TM1∼158 K, consistent with the previous\nreports.43The Ca-doping significantly enhances the electrical conductivity (i.e . reduces the\nresistivity). In particular, the x=0 and 0.01 samples are insulators, while the x=0.05 and\n0.10 samples show a metal-insulator (MI) transition at about 150 K an d 100 K, respectively.\n0.0010.010.11101001000ρ (Ω m)\n300 200 100 0\nT (K)x=0    \nx=0.01\nx=0.05\nx=0.102345670.001\nρ (Ω m)\n68\n102468\n1002\nT (K) x=0.05\n x=0.10\nTMI=150K\nTMI=100K\nFIG. 5. Temperature dependence of electrical resistivity f or all (Y 1−xCax)2Ir2O7samples. Inset:\nmagnification of the data for the x=0.05 and 0.10 samples.\nThe doping of Ca2+has two effects. First, it increases the A-site ionic radius, which cou ld\nenhance the conductivity by reducing the trigonal compression on the IrO 6octahedra.6,49\nIndeed, earlier studies have shown that, upon the increase of the A-site radius, the trans-\nport properties of A-227 change successively from insulating beha vior,21,34,39,50,51to MI\n10transition,12,13,23,34,51and finally to metallic behavior.26,29,34,50,51Ca2+has an ionic radius\nof 1.12˚A52which is slightly larger than that of Y3+(1.019˚A).52With a doping of x=0.05,\nthe average A-site radius is increased to 1.024 ˚A, which is slightly smaller than that of Gd3+\n(1.053˚A).52We note that Gd-227 is still insulating over the entire temperature r ange (5 -\n300 K),34,50so the high temperature metallic behavior of our Ca-doped samples is unlikely\nto be due to the increase of A-site radius. The second effect of Ca- doping is the inducing of\ncharge carriers. As discussed earlier, the substitution of Y3+by Ca2+increases the valence\nstate of Ir from 4+ to 5+. In the stoichiometric A-227 compounds, the Ir4+has an unpaired\nJeff=1/2 electron that is localized due to the electron-electron interac tion.1,2,4In the doped\ncompounds, the Ir5+has an empty Jeff=1/2 level which allows the hopping of the Jeff=1/2\nelectron from the nearby Ir4+, leading to the delocalization of electrons and enhancement of\nelectrical conductivity.\nTo characterize the valence state of Ir, we performed XPS measu rements on the x=0\nand 0.10 samples. A set of Ir 4 fspectra are shown in Fig. 6and another set of spectra\ntaken at a different depth are shown in Fig. S4 in Supplemental Mater ial to demonstrate\nthe consistency of spectra through depth. The apparent featu re of multi-peaks indicates the\ncoexistence of different valence states of iridium for both samples. We fitted the spectra\nusing a standard software package called CasaXPS. It is worth not ing that the screening\neffect observed in the metallic IrO 255is negligible in our samples due to their much higher\nresistivities,55,56-therefore all the fitting components are considered to be symme tric with\na Voigt shape. As shown in Fig. 6, the experimental data can be fitted well using three\nsets of iridium components. Taking the 4 f7/2spectra as an example, the component with\nthe lowest binding energy of 60.9 eV (blue) is determined to be Ir,53consistent with the\nexistence of Ir impurity in the XRD pattern (see Fig. S1 in Supplement al Material).41The\nintermediate component (62.4 eV, red) is Ir4+54which is the nominal valence state of Ir\nin the undoped sample. The component with the highest binding energ y of 65 eV (green)\nshould be attributed to a valence state that is higher than 4+, i.e. 5+ or 6+. Since this\nhigh valence component is found to increase by ∼8% with ∼10% doping of Ca2+, it\nis likely to be Ir5+instead of Ir6+. Moreover, as discussed above, the coexistence of Ir5+\nand Ir4+can give rise to delocalization of Jeff=1/2 electron and enhancement of electrical\nconductivity, while the same effect is not applicable for a mixed valence states of 6+ and\n4+. The existence of Ir5+in the undoped sample should be attributed to non-stoichiometry,\n11e.g. deficiency of metal elements (Y or Ir) or excess oxygen.\nInt. (arb. unit)\n72 68 64 60 56\nBinding Energy (eV)Ir 4f XPS4f7/2 4f5/2\nx=0.0  \nx=0.10 Ir\n Ir4+\n Ir5+\n bg\n fit.\nFIG. 6. Typical Ir 4 fXPS spectra of Y 2Ir2O7and (Y 0.9Ca0.1)2Ir2O7. The black solid curve is the\nfitted envelope using three individual components, i.e. Ir, Ir4+and Ir5+which are plotted in blue,\nred and green colors, respectively. The black dashed curve d enotes the background.\nNow we briefly discuss the possible origin of the observed ferromagn etism based on the\nknown valence state of Ir. In the pyrochlore iridates, there are t hree major magnetic\ninteractions, namely the Heisenberg AFM interaction, single-ion anis otropy and the D-\nM interaction.4,35,36Theoretical calculations predicted that the interplay and competit ion\namong these three interactions stabilize the all-in/all-out antiferro magnetic phase.4We note\nthat these studies were based on the assumption that all the Ir ar e in the 4+ valence state.\nHowever, according to our XPS measurements, there are some Ir5+in both the undoped\nand doped samples. In contrast to Ir4+which has a magnetic moment of 1/3 µB,20the Ir5+\nhas an empty Jeff=1/2 level along with fully occupied Jeff=3/2 states, and hence has no\nnet moment. As a result, the replacement of Ir4+by Ir5+could modify the competition of\nthose three magnetic interactions between the Ir4+ions in the vicinity of Ir5+and may favor\na ferromagnetic state. Furthermore, the mixed-valence states of Ir may lead to a double-\nexchange interaction: the O2 pelectron hops to the empty Jeff=1/2 orbital of Ir5+, and then\n12theJeff=1/2 electron on the nearby Ir4+hops to the O2 porbital, giving rise to electrical\nconductivity as discussed earlier. This double-exchange interactio n between Ir5+and Ir4+\nthrough the oxygen 2 porbital is similar to the Mn3+-O-Mn4+interaction in the well-known\nmanganites,57and may give rise to ferromagnetism. We note that future theoret ical cal-\nculations considering Ir5+will help understand the precise nature of the ferromagnetism in\nthese compounds.\nIV. CONCLUSIONS\nInthis work, we performed systematic studies of themagnetic pro perties of a prototypical\npyrochlore iridate Y 2Ir2O7and its hole-doped compounds. We have demonstrated the exis-\ntence of weak ferromagnetism in the undoped compound through a combination of magnetic\ncharacterizations and electron spin resonance studies. Ca-dopin g leads to the enhancement\nof ferromagnetism and improvement of electrical conductivity. We have also observed a ver-\ntical shift in the M-Hcurves, which suggests a strong coupling between the ferromagn etic\nphase and the large antiferromagnetic background. The XPS char acterization shows the\nexistence of Ir4+and Ir5+in both the doped and undoped samples, and the amount of Ir5+\nincreases with Ca-doping, which highlights the possible origins of the w eak ferromagnetism\nassociated with the formation of Ir5+.\nACKNOWLEDGMENTS\nWe thank Professors Herb Fertig, Gerardo Ortiz, Kai Sun and Luis Brey for helpful\ndiscussions, and Dr. D. Williams, W. Tong, L. Pi, and L. Ling for experim ental assistance.\nS.X.Z. and B. S. would like to thank Indiana University (IU) College of Ar ts and Sciences for\nstart-up support. F. Y. 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Rev. 82, 403 (1951).\n16"    },    {        "title": "1210.2609v3.Micromagnetic_theory_of_spin_relaxation_and_ferromagnetic_resonance_in_multilayered_metallic_films.pdf",        "content": "arXiv:1210.2609v3  [cond-mat.mtrl-sci]  19 Oct 2012Micromagnetictheoryof spinrelaxation andferromagnetic resonance in\nmultilayered metallicfilms\nSergey Bastrukov,1,a)JunYongKhoo,1Boris Lukiyanchuk,1andIrinaMolodtsova2\n1)Data StorageInstitute(DSI), Agency for Science, Technolo gyand Research (A*STAR),\n5 EngineeringDrive1, 117608Singapore\n2)Laboratory of Informational Technologies, Joint Institut e for Nuclear Research,\n141980Dubna,Russia\n(Dated: 15June 2021)\nSpin relaxation in the ultrathin metallic films of stacked mi croelectronic devices is inves-\ntigatedonthebasisofamodifiedLandau-Lifshitzequationo fmicromagneticdynamicsin\nwhich the damping torque is treated as originating from the c oupling between precessing\nmagnetization-vector and the introduced stress-tensors o f intrinsic and extrinsic magnetic\nanisotropy. Particular attention is given to the time of exp onential relaxation and ferro-\nmagnetic resonance linewidth which are derived in analytic form from the equation of\nmagnetization energy loss and Gabor uncertainty relation b etween the full-width-at-half-\nmaximum in resonance-shaped line and lifetime of resonance excitation. The potential of\ndevelopedtheory isbriefly discussedinthecontextofrecen t measurements.\na)Electronicmail: Sergey B@dsi.a-star.edu.sg\n1I. INTRODUCTION\nAn understanding of relaxation processes in ultrathin films of ferromagnetic metals is crucial\ntothedesignandconstructionofmicroelectronicdevices1,2,likemagneticrandomaccessmemory\n(MRAM) and spatial light modulators (SLM). The main source o f information about relaxation\nprocesseshasbeenandstillistheferromagneticresonance (FMR)measurementsaimedatreveal-\ning the frequency dependence of the full-width-at-half-ma ximum in FMR spectral line3,4. Tradi-\ntionally,theresultsoftheseexperimentsaretreatedwith intheframeworkofthephenomenological\nLandau-Lifshitz-Gilbertmodel5–7,describingtheFMRresponseintermsofuniformprecession of\nmagnetization M(t)with the preserved in time magnitude |M(t)|=MswhereMsis the satu-\nration magnetization. The dynamical equation governing pr ecessional motions of M(t)can be\nconvenientlywrittenin thefollowinggeneral form\n˙M(t) =−γµ0T(t)−R(t), (1)\nwhereγis the electronic gyromagnetic ratio and µ0is the magnetic permeability of free space;\nSI units are used throughout this paper. The vector-functio n,T(t) = [M(t)×H], represents the\nmagnetictorquethat drivesthefree Larmor precession of M(t)about the axis of thedc magnetic\nfield,H=constant, intheprocess ofwhich theZeeman magnetizatione nergy,Wm(t) =−µ0H·\nM(t), is conserved: ˙Wm(t) = 0. Central to understandingthe relaxation process is the rel axation\nfunctionR(t)definingtherateofmagnetizationenergy loss\n˙Wm(t) =−µ0H·˙M(t) =µ0H·R(t). (2)\nInthisworkwefocusonLandau-Lifshitz(LL)formofthisfun ctionRin(t) =λin[M(t)×[M(t)×\nH]]which provides geometrically transparent insight into the magnetization-vectormotion in the\nprocess of aligning MwithH. The material-dependent parameter λincan be thought of as de-\nscribingthestrengtheffectoftheintrinsicanisotropyon therelaxationdynamicsofmagnetization\nprecession which is constrained by the conditions M(t)·˙M(t) = 0andM(t)·R(t) = 0. In this\npaper we consider an alternative micromagnetic treatment o fR(t)according to which the origin\nof the damping torque responsible for spin relaxation in mul tilayered metallic films is attributed\nto the coupling between the uniformly precessing magnetiza tion-vector and the stress-tensors of\nintrinsicand extrinsicmagneticanisotropy.\n2II. STRESS-TENSOR REPRESENTATION OFMICROMAGNETICDAMPIN G\nTORQUE\nThe equilibrium magnetic anisotropy exhibited in the easy a nd hard axes of magnetization di-\nrection8is a hallmark of ultrathin films of ferromagnetic metals. Vie wing this property from the\nperspective of the macroscopic electrodynamics of magneti c continuous media9,10, it seems quite\nnatural to invoke the stress-tensor description of magneti c anisotropy, namely, in terms of sym-\nmetrictensorsofmagnetic-field-dependentstresses. Inso doingweadoptthefollowingdefinition\nofstress-tensorofintrinsicanisotropy σin\nlk(generictobothmonolithicandmultilayeredferromag-\nneticfilms)andstress-tensorofextrinsicanisotropy σex\nik(arisingfromimpuritiesandimperfections\nofthefilmcrystallinelattice)\nσin\nlk=µ\n2[(MnHn)δkl−(MlHk+MkHl)], (3)\nσex\nik=µ\n2/bracketleftbig\nH2δkl−(HlHk+HkHl)/bracketrightbig\n, (4)\nwhereδikis the Kronecker symbol and µstands for the effective magnetic permeability which is\nderived from the magnetization curve according to the rule11:µ= ∆B/∆H. Hereafter Hrefers\nto the total (applied and internal effective) field. It is wor th noting that the above stress-tensor\ndescription of intrinsic and extrinsic magnetic anisotrop y is consistent with the definition of the\nenergy densityofmagneticfield storedin aferromagneticfil m12\nu=1\n2B·H,B=µ(H+M) (5)\ninthesensethattherelationbetweenthestress-tensorofc ombinedintrinsicandextrinsicmagnetic\nanisotropy, σlk=σin\nlk+σex\nlk, and theenergy density uisdescribed by\nu=Tr[σlk] =σll=µ\n2(MH+H2), (6)\nwhereTr[σlk]stands for the trace of tensor σlk. In what follows we focus on the effect of above\nmagnetic stresses on the precessing magnetization vector w hose mathematical treatment is sub-\nstantiallyrelied on thesymmetrictensor\nγik= [M2\nsδik−MiMk], γik=γki, (7)\nhaving,inappearance,somefeaturesincommonwiththatfor isotropicmagneto-strictionstresses13.\nIt can be verified by direct calculation that the stress-tens or representation of the intrinsic relax-\n3ationfunctionisidenticalto theLLrelaxation function\nRin\ni=2λin\nµM2sγikMlσin\nkl=λin[Mi(MkHk)−HiM2\ns], (8)\nRin=λin[M(t)×[M(t)×H]].\nIn choosing the above form of the tensor γikwe were guided by previous investigations14of the\ndamping terms in ferro-nematic liquid crystals dealing wit h the tensor constructions of a similar\nform. Fortheextrinsicrelaxationfunction,owingitsorig intothecouplingof Mlwithσex\nlk,weuse\nthefollowingstress-tensorrepresentation\nRex\ni=−λex\nµΩγikMlσex\nlk (9)\n=−λex(MnHn)\nΩ[Mi(MkHk)−HiM2\ns].\nThe minus sign means that extrinsic damping torque countera cts the damping torque originating\nfrom theintrinsicstresses. Thevectorform ofextrinsicre laxationfunction(9)reads\nRex(t) =−λex(M(t)·H)\nΩ[M(t)×[M(t)×H]]. (10)\nAs isshownbelow,theparameter-free frequencyofthetrans ientmagnetizationconfiguration\nΩ =ω\n1−(ωM/ω)1/2, ωM=γµ0Ms, ω= 2πf, (11)\nprovides correct physical dimension of the extrinsic dampi ng torque and proper account for the\nempiricaldependenceoftheFMRlinewidth ∆Hupontheresonance frequency f. Makinguseof\nargumentofphysicaldimensionitiseasytoshowthatthemat erial-dependentparameters λin>0\nandλex>0(measuring strength of intrinsic and extrinsic stresses on the relaxation process in\nmultilayeredfilm)canberepresentedintermsofdimensionl essdampingconstants αandβ(whose\nmagnitudesare deduced fromtheempiricalfrequency depend enceofFMRlinewidth)as follows\nλin=αγµ0\nMs, λex=β/parenleftbiggγµ0\nMs/parenrightbigg2\n. (12)\nThe net outcome of the above outlined procedure of computing the combined (intrinsic plus ex-\ntrinsic)dampingtorque\nR=Rin+Rex (13)\n=/bracketleftbigg\nλin−λex(M(t)·H)\nΩ/bracketrightbigg\n[M(t)×[M(t)×H]],\n4entering the basic equation of micromagnetic dynamics (1) i s the following Modified Landau-\nLifshitz(MLL)equation\n˙M=−γµ0[M×H] (14)\n−/bracketleftBigg\nαγµ0\nMs−β/parenleftbiggγµ0\nMs/parenrightbigg2(M·H)\nΩ/bracketrightBigg\n[M×[M×H]],\nwhich obeys all constraints of the canonical LL equation. On e sees that unlike the linear-in-\nmagnetic-field intrinsic damping torque, the extrinsic dam ping torque is described by quadratic-\nin-magnetic-field relaxation function. At this point it see ms noteworthy that the need in allowing\nfor the quadratic-in- Hdamping terms has been discussed long ago15. The above scheme can be\nregarded,therefore, as adevelopmentofthislineoftheore ticalinvestigations. In termsoftheunit\nvector of magnetization m(t) =M(t)/Msand Larmor frequency ω=γµ0Hthe last equation\ncan beconvertedto (see16forcomparison)\n˙m= [ω×m]−/bracketleftbigg\nα−β(m·ω)\nΩ/bracketrightbigg\n[m×[m×ω]]. (15)\nIt can be seen that the obtained MLL equations (14) and (15) ar e reduced to the standard LL\nequationwhen theeffect ofextrinsicstressesis ignored(i .e.β= 0).\nIII. VARIATIONMETHOD OFCOMPUTINGRELAXATION TIMEAND FMR\nLINEWIDTH\nTherelaxationtimeisamongsttheprimarytargetsofcurren tFMRexperiments. Inthissection,\nwe present variational method of analytic computation of th e FMR linewidth which is quite dif-\nferent from thewell-knownsolutionof thesusceptibilitys olutionofLL equation7. At the baseof\nthe variation method under consideration lies the equation of the magnetization energy loss from\nwhich the exponential relaxation time τas a function of FMR frequency f=ω/(2π)is derived.\nTheFMRlinewidth, ∆ω=γµ0∆H,iscomputedfromthewell-knownGaboruncertaintyrelatio n\n(e.g.17,Sec.11.2)\n∆ωτ= 1 (16)\nbetween thefull-width-at-half-maximumin the resonance- shaped spectral line ∆ωand lifetime τ\nofresonanceexcitation.\n5A. FMR linewidth causedby intrinsicdamping torque\nFortheformer we considerrelaxation process brought about by intrinsicdampingtorque. Our\napproachisbasedontheobservationthattheequationofmag netizationenergy lossintheprocess\nofauniformprecessionofmagnetizationinadcmagneticfiel d\ndWm\ndt=−µ0HMsd(cosθ(t))\ndt=µ0HiRin\ni(t), (17)\nRin\ni=2λin\nµM2sγikMlσin\nkl, (18)\nis reduced to theequation forthecosinefunction u(t) = cosθ(t)ofangleθ(t)betweenM(t)and\nH,namely\ndu(t)\ndt=−αω[u2(t)−1], ω=γµ0H. (19)\nThe right hand side of (19) suggests that there are two equili brium configurations, namely, with\nu(0) = 1corresponding to M↑↑Handu(π) =−1corresponding to M↑↓H. The stability\nof theseconfigurations can be assessed by the standard proce dure of introducingsmall-amplitude\ndeviations δu(t)fromtheequilibriumvalues u(0) =u0=±1. Onsubstituting u(t) =u0+δu(t),\ninto (19) with u0= 1and retaining first order terms in δu(t)we obtain equations describing\nexponentialrelaxation ofmagnetizationtothestateofsta blemagneticequilibrium:\ndδu(t)\ndt=−(2αω)δu→δu(t) =δu(0)e−t/τ, (20)\nτ−1= 2αω, ω=γµ0H. (21)\nThe second stationary state, with u0=−1, is unstable, since in this case the resultant linearized\nequation, δ˙u= (2αω)δu,havingthesolution, δu(t) =δu(0)et/τ, describes anon-physicalbehav-\nior ofδuas the time is increased. Inserting (21) in (16), we arrive at the basic prediction of the\nstandardmicromagneticmodel\n∆H(f) =Af, A =4πα\nγµ0. (22)\nThis last equation provides a basis for discussion of empiri cal linewidth-frequency dependence\n∆Hexp= ∆H0+∆Hexp(f)with∆Hexp(f) =Aexpf. Central to such a discussionis the identi-\nfication of theoretical and experimental linewidths, ∆H(f) = ∆Hexp(f), from which the magni-\ntude ofαis deduced and applied to (21) for obtaining numerical estim ates of the relaxation time\nτ.\n6B. FMR linewidthcaused by both intrinsicandextrinsic damp ing torques\nIn this case the starting point is the equation of magnetizat ion energy loss with the combined\nrelaxationfunction\ndWm\ndt=−µ0HMsd(cosθ(t))\ndt=µ0HiRi(t), (23)\nRi=2λin\nµM2\nsγikMlσin\nkl−λex\nµΩγikMlσex\nlk, (24)\nwhichafter somealgebraisconverted intoequationfor uhavingtheform\ndu(t)\ndt=−/bracketleftbigg\nαω−βω2\nΩu(t)/bracketrightbigg\n(u2(t)−1). (25)\nTherighthand sideofthisequationsuggeststhat therearet hreestationarystatecharacterized by\nu0(θ= 0) =±1, u0(θ=θM) =α\nβΩ\nω. (26)\nApplying to (25) the standard linearization procedure u(t) =u0+δu(t)in (25) one finds that\nresultant equation is equivalent to the equation of exponen tial relaxation, δ˙u=−τ−1δu, if and\nonlyiftheparameter\nτ−1=/bracketleftbigg\n2/parenleftbigg\nαω−βω2\nΩu0/parenrightbigg\nu0+βω2\nΩ(1−u2\n0)/bracketrightbigg\n(27)\nis a positive constant. It is easy to see that this is the case f oru0(θ= 0) = 1 andu0(θ=θM)\ngiven by rightmost of equations (26). This latter u0corresponds to a quasi-stationary transient\nconfiguration of precessing magnetization owing its existe nce to the coupling of magnetization\nwith extrinsic stresses of magnetic anisotropy. The state w ithu0=−1, is unstable. For the total\nrelaxationtime τ−1=τ−1(θ= 0)+τ−1(θ=θM)andtheFMRlinewidth(followingfromGabor\nuncertaintyrelation ∆H= [γµ0τ]−1)weobtain\nτ−1= 2αω−βω2\nΩ(1+cos2θM), (28)\n∆H=2ω\nγµ0/parenleftbigg\nα−βω\n2Ω/bracketleftbig\n1+cos2θM/bracketrightbig/parenrightbigg\n(29)\n=4πf\nγµ0/bracketleftBigg\nα−β\n2/parenleftBigg\n1−/parenleftbiggγµ0Ms\n2πf/parenrightbigg1/2/parenrightBigg\n(1+cos2θM)/bracketrightBigg\n.\nIt is worth emphasizing that the expounded micromagnetic me chanism of the magnetization\nprecession damping (due to magnetization-stress coupling ) presumes that the process of spin-\nrelaxationisnotaccompaniedbygenerationofspin-waves( magnons),becausethemagnetization\n70 20 40 60 80\nf [GHz]050100150200250300350∆H [Oe]Present Work\nα = 0.0097, β = 0.0103\nLandau-Lifshitz\nα = 0.007\n5 10 15 20 25 30\nf [GHz]00.51.01.52.02.5τ [ns]\nFIG. 1. TheFMR linewidth ∆Hand relaxation time τas functions of the FMR frequency f, computed on\nthe basis of the standard and modified in the present work Land au-Lifshitz equation.\nMis regarded as a spatially-uniform vector across the multil ayered film. At this point the con-\nsidered regimeof themagnon-free spin relaxation (in which the wavevector of spin wave k= 0)\nis quitedifferent from spin relaxation caused by two-magno n scattering18. The most conspicuous\nfeature of this (substantially macroscopic) mechanism, re sponsible for the non-linear frequency\ndependence of FMR linewidth,is thetransient magnetizatio nconfiguration owingits existenceto\nthe extrinsic stresses generic to the multilayered films. Su ch a configuration is absent in perfect\nmonolayered films (without impurities and defects of crysta lline lattice) of pure ferromagnetic\nmetals (Ni, Co, Fe) whose ferromagnetic properties are domi nated by intrinsic stresses of mag-\nneticanisotropy.\nIV. DISCUSSION ANDSUMMARY\nIn approaching the interpretation of FMR measurements in te rms of presented theory, in the\nremainder of this work, we focus on a case of in-plane configur ation (M↑↑H) which is of\nparticular interest in connection with the recent discover y of non-linear frequency dependence of\nFMRlinewidth4,19,20. In thiscase, thelastequationfortheFMRlinewidthtakes t heform\n∆H=4πf\nγµ0/bracketleftbig\nα−β(1−(γµ0Ms/2πf)1/2)/bracketrightbig\n. (30)\nTo illuminate the difference between predictions of the sta ndard and modified LL models, in\nFig.1 we plot τand∆Has functions of the FMR frequency fcomputed with the pointed out\n8FIG. 2. Theoretical fit, equation (30), of the empirical non- linear frequency dependence of FMR linewidth\ndetected inthe FMRmeasurements19on multilayered metallic nanostructures Pd/Fe/GaAs.\nFIG.3. SameasFig.2, but for the thin filmof Fe/Vmeasured in20.\nparameters of αandβ. In computation based on the standard micromagnetic model, equation\n(22),wehaveusedoneandthesamevalueofparameter αasin3reportingtheFMRmeasurements\nonultrathinfilmsofPermalloy. ThepresentedinFig.1value sofαandβhavebeen deducedfrom\nfitting,equation(30),ofthenon-linearfrequencydepende nceofFMRlinewidthdiscoveredinthe\nFMR measurements19. The result of this fit is shown in Fig.2. In Fig.3, we plot our fi t of the\nFMR linewidth measurements20on multilayered samples of Fe/V. A more detailed discussion of\nconsequences of considered micromagnetic mechanism of spi n relaxation will be the subject of\nforthcomingarticle.\n9REFERENCES\n1F. Xu,S. Liand C.K. Ong,J. Appl.Phys.109(2011)07D322.\n2J.W.Lau and J.M.Shaw,J.Phys.D: Appl.Phys.44 (2011)30300 1.\n3S.S. Kalarickal, P. Krivosik, M. Wu, C.E. Patton M.L. Schnei der, P. Kabos, T.J. Silva and J.P.\nNibarger, J.Appl.Phys.99 (2006)093909.\n4K.Baberschke, J.Phys. Conf. Ser. 324(2011)012011.\n5H. 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Patton,Phys.Rev.B 11(1975)2668.\n16G.V.Skrotskii,Sov.Phys.Usp.27 (1984)977.\n17M.J.Buckingham, Noisein electronicdevicesand systems (Wiley,New York, 1983).\n18R. Arias andD. L. Mills,Phys.Rev.B 60(1999)7395.\n19G.WoltersdorfandB. Heinrich,Phys.Rev. B 69(2004)184417.\n20K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, A. Janossy , Phys. Rev. B 73(2006)\n144424.\n10"    },    {        "title": "1410.4789v2.Ferromagnetic_resonance_in__ε__Co_magnetic_composites.pdf",        "content": "Ferromagnetic resonance in \u000f-Co magnetic composites\nKhattiya Chalapat,1Jaakko V. I. Timonen,2Maija Huuppola,3Lari Koponen,2Christo\u000ber Johans,3\nRobin H. A. Ras,2Olli Ikkala,2Markku A. Oksanen,4Eira Sepp al a,4,\u0003and G. S. Paraoanu1\n1O. V. Lounasmaa Laboratory, Aalto University, P.O. Box 15100, FI-00076, Finland\n2Department of Applied Physics, Aalto University, P.O. Box 14100, FI-00076, Finland\n3Department of Chemistry, Aalto University, P.O. Box 16100, FI-00076, Finland\n4Nokia Research Center, P.O. Box 407, 00045 NOKIA GROUP, Finland\nWe investigate the electromagnetic properties of assemblies of nanoscale \u000f-cobalt crystals with\nsize range between 5 nm to 35 nm, embedded in a polystyrene (PS) matrix, at microwave (1-12\nGHz) frequencies. We investigate the samples by transmission electron microscopy (TEM) imaging,\ndemonstrating that the particles aggregate and form chains and clusters. By using a broadband\ncoaxial-line method, we extract the magnetic permeability in the frequency range from 1 to 12\nGHz, and we study the shift of the ferromagnetic resonance with respect to an externally applied\nmagnetic \feld. We \fnd that the zero-magnetic \feld ferromagnetic resonant peak shifts towards\nhigher frequencies at \fnite magnetic \felds, and the magnitude of complex permeability is reduced.\nAt \felds larger than 2.5 kOe the resonant frequency changes linearly with the applied magnetic \feld,\ndemonstrating the transition to a state in which the nanoparticles become dynamically decoupled.\nIn this regime, the particles inside clusters can be treated as non-interacting, and the peak position\ncan be predicted from Kittel's ferromagnetic resonance theory for non-interacting uniaxial spherical\nparticles combined with the Landau-Lifshitz-Gilbert (LLG) equation. In contrast, at low magnetic\n\felds this magnetic order breaks down and the resonant frequency in zero magnetic \feld reaches a\nsaturation value re\recting the interparticle interactions as resulting from aggregation. Our results\nshow that the electromagnetic properties of these composite materials can be tuned by external\nmagnetic \felds and by changes in the aggregation structure.\nI. INTRODUCTION\nThe realization of novel materials with designed and tunable optical, mechanical, thermal and electromagnetic\nproperties is a major goal in nanotechnology. A promising direction is the study of nanocomposites, consisting of\nnano-scale particles (silica, Fe, Co, Ni, Al, Zn, Ti, as well as their oxides) or other low-dimensional structures (carbon\nnanotubes, graphene, DNA) embedded in a bulk matrix or polymer1. Nanocomposites with magnetic properties,\nwhich use magnetic nanoparticles inserted in dielectric matrices, are of considerable technological importance, due\nto the simplicity of fabrication and potential applications for radio-frequency and microwave antennas, for GHz-\nfrequency data transfer components, and for electromagnetic shielding2,3. On the scienti\fc side, these materials are\nintriguing because concepts such as ferromagnetic resonance (FMR), predicted almost a century ago4{6and later\ndemonstrated experimentally in bulk materials7,8, can now be applied to particles with nano-scale dimensions and are\nrelevant for understanding the properties of the resulting composite materials. The standard measurement technique\nin FMR is narrow-band: the material is placed in a microwave cavity whose response is monitored while sweeping the\nexternal magnetizing \feld until the lowest quality factor (maximum of absorption) is observed. However, for modern\napplications a broadband characterization of the electromagnetic properties of the material is required. Recently,\nnon-resonant transmission line methods have been employed to study magnetic nanoparticles, but, with the notable\nexception of magnetic \ruids in the 1990's9,10, only in zero external magnetic \feld11{16. There is to date no systematic\ncharacterization of the nano-scale and magnetic properties (including the e\u000bects of sizes, shapes, magnetic domains,\naggregation, etc.) of composite materials containing magnetic nanoparticles.\nIn this paper we present a comprehensive study of the broadband microwave properties and FMR resonance of\ncomposite materials comprising chemically synthesized \u000f-cobalt nanoparticles with sizes between 5 nm and 35 nm\nembedded in polystyrene, both in zero and in a \fnite magnetic \feld, including information from superconducting\nquantum interference device (SQUID) measurements. The particles are synthesized by two di\u000berent methods, the\nhot-injection method17and the heating-up method18. At macroscopic scales, bulk cobalt has only two forms of lattice\nstructures, namely hexagonal-closed-packed (hcp) and face-centered-cubic (fcc). For nanoparticles, a cubic \u000f-cobalt\nphase, with a structure similar to \f-manganese, has been observed in addition to the hcp and fcc phases19,20, and\nmore recently developed chemical syntheses have allowed the production of \u000f-cobalt particles at speci\fc diameters\n(narrow size distribution)21,22. The resonance condition of the cobalt nanocrystals is found by combining Kittel's FMR\ntheory5,6with the Landau-Lifshitz-Gilbert (LLG) equation4,23and various e\u000bective-medium models. Kittel's theory\nis a valid approach under the assumption that the eddy current skin depths are larger than the typical dimension\nof the particles. In Co, the skin depths in the frequency range between 1 GHz and 10 GHz are of the order of 100\nnm, therefore the nanoparticles used in our samples satisfy this requirement. We identify two distinct magnetizationarXiv:1410.4789v2  [cond-mat.mtrl-sci]  6 Nov 20142\nregimes: at high external magnetic \felds, the FMR resonance changes linearly with the \feld and the composite is well\ndescribed by the Kittel-LLG model, while at low magnetic \feld the FMR resonance saturates to a constant value.\nThis is explained by the existence of particle-particle magnetic interactions, which become the dominant e\u000bect at\nsmall and zero external magnetic \feld. Transmission electron microscopy (TEM) imaging con\frms the formation of\nparticle aggregates in the composite.\nII. METHODS\nA. Synthesis and TEM imaging of \u000f-cobalt nanoparticles\nSpherical\u000f-cobalt nanoparticles were synthesized by thermally decomposing 1080 mg of dicobalt octacarbonyl\n(STREM, 95%) in presence of 200 mg of trioctylphosphine oxide (Sigma-Aldrich, 99%) and 360 mg of oleic acid\n(Sigma-Aldrich, 99%) in 30 ml of 1,2-dichlorobenzene (Sigma-Aldrich, 99%) under inert nitrogen atmosphere18,21.\nThe cobalt precursor was either injected into the surfactant mixture at 180oC17or mixed with the surfactant solution\nat room temperature and heated up to 180oC18. No di\u000berence in particles produced by the hot-injection or heating-up\nmethods was observed. The particle size was adjusted by the temperature kinetics during the reaction18,i.e.by using\na higher heating rate in the heating-up method or a higher recovery rate in the hot-injection method to increase the\nnucleation rate and to produce smaller particles. The excess surfactants were removed after synthesis by precipitating\nthe particles by adding methanol, followed by redispersing the particles into toluene. The composites were fabricated\nby dissolving desired amounts of polystyrene (Sigma-Aldrich) in the nanoparticle dispersion, followed by evaporation\nof the solvent to yield black solid composite materials. The samples can be regarded as a two-component composite,\nmade of magnetically-active Co particles (density \u001aCo= 8:63 g/cm3) in a magnetically-inert polystyrene bulk (density\n\u001aPS= 1:05 g/cm3). The amount of oleic acid left is small and its density is anyway close to that of polystyrene\n(\u001aoleic acid = 0:89 g/cm3). Both the pure nanoparticles and thin cross-sections of the composite materials were\nanalyzed by transmission electron microscopy (Tecnai T12 and JEOL JEM-3200FSC). The images showed that our\n\u000f-cobalt nanoparticles typically consists of a single crystal core. For the high-frequency measurements, the composite\nmaterials were compression molded at 150oC into circular pellets with a diameter of 7.0 mm and thickness of 4.0\nmm. A circular hole of 3.0 mm was drilled through each pellet to accommodate the inner conductor of the coaxial\nline.\nB. Theoretical model\nThe phenomenon of ferromagnetic resonance (FMR) was predicted by Landau and Lifshitz in 19354and was\nobserved experimentally more than a decade after by Gri\u000eths7, and then by W. A. Yager and R. M. Bozorth8.\nGri\u000eth also found that the ferromagnetic resonance does not occur exactly at the electron spin resonance of frequency\n~!0=ge\u0016BHor!0=\rH, wheregeis the electron g-factor, \u0016B=e~=2meis the Bohr magneton, His the internal\nmagnetizing \feld, and \re=gee=2meis the electron gyromagnetic ratio, \re=2\u0019= 2.7992\u00021010Hz/T. Immediately\nafter, Kittel5,6proposed that the ferromagnetic resonance condition should be modi\fed from the original Landau-\nLifshitz theory by taking into account the shape and crystalline anisotropy through the demagnetizing \felds.\nIn the case of disordered magnetic composites, the physics is expected to be very complicated, due the interplay\nof geometry, anisotropy, interparticle interactions, and sample inhomogeneities. In principle, one can attempt to give\na microscopic description of these e\u000bects and average over several realizations; however this approach is likely to be\ncomplicated. Here we propose a simple phenomenological description of the composites. Similar to e\u000bective-\feld\ntheories in physics, the idea is to construct a model that incorporates the basic known physical processes - in this\ncase, magnetization precession in an applied magnetic \feld and dissipation - and solve the problem using very general\nmethods - linear response theory in our case. This allows us on one hand to extract the electromagnetic parameters\nthat are relevant for technological applications (high-frequency magnetic permittivity and permeability), and on the\nother hand to give an e\u000bective quantitative description of the ferromagnetic resonance observed.\nThe starting point of the model is the observation that if the external magnetic \feld Happlied is large, then the\nparticles will get strongly magnetized in the direction of the applied \feld. As a result, irrespective to the con\fguration\nof the sample or on the geometry of particle anisotropy, the sample behaves as a collection of noninteracting domains\nwith magnetization rotating in synchronization around a total e\u000bective magnetic \feld He\u000b=H+~H. In this regime,\none then expects a linear law for the resonance frequency,\n!0=\rHe\u000b=\r(H+~H); (1)3\nwhere ~Hincorporates the e\u000bects mentioned above. As we will see later, the main contribution in ~Hcomes from an\ne\u000bective magnetic anisotropy \feld HA.\nThe rotation of the magnetic domains is also accompanied by energy loss, which can be accounted for through\nthe Landau, Lifshitz4and Gilbert23formalism. The dynamics of magnetization is described by the Landau-Lifshitz-\nGilbert (LLG) equation\nd~Mtot\ndt=\u0000\r~Mtot\u0002~Htot+\u000b\nMs~Mtot\u0002d~Mtot\ndt: (2)\nThe \frst term on the right hand side represents the precession of magnetization. The energy loss is taken into account\nby the second term via a single damping constant \u000b.~Mtotis the magnetization of the system, and ~Htotis the total\nmagnetizing \feld which includes a small microwave probe \feld ( hei!t),\n~Htot=~He\u000b+~hei!t: (3)\nDemagnetizing \felds can be introduced as well in Eq. (3) and it can be checked that for spherical symmetry they\ncancel out from the \fnal results. Similarly, the magnetization at a given time includes the magnetization of the\nparticle\u0019~Msand the time-varying term,\n~Mtot=~Ms+~ mei!t; (4)\nwhere~Msis the saturation magnetization.\nTo solve the equation, it is convenient to take the coordinate zalong the direction of the \feld ~He\u000b, which in the limit\nof large \felds discussed above coincides with the direction of the magnetization Ms. The magnetic susceptibility is\ngiven by\u001f?=@mx=@hx=@my=@hy, with the magnitude hof the probe microwave \feld assumed much smaller than\nthe static \felds, h\u001cHe\u000b. This yields a simple analytical expression for the complex susceptibility \u001f?=\u001f0\n?\u0000i\u001f00\n?,\n\u001f0\n?=\rMs!0[!2\n0\u0000!2(1\u0000\u000b2)]\n[!2\n0\u0000!2(1 +\u000b2)]2+ 4!2!2\n0\u000b2; (5)\n\u001f00\n?=\u000b\rM s![!2\n0+!2(1 +\u000b2)]\n[!2\n0\u0000!2(1 +\u000b2)]2+ 4!2!2\n0\u000b2; (6)\nwhere!0=\r(H+~H).\nThe measured resonance frequency in the presence of dissipation can be obtained by searching for the maximum\nof Eq. (6). We have solved the equation @\u001f00\n?=@! = 0 analytically, using Mathematica. The polynomial resulting\nfrom the derivative has 6 roots, of which we select the physically relevant one (real and positive), and check that it\ncorresponds to a maximum. This solution is\n!r= 2\u0019fr=!0p\n1 +\u000b2=\rp\n1 +\u000b2(H+~H): (7)\nThis solution is also intuitively appealing, as it corresponds to a zero in the \frst term in the denominator of Eq. (6),\na term which tends to increase fast if the frequency is detuned only slightly from the resonance.\nIn general, the total susceptibility is \u001f(!) =\u001f(!)0\u0000i\u001f(!)00and the relative permeability \u0016Co(!) = 1 +\u001f(!) is\ncomposed of both parallel \u001fk, and perpendicular \u001f?components. The total susceptibility is an average of these\ncomponents, with\n\u001f(!) =1\n3[\u001fk(!) + 2\u001f?(!)]: (8)\nThe perpendicular component can be well described by the LLG-Kittel theory, i.e. Eq. (5)-(6). The parallel\ncomponent is purely relaxational and usually assumed to be described by the Debye model24,\n\u001fk(!) =\u001fk(0)\n1 +i!\u001ck: (9)\nHere\u001ck=\u001c0\u001b, where\u001c0= (\r\u000bkHA)\u00001,\u001b=KV=k BT,Vis the average particle volume, and the e\u000bective anisotropy\n\feldHAand the e\u000bective anisotropy constant Kcan be estimated from SQUID measurements (see subsection II C).4\nFIG. 1: The magnetic response of \u000f-cobalt nanoparticle composite measured by a SQUID magnetometer. a) Magnetization\nMversus applied magnetic \feld showing a saturation magnetization of about 77 emu/g. The inset picture is the result of\nanother section from the same sample. b) The ZFC-FC curve of the same sample ( mdenotes the magnetic moment) showing\nthe blocking temperature above room temperature.\nFor materials with inclusions, mixing rules such as Bruggeman equation25and the Maxwell-Garnett model26,27\ngive good results when the volume fraction of the inclusions is not too large. In the case of spherical inclusions, the\nMaxwell-Garnett formalism gives the following expression for permeability\n\u0016r=2(f\u00001)\u00162\nM\u0000(1 + 2f)\u0016Co\u0016M\n(f\u00001)\u0016Co\u0000(2 +f)\u0016M; (10)\nwhile the Bruggeman equation reads\n\u0016r=1\n4[(3f\u00001)\u0016Co+ (2\u00003f)\u0016M (11)\n\u0006p\n8\u0016Co\u0016M+ ((3f\u00001)\u0016Co+ (2\u00003f)\u0016M)2];\nwherefis the particle volume fraction, \u0016Cois the permeability of the cobalt particles, and \u0016Mis the permeability of\nthe insulating material.\nWe now show that the formula for the resonance frequency Eq. (7) remains valid also for composites, no matter\nwhat the mixing rule is, under the condition that the material used for the matrix is magnetically inert. This condition\nis certainly satis\fed for our samples. To prove this, let us consider an arbitrary function of the two components \u0016M\nand\u0016Co,\u0016r=\u0016r[\u0016M;\u0016Co]. Then\n@\u0016r\n@!=@\u0016r\n@\u0016Co@\u0016Co\n@!; (12)\nwhere we have used the assumption above about the matrix material, namely that its spectrum is \rat, @\u0016M=@!= 0.\nThus, the zeroes of @\u0016r=@!will coincide with the zeroes of @\u0016Co=@!, and Eq. (7) can also be used for the composite.\nThis is intuitively very clear: since the Co is the only material in the composite that has some magnetic properties\n(the ferromagnetic resonance in this case), one expects that these properties and only these will be responsible for\nany structure in the spectra of the composite as well.\nC. SQUID magnetometry\nThe e\u000bective anisotropy \feld Kand the saturation magnetization Mscan be determined from magnetic mea-\nsurements done with a superconducting quantum interference device (SQUID) Quantum Design MPMS XL7 mag-\nnetometer. These measurements provide as well an estimate for the e\u000bective anisotropy \feld HA, for which we use\nthe standard result of the Stoner-Wohlfart model HA= 2K=\u0016 0Ms(see e.g. Ref. [3] for a simple derivation). The5\nmeasurements were performed on small sections from the actual sample, see Fig. 1a). Each section has a volume\nof approximately 1 mm3. For convenience, we will express the saturation magnetization in the standard units of\nmagnetization per unit mass (emu/g), which is obtained from the measurements of the weights of all the samples and\ntheir volume fractions. When used in the theoretical model, the magnetization is converted into units of magnetic\nmoment per volume (A/m) by using the density of cobalt, \u001aCo= 8:6 g/cm3.\nTo determine the e\u000bective anisotropy constant K, we measure the ferromagnetic-superparamagnetic blocking tem-\nperatureTBfrom the zero-\feld cooled (ZFC) and \feld-cooled (FC) magnetization curve. Fig. 1b shows a typical\nZFC/FC curve of a \u000f-cobalt nanoparticle composite. The measurement was realized by \frstly cooling the small piece\nof sample to 2 K without an external magnetic \feld for a ZFC measurement. At the end, the sample was freezed\nwith no net magnetization, due to the random magnetization at room temperature. Next, a small \feld (100 Oe) was\napplied, and the magnetization ( M) of the sample was measured at di\u000berent temperature from 2 to 400 K. As the\ntemperature increases, more particles go from the ferromagnetic (blocked) to the paramagnetic phase, and align with\nthe applied \feld. The magnetization reached a maximum when the blocking temperature was reached.\nD. Microwave measurements\nStandard experiments on FMR such as those analyzed by Kittel were done by monitoring the on-resonance response\nof a microwave cavity while sweeping the external magnetizing \feld. In contrast, our setup is broadband, allowing\nto monitor the response at all frequencies. The complex magnetic permeability of the cobalt samples were measured\nover a frequency range between 1 and 12 GHz by a transmission and re\rection method. Prior to the measurements,\na sample was inserted inside the coaxial line (7-mm precision coaxial air line) which was placed between the poles\nof an electromagnet (the axis of the line being perpendicular to the magnetic \feld). The line was connected to the\nports of a vector network analyzer (VNA) by Anritsu 34ASF50-2 female adapters. The transmission/re\rection signals\nwere measured and used as the input parameters of the reference-plane invariant algorithm to obtain the complex\npermittivity and permeability28. Similar results are obtained using the short-cut method29.\nIII. RESULTS AND DISCUSSION\na. Measurements in externally applied magnetic \felds. We start by presenting the e\u000bects of an external magnetiz-\ning \feld on the FMR spectra of composites made with spherical \u000f-cobalt nanoparticles synthesized by the hot-injection\nmethod21. We call SET-1 this \frst set of samples (and similar notations will be used for the rest of the sample sets, see\nTable I). We measured the relative complex magnetic permeability ( \u0016r=\u00160\nr\u0000i\u001600\nr) of these samples over a wide range\nof frequencies for various non-zero external static \felds H. The material exhibits a broad resonance peak around 4\nGHz in a zero external \feld, see Fig. 2a. The application of an external magnetic \feld causes the resonance to shift\ntowards higher frequencies, accompanied by a reduction of the magnetic loss peak. Interestingly, in the small \feld\nregime (below 1.5 kOe), the magnetic absorption remains almost independent of the \feld at frequencies above 8 GHz.\nFig. 2b presents a comparison between the theoretical models and experimental results. The theoretical spectra\n(\u001600\nr) were calculated by combining the phenomenological models: LLG-Kittel4,6and Debye/LLG-Kittel24with the\nBruggeman e\u000bective medium model25. The simulation was done with a particle volume ratio of 0.1, and an average\nparticle diameter dof 16 nm. The magnetizing \feld corresponds to the experimentally-measured magnetic \feld at\nthe sample (using a calibrated gaussmeter) and ranges from 2.5 to 5.0 kOe. The dotted lines show the prediction of\nthe LLG-Kittel model, Eq. (1), Eq. (6) and Eq. (12), with parameters Ms= 64 emu/g, TB= 400 K, and \u000b= 0:37.\nThe solid and dashed lines show the prediction of the Debye/LLG-Kittel model, Eq. (1), Eq. (6), Eq. (9) and Eq.\n(12). The Debye model (parallel susceptibility) was calculated by setting \u000bk= 5\u000210\u00004and\u001fk(0) = 104. The solid\nlines present the theoretical prediction with Ms= 74 emu/g, and \u000b= 0:37. An even better \ftting with the data can\nbe obtained (dashed lines) by allowing for a small magnetic-\feld dependence of the damping \u000b: 0.44 (H= 2:5 kOe),\n0.40, 0.36, 0.32, 0.28 and 0.24 ( H= 5 kOe) with Ms= 84 emu/g.\nThe analysis presented in Fig. 2b suggests that both LLG-Kittel and Debye/LLG-Kittel models can approximately\npredict the FMR spectra in the regime where the magnetizing \feld is large. The LLG-Kittel (dotted lines) predicts\nthat the imaginary part of magnetic permeability \u001600should go to zero at very low frequencies. In the experiment\nhowever we see that the decrease in the permeability \u001600does not continue inde\fnitely at low frequencies, but instead\nstarts to increase again below 2 GHz. This large absorption at the low-frequency range is governed by non-resonant\nrelaxation processes, which can be estimated by the Debye/LLG-Kittel model (solid lines).\nNext, we present the permeability data in the magnetizing-\feld domain. Four sets of data that show nearly-\nLorenzian curves are plotted in Fig. 3 to show the FMR peaks. The absorption spectra shown in Fig. 3 provide a6\nFIG. 2: a) Imaginary magnetic permeability of a composite of polystyrene (PS) and \u000f-cobalt particles, at a volume fraction\nof 10%. b) Theoretical predictions: LLG-Kittel (dotted lines) and Debye/LLG-Kittel (solid and dashed lines), shown in\ncomparison with the experimental data (discrete symbols) at high magnetic \felds ( \u00152:5 kOe). The average particle diameter\nwasd= 16 nm. Inset: theoretical prediction of LLG-Kittel for a wider range of \feld values.\nFIG. 3: The imaginary part of magnetic permeability as a function of the applied magnetic \feld H. Each color represents the\npeak at one microwave frequency in the external \feld domain. The measurement shows that the resonance shifts towards a\nhigher magnetic \feld when the microwave frequency is increased. The inset picture shows the linear dependence between the\nresonance frequency frof the composite and the external magnetic \feld H.\ndirect proof of the linear relation between the resonant microwave frequency and the magnetizing \feld (inset graph\nof Fig. 3). According to Eq. (7) the slope of the fr\u0000Hpredicted by the LLG-Kittel model is \r=2\u0019p\n(1 +\u000b2) = 2:62\nGHz/kOe for \u000b= 0:37 and\r=\re. From the data, we can estimate an average slope of 2 :4 GHz/kOe, remarkably\nclose to the ideal value.\nAt low magnetic \felds, discrepancies with respect to the LLG-Kittel theory start to appear. Fig. 4 includes the low-\n\feld dependence of the ferromagnetic resonance, showing that the linear relation Eq. (7) valid at high \felds in region\n(III) does not hold anymore in region (I). Extrapolating the linear dependence !r=\r(H+~H) of region (III) to low \feld\nvalues and estimating ~H\u0019HAstill yields a good estimate for the e\u000bective magnetic anisotropy constant K(energy\nper unit volume) of the order of 104J/m3, withMs= 84 emu/g as used in the Debye/LLG-Kittel model (dashed lines)\nin Fig. 2b. However, in the low \feld (I) regime, the resonance occurs at a slightly higher frequency. To understand the\norigin of this frequency shift, we imaged the \u000f-cobalt nanoparticles with a transmission electron microscope (TEM),\nand found that they tend to agglomerate in clusters, see Fig. 6. When the particles aggregate, the distance between7\nFIG. 4: Diagram showing the relation between the resonance frequencies ( !0) and the applied magnetizing \feld ( H). The\nresonant frequencies are obtained for each applied external \feld from the \ftting with the LLG/Kittel-Bruggeman equations,\nand plotted as square markers. In the low \feld regime (I), the magnetization of the particles is random and aligns along local\nanisotropy \felds (denoted by Ha). When chains or clusters are present in the composite, the local \felds cause an increase of\nthe FMR resonance frequency (continuous line). In the intermediate regime (II) and in especially in the high \feld regime (III),\nthe magnetization of the particles is directed along the external magnetizing \feld, and Kittel's FMR theory predicts correctly\nthe resonance absorption spectra (dotted line).\n0 2 4 6 8 10 1200.511.522.533.5\nf (GHz)μ’’no interaction\ndipolar coupling\nFIG. 5: Simulated imaginary part of the permeability for two monodisperse \u000f-Co nanoparticles (10 nm) in random 3D aggregates\nwith surface-to-surface separation of magnetic cores of 5 nm and damping \u000b= 0:05.\nthem might be small enough to create a sizable magnetic interaction. In each cluster, the dipole interactions between\nparticles will generate additional local \felds Ha, which are not accounted for in the noninteracting model, see e.g.Eq.\n(1). These interaction e\u000bects become dominant in the regime of small magnetic \felds (I). This magnetostatic dipolar\ninteraction between particles gives an additional component to the total anisotropy, now proportional to Ms=d3where\ndis the interparticle spacing. The e\u000bect of this interaction can be obtained by solving numerically the LLG equation\nusing a micromagnetic simulation package30,31for two particles. When the dipolar coupling is not enabled the free\nparticle resonance is obtained. When dipolar coupling is turned on, the resonance shows signi\fcant broadening and8\nFIG. 6: TEM image of Co nanoparticles within SET-1 before mixing with polystyrene. The average diameter is in the order\nof 16 nm (with most of the particles between 10 and 20 nm).\nshifting toward higher frequencies due to the formation of collective resonance modes (see Fig. 5). Although this\nis a simpli\fed model with only two particles, the result supports the experimental observation that the resonance\nfrequency is shifted to higher values (see Fig. 4). This is also in agreement with the result of Ref. [32]. The appearance\nof collective behavior due to dipolar magnetic percolation and the formation of correlated agglomerates has been also\nobserved experimentally by magnetic force microscopy in two-dimensional Co layers33.\nb. Measurements at zero magnetic \feld. For technological application it is not always possible to apply large\nmagnetic \felds, thus one should provide a systematic classi\fcation of the composites. Since we cannot control the\naggregation patterns of the particles, we have chosen instead to modify the structure, distribution, and morphology\nof the constituent nanoparticles. The measurements show that this does not produce qualitative changes in the\nspectra, and the results are compared with the predictions of the LLG-Kittel complemented with the Bruggeman and\nMaxwell-Garnett mixing rules. These measurements are also very useful for technological applications, where precise\nknowledge of the electromagnetic parameters of these composite materials is needed. The particle-level properties of\nthese composites are summarized in Table I.\nsample set volume fraction, f [%] diameter,w\u0006\u001b[nm] dispersion, 100 \u0001\u001b=w [%]synthesis method\nSET-1 10 % 16\u00065 nm 31% (polydisperse) hot injection\nSET-2 4% and 11 % 13.9\u00060.6 nm 4% (monodisperse) heating-up\nSET-3A 4.3% 8.6\u00061.4 nm 16% (polydisperse) heating-up\nSET-3B 1.9% 27.7\u00062.7 nm 10% (intermediate) heating-up\nSET-3C 1.6% 32.1\u00067.9 nm 25% (polydisperse) heating-up\nSET-4A N/A 5\u00060.6 nm 12% (intermediate) hot injection\nSET-4B 3.3%, 6.5%, and 6.9% 8\u00060.8 nm 10% (intermediate) hot injection\nSET-4C 2.9% and 6.7% 11\u00060.9 nm 8% (intermediate) hot injection\nSET-4D 7.7% and 8.5% 27\u00063.5 nm 13% (intermediate) hot injection\nTABLE I: Systematic of the nanoparticle properties of various samples fabricated and measured. Note: very small particles\nwere observed to be present in all solutions; they were not included in the calculation of the average diameter, and they are\ntherefore not subsequently included in the estimations of the e\u000bective anisotropy, as these particles are magnetically inert.\nThe monodispersed \u000f-cobalt particles of SET-2 are fabricated by the heating-up chemical synthesis method18. The\nTEM images of these particles show that the particles have an average diameter of 14 nm, see Fig. 7. We measured the\npermeability spectra of two samples with di\u000berent volume fractions. The results are shown in Fig. 8a. We notice that\nincreasing the volume fraction not only increases the magnitude of \u001600\nr, but also shifts the resonant peak towards lower\nfrequencies. The calculations based on e\u000bective medium models show that the Bruggeman equation also predicts that\nthe resonance absorption ( \u001600peak) shifts towards lower frequencies when the volume fraction is increased (see Fig.\n8b). In Fig. 8, the theoretical graph was determined with an e\u000bective blocking temperature of 400 K, reproducing\ncorrectly the resonant peak around 3 GHz. An estimation of the e\u000bective magnetic anisotropy Kfrom the exponent9\nFIG. 7: TEM images of cobalt nanoparticles in SET-2. The main \fgure shows a single crystalline core inside a particle with\nan average diameter of about 14 nm. Note the high quality of the particle: a single crystal, with no grain boundaries, no\ndislocations or other defects. a) The \frst inset shows the particles after being left dry on a TEM grid. The size distribution of\nthe particles is very narrow (almost equal size). b) The second inset shows the arrangement of particles inside a cluster in the\ncomposite; superlattices (closed-pack array of particles) are seen in many parts of the cluster.\nof the N\u0013 eel-Arrhenius law TB\u0019KV=k B= 400 K yields K\u00193.84\u0002103J/m3= 3.84\u0002104erg/cm3. This e\u000bective\nmagnetic anisotropy is lower than the anisotropy constant of \u000f-cobalt extracted from magnetic measurements34. This\nis an expected limitation of our simpli\fed model, in which a single e\u000bective superparamagnetic blocking temperature\nis used as a parameter to \ft the permeability spectra. For our samples, di\u000berent parts of the composite may have\ndi\u000berent blocking temperature, depending on how clusters are formed. For example, it is known that in ferromagnetic\n\flms the blocking temperature rises when the grains coalescence, and also when the magnetic domains grow35.\nFIG. 8: a) Complex magnetic permeability for SET-2. b) Theoretical spectra calculated using Maxwell-Garnett and Bruggeman\ne\u000bective medium models with the volume fraction of 0.11. The LLG/Kittel modelling was done with d= 14 nm,Ms= 150\nemu/g,TB= 400 K,\u000b= 0.12, and ~H=HA+0:3 kOe (or\u00160~H=\u00160HA+30 mT), with HAestimated from HA= 2K=\u0016 0Msand\nK\u0019kBTB=V, whereVis the average particle volume. The Debye part was calculated using \u000bk= 2\u000210\u00003and\u001fk(0) = 104.\nc. Particle-size and morphology e\u000bects. Next, we investigate how the particle size and particle morphology a\u000bects\nthe microwave absorption properties of a composite made from \u000f-cobalt particles. First, we study three new composites\n(SET-3A, SET-3B, SET-3C), from a third set of samples (SET-3). The TEM images of the particles within these\ncomposites are shown in Fig. 9. We see that sample-A contains spherical nanoparticles with the average diameter of10\nFIG. 9: TEM images of \u000f-cobalt nanoparticles in SET-3. a) SET-3A contains particles with the smallest average diameter\n(8.6\u00061.4 nm) showing some aggregation in the form of chains b) SET-3B contains particles with the average diameter of\n27.7\u00062.7 nm, and c) SET-3C contains particles of the average diameter of 32.1 \u00067.9 nm.\n8:6\u00061:4 nm, while sample-B and sample-C represent the composites made from bigger particles: 27 :7\u00062:7 nm and\n32:1\u00067:9 nm, respectively. The images also suggest that the heating-up synthesis method18can produce \u000f-cobalt\nnanoparticles exhibiting facet-structures if the particle sizes are bigger than 25 nm.\nThe magnetic permeability of these samples (Fig. 10) demonstrate that faceted particles (non-spherical) exhibit a\nsimilar FMR absorption peak as the smaller spherical particles. The absorption is highest between 2 and 5 GHz. Note\nthat the samples that are made from bigger (25-40 nm) \u000f-cobalt particles tend to have a wider absorption bandwidth.\nTEM imaging con\frms that nanoparticles larger than 25 nm form facet structures, see Fig. 11d. These non-spherical\nparticles (Fig. 9b-c and Fig. 11d) cause their composites to exhibit a small absorption at 1 GHz ( \u001600\nris less than 0.1)\nand a large absorption (resonance) over a wide frequency range above 1.5 GHz.\nFrom the measurements presented in Fig. 12, we notice that composites made of low-dispersion 11-nm particles\n(SET-4C), see Fig. 11c, have relatively large absorption at frequencies around 1 GHz, compared to the absorption\ngiven by larger-size particles (27 nm, SET-4D) see Fig. 11d. This may be due also to the large fraction of closed-pack\nstructures and aggregates. We also found that smaller particles, with an average diameter of 5 nm (SET-4A in Fig.\n11a), do not exhibit signi\fcant absorption in this range.\nThe magnetic permeability of SET-4 shows that a composite made from particles with diameter smaller than 8 nm\nis a weak absorber compared to composites made from bigger (11 to 35 nm) particles (Fig. 12). This observation\nsuggests that the microwave absorption of SET-3A (Fig. 9a) is associated with the FMR of minority particles that\nhave bigger sizes (those which form chain structures). In principle, chains can form during the wet-chemical synthesis\nbecause of the presence of magnetic dipole interaction. Aggregation can change the anisotropic energy and so cause\nthe unexpected (small) change of microwave absorption spectra. From Fig. 12, we \fnd that the composites with the\naverage particle diameter of 27 nm exhibit high permeability ( \u00160\nr\u00191:7) at 1 GHz, with the magnetic loss \u001600\nrof less\nthan 0.1.11\nFIG. 10: The imaginary part of the magnetic permeability of (a) SET-3A, (b) SET-3B, and (c) SET-3C.\nFIG. 11: TEM images of \u000f-cobalt nanoparticles in SET-4. a) SET-4A contains nanoparticles with average diameter of 5 nm.\nb) SET-4B contains nanoparticles with average diameter of 8 nm, c) SET-4C contains nanoparticles with average diameter of\n11 nm, and d) SET-4D contains bigger facet-particles with average size of 27 nm.12\nFIG. 12: The real and the imaginary part of the magnetic permeability for SET-4B, SET-4C, and SET-4D.\nIn consequence, for the applications in the area of low-loss devices/materials, the experiments show that it is suf-\n\fcient to use \u000f-cobalt facet-nanoparticles with sizes of about 20 nm. For technological applications in the area of\nmicrowave absorbers we \fnd that the absorption spectra are also sensitive to the way particles arrange themselves\ninside the composite materials. The formation of superlattices in monodispersed particles could result in large absorp-\ntion over a broader bandwidth. Ideally, one should also aim at controlling the particle-particle interaction (distance)\nas well as at creating regular (periodic) structures, which can be in principle done by novel self-assembly and poly-\nmerization techniques. One possibility would be to use in the synthesis a surfactant with an acid group at one end\nand a double bond at the other end (for example docos-21-enoic acid), in order to obtain particles with a double bond\nfunctionality. After mixing the particles with styrene and polymerization, the interparticle distance will be limited to\nthe same value for all particles.\nd. E\u000bects of aging. Due to the natural oxidation of cobalt particles, the magnetic permeability of a composite\nsample will decrease in time. This reduction is associated with the reduction of saturation magnetization. Fig. 13\nshows the magnetization (zero-\feld-cooled curve) of SET-2 measured by SQUID magnetometry \fve months after the\nsynthesis. We observe a change in TBfrom 400 K to 350 K. According to the LLG-Kittel theory, this will not change\nthe permeability spectra that much. However, the decrease of saturation magnetization Msdue to oxidation can\ncause a signi\fcant reduction of \u00160values.\nIV. CONCLUSION\nWe have experimentally studied the wideband microwave absorption of composite materials made from \u000f-cobalt\nnanoparticles and polystyrene. The experiments show that the randomly-oriented spherical nanoparticles inside a13\nFIG. 13: The ZFC-FC curve of SET-2, \fve months after synthesis. The average particle diameter is 14 nm. The inset \fgure\nshows that the saturation magnetization is below 20 emu/g.\ncomposite induce ferromagnetic resonant absorption at microwave frequencies (1 to 12 GHz). Composites consisting\nof\u000f-cobalt nanoparticles within the size range between 8 and 35 nm, either monodispersed or polydispersed, exhibit\nresonant absorption which peaks in the frequency range between 2 and 6 GHz. Particles of smaller sizes, especially\nthe ones below 5 nm, do not have signi\fcant response to the microwave \feld in this frequency range. The permeability\nspectra is well described, especially at high-magnetizing \felds, by the LLG-Kittel equation and the e\u000bective medium\nmodel (Bruggeman or Maxwell-Garnett). At zero-magnetizing \feld, the LLG-Kittel equation and the e\u000bective medium\nmodel predicts magnetic resonance at lower frequencies compared to the experimental values. We analyze this \fnding\nqualitatively by dividing the magnetic response of nanoparticles into three regimes, namely low-\feld, intermediate-\n\feld, and high-\feld. In case of polydispersed \u000f-cobalt particles with sizes of about 10 nm, the high \feld regime begins\nat a magnetizing \feld of about 3 kOe.\nFor technological applications, our results demonstrate that the absorption spectra of composites made with either\nmonodispersed and polydispersed particles can be tuned by the application of an external magnetic \feld. Also, for\nhigh-\u0016low-loss applications around 1 GHz a good choice are faceted (non-spherical) particles with sizes of about 27\nto 35 nm. Medium-size monodispersed spherical particles (10-20 nm) are good as microwave absorbers, particularly\nif aggregates are formed. Our measurement results show that in this case the absorption capability of the material\nis increased, thus decreasing the amount of material needed for fabrication of absorbers and \flters. This suggests an\nalternative route to functionalizing these materials through the control of the arrangement of particles, which could\nbe used for engineering tunable microwave circuits and RF components without the need of a magnetic \feld.\n\u0003Current a\u000eliation: Spinverse Oy, Tekniikantie 14, 02150 Espoo, Finland\n1Ajayan P M, Schadler L S and Braun P V 2003 Nanocomposite Science and Technology (Weinheim, Germany: Wiley-VCH)\n2Chen L F, Ong C K, Neo C P, Varadan V V and Varadan V K 2005 Microwave Electronics: Measurement and Materials\nCharacterization (Chichester, UK: John Wiley & Sons, Ltd)\n3Timonen J V I, Ras R H A, Ikkala O, Oksanen M, Sepp al a E, Chalapat K, Li J and Paraoanu G S 2010 Magnetic\nnanocomposites at microwave frequencies , inTrends in nanophysics: theory, experiment, technology , ed V. Barsan and A.\nAldea, Engineering Materials Series, (Berlin: Springer-Verlag) pp 257-285\n4Landau L and Lifshitz E 1935 Phys. Zeitsch. der Sow. 8153\n5Kittel C 1947 Phys. Rev. 71270\n6Kittel C 1948 Phys. Rev. 73155\n7Gri\u000eths J H E 1946 Nature 158670\n8Yager W A and R. M. Bozorth R M 1948 Phys. Rev. 7280\n9Fannin P C, Charles S W and Relihan T 1996 J. Magn. Magn. Mater. 162319\n10Fannin P C and Charles S W 1999 J. Magn. Magn. Mater. 196-197 586\n11Wu L Z, Ding J, Jiang H B, Neo C P, Chen L F, and Ong C K 2006 J. Appl. Phys. 99083905\n12Hong Z, Yong Y, Fu-Sheng W, Hai-Bo Y, Dong Z, and Fa-Shen L 2009, Chin. Phys. Lett. 26017501\n13Neo C P, Yang Y, and Ding J 2010 J. Appl. Phys. 10708390614\n14Yang R B, Liang W L and Lin C K 2011 J. Appl. Phys. 10907D722\n15Longgang Yan Y, Wang J, Han X, Ren Y, Liu Q and Li F 2010 Nanotechnology 21095708\n16Yong Y, Cai-Ling X, Liang Q, Xing-Hua L, and Fa- Shen L 2010 Chin. Phys. Lett. 27057501\n17Puntes V F, Krishnan K M and Alivisatos A P 2001 Science 2912115\n18Timonen J V I, Sepp al a E, Ikkala O, and Ras R H A 2011 Angew. Chem. Int. Ed. 502080\n19Dinega D P and Bawendi M G 1999 Angew. Chem. Int. Ed. 381788\n20Sun S and Murray C B 1999 J. Appl. Phys. 854325.\n21Puntes V F 2001 Appl. Phys. Lett. 782187\n22Xia Y, Xiong Y, Lim B, and Skrabalak S E 2009 Angew. Chem. Int. Ed. 4860103\n23Gilbert T L 2004 IEEE Trans. Magn. 403443\n24Fannin P C, Relihan T, and Charles S W 1997 Phys. Rev. B 5514423\n25Bruggeman D A G 1935 Ann. Phys. 416665\n26Garnett J C M 1904 Phil. Trans. R. Soc. Lond. 203385\n27Mallet P, Guerin C A and Sentenac A 2005 Phys. Rev. B. 72, 014205\n28Chalapat K, Sarvala K, Li J and Paraoanu G S 2009 IEEE Trans. Microw. Theory Tech. 572257\n29Veps al ainen A, Chalapat K and Paraoanu G S 2013 IEEE Trans. Instrum. Meas. 622503\n30Fischbacher T, Franchin M, Bordignon G, and Fangohr H 2007 IEEE Trans. Magn. 432896\n31Timonen J V I 2013, Collective dynamics in near-\feld coupled magnetic nanocrystals at microwave frequencies, submitted.\n32Buznikov N A, Iakubov I T, Rakhmanov A L and Sboychakov A O 2005 J. Magn. Mag. Mater. 293938\n33Puntes V F, Gorostiza P, Aruguete D M, Bastus N G and Alivisatos A P 2004 Nat. Mat. 3263\n34Puntes V F and Krishnan K M 2001 IEEE Trans. Magn. 37, 2210\n35Frydman A and Dynes R 1999 Solid State Commun. 110485490\nAcknowledgements\nWe thank M. Sarjala, W. Skowronski, S. van Dijken, A. Savin, and J. Seitsonen for discussions and technical\nadvice. This project was supported by Thailand Commission on Higher Education, the Academy of Finland (projects\n118122, 141559 and 135135), and by the Finnish Funding Agency for Technology and Innovation (TEKES) under the\nNanoRadio project. This work was done under the Center of Excellence \"Low Temperature Quantum Phenomena\nand Devices\" (project 250280) of the Academy of Finland, and it used the Aalto University Nanomicroscopy Center\n(Aalto-NMC) and the Cryohall (Low Temperature Laboratory) premises."    },    {        "title": "1408.5921v2.Spin_Scattering_Rates_in_Metallic_Thin_Films_Measured_by_Ferromagnetic_Resonance_Damping_Enhanced_by_Spin_Pumping.pdf",        "content": "US government work. Not protected by US copyright.  Spin -Scattering  Rate s in Metallic Thin Films  Measured by  Ferromagnetic \nResonance Damping Enhanced  by Spin -Pumping  \n \nC. T. Boone, J. M. Shaw, H.  T. Nembach, T.  J. Silva  \nNational Institute  of Standards and Technology, Boulder, CO 80305  \n \nAbstract  \nWe determined the spin -transport  properties of Pd and Pt thin  films by measuring \nthe increase in ferromagnetic resonance damping due to spin -pumping  in \nferromagnet ic (FM) -non ferro magnetic  metal (NM) multilayers with varying NM \nthicknesses.  The increase in damping with NM thickness depends strongly on both \nthe spin - and charge -transport properties of the NM, as modeled by diffusion \nequations that include both momentum - and spin -scattering parameters .  We use \nthe analytical solution to the spin -diffusion equations to obtain spin -diffusion  \nlengths for Pt and Pd.  By measuring the dependence of conductivity on NM \nthickness, we correlate the charge - and spin -transport  parameters , and validate t he \napplicability of various models for momentum -scattering and spin -scattering rates \nin these sy stems : constant, inverse -proportional (Dyakanov -Perel), and linear -\nproportional (Elliot -Yafet) . We confirm previous reports that the spin -scattering  \ntime appear s to be shorter than the momentum scattering time in Pt, and the \nDyakanov -Perel -like model is  the best fit to  the data.  \nI. INTRODUCTION  \nSpin -transport  is currently a topic of enormous interest . New devices based \non spin -transport are frequently proposed ; such devices  utilize phenomena such as  2 pure  spin -currents  [1], the spin Hall effect  [2,3,4, 5, 6, 7], as well as giant - and \ntunneling -magnetoresist ance [ 8, 9, 10].  A fundamental understanding of how spins \npropagate in metals , as well as how they are absorbed and transmitted at NM/FM  \ninterfaces , is vital for the exploitation of  phenomena such as spin -pumping [ 11, \n12,13, 14, 15, 16] and spin -torque  from  the spin Hall effect  (SHE)  [17 ,18, 19, 20, 21, \n22, 23,24] for technological applications  such as magnetic random -access memory, \nmagnetic data storage, and spin -based logic .  The effects of charge scattering , \nproximity -induced  polarization at the NM/FM interface,  and magnetic “dead ” layers  \non spin -diffusion  are subject s of ongoing debate . \nIn this work,  we use a spectroscopic method  to address  how the momentum - \nand spin -scattering rates for electrons in  metals with  strong  spin -orbit coupling , e.g., \nPt and Pd, affect the diffusive transport of pure spin -currents. We find  strong \nevidence to support the con jecture that momentum - and spin -scattering are indeed \ncoupled processes in  thin -films  of Pt and Pd , though the coupling process is more \nsubtle than would be presumed from the simple Elliot -Yafet picture for spin -flip \nprocesses that affect charge carriers . \nAn outstanding  question remains as to  how spin - and charge -transport are \nrelated in high -Z metals  such as Pt and Pd  [25,26,27,28,29,30].  In the standard spin -\ncharge diffusion equations, spin -transport  is dependent on the charge conductivity , \nas can be seen in the standard time -independent , steady -state equation s that \ndescribe spin -diffusion  [31, 32, 33, 34,35],  3  \nˆˆ 2\n2\nˆˆ 2\nsf,2\n1,ss\nssQe\n \n\n (1) \n  \nwhere 𝑄⃗ 𝑠̂ is the spin -current dyadic , \nˆs is spin -accumulation  for spins polarized \nalong the 𝑠̂ direction , σ is the electrical conductivity , \n is the reduced Planck’s \nconstant, and λsf is the spin diffusion length .  Here, we have omitted the precessional \nterm of the form 𝑠̂×𝐻⃗⃗ , where 𝐻⃗⃗  is the net magnetic field , because spin -\ndepolarization  in metals occurs on much faster timescales (fs) than spin -precession \n(ps).  \nThe spin -diffusion  length λsf and electron mean  free  path  \n are oft en treated \nas indepen dent  for the purposes of fitting  experimental data  [19, 30, 36, 37], but \ntheor y suggest s they are related. In particular, the Eliot -Yafet (E-Y) scattering  \nmechanism  [38, 39], whereby each momentum -scattering event has a certain \nprobability P of being a spin -scattering  event, suggests that the spin -flip time is \ngiven by\nsf P , where τ is the momentum -scattering time .   \nThe complementary relation between spin - and charge -scattering  is the \nDyakanov -Perel  (D-P) mechanism  [40, 41], where by spins continuously de-phase \ndue t o the combination of spin -orbit coupling and crystal -lattice inversion -\nsymmetry breaking  until a momentum -scattering event  occurs . In this picture, the \nspin -flip time is inverse ly proportional to the momentum -scattering time , i.e., \n1\nsf\n. Strictly speaking, the  D-P mechanism is not operative in materials  with 4 cubic symmetry  because they are inversion -symm etric . However, the recent work of \nJiao and Bauer  [42] found that the conventional theories for spin -pumpin g, spin -\ndiffusion, and the spin -Hall effect could be successfully employed to fit dc voltages \ndue to spin -pumping and the inverse spin -Hall effect in Permalloy /Pt bilayers for \nvarying Pt thickness if they used measured values  for thickness -dependent \nconductivity  \nL , but kept a constant value of \nsf1.3  nm. A constant spin \ndiffusion length with thickness -varying conductivity also explained recent spin Hall \nmagnetoresistance data [ 43]. The invocation of a constant \nsf  but varying \n  has a \nsubtle but  important logic al consequence; if we invoke the diffusion relation \nsf sf D\n, where D is the diffusion constant,  the Einstein relation \n2De , \nwhere  \n is the density of states,  and the Drude model for conducti vity \n2ne m\n, where n is the conduction electron density and m* is the effective mass,  we then \nobtain  \n \n2\nsf\nsf1.m\nn\n (2) \nHence, the fitting procedure used in Ref. [42] is phenomenologically equivalent to D-\nP insofar as it implies that \nsf1 . At face value, the implication of the procedure \nused in Ref. [ 42] would appear to explain various experimental data that sugges t \nsf\n for samples where  the NM thickness is comparable to, or even smaller than,  \nthe bulk -value for \n  [36, 44]. If \n  and, therefore, \n  are sufficiently reduced at small \nNM thicknesses, then it is possible that \nsf , if \n2\nsfmn   . This would satisfy  5 an important requirement for the application of diffusion theory for spin transport  \nin nonferro magnetic  metals . In principle, both D -P and E -Y can exist for different \nscattering sites, leading to a more complicated relation between τ and τsf [45].   \nThe reported data for \nsf  in Pt and Pd span a significant range (0.5 to 14 nm \nfor Pt  [19, 46, 47], 2 to 12 nm for Pd  [36, 44, 48, 49, 50], with several reported \nvalues  smaller than the commonly -assumed mean -free -path  for these materials . \nBecause most measurements are performed on sputtered thin films , it has been \nsuggested that the strong thickness - and growth -dependence of the conductivity  \nand interfacial properties  for thin -films  could explain such a wide range of reported \nvalues  [51]. Much of the previous work had been performed at low temperature, \nmaking extrapolation to room temperature difficult. An alternative explanation rests \non the fact that Pt and Pd are “almost” ferromagnetic according to Stoner theory \n[52]. As such, Pt and Pd are magnetically polarized when in direct contact with  a \nferromagnet  [53, 54], which could  affect the spin -diffusion  length , and even the \nfunctional form of spin -transport [44, 50]. \nWe use the phenomenon of spin -pumping to determine diffusive spin -\ncurrent flow in metallic multilayers : when a FM is adjacent to a NM metal and the \nmagnetization is out of thermal equilibrium, spin -accumulation forms at the FM/NM \ninterface as a result of spin -pumping [ 32, 34].  As Fig. 1(a) illustrates, the spin -\naccumulation diffuses away from the interface, creating a spin -current  𝑄⃗ 𝑠̂, which  \ndecays with a characteristic length λsf, as described by Eq.  (1).  This spatial flow of \nangular momentum manifests itself as an increase in the Gilbert damping parameter 6 of the FM [ 11, 32, 55]. We measure  the change in Gilbert damping as a function of \nlayer thicknesses in ferromagnetic multilayers in order to infer the spin -current \nflowing away from the FM/NM interface . Fitting of  the spin -pumping data vs. film \nthickness allows us to extract \nsf . This  room -temperature, spectroscopic method \ndoes not require  patterning or  electrically  conductive contacts  and therefore is free \nof artifacts related to edge defects and contact resistances.   \nBy determin ing both λsf and σ for the same samples, we can ascertain  \nwhether  there is a n inter dependence of τsf and τ, thereby testing various models for \nthe spin -scattering. In so doing, w e present experimental evidence that  \nsf\n  for \nall but the smallest NM thicknesses, and that the thickness -dependence of \n  cannot \nresolve this apparent conundrum.  \n We study spin -transport with two different general categories of \nmultilayers: (1) those with either the NM = Pt or Pd in direct contact with the FM  = \nNi80Fe20 (Permalloy,  “Py”); and (2) those with Cu inserted between the NM and t he \nFM. We find that the spin absorption properties for a NM in direct contact with a FM \nand with a spacer are different ; it appears that the polarization of the NM at the \ninterface affect s the spin -transport . This suggests  that the usual spin/charge \ndiffusion equations may require further augmentation to adequately describe \ntransport in systems where the exchange splitting of the bands is not purely local.  7 II. THEORY  \n When a FM , with instantaneous magnetization unit vector \nˆm , is adjacent to \na NM, magnetization dynamics in the FM effectively  act as a transverse spin -\npotential source, so that \n2\nˆˆ2ˆˆ 2 (0) (0)\nRe[ ]ssem dm dt Q\nG  \n  at the FM/NM \ninterface  [32].  Spin current  of polarization \nˆs  is thus sourced at the FM/NM interface \nthrough a conductance \n2Re[ ] G , beyond  which it obeys  Eq. (1).  This is shown \ndiagrammatically in Fig. 1a.   \nFor uniform excitation of the magnetization , the problem reduces to the one -\ndimensional case with  an N-layer NM stack adjacent to the FM layer . We  can treat \neach layer  i, with conductivity σi, thickness Li, and spin diffusion length λsf,i, as a \nspin -resistor l adder network  in series with a spin -resistor at the FM/NM interface \n(i.e., the reciprocal of the spin -mixing conductance \nG ), shown in Fig. 1b, where the \ntwo parallel -resistors -to-ground \np\niR  for the ith layer have the value \n ,, coth 2p\ni sf i i i sf iRL  \n and the series -resistor \ns\niR  has the value \n ,, sinhs\ni sf i i i sf iRL  \n.  Thus, e lementary circuit theory allows us to relate  the \nspin -current \n0\nˆsQ  at the FM/NM inte rface  and the transverse spin -potential in the FM  \nvia matrix multiplication,  8  \n0\nˆ\n0\nˆ ˆ,0s\nN\ns sQ\n    \n   \n  \n (3) \nwhere  \nˆN\ns  is the spin -potential at the far edge of  the Nth (i.e. , last) layer of the NM \nstack and  \n \n1 1,1 1,2\n2,1 2,2N\ni\niT\n \n\n (4) \nand \n \n, , ,\n,\n,,.\ncosh sinh\nsinh coshi\ni i i\nsf i sf i sf i\nsf i ii\ni sf i sf iT\nLL\nLL\n  \n\n  \n             \n               \n (5) \nBy use of Eq. (3), we can then solve for the sourced spin -current as  \n \n2\n0\nˆ 2\next2ˆˆ ,4 21sGQ m dm dte G\nG\n\n\n (6) \n  \n   \nwhere  9  \n1,2\next\n1,1. G\n (7) \nThe ratio \next 2GG  is sometimes referred to as the backflow factor [ 34, 36]. The \noutlined matrix formalism can also accommodate interfacial spin -flip between any \ntwo la yers by insertion of a fictitious  jth layer between any two layers to represent \nany interface, with the substitutions \n, j j sf jL  and \n, j j j j sf j jAR L     , \nwhere \nj  is the interfacial spin -flip parameter, and \njAR  is the interfacial resistivity, \nas defined in Ref . [56]. \nThe flow of transverse spin -current across the FM/NM interface is an \nadditional source of damping  for the magnetization dynamics in the FM, which is \ncomputed via the general relation  \n \n\n2\n2 eff\nFM\neff\nextRe 2 ;4\n22,\n12sGe M L\nGG\nGG\n\n\n\n\n (8) \n   \nwhere  is the gyromagnetic ratio, e is the electron charge, Ms is the saturation \nmagnetization of the FM, and LFM is the FM layer thickness. \neffG  is the effective spin -\nconductance that restricts the flow of transverse spin -current due to the series \narrangement of \nG  and \nextG . The combination of Eqs. (3-8) allows us to fit data for  \nthe spin -pumping contribu tion to the damping for any arbitrary N-layer NM stack , \ninclusive of interfacial spin -flipping . 10  \nWe note that the application of different hypotheses of how NM transport \nproperties are correlated  with other film properties  can signific antly affect the data  \nfits to the  diffusive spin -current model , and the wide variability of fitted parameters , \nsuch as spin -diffusion  length , reported in the literature for materials such as Pt , may  \n \n \nFigure 1: (a) Sketch of the spin -pumping mechanism. (b) Schematic of \nequivalent ladder -circuit element to describe spin transport for the ith layer, \nfrom  which any multilayer stack can be built. \nsR  and \npR  are equivalent spin \nresistances that are defined in the main text.  (c) Calculated dependence of \neffective spin -mixing conductance of a FM/NM bilayer for two cases: 1. An \nunphysical constant conductivity independent of NM thickness (solid red). 2. A \nreali stic NM -thickness -dependent conductivity (solid blue).  If we fit the curve \nfor case 2, but erroneously assumes a constant conductivity, the resultant fit is \nshown as the dashed green curve. While the fit captures the salient qualitative \nfeatures of the th ickness -dependence, the fitted parameters suffer from errors \nof approximately 50%.  \n11 be partially explained by this.  For example, if the conductivity of a NM layer is \nassumed  to be constant, but is in reality decreasing  with decreasing  layer thickness, \nthe fitted spin -diffusion  length will incur a systematic inaccuracy . Figure 1 (c) \nillustrate s how this can happen by comparing calculations, for the illustrative case of \na simple FM/NM bilayer, of the expected dependence of Geff on NM thickness, where \nwe consider  one material  with  constant  conductivity  and another with the \nexperimentally determine d thickness -dependent conductivity  we observe for Pt, as \ndiscussed in more depth later in this paper. Here, we  assum e sf = 5 nm and \n1510 G\n -1 m-2.  An important point is that a model that assumes a constant value \nof conductivity provides reasonable fits to the data regardless of whether the \nconductivity is constant or contains a thickness -dependence.  In the case where the  \nsimulated data have a thickness -dependent conductivity,  the fitted parameter s sf = \n7.5 nm and \n151.5 10 G  -1 m-2 are 50  % large r than the parameters used to \ngenerate the simulated data . Thus, fits of spin -pumping data with a model that \nignores the thickness -depende nce of the conductivity provide , at best, an  upper \nboun d on\nsf  and\nG . \n 12 III. EXPERIMENT  \nA. dc conductivity measurements  \nTo obtain the net dc resistivity \nNM  of the  upper  NM layer , we measure the \nfour -probe, current -in-plane sheet -resistance of the multilayer as a functi on of NM \nthickness.  Interfacial charge scattering results in a resistivity component that scales \ninversely with NM thickness \nNML  [57, 58, 59],  \n \ns\nNM b\nNM,L\n (9) \n  \nwhere \nb  is the bulk resistivity, and \ns  is the interfacial resistivity coefficient. Eq. \n(9) is numerically consistent with models for the thickness -dependent resistivity of \ngranular thin -films  given by solution of the Boltzmann equation , under the \nassumption of  diffus ive surface - and grain -boundary  scattering [ 60].  Small \ndeviations from this functional for m may occur at the smallest thicknesses. The \ncontribution of all the other layers in the multilayer stack (i.e. , the “under -layer ”) to \nthe resistivity are accounted for by assuming that the under -layer resistivity is \nconstant for a given structure , and that the under -layers  together with the NM layer \nact as  parallel resistors so that  \n \n1\nUL NM\nsheet\nUL NMLLR\n\n (10) \n  13 where  \nsheetR  is the sheet  resistance  of the entire stack , \nUL  and \nULL  are the under -\nlayer resistivity and thickness, respectively.  \nFor the purposes of determining the thickness -dependence of NM resistivity \nin representative multilayer stacks, we sputter -deposited samples of Ta(3 nm)/Py(3 \nnm)/NM( LNM) onto oxidized Si substrates in a chamber of base pressure of  10-7 Pa \n(10-9 Torr ). Deta ils of the sputtering chamber and representative film roughnesses \nwith Ta seed layers (<0.5 nm) are described in Shaw, et al. [ 61]. In Figure 2(a), we \npresent  the resistivity of a Ta(3 nm)/Py(3 nm)/Pt( LNM) multilayer s as a function of  \nNML\n, along with a fit to Eq s. (9) - (10) . The model fits the experimental data very \nwell over the entire range of NM-thicknesses  that w as prepared: 0.5 to 30 nm .  The \nextracted bulk - and surface -resistivities of the NM layers are shown in Table 1. The \nbulk resistivities that we obtain for Pd and Pt are somewhat higher than tabulated \nroom -temperature values (\n71.05 10   m and \n71.04 10   m for Pd and Pt, \nrespectively) , but are similar to previously -reported measured values  of the thin -\nfilm resistivity  in other studies of the spin diffusion length for Pt  (\n71.6 10 m for \nRef [ 19];\n71.8 10  m for Ref  [37]; \n72 10 m for Ref [ 62]). \nNM Underlayer  b (10-7  m) s (10-16  m2) \nPd Ta(3 nm) /Py(3 nm)  1.36 ± 0.03 4.4 ± 0.3 \nPd Ta(3 nm) /Py(3 nm) /Cu(5 nm)  1.38 ± 0.04 2.1 ± 0.1 \nPt Ta(3 nm) /Py(3 nm)  1.70 ± 0.03 3.3 ± 0.3 14 Pt Ta(3 nm) /Py(3 nm) /Cu(5 nm)  1.71 ± 0.03 2.9 ± 0.2 \n Table 1: Resistivities of the Pd and Pt for the structures studied here.  \n \nB. Ferromagnetic resonance  \nWe used ferromagnetic resonance (FMR) to study changes in damping  due to \nspin -pumping  from Py into  the NMs Pd or Pt.  The metals Pd and Pt are of interest \nbecause they are “almost” ferromagnetic , i.e., the product of the density of states and \nthe exchange integral almost satisfy the Stoner criterion for itinerant \nferromagnetism in Stoner theory [ 52]. In addition, Pd and Pt are well known to \nexhibit interfacial spin -polarization  when in contact with metallic ferromagnet s [53, \n54]. Finally, the substantial spin -orbit coupling in these materials gives rise to \nsignificant spin -Hall a ngles [ 63], making them technologically useful for future \nspintronic applications .  For the FMR measurements, m ultilayers  of \nTa(3 nm)/Py(3 nm)/NM(X ) and Ta(3 nm)/Py(3 nm)/Cu(5 nm)/NM(X) were sputter -\ndeposited  onto oxidized Si substrates in a chamber with a base pressure of 10-7 Pa \n(10-9 Torr) , and subsequently coated with PMMA .  We used a Cu spacer layer to \ndetermine the role of interface polarization  in spin -pump ing measurements.  A 5 \nnm-thick  Cu spacer is sufficiently thick to prevent direct exchange -coupling between \nPermalloy and the NM,  but sufficiently thin  to prevent loss of spin -accumulation  \nwith in the Cu  [64].  Deposition rates  were calibrated with  x-ray reflectometry , and \nthicknesses are accurate to within 2  %, which is included  in our estimated error \nbars .  The Ta seed -layer is used to promote a  (111) texture of the Permalloy . 15 Using  a standard broadband (5 -30 GHz) ferromagnetic resonance technique , \nwe measure d for each sample  the damping α, the spectroscopic g-factor  gL, and the \neffective magnetization  \n  eff Py 0 Py 2s s s M t M K M t   , where Ks is the net \nperpendicular interfacial anisotropy  with units of J m-2, \nPyt is the thickness of the Py \nlayer, and Ms is the saturation magnetization  [65]. We used an FMR -geometry with \nthe saturating magnetic field applied perpendicular to the sample plane to eliminate \ntwo-magnon scattering as an additional source of linewidth broa dening [ 66]. The \nbroadband measurement allows for the accurate extraction of the Gilbert -like \ncomponent of damping. This geometry also results in precessional motion that is \nboth low amplitude  and circularly polarized .  Figure 2 (b) shows a represe ntative \nresonance curve, with  a fit to the Polder susceptibility  that allows for extraction of  \nthe resonance field  and field -swept linewidth .  The extrapolation method developed \nin Ref. [67] is used to determine \nLg  and Meff, which are then u sed for the extraction \nof \n  from a fit of the linewidth to \n  004 H H f         , where f is the \nmeasurement frequency  and Δ H0 is the inhomogeneous linewidth  broadening . \nRepresentative data for Δ H vs. f are shown in Figure 2(c) .  All linewidth data exhibit a \npurely linear dependence on f.  By performing FMR measurements with \nTa(3 nm)/Py( tPy)/Pd(10nm)  for different Permalloy thicknesses  tPy, and subsequent \nfits to \n 0 Py t     and \n eff PyMt , we obtain the intrinsic damping \n00.0044 0.0004\nand saturation magnetization  μ0Ms = 1.068 ± 0.007 T , in \nagreement with previously reported values  for bulk Permalloy [68]. These results 16 confirm an interfacial component of damping and anisotropy , as previously \nobserved in similar systems [ 49, 50].  Fig. 2(d) shows t he data for the spectroscopic  \ng-factor for all samples  with tPy = 3 nm. We find that gL shows no discernable \ndependence  on NM thickness,  and an inconclusive  dependence on details of the NM \nstack , with an average value of  gL = 2.08  ± 0.01 . \n \nC. Data fitting and analysis  \nWe consider three different fitting -models to interpret our data:   \n \nFigure 2: (a) Measured multilayer resistivity as a function of Pt -layer -thickness, \nwith fit to Eqs. (9) and (10) showing the reciprocal -thickness -dependence of the \nresistivity.  Inset shows the extracted mean free paths as a function of thickness.   \n (b) Representative resonance curve, Ta(3nm)/Py(3nm)/Pt(7nm) sample, and \nfits to the Polder susceptibility, from which the linewidth and resonance field are \nextracted.   (c) Representative linewidths versus frequency, showing the linear \ndependence across samples, from which α can be precisely extracted (d) g -factor \nversus NM thickness for all series, showing little variation and no discernable \nthickness dependence.  \n17 (1) Both \nsf  and \n  are independent of NM film thickness . This is equivalent to a \nconstant \nsf  and a constant bulk -value for  σ, similar to prior work.  The fitting \nparameters are \nG  for the FM/NM interface, \nsf  for the top NM layer, and \n  for \nthe Cu/NM or the Py /NM interface, depending on whether the sample is a bilayer or \ntrilayer stack.   \n(2) \nsf1 . This is equivalent to a constant  \nsf, but a  thickness -dependent  σ. \nAs discussed in detail earlier, t his is identical to the D-P-like model  used  by Jiao and \nBauer  to fit inverse spin Hall effect data in Ref.  [42]. We use the fitted function of the \nthickness -dependent \n  obtained from our four -probe measurements with the \naforementioned multilayers. The fitting parameters  are the same as in Model 1 .  \n(3) \nsf . This is the E -Y mechanism and is equivalent to \nsf , as \ndescribed earlier. As for Model 2, we use the fitted function of the thickness -\ndependent \n .The fit ting parameters are \nG  for the FM/NM interface,\nsf\n  for \nthe top NM layer, and \n  for the Cu/NM or the Py/NM interface, depending on \nwhether the sample is a bilayer or trilayer stack.  \nFor all models, when considering interfacial spin flip, w e use \n15 20.1 10   m AR  \n based upon previously reported values for high -quality \nmetallic interfaces [69, 70]. Deviations of AR from this value will affect the exact \nfitted value of \nG but do not affect our other conclusions. Interfacial spin -flip is \nneglected for the  the Py/Cu  interface  due to the absence of  damping enhancement 18 for the case of a FM/Cu/Ta trilayer stack [ 36], indicative that both the FM/Cu and \nthe Cu/Ta interfaces exhibit negligible spin -flip.  \nFor our 5 nm thick Cu spacer, we measure d σCu/2L Cu =3.6 × 1015 (Ω-1 m-2).  \nEqs. (3)-(8) remain valid f or L Cu→0, i.e. , no spacer layer, with \nG , δ, and AR I \nchanging for the different FM/NM interfaces.  \nTo calculate the momentum -scattering -times \n  and mean -free -paths \n  for Pd  \nand Pt based upon our measured resistivity values, w e use the following  values \nobtained from de Haas -van Alphen measurements:  a) for Pd,  electron density  n = \n0.376 electrons/atom, average effective mass \n2.1e mm , and  average Fermi \nveloci ty \n61.1 10Fv  m s-1 [71]; b) for Pt, n = 0.426  electrons/atom , \n2.4e mm , and \n5\nF8.8 10v\n m s-1 [72] and atomic densities nat,Pd  = 6.8 × 1022 cm-3 and nat,Pt = 6.62 × \n1022 cm-3 [73]. Using  the Drude  equation for resistivity \n2m ne , we obtain \nbulk values of \nPd 21 fs and \nPt  18 fs. From the definition of the mean -free -path \nFv\n, we obtain bulk mean -free -paths  \nPd\n  23 nm and  \nPt\n  16 nm. \nThe results of all fits are tabulated  in Table 2.  Figure s 3(a)-3(d)  show the \ndata of  α vs. NM thickness x for Py/Cu/Pd( x), Py/Cu/Pt( x), Py/Pd( x), and Py/Pt( x), \nrespectively, along with fits to Models 1 , 2 and 3 .  \n  19  \nMultilayer  Model  G↑↓(1015  m-2) sf (nm) for \nmodels 1 \nand 2, or  \nfor model 3  \n δ \nTa/Py/Cu/Pd  1 0.7± 0.14  4.8 ± 0.4  0 \n 2 0.49 ± 0.12  2.79 ± 0.09  (2±2)×10-9 \n 3 0.46 ± 0.56  0.2± 0.4 0.01± 0.68  \nTa/Py/Pd  1 2.8± 0.5  5.2 ± 0.5 0.21 ± 0.02  \n 2 1.33 ± 0.12  2.7 ± 0. 4 0.24± 0.0 2 \n 3 37 ± 278  0.3 ± 0.1 0.17 ± 0.1 5 \nTa/Py/Cu/Pt  1 0.47 1 ± .009  1.18 ± 0.0 7 0.01± 0.02  \n 2 0.44 ± .0 2 0.8 ± 0.4  0.36± 0.0 9 \n 3 0.72± 0.0 5 0.18  ± .05 (2±4) × 10-8 \nTa/Py/Pt  1 4.9± 0.3  1.2 ± 0.1  0.30± 0.0 6 \n 2 2.65± 0.11 0.37± 0.09 0.30± 0.07 \n 3 12± 40  0.2 ± 0.3 0.42 ± 0.40  \nTable 2: Fitt ing parameters obtained by use of the three different models for spin \nrelaxation rate vs. conductivity described in the text . \n 20 For all four sample types, Model 3 is a poor fit to the data, though it is \nmarginally acceptable for Py/Cu/Pt with respect to the data error bars. \nNevertheless, we see  clear ly that the assumption that both \n  and \nsf  are unaffected \nby sample thickness does not agree with the qualitative dependence of \n  on \nNML , \nat least within the context of conventional diffusive spin -transport theory.  \nFor samples with Cu spacer l ayers , Model 1 provides a n adequate  fit, but  the \naccuracy of the fit to Model 2  is markedly improved . The interfacial spin -loss \nparameter is negligible  for all cases  with Pd . For Py/Cu/Pt (x), the fitted valu e for  is \nno longer negligible . \nOmission of the Cu spacer layer has a strong effect on the fitted parameters. \nThe fits yield  similar  values of  \nsf for Py/Pd(x) compared to Py/Cu/Pd( x), but \nG  is \nsubstantially enhance d (3 -4 times) for the Py/Pd interface relative to that of the \nPy/Cu interface . The fitting result  for Model 3 is anomalous, with a non physical \nvalue for \nG when compared to first -principles calculations [ 74]. The fitted values \nfor \n  are no longer negligible  for Py/Pd , suggesti ng that proximity polarization of \nPd might be a source of interfacial spin -flip. For all three models, the fitted values \nfor \nsf  are substantially smaller  than the bulk Pd mean -free -path of 23  nm. \nAs is the case for Py/Pd(x), the fits for Py/Pt( x) yielded similar values for \nsf , \na marked increase in the values for \nG  compared to samples with a Cu spacer , and \nnon-negligible results for \n . Again, t he latter result is consistent with the \nhypothesis that proximity polarization  plays a role in the interfacial spin -flip 21 proces s, though the significant value of \n  for Py/Cu/Pt( x) suggests that proximity \npolarization is not the only factor to consider.  Again, the fitted value for \nG  \nobtained with  Model 3 is unphysically large, as seen  in the case of Py/Pd( x). \nThis final r esult deserves some explanation. I n the context of conventional diffusive  \n \n \n \nFigure 3: (a) Measured damping α (black dots) as a function of Pd thickness \nfor Py/Cu/Pd( x), together with fits to Models 1 (blue), 2 (red) and 3 (green).  \nTop panel zooms in on smaller thicknesses while bottom panel shows data \nand fits over the entire range of Pd thicknesses that were measured.  3(b -d) is \nthe same as 3( a), but for Py/Cu/Pt( x), Py/Pd( x), and Py/Pt( x), respectively. In \nall four sample systems, Model 3, with an Elliot -Yafet -like proportionality \nbetween \n  and \nsf , was grossly inadequate as a quantitative description of \nthe data. Conversely, Model 2, with a Dyakanov -Perel -like proportionality \nbetween \n  and \nsf , provided the superior fitting results, though Model 1 is \nnever inadequate.  \n \n22 spin -transport theory, the spin -pumping contribution to the damping in Model 3 for \nsamples without a Cu spacer appears to be limited by the bulk spin -conductivity \nsf\n of the NM , such that no amount of increase in the value of \nG  can give rise to \nfurther increase in \n . Given that \nsf  in Model 3, the reduction in \n  for small \nNM thickness does not change the insensitivity of \n  to \nG . Hence, the \nnonphysical fitted values of \nG  for Model 3 ar e related to the general inadequacy \nof Model 3 to fit any of the data.  \nAll the fits for NM = Pt yield \nsf1.5  nm, a value that is vastly smaller than \nthe bulk mean -free -path of \nPt16\n  nm. Other workers have  suggested  that fits of \nthe thickness -dependence  for damping  in spin -pumping measurements are subject \nto artifacts if interfacial spin -flip is not properly accounted for  [75, 76]; by this \nargument , the spin -diffusion length for bulk Pt is actual ly longer  than the fitted \nvalue , but the sensitivity of the fit s to the bulk \nsf  is negligible because the majority \nof spins -flips occur at the interface. This would lead to only a small  net increase in \nthe damping for NM thicknesses greater than that required to completely form the \nPy/NM interface. However, we still observe a small  but statistically significant \nincrease in \n  for Pt thicknesses up to 2 nm for both Py/Cu/Pt and Py/Pt , which is \nconsistent with fitted val ues of \nsf1.5  nm for Pt, even when interfacial spin -flip is \naccounted for.  Hence, we conclude that the observed increase in damping with \nincreasing Pt thickness indeed reflects a bulk spin -flip process, and not an interfacial 23 effect.  Our extracted value of λ sf~0.8 nm for Pt is also consistent with recent fitting \nof first -principles calculations to the diffusive model [ 74]. \nTo compare spin rel axation times  \nsf with momentum scattering times  \n in \nthe context of Model 2 , we use values for the electronic density of states  at the Fermi \nsurface  \nDOS , once again obtained via de Haa s-van Alphen measurements for Pd and \nPt: \nDOS, Pd 3.97  eV-1 atom-1, \nDOS,Pt  2.79  eV-1 atom-1 [71,72]. We use the Einstein \nrelation  for the diffusion parameter  \n2\nDOS at 1D e n  , where \natn  is the atomic \ndensity, and the diffusion equation \nsf sf D , as well as  the fitted parameters  \nobtained in Table 2 with  Model 2 , to calculate the dependence of the spin -relaxation \ntime on NM thickness  for all samples , as shown in Fig. 4.  \nFor all samples with Pd, we find that the asymptotic fitted spin -relaxation \ntime at large Pd thickness  of \n45 fs is indeed longer than the momentum scattering \ntime  of \n21 fs, consiste nt with the supposition of an E -Y spin -flip process for bulk \nPd. This is in contrast to the conclusion  presented in Ref . [77], where Foros, et al.  \nclaimed the observation of a spin -diffusion le ngth that was equal to the mean free \npath (~ 9 nm) was indicative of a paramagnon -mediate d spin -decoherence process. \nHowever, the relative scale of the mean free path and the spin -diffusion length  is not \nthe appropriate comparison when attempting to ascertain the mechanics of spin -\nscattering : the relevant parameters are the spin scattering time and momentum \nscattering time [ 34, 42]. Indeed, as we show here, while the  mean free path of \n23\nnm is in actuality longer than the spin -diffusion length \n2.8 nm, the momentum -24 scattering time is still shorter than that for spin -scattering.  The bulk spin lifetime in \nPt, as calculated assuming the diffusive theory, is shorter than the  bulk  momentum \nscattering time , as seen in Fig. 4(b) and 4(d) . The diffusive theory  based on a \nBoltzmann treatment  [78] requires that the spin lifetime  be greater than the \nscattering time. For Pt, therefore, either the simple diffusive theory must be \ninadequate , requir ing theoretical extensions , or the decay length observed in the \ndamping and some inverse spin Hall measurements cannot be interpreted as a spin \ndiffusion length . Further work is therefore needed to determine the dominant \nmechanism of spin transport in such a material.  \nThe failure of  the diffusive model for Pt does not discount the importance of \nthis work in terms of measuring  spin diffusion lengths and scattering times for more \nordinary metals. Indeed, verification of spin diffusion lengths and interfacial spin \nloss parameters indepe ndent from spin valve measurements are needed, especially \nat room temperature [ 79]. Our method can  also  be extended to low temperatures to \ncompare with lateral spin valve measurements similar to those in Ref. [ 46].  \nWhile the data  of Model 2  clearly provide  the best fit, the question remains as \nto how such phenomenology can be appropriate for isotropic , transition metal \nalloys, given that  the Dyakanov -Perel mechanism  is disallowed  in these materials  \nsince  they lack the necessary translational symmetry breaking.  Possibly, the fact \nthat the change in conductivity in our samples is dominated by interfacial and grain \nboundary scattering, which do break translational symme try, could contribute to \nthe Dyakanov -Perel -like nature of the observed dependences. Future work studying \nimpurity scattering and thermal dependence can be used to check this hypothesis.  25 An alternative understanding of the data presented here is that the s pin-\nrelaxation rate might be reduced at smaller film thicknesses for reasons that are \nindependent of the enhanced resistivity. For example, electron spin -relaxation rates \ncan be reduced for extremely small (< 2 -3 nm) metallic nanoparticles when \nmeasured wi th electron spin resonance  [80,81,82]. Such a reduction is to be \nexpected if the electron energy level spacing \n  is larger than the energy \nbroadening due to spin -flip processes \nsf\n  [83]. If the condition \nsf\n  is \nsatisfied, the spin -flip rate is renormalized, such that  \n \n1\nsf\n0\nsf sf111.\n\n (11) \n  \nGiven that the NM layer thickness is indeed thinner than the mean free path for \nmost of the samples that we have measured, there is the possibility that lateral \nconfinement has a nontrivial effect on both the momentum - and spin -relaxation \nprocesses. In the  case of thin films, such considerations do not generally apply \nbecause the lateral component of the wave  vector is not quantized. However, if the \nspin -flip process is highly anisotropic, such that the spin relaxation rate is strongly \nenhanced when scatter ing between  momentum states perpendicular to the film \nplane, then quantization in the perpendicular direction could still lead to a \nsuppression of the spin -relaxation rate. In other words, quantum confinement for a \nthin film can be important  only  if the in tra-band scattering matrix element is zero, \ni.e., only momentum perpendicular to the film is lost in a spin -scattering event. In \nprinciple, this might be true if the spin -scattering process in the NM is renormalized 26 by the spin -Hall effect, whereby a pure spin -current flowing perpendicular to the \nFM/NM interface is converted into a charge current flowing parallel to the interface. \nIf this is true, in -plane and out -of-plane FMR measurements using our technique will \nresult in different spin diffusion lengths for the same material. Both Pt an d Pd \nexhibit a significant spin -Hall effect [ 63], which suggests that such specul ation is not \nwithout physical basis. If we use the well -known result \n2\nFk m L\n  for one-\ndimensional  confinement of a free electron at the Fermi level, where L is the film \nthickness, then quantum confinement is important for \n4.3L  nm in the case of Pt, \nand \n74L  nm in case of Pd. The increase in \nsf  with decreasing Pt -thickness in Fig. \n4 appears to occur  close to the critical thickness for quantum confinement, but the \nestimate f or Pd vastly overestimates what is observed.  27  \nD. Conclusions  \n \nWe have measured spin - and charge -transport properties of Pd and Pt using  \nFMR measurements. In particular, we have used the increase in damping as a result \nof spin -pumping  to probe the flow of spin -current from the FM probe layer . We \npresented a matrix method to facil itate rapid formulation of the spin -back -flow \nparameter in arbitrary, one -dimensional multilayer stacks. We used precise \nmeasurements of the NM conductivity  in conjunction with the FMR me asurements  \nto determine  spin -diffusion  length , the FM/NM  spin -mixing  conductance, and the \ninterfacial spin -flip parameter, all in the context of three different models for the \ncorrelation of charge - and spin -transport properties. All the data are most \naccu rately fitted with a model where \nsf1 , in sharp contrast to the usual  \n \nFigure 4: Calculated spin (red) and momentum (blue) scattering times for the \nexperimental structures and extracted parameters from Model 2, for (a) Pd on \nCu, (b) Pt on Cu, (c) Pd in direct contact with Py , and (d) Pt in direct contact \nwith Py.  \n28 presumption that E -Y processes should dominate for spin -relaxation in transition \nmetals.  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In our experiments we apply a static m agnetic \nfield at an angle θ with respect to the wire, while the microwave current, which is responsible for driving \nFMR, is always flowing along the wire. For different θs we find a very similar behavior for both microwave -\nabsorption and electrically -detec ted FMR – the resonance magnetic field follows a simple “1/cos( θ)\" \ndependence. This simple behavior highlights the importance of the relative orientation between the driving \ncurrent and magnetic field. We also investigated the dependence of the electricall y detected FMR on dc and \nrf (microwave) current magnitudes. As expected, the resonance signal increases linearly with both the applied \ndc current and the microwave power.   \nI. Introduction  \nFerromagnetic resonance (FMR) is a powerful method to study magnetic dy namics in  various  \nmedia, from bulk magnetic materials to nanoscale heterostructures [1]. The resonance occurs when \nthe natural precession frequency of magnetization in the media matches the frequency of an \nexternally applied rf magnetic field. The latter is often generated by an applied rf (microwave ) \ncurrent flowing through the media. That makes the direction of this driving current an impo rtant \nparameter in FMR experiments.  \nIn this paper we study the dependence of FMR on the relative orientation between the driving \ncurrent and applied magnetic field by two detection techniques. First, FMR is detected traditionally \nby measuring the absorpti on of applied microwaves. Second, FMR is detected electrically. Here \nthe precession of magnetization driven by microwaves produces variations in the media’s \nresistance via mechanisms like anisotropic magnetoresistance (AMR), anomalous Hall effect \n(AHE), tu nneling magnetoresistance (TMR), or inverse spin -Hall effect (ISHE) [2, 3]. The se \nresistance variations , in turn,  produce a rectified dc voltage (photovoltage) that  can be detected \nelectrically . The latter  enabl es the electrical detection of FMR , which  has been widely used to \nstudy magnetization and spin dynamics in magnetic nanostructures over the past decade [4, 5].   \nII.  Methods  \nIn our experiments we use a Ni 36Fe64 wire with diameter 50 μm  (Goodfellow FE025100 ). The \n1.3 mm long wire terminates a coaxial cable used to deliver dc and rf (microwave) current s to the \nwire. The microwave current produces  a circumferential  (rf) Oersted  field that generates a torque \non the sample’s magnetization and drive s FMR  [6, 7] . In our experimental setup b oth dc (current \nsource  and voltmeter) and rf (microwave generator  and power sensor ) electronics were connected 2 \n to the wire using a bias tee as schematically shown in Fig. 1 . The power  sensor ( Keysight U2002A ) \nis used to detect the microwave power  reflected from the  wire. The dc voltmeter (Keithley 2182A ) \nis used to detect the dc voltage across the wire, which  includes a small rectified voltage  \n(photovoltage ) produced by microwaves . Both the microwave power  and dc voltage were \nmeasured as a function of external magnetic field (up to 0. 7 T) applied at an angle θH with respect \nto the wire  (current direction) . The angle was varied from 0 -360 degrees .  \nIII. Modeling  \nKittel’s model [1] is routinely used to describe FMR in ferro magnetic media  of different \ngeometry . Our wire diameter (50 μm ) is much larger than the electromagnetic skin depth  (~1 μm) \n[8]. Therefore, it is safe to assume that  the microwave  current is confined to a thin layer under the \nwire surface . For an external field applied along the wire, t his is equivalent to the case of a thin \nfilm in a parallel magnetic field  for which the Kittel’s resonance condition is [1, 8]:  \n𝜔\n𝛾=√𝐻𝑟𝑒𝑠(𝐻𝑟𝑒𝑠+4𝜋𝑀𝑠)  (1) \nwhere  is the rf frequency,  the gyromagnetic ratio , Ms the saturation magnetization , and Hres \nthe resonance field. For an external field applied perpendicular to  the wire, the resonance condition \nbecomes :  \n𝜔\n𝛾=𝐻𝑟𝑒𝑠−4𝜋𝑀𝑠  (2) \nFinally, f or a static field applied at an arbitrary angle θH to the wire , an elaborated Kittel’s model \n[9] predicts  the following dispersion relation between  and Hres:  \n(𝜔\n4𝜋𝛾𝑀𝑠)2\n=[(𝐻𝑟𝑒𝑠\n4𝜋𝑀𝑠)𝑐𝑜𝑠(𝜃𝐻−𝜃𝑀)+𝑐𝑜𝑠(2𝜃𝑀)][(𝐻𝑟𝑒𝑠\n4𝜋𝑀𝑠)𝑐𝑜𝑠(𝜃𝐻−𝜃𝑀)−𝑠𝑖𝑛2(𝜃𝑀)] (3) \nwhere  θM is the angle between Ms and the wire (see Fig. 1 ), which  can be found from : \n(𝐻\n4𝜋𝑀𝑠)𝑠𝑖𝑛(𝜃𝐻−𝜃𝑀)=𝑐𝑜𝑠(𝜃𝐻)𝑠𝑖𝑛(𝜃𝑀) (4) \nUsually,  the magnetization angle 𝜃𝑀 lags behind the applied magnetic field angle 𝜃𝐻, except in \nthe two special cases:  θM = θH = 0° (Eq. 1) and θM = θH = 90°  (Eq. 2) where they are aligned with \neach other.  \nAlternatively, we will use  a simple ‘cos’ model to describe FMR, which highlights the \nimportance of the driving current direction . In our experiment  (Fig. 1), the microwave current Irf \nis flowing along the wire and produces a circumferential  (rf) magnetic field hrf, which drives FMR \nand is always perpendicular to the wire. We assume that the parallel pumping is negligible and  \nonly the perpendicular (to hrf) component of the wire magnetization can be driven into FMR. We \nfurther assume that the magnetization is always aligned with the applied  magnetic field , i.e., θM = \nθH. Then only the perpendicular to hrf (parallel to wire) component of the magnetic field ( HcosθH) \nwill contribute to FMR and the resonance condition  reduce s to Eq. 1 with Hres replaced by HcosθH. \nTherefore, we refer to such a model as the ‘ cos’ model.  3 \n Figure 2 show s the FMR dispersion relations at different θH predicted by the two models  – \nKittel’s model and ‘cos’ model . The dispersions  predicted by two models are quite different for  \nmagnetic fields larger than the saturation magnetization (𝐻𝑟𝑒𝑠\n4𝜋𝑀𝑠>1). However, at relatively small  \nfields (𝐻𝑟𝑒𝑠\n4𝜋𝑀𝑠<1), the predictions are  very close  to each other.  \nThe electrical detection of FMR [6, 7, 10 ] is based on rectifi cation  and frequency mixing \nproperties of our wire . Its current -dependent  mixing characteristics can be described by assuming \nthat the current I = I dc + irf cos(ωt) through the wire has a dc and rf (microwave ) components. Then \nthe rectification properties of the wire can be  found by expanding the resulting voltage V(I) across \nthe wire about the bias current Idc. Mixing of a time -dependent  component of the wire resistance \nR=V/I with the microwave current irf cos(ωt) contributes a dc (photovoltage) term to V(I): \n𝑉~𝑖𝑟𝑓2(𝑑2𝑉𝑑𝐼2⁄)𝐼𝑑𝑐  (5) \nwhich in the simplest case is ~𝑖𝑟𝑓2𝐼𝑑𝑐 [10] and suggests that the resulting photovoltage should \nincrease linearly with the applied rf power ( ~𝑖𝑟𝑓2) and the applied dc bias current Idc. \nIV. Results and Discussion  \nFigure 3a shows FMR absorption  and photovoltage spectra at θH=0° for a constant applied \nfrequency  (10.25 GHz ) and power (17 dBm ) at the source . The green trace is the raw data of the \nmicrowave power reflected from the wire as a function of magnetic field. It displays a dip in the \npower at ±0.1 T which corresponds to the maximum absorption of microwaves in the wire at FMR. \nThe blue trace is the dc photo voltage V induced by the microwaves  (Idc= -10 mA ). It is the \ndifference between dc voltage s across the wire with V(I dc + irf) and without V(I dc) microwaves. \nThe voltage signal  V(I dc) without microwaves is shown in black and combines an AMR peak at \nzero field and linear magnetoresistance at higher fields. The absorption  and photovoltage  spectra \nin Fig. 3a are very similar with some differences at low fields and a higher level of noise in the \nvoltage data. The dip in reflected power at ±0.1 T correlates well with the minimum in \nphotovoltage and can be described by the Kittel’s FMR  condition  at θH=0° (Eq. 1) assuming \n4Ms=1.33 T and =27.8 GHz/T.  \nFigure 3 b shows the FMR dispersion  relation between the applied frequency and the resonance \nmagnetic field . Solid symbols show the experimental data for different θH = 0, 30, 40, 50, 60, 74, \n80, 84, 86° (color coded). Dashed curves are the ‘cos’ model  fits. The fitting angles (θ = 0, 28, 37, \n46, 55, 69, 76, 80, 83°) are within a few degrees of θH that is consistent with the experimental \naccuracy of  determining  θH=0° ( a few degrees)  and may also suggest that the magnetization \ndirection (θM) lags behind the applied magnetic field  by 2-4°. \nWe have investigated the dependence of the electrically detected FMR on dc and rf \n(microwave) current magnitudes. Figure 4 shows  the peak FMR photovoltage vs dc current at a \nfixed rf power P=21 dBm (Fig. 4a) and vs rf power at a fixed Idc=10 mA  (Fig. 4b) . The linear fits \n(dashed lines ) confirm that the resonance signal increases linearly with both the applied dc current \nand the microwave power  as expected from Eq. 5 .  4 \n Figure 5 shows the angular dependence of FMR. 2D gray -scale plots show the FMR ab sorption \n(Fig. 5a) and photovoltage ( Fig. 5b) spectra as a function of θH. Lighter color indicates higher \npower/voltage. The blue and green lines indicate positions of the θH=0 spectra from Fig. 3a. The \nred curve is the ‘ cos’ model fit. Both absorption and photovoltage spectra  show very similar \nbehavior s as a function of θH. The similarities of the absorption  and photovoltage spectra in Fig. \n3a and their respective angular dependencies in Figs. 5a and 5b suggest that the electrical detection \nof FMR is essentially equivalent to the traditional absorption measurements.  \nFigure 6 highlights  the angular dependence of the resonance field . The field increases with \nincreasing θH. Both the Kittel's model fit (solid curve  in Fig. 6 ) and the ‘cos’ model fit (dashed \ncurve)  are consistent with this behavior and display  some deviations from the  experimental data \n(red symbols)  only for very large angles (close to θH=±90°). This result suggests that the simple \n‘cos’ model can capture the essence of the angular dependence  by assu ming that only the parallel \n(to the wire ) component of magnetic field ( HcosθH) contributes to FMR . In contrast , the Kittel’s \nmodel  expla ins the increase of the resonance field by an opposing demagnetizing field , which \nappears when the magnetization has a  component  perpendicular to the wire. In order to test  this \nhypothesis experimentally we have performed FMR measurements in an alternative geometry , \nwhere the demagnetizing field is expected to have a minimal effect on the resonance field.  \nFigure 7 shows  the angular dependence of the current -driven FMR  in a 0.1 thick Fe foil . The \nrectangular s hape of the foil (1.3 mm  0.7 mm)  was chosen  to (i) minimize the demagnetizing \neffects  and (ii) limit the current’s spreading . The applied magnetic field was rotated in the plane \nof the foil (see insert to Fig. 7). The 2D gray -scale plot in Fig. 7 shows the absorption spectra as a \nfunction of θH and the ‘cos’ model fit  (red curve) . An increase of t he resonance field for angles \nclose to θH=±90° is obvious  despite a significant broadening of the resonance  due to the current \nspreading  (compare with Fig. 5a) . In this rectangular geometry  the demagnetizing effects  are \nexpected to be much smaller  than tho se in the wire , that plays in favor of the ‘cos’ model  and \nhighlights the importance of the relative orientation between the driving current and magnetic \nfield.  \nV. Summary  \nWe have  experimentally investigate d the angular dependence of the electrically  driven FMR \nin Ni36Fe64 wires. Two FMR detection techniques were used: traditional microwave -absorption \nand electrical  detection . Both techniques showed very similar results both in terms of the FMR \nline shape and the angular dependence of the resonance field. The resonance field was found to \nincrease significantly when the field direction approaches the perpendicular -to-wire geometry \n(θH=±90°). We have exploited two models – the Kittel's model and the ‘cos’ model  – to fit the \nexperimental data. Both models are consistent with experimental observations and display  some \ndeviations from the  data only for very large angles (close to θH=±90°). We have performed a test \nexperiment with a rectangular Fe foil to distinguish between the models. The resulting ‘cos’-like \nangular dependence supports the ‘cos’ model  and highlights the importance of the relative \norientation between the driving current an d magnetic field .  5 \n References  \n \n[1] C. Kittel, Introduction to Solid State Physics , 8th ed (Wiley, Hoboken, NJ, 2005).  \n[2] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C. -M. Hu, Analysis of the Line Shape of \nElectrically Detected Ferromagnetic Resonance , Phys. Rev. B 84, 054423 (2011).  \n[3] Y. Zhang, X. S. Wang, H. Y. Yuan, S. S. Kang, H. W. Zhang, and X. R. Wang, Dynamic \nMagnetic Susceptibility and Electrical Detection of Ferromagnetic Resonance , J. Phys.: \nCondens. Matter  29, 095806 (2017).  \n[4] M. Tsoi, A. G. M. Jansen, and J. Bass, Generation and Detection of Phase -Coherent \nCurrent -Driven Magnons in Magnetic Multilayers , Nature 406, 46 (2000).  \n[5] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, \nand D. C. Ralph, Microwave Oscillations of a Nanomagn et Driven by a Spin -Polarized \nCurrent , Nature 425, 380 (2003).  \n[6] T. Staudacher and M. Tsoi, Spin-Torque -Driven Ferromagnetic Resonance in Point \nContacts , J. Appl. Phys. 109, 07C912 (2011).  \n[7] C. Wang, H. Seinige, and M. Tsoi, Ferromagnetic Resonance Driven by an  Ac Current: A \nBrief Review , Low Temp . Phys . 39, 247 (2013) . \n[8] L. Kraus, Ferromagnetic resonance in individual wires: from micro - to nanowires  (pp 449-\n486, in “Magnetic Nano - and Microwires: Design, Synthesis, Properties and Applications ”, \nEditor: Manuel  Vázquez , Elsevier , 2015).  \n[9] K. Ando, Y. Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh, \nAngular Dependence of Inverse Spin –Hall Effect Induced by Spin Pumping Investigated in a \nNi81Fe19/Pt Thin Film , Phys. Rev. B 78, 014413 (2008).   \n[10] H. Seinige, C. Wang, and M. Tsoi, Ferromagnetic resonance detection by a point -contact \nbolometer , Proc. SPIE  8813 , 88131K (2013).  \n  6 \n Figure Captions  \n \nFigure 1 . Experimental setup. DC (current source and voltmeter) and RF (microwave generator \nand power sensor) electronics are connected to the Ni36Fe64 wire using a bias tee .  \nFigure 2 . FMR dispersion. The comparison of the predictions by the Kittel's model (solid c urves) \nand the ‘cos’ model (dashed curves) for different θH from 0 -89. \nFigure 3 . (a) FMR photovoltage (blue) and absorption (green) spectra at θH=0. The black trace \nshows the dc voltage across the wire with out microwaves . (b) FMR dispersion. \nExperimental data (solid symbols) and corresponding ‘cos’ model  fits (dashed curves) \nfor different θH (color coded) .  \nFigure 4 . The peak FMR photovoltage vs (a) dc current at a fixed rf power P=21 dBm and ( b) vs \nrf power at a fixed Idc=10 mA. The insert shows the same data vs power in dBm. The \ndashed lines are linear fits.   \nFigure 5. 2D gray -scale plots show the FMR absor ption ( a) and photovoltage ( b) spectra as a \nfunction of the magnetic field angle θH. Lighter color indicates higher power/voltage. \nThe blue and green lines indicate positions of the θH=0 spectra from Fig. 3a. The red \ncurve is the ‘ cos’ model fit.  \nFigure 6. The angular dependence of the resonance field:  Kittel's model (solid curve), ‘ cos’ \nmodel (dashed curve), and experimental data (red symbols).  \nFigure 7. Angular dependence of FMR in Fe foil . 2D gray -scale plot show s the absorption spectra \nas a function of θH. Lighter color indicates higher power.  The red curve is the ‘ cos’ \nmodel fit. The insert shows experimental s chematic : a 0.1 mm thick  Fe foil (1.3  mm  \n0.7 mm) with an in-plane magnetic field applied at an angle θH with the rf current .  \n \n  7 \n  \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 \n \n \n \n \n8 \n  \n \n \n \n \n \n \n \n \n \nFig. 2 \n \n \n \n \n \n9 \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3 \n \n \n \n \n \n10 \n  \n \n \n \n \n \n \n \n \n \n \n \nFig. 4 \n \n \n \n \n \n \n \n11 \n  \n \n \n \n \n \n \n \n \n \n \n \nFig. 5 \n \n \n \n \n \n12 \n  \n \n \n \n \n \n \n \n \n \nFig. 6 \n \n \n \n \n \n13 \n  \n \n \n \n \n \n \n \n \nFig. 7 \n \n \n \n"    },    {        "title": "1602.08377v1.Tuning_the_magnetocrystalline_anisotropy_in__R_CoPO_by_means_of__R__substitution__a_ferromagnetic_resonance_study.pdf",        "content": "Tuning the magnetocrystalline anisotropy in RCoPO by means of Rsubstitution: a\nferromagnetic resonance study.\nG. Prando,1, 2,\u0003A. Alfonsov,2A. Pal,3, 4V. P. S. Awana,3B. B uchner,1, 2and V. Kataev2\n1Center for Transport and Devices of Emergent Materials,\nTechnische Universit at Dresden, D-01062 Dresden, Germany\n2Leibniz-Institut f ur Festk orper- und Werksto\u000bforschung (IFW) Dresden, D-01171 Dresden, Germany\n3National Physical Laboratory (CSIR), New Delhi 110012, India\n4Department of Physics, Indian Institute of Science, Bangalore 560012, India\n(Dated: June 19, 2018)\nWe report on broad-band electron spin resonance measurements performed within the itinerant\nferromagnetic phase of RCoPO (R= La, Pr, Nd and Sm). We reveal that the Rsubstitution is\nhighly e\u000bective in gradually introducing a sizeable easy-plane magnetocrystalline anisotropy within\nthe Co sublattice. We explain our results in terms of a subtle interplay of structural e\u000bects and of\nindirect interactions between the fanddorbitals from Rand Co, respectively.\nPACS numbers: 75.50.Cc, 76.30.-v, 76.50.+g\nI. INTRODUCTION\nThe interest for RMX O oxides (R,MandXbeing\nrare-earth, transition metal and pnictide ions, respec-\ntively) has arisen dramatically after the recent discov-\nery of high- TcSC (superconductivity) in this class of\nlayered materials.1{3The prototype RFeAsO 1\u0000xFxsys-\ntems reach remarkable Tcvalues higher than 50 K,3{5\nwhile lower Tc's are typically achieved for the di\u000berent\ncompositions RFe1\u0000xCoxAsO.5{12Both the O 1\u0000xFxand\nthe Fe 1\u0000xCoxdilutions nominally introduce one extra-\nelectron per substituted ion (however, see Refs. 13 and\n14) leading to SC for values x\u00180:05\u00000:1.5Still, it is\nvery interesting to consider how the electronic ground\nstate evolves in the opposite limit x!1, where SC\nis completely suppressed. An itinerant FM (ferromag-\nnetic) phase is achieved in RCoAsO and RCoPO, with\nan ordered magnetic moment per Co ion in saturation\nstrongly suppressed if compared to its value in the para-\nmagnetic regime.15While itinerant ferromagnetism can\nbe predicted for these materials by means of ab-initio\ncomputations,15,16a detailed investigation of their prop-\nerties can possibly lead to interesting insights also in the\nsuperconducting state in view of the closeness of these\ntwo ground states in the phase diagram.\nIn particular, this is the case for the impact of di\u000berent\nRions on the whole electronic properties of the systems.\nPreviously,17,18we showed by means of \u0016+SR (muon spin\nspectroscopy) that the felectronic degrees of freedom as-\nsociated with Rions do not play an active role in RCoPO\nbut, on the contrary, Rions should be thought as \\pas-\nsive\" sources of chemical pressure which ultimately tune\nTC, i. e., the FM transition temperature.17,18As further\ncon\frmation of the crucial importance of structural ef-\nfects, we also demonstrated the full equivalence of chemi-\ncal and external pressures on a quantitative level as long\nasTCis considered.17,18These results can be interest-\ning in view of the analogy with superconducting samples\nand, in particular, with the strong dependence of the Tcvalue on the actual Rion at optimal doping.4,19It should\nbe stressed that we could not demonstrate a full analogy\nbetween chemical and external pressures for supercon-\nducting samples, as here quenched disorder contributes\nin a complicated and non-negligible way as well.20\nIn this paper we report on ESR (electron spin res-\nonance) measurements performed in the FM phase of\nRCoPO (R= La, Pr, Nd and Sm). We analyse the ESR\nsignal in a wide range of temperature ( T), magnetic \feld\n(H) and frequency of the employed microwave electro-\nmagnetic radiation ( \u0017). We observe for all the samples a\nclear crossover from a high- Tparamagnetic region, where\nthe ESR line shows a Dysonian distortion, to a low- Tre-\ngion, where the ESR line arises instead from the macro-\nscopic magnetization of the whole sample (FMR, ferro-\nmagnetic resonance). Remarkably, within the FM phase,\nwe unambiguously detect the gradual development of a\nsizeable easy-plane magnetocrystalline anisotropy upon\nincreasing chemical pressure. We discuss our experimen-\ntal results in the light both of the distortion of the local\ntetrahedral crystalline surroundings of Co ions and of the\nanisotropic properties introduced by the strong indirect\ninteraction between fanddelectronic degrees of freedom\nfromRand Co orbitals, respectively.\nII. EXPERIMENTAL DETAILS\nA. Samples' characterization\nWe reported details about the synthesis of polycrys-\ntallineRCoPO (R= La, Pr, Nd, Sm) in our previous\npublications, together with thorough investigations of the\nconsidered samples by means of dc magnetometry and ZF\n(zero-\feld) \u0016+SR under pressure.17,18In this paper we\ndiscuss ESR measurements performed on the same sam-\nples already studied by means of the other techniques\nmentioned above. In the whole text, we refer to ground\npowders composed to a \frst approximation of sphericalarXiv:1602.08377v1  [cond-mat.str-el]  26 Feb 20162\ngrains with similar dimensions for all the samples. The\npowders were embedded in Double Bubble 2-part epoxy\n(Loctite) for the aim of avoiding sample movement and\ngrain re-orientation triggered by H, i. e., preserving con-\nstant powder-average properties of the ESR signal for all\nthe accessed experimental conditions.\nWe measured M(macroscopic magnetization) for the\nfour samples as a function of Tat \fxed sample-dependent\nvalues ofHby means of a Magnetic Property Measure-\nment System based on a superconducting quantum inter-\nference device (by Quantum Design).\nB. Electron spin resonance\nWe performed continuous wave ESR measurements at\n\fxed\u0017while sweeping H. For this aim, we employed two\ndi\u000berent experimental con\fgurations.\n1. X-band regime\nWe accessed the low-frequency regime ( \u0017'9:56 GHz)\nby means of a commercial Bruker EMX X-band spec-\ntrometer equipped with an Oxford Instruments ESR900\ncontinuous4He \row cryostat ( T= 4:2\u0000300 K). Mea-\nsurements were always performed upon gradually warm-\ning the sample from the lowest accessed Tvalue after\na zero-\feld cooling protocol. Standing electromagnetic\nmicrowaves were induced in a rectangular cavity (Bruker\nX-band resonator ER4104OR, TE 102mode). The sample\nwas placed in the cavity's centre where the Hcomponent\nis maximum and we measured the P(power) resonantly\nFIG. 1: (Color online) Representative result for the \frst-\nderivative X-band ESR measurements (squares) together with\nthe \ftting curve (black continuous line) according to Eqs. (1),\n(2) and (3). First-derivative data are numerically-integrated\nto obtain the actual P(H) behaviour (red continuous line),\nwhich is enlarged in the inset showing the de\fned resonance\n\feldHrand the two half-height \feld values HlandHh.absorbed by it as a function of H(t) =H0+Ha(t). Here,\nthe quasi-static component H0was in the range 0 \u00009\nkOe and it was swept with a typical rate \u001850 Oe/s. Si-\nmultaneously, the t(time) dependent \feld Ha(jHaj\u001420\nOe) was sinusoidally-modulated with frequency 100 kHz\nand superimposed to H0. By means of a lock-in detec-\ntion at the modulation frequency, we directly measured\nthe \frst derivative d P/dHrather than P(see Fig. 1 for\na representative experimental curve).\nWe \ftted the d P/dHdata by means of the expression\nf(H) =pdfAbs\nL(H)\ndH+ (1\u0000p)dfDisp\nL(H)\ndH+rH+q;\n(1)\nwhere the coe\u000ecients randqallow for a small linear\nbackground while\nfAbs\nL(H) =AL\n\u0019\"\n\u0000L\n\u00002\nL+ (H\u0000HrL)2\n+\u0000L\n\u00002\nL+ (H+HrL)2#\n(2)\nand\nfDisp\nL(H) =AL\n\u0019\"\n(H\u0000HrL)\n\u00002\nL+ (H\u0000HrL)2\n+(H+HrL)\n\u00002\nL+ (H+HrL)2#\n(3)\nare the absorptive (Abs) and dispersive (Disp) compo-\nnents of the employed Lorentzian model (hence the sub-\nscript L) weighted by the parameter 0 \u0014p\u00141.21{24\nHere,ALis the signal amplitude and HrLthe resonance\n\feld, while for the linewidth the relation \u0000 L= \u0001H=2\nholds with \u0001 Hrepresenting the FWHM (full width at\nhalf maximum). Eqs. (2) and (3) already incorporate the\ncontribution from negative magnetic \felds arising from\nthe linear polarization of the electromagnetic radiation\nin the cavity.23,25This correction is mostly relevant for\nbroad ESR lines, namely whenever \u0000 L&HrL.\nThe choice of Eqs. (1), (2) and (3) gives excellent \ft-\nting results in LaCoPO at all Tvalues, except for a\nnarrow region around the onset of the long-range or-\ndered FM phase where the signal is slightly distorted.\nSimilar distortion e\u000bects around the ordering tempera-\ntures of magnetic phases have been reported before for\nother materials.26For highTvalues, \fts by Eqs. (1), (2)\nand (3) still yield to excellent results also in the case of\nPrCoPO, NdCoPO and SmCoPO (see in Sect. III). How-\never the situation for these materials is di\u000berent in the\nwhole low-TFM regime, where the signal is always so dis-\ntorted that it can not be \ftted properly. For this reason,\nwe took an alternative empirical approach to data analy-\nsis. In particular, we numerically-integrated the d P/dH\ndata to give the actual P(H) behaviour, from which we3\nextracted important quantities such as\nI(T) =Z+1\n0PT(H)dH; (4)\nnamely the integrated intensity of the ESR signal at \fxed\nT, and the characteristic \feld values Hr(resonance \feld),\nHlandHh(half-height \felds) de\fned as shown in the\ninset of Fig. 1. Accordingly, we de\fned the FWHM as\n\u0001H\u0011Hh\u0000Hland introduced the empirical parameter\n\u0011\u0011Hh\u0000Hr\nHr\u0000Hl(5)\nto quantify the half-width asymmetry of the ESR line. In\nparticular, \u0011= 1 corresponds to a symmetric line with\nrespect toHr, while\u0011 > 1 is found when experimental\nlines are broadened on the high-\felds side.\nAs is well-known, an asymmetry ( \u0011 > 1) of the ESR\nline may have di\u000berent physical origins. One possibility\nis the so-called Dysonian distortion typical of metallic\nsamples.27Here, the impinging electromagnetic radiation\nis mostly screened and it penetrates the material over\nthe skin-depth \u000es.28Accordingly, the resonance process\nonly takes place in the non-screened fraction of the sam-\nple, namely within \u000es. The resonance signal may arise\nboth from localized magnetic moments interspersed in\nthe metallic background and from conduction electrons\nthemselves. In the latter case, two main characteristic\ntimes govern the resonance process, i. e., the intrinsic\ntransverse relaxation time of electrons Tesand the so-\ncalled di\u000busion time TD=\u000e2\ns=D.27Here,\u000esis the length-\nscale of interest for the electron di\u000busion while Drepre-\nsents a constant characteristic of the process. When the\nelectron di\u000busion can be neglected, i. e., when\n1\nTes\u001dD\u000e\u00002\ns; (6)\nthe absorbed Pcan be conventionally expressed in terms\nof the sample impedance. In the ideal case of metallic\nspherical grains with diameter d, the condition \u000es\u001dd\nimpliesP\u0018\u001f00(bulk-impedance limit) while P\u0018\n(m\u001f0+n\u001f00) holds with m=nin the opposite limit\n\u000es\u001cd(surface-impedance limit), with \u001f0(\u001f00) the real\n(imaginary) component of the magnetic susceptibility.27\nWhen a conventional Lorentzian relaxation process is be-\ning considered, the former condition implies A=jBj'1\nfor the ratio of the two quantities de\fned in the main\npanel of Fig. 1, while the latter condition typically re-\nsults inA=jBj'2:55 and, accordingly, in a broadening\nofP(H) on the high-\felds side.27,29\nOn the other hand, anisotropic magnetic properties\ngenerally cause an inhomogeneous broadening of mag-\nnetic resonance lines for randomly-oriented powders as\nwell.30It should be recalled that only the anisotropy-\nbased distortion would still be detected in case the exper-\nimental apparatus allowed one to independently measure\n\u001f0and\u001f00. While this is not feasible with our X-band in-\nstrumentation, we could successfully disentangle the two\nsignals by means of a di\u000berent setup, as discussed below.2. High-frequency/high-\feld regime\nWe performed ESR measurements at higher \u0017andH0\nat selected Tvalues by means of a home-made spec-\ntrometer based on a PNA network analyser N5227A\n(Keysight Technologies), generating and detecting mi-\ncrowaves with broad-band tunable frequency \u0017= 10 MHz\n\u000067 GHz. We extended the upper \u0017limit to 330 GHz by\nmeans of complementary millimiter-wave modules (Vir-\nginia Diodes, Inc.). We also accessed the 20 GHz \u0000\n30 GHz regime by means of a home-made spectrometer\nbased on a MVNA vector network analyser (AB Millime-\ntre). We performed measurements at selected Tvalues in\na transmission-con\fguration31by exploiting gold-plated\ncopper mirrors, German silver waveguides and brass con-\ncentrators to properly focus the radiation on the sample.\nWe could generate quasi-static H0values up to 160 kOe\n(with a typical ramping rate \u0018150 Oe/s) by means of a\nsuperconducting solenoid (Oxford Instruments) equipped\nwith a4He variable temperature insert.\nThis experimental setup allowed us to directly measure\nthe complex impedance of the whole system (sample and\nwaveguides) and to associate anomalies induced by H0\nto the resonant Pabsorption in the sample. Di\u000berently\nfrom the X-band setup, the network analyser allowed us\nto disentangle real and imaginary components of the sig-\nnal, i. e., dispersive and absorptive components of the\nsample's uniform magnetic susceptibility \u001f(see Fig. 2\nfor a representative experimental curve). After a proper\nbackground-subtraction and phase-correction, we de\fned\nHr,HlandHhfrom the absorptive component, analo-\ngously to the case of X-band (see the inset of Fig. 1).\nFIG. 2: (Color online) Representative results for high-\n\feld ESR, obtained after proper background-subtraction and\nphase-correction. The continuous lines are relative to a si-\nmultaneous best-\ft to the dispersive (Disp.) and absorptive\n(Abs.) components according to a Lorentzian model.4\nIII. RESULTS\nA. Summary of the main magnetic properties of\nRCoPO\nRCoPO materials display a metallic behaviour for the\nTdependence of the electrical resistivity (typical values\n\u00181\u000210\u00001m\n cm) with negligible qualitative and quan-\ntitative dependences on the actual Rion.15,32They ex-\nhibit interesting magnetic properties with the appearance\nof an itinerant FM phase below a characteristic critical\ntemperature TC. This FM state is understood in terms\nof a conventional Stoner criterion after computing D(EF)\n(density of states at the Fermi energy) which, as a result,\nis mainly of dcharacter and arising from Co orbitals.15,18\nIn a simple covalent picture, the valency of Co ions is 2+\nand the measured value for the ordered magnetic moment\nper Co ion is\u00180:3\u0016Bfor LaCoPO. This value slightly\ndecreases with decreasing rI(ionic radius of the Rion)\nor, equivalently, the equilibrium unit cell volume V{ see\nlater on in Tab. I. While density functional theory cal-\nculations are able to reproduce this trend, the absolute\nvalues typically overestimate the experimental ones by\na factor\u00181:7.15,18This is highly reminiscent of what is\nobserved for the isostructural oxides based on Fe, as asso-\nciated to the di\u000eculties in describing these materials only\nfrom a fully-itinerant or a fully-localized perspective.33{36\nWe observed a linear relation for the TCvs.Vtrend\nand, as already discussed based on ZF- \u0016+SR measure-\nments, we quantitatively veri\fed this dependence also\nFIG. 3: (Color online) Phase diagram of RCoPO after \u0016+SR\nin zero magnetic \feld.17,18Values ofTC(squares) and TN\n(circles) are reported as a function of the equilibrium unit cell\nvolume. The upper x-axis indicates external pressure and it\nis referred to TCvalues only. The continuous red curve is\na best-\ftting linear function to TCdata. The dashed blue\ncurve is a guide to the eye. The black hatched area denotes\nuncertainty about the emergence of the AF phase.with further decreasing Vby means of hydrostatic pres-\nsure, pointing to a one-to-one correspondence between\nchemical and external pressures in these materials (see\nFig. 3).17,18According to this picture, the active role of\nRions is limited to the generation of chemical pressure as\nlong as the itinerant FM phase is concerned. Otherwise\nsaid, thefelectronic degrees of freedom localized on the\nRions do not in\ruence TCsigni\fcantly.\nUpon gradually increasing the chemical pressure, a sec-\nond magnetic phase appears at lower Tvalues, below the\ncritical temperature TN. Here, the Co sublattice enters\nan AF (antiferromagnetic) phase, as marked by the sud-\nden vanishing of the macroscopic magnetization and by\nclear modi\fcations in the Mvs.Hhysteresis curves.18,32\nThe AF phase is observed in NdCoPO and SmCoPO but\nnot in LaCoPO and PrCoPO. Since the localized mag-\nnetic moments on Pr3+and Nd3+ions are comparable,37\nthis observation provides further evidence for the ine\u000bec-\ntiveness offelectronic degrees of freedom in driving the\noverall magnetic properties of RCoPO. Once in the AF\nstate, we observed a gradual orientation of the Nd3+and\nSm3+magnetic moments, giving rise to a peculiar Tde-\npendence of the internal magnetic \feld at the \u0016+site.18\nB. ESR. Low-frequency regime (X-Band)\nWith decreasing Tand for all the investigated com-\npounds, the onset for the detection of a well-de\fned ESR\nsignal isT\u001890\u0000120 K, i. e., well above the TCvalues\nestimated in zero magnetic \feld by means of \u0016+SR.\n1. Signal intensity\nWe show the behaviour of the integrated intensity I(T)\nfor the four samples in comparison to Min Fig. 4. We\nmeasured the latter quantity at sample-dependent Hval-\nues comparable to those of the central resonance \feld Hr\n(see later on). The good agreement between I(T) and\nMis an indication that the ESR signal is indeed intrin-\nsic for every sample and not associated to, e. g., extrinsic\nmagnetic impurities. In particular, we notice that I(T) is\nmonotonously increasing with decreasing Tin LaCoPO\nand PrCoPO. On the other hand, both MandI(T) go\nthrough a maximum at around \u001815 K and\u001840 K\nfor NdCoPO and SmCoPO (respectively), i. e., in cor-\nrespondence to the TNvalues detected by ZF- \u0016+SR. The\nfast suppression of I(T) in the AF phase for NdCoPO\nand, in particular, for SmCoPO is a clear indication that\nESR is actually probing the signal associated to the FM\nphase. For this reason, we will refer to FMR38,39rather\nthan ESR from now on. Moreover, in view of the gen-\neral arguments discussed above about RCoPO and after\nconsidering the fact that Co2+is the only source of mag-\nnetism in LaCoPO, we are con\fdent to assign the ob-\nserved signal to Co electrons for all samples. We notice\nthat a quantum mechanical treatment of the Co2+ion5\nFIG. 4: (Color online) A comparison of the ESR line intensity (squares) and dc magnetization (diamonds) is presented for each\nsample. The estimates of ZF- \u0016+SR data for TCandTNare indicated by the vertical dashed lines.17,18\nin a tetrahedral crystalline environment (strong-ligand-\n\feld approach) would lead to an orbital singlet ( S= 3=2)\nassociated with the upper t2gtriplet.40\nWe need to discuss SmCoPO data further. As men-\ntioned in Sect. II, P(H) curves are distorted for T <T C,\nmaking \ftting attempts according to Eqs. (1), (2) and\n(3) unsuccessful in this whole Tregion. Moreover, for\nT <T N, not only is I(T) sharply suppressed but the sig-\nnal is also sizeably shifted to low Hvalues. This means\nthat a non-negligible amount of the signal is located in\nnegative \felds for T <T Nand, as such, it is simply not\nmeasured (see Fig. 5). This e\u000bect implies that a numeri-\ncal integration of d P/dHdata is also no longer possible\nin this limit, as a wide section of the resulting P(H) curve\nwould take unphysical negative values for H > 2 kOe ir-\nrespective of any attempted background subtraction. We\nalso notice that the FMR signal for T <T Nlikely arises\nfrom an inhomogeneous state, where FM islands inter-\nspersed in an AF background still give a non-zero contri-\nbution to the overall FMR signal. This may introduce an\nadditional not-controllable signal distortion. For all the\nreasons mentioned above, our data analysis for SmCoPO\nis limited to 35 K as lowest temperature.\nFinally, we stress that in SmCoPO a second signal de-velops forT <T Nat highHvalues, as is shown in Fig. 5.\nThis is strongly shifting to even higher Hvalues with de-\nFIG. 5: (Color online) d P/dHcurves for SmCoPO for T <\nTN, displaying the appearance of a second signal possibly be-\nlonging to an AF resonance branch. Data in the inset are\nvertically shifted for the aim of better visualization.6\nFIG. 6: (Color online) Colour plots presenting the Tdependence of the P(H) curves normalized with respect to the maximum\nvalue observed at Hrfor the four investigated samples. The black dots denote the actual position of Hr.\ncreasingTand it is completely lost for T < 25 K. We\ninterpret this as a signature of the AF phase, possibly\nbelonging to an AF resonance branch. This signal will\nnot be discussed in the rest of the manuscript.\n2. Asymmetry of the FMR line\nAfter estimating I(T), we normalized the P(H) curves\nwith respect to the relative maximum value observed at\nHr. The results are shown for the four samples in Fig. 6,\nglobally displaying all the main features of the FMR line\nto be discussed in detail below. We \frst focus on the\nline asymmetry quanti\fed by the parameter \u0011, whose be-\nhaviour for the four samples is displayed in Fig. 7. In the\nhighTparamagnetic limit, i. e., for T\u001dTC,\u0011takes\nvalues\u00181:2\u00001:3 independently on the actual sample.\nAccordingly, the origin of the slight asymmetry in this\nlimit must be a common mechanism for all RCoPO ma-\nterials which, we argue, is associated with an incomplete\nDyson-like distortion [Eqs. (1), (2) and (3) lead indeed\nto excellent \ftting results for all the samples in this T\nregion]. On more physical grounds, as main argument\nin favour of the given interpretation, we notice that the\npowders' morphology is identical for the four samples\nand, moreover, no substantial di\u000berences are noticed for\ntheTdependence of the electrical resistivity.32As a con-\nsequence,\u000esanddhave similar values for all the samples.\nA quanti\fcation of the A=jBjratio (see Sect. II) leads to1:25\u00001:4 as typical values, which are quite far from 2 :55\nas expected in the case of full Dyson-like distortion with\na negligible e\u000bect of electrons' di\u000busion [or, equivalently,\nassuming that Eq. (6) is satis\fed]. We conclude that our\nsamples are in an intermediate regime \u000es.d. By con-\nsidering\u001a'0:2 m\n cm as typical value for the electrical\nFIG. 7: (Color online) Tdependence of the asymmetry pa-\nrameter.\u0011= 1 corresponds to a symmetric line with respect\nto its central resonance \feld Hr, while values \u0011 >1 is corre-\nsponding to lines broadened on the high-\felds side.7\nFIG. 8: (Color online) d P/dHnormalized data for the four\nsamples at the common value T= 45 K, safely above the AF\nphase for both NdCoPO and SmCoPO. In spite of the qualita-\ntive resemblance to Dysonian lines, none of the experimental\ncurves for PrCoPO, NdCoPO and SmCoPO can be precisely\n\ftted by Eqs. (1), (2) and (3).\nresistivity at 100 K,32we deduced&\u000es\u001810\u0016m, which\nshould be considered as a reasonable order of magnitude\nford.\nTheTdependence of \u0011for LaCoPO evidences a sharp\ncrossover at around T'55 K. In particular, below this\ntemperature, the FMR line gets perfectly symmetric with\n\u0011'1 (as also observed by eye in Fig. 1). We argue that\nthe signal for T&55 K arises from a set of moments with\nFM correlations, hence the applicability of the Dyson's\ntheory and the resulting distortion, though partial. On\nthe other hand, for T.50 K, the resonance signal is\nof collective nature, associated with an isotropic macro-\nscopic magnetization of the sample which is not subject\nto the microscopic origin of the Dysonian distortion. This\nis in agreement with previous reports on itinerant ferro-\nmagnets with sizeable magnetization values.29We also\nnotice that the onset of the FM state is not straightfor-\nward to locate precisely from these data alone but would\nbe for sure higher than TC= 33:2 K estimated by means\nof ZF-\u0016+SR. Considering that the current measurements\nare performed with a non-zero Hvalue, this result is con-\nsistent with the development of a FM phase.\nFor PrCoPO, NdCoPO and SmCoPO, decreasing T\nalso results in a clear departure from the \u0011\u00181:2\u00001:3\ncondition. However, for these materials, the behaviour\nis opposite if compared to LaCoPO, i. e., the FMR line\nasymmetry strongly increases. This is further displayed\nin Fig. 8 where we report d P/dHcurves for all the sam-\nples at the common value T= 45 K, i. e., safely above the\nAF phase for both NdCoPO and SmCoPO. As already\nmentioned above, there is no sign in the Tdependence of\nthe electrical resistivity for the four samples which could\nexplain the origin of this strong asymmetry in terms of\nDyson-like distortions. Moreover, despite the qualitative\nFIG. 9: (Color online) Temperature dependence of the Hr\n(empty symbols) and HrL(full symbols) values for the reso-\nnance \feld of the four samples.\nresemblance to Dysonian lines, none of the experimental\ncurves for PrCoPO, NdCoPO and SmCoPO can be pre-\ncisely \ftted by Eqs. (1), (2) and (3). We rather argue that\nthis e\u000bect should be associated to an intrinsic magne-\ntocrystalline anisotropy gradually developing within the\nFM phase of PrCoPO, NdCoPO and SmCoPO, leading\nto an inhomogeneous broadening of the FMR line.\n3. Resonance \feld, e\u000bective gfactor and linewidth\nAs is evident after inspecting Fig. 6, the central reso-\nnance \feld Hris not shifting for LaCoPO in the whole\naccessed experimental range but its Tdependence be-\ncomes gradually more and more marked when consider-\ning PrCoPO, then NdCoPO and \fnally SmCoPO. These\narguments are made clearer in Fig. 9. The origin of the T\ndependence of Hrcannot be ascribed to the development\nof an internal magnetic \feld within the FM phase, as this\ncannot explain the almost complete lack of any shift for\nLaCoPO. In this respect, it should be further stressed\nthat in LaCoPO the ordered magnetic moment per Co\nion even takes its strongest value within the considered\nsamples' series (see later on in Tab. I). Overall, this is\nanother indication that an increasing magnetocrystalline\nanisotropy is developing in RCoPO compounds while de-\ncreasing the volume of the unit cell V.\nA more detailed data analysis is needed in the param-\nagnetic regime where the signal distortion is arising af-\nter a Dyson-like mechanism. As is well known, a simple\nestimate of characteristic \felds as done in the inset of\nFig. 1 is indeed not accurate in the presence of a Dyso-\nnian distortion. In particular, this analysis introduces\nsystematic shifts in Hrwhich should be merely consid-\nered as artefacts.29,41A proper way of accounting for\nthese e\u000bects is to perform a conventional \ftting proce-\ndure of the Dysonian line in the high- Tregion by means8\nFIG. 10: (Color online) Temperature dependence of the \u0001 H\n(empty symbols) and 2\u0000 L(full symbols) values for the FWHM\nof the four samples. Vertical arrows de\fne the Tminvalues\ndiscussed in the text.\nof Eqs. (1), (2) and (3). Accordingly, with decreasing T,\nwe followed this strategy down to the point where the\ncontribution of the magnetocrystalline anisotropy starts\nto introduce a severe distortion in the FMR line. With\nfurther decreasing T, the line \ftting is no longer possible\nand we then refer to the more empirical data analysis\ndescribed in the inset of Fig. 1. As already mentioned\nin Sect. II, in LaCoPO a sizeable line distortion is only\nobserved around the onset of the FM phase, i. e., in the\nnarrow range 35 K .T.45 K and, accordingly, the two\n\ftting approaches are equivalent for T < 35 K. Still, we\nconsider the empirical approach of Fig. 1 (inset) in this\nTregion for consistency with the other samples.\nResults of both the analyses are presented in Fig. 9. A\ndiscrepancy between HrandHrLdata is con\frmed in the\nparamagnetic regime. Here, while Hrshows a marked de-\npendence on T,HrLtakes indeed a constant value for La-\nCoPO, PrCoPO and NdCoPO. The latter result re\rects\nthe intrinsic physical behaviour and it allows us to derive\nthe e\u000bective ge\u000bfactor values 2 :08\u00060:005, 2:05\u00060:005\nand 1:995\u00060:005 for Co2+in LaCoPO, PrCoPO and Nd-\nCoPO, respectively. All these values are far from the re-\nportedg0= 2:25\u00002:30 for Co2+in tetrahedral crystalline\nenvironments.40This estimate cannot be performed for\nSmCoPO in the accessed Trange, where HrLis still show-\ning a strong Tdependence in the paramagnetic regime,\nsuggesting a \rattening only at higher T.\nTheTdependence of the FWHM is displayed in Fig. 10\nfor the four samples. Similarly to Fig. 9, we report data\nfrom both the analysis procedures described above with\nthe same meaning of symbols (however, only 2\u0000 Ldata\nare reported in the paramagnetic regime for the aim of\nclarity). In LaCoPO, a fast decrease is observed with\ndecreasing Tin the paramagnetic regime until a mini-\nmum value \u0001 Hminis reached at T=Tmin. With further\ndecreasingT, the linewidth increases again with a much\nlower rate than in the paramagnetic regime. The ob-\nFIG. 11: (Color online) The \u0001 Hand 2\u0000 Ldata (already pre-\nsented in Fig. 10) are here reported after normalizations by\nTminand \u0001Hminvalues on the x-axis and on the y-axis, re-\nspectively. Inset: dependence of Tminfor the four samples as\na function of the TCvalues estimated by means of ZF- \u0016+SR.\nThe dashed line is a linear guide to the eye.\nserved result is in qualitative agreement with previous\nobservations in itinerant compounds with diluted mag-\nnetic moments even if, in these systems, the observed\nrates are opposite (i. e., slow decrease and fast increase\nabove and below Tmin, respectively).27,29,41A qualita-\ntively similar Tdependence of the FWHM is observed\nalso for PrCoPO, while a new feature emerges for Nd-\nCoPO. Here, below T'55 K, \u0001His further suppressed\nupon decreasing Tgiving rise to a local maximum. We\nargue that this additional feature is associated to the\nincreased magnetocrystalline anisotropy and, possibly,\nalso to an additional dynamical contribution associated\nwith the onset of antiferromagnetic correlations prelud-\ning to the AF phase. Finally, we stress that a similar\ne\u000bect is observed for SmCoPO as well. However, in this\ncompound, the strong e\u000bects of the magnetocrystalline\nanisotropy (and, possibly, of additional dynamical con-\ntributions) set in at much higher Tvalues, making the\noverall \u0001Hvs.Tbehaviour qualitatively di\u000berent from\nthe ones discussed above. Still, an in\rection point can\nbe distinguished at Tmin\u001890 K.\nIn Fig. 11, we report the data already presented in\nFig. 10 after normalization by Tminand \u0001Hminvalues\non thex-axis and on the y-axis, respectively (the mean-\ning of the used symbols is preserved). Remarkably, the\nnormalized experimental points collapse onto one single\nwell-de\fned trend for T=T min&1. At the same time,\nas shown in the inset, we notice that Tminlinearly cor-\nrelates with the TCvalues estimated by means of ZF-\n\u0016+SR. Accordingly, we deduce that the FWHM is inti-\nmately governed by the growing ferromagnetic correla-\ntions within the Co sublattice for T&Tminand that\nthese latter show similar properties for all the samples.\nAs already commented above, the deviations observed9\nFIG. 12: (Color online) Experimental FMR lines at compara-\nble frequencies for the four investigated samples at di\u000berent\nTvalues safely within the FM phase. The vertical dashed line\ndenotes the position of Hrfor LaCoPO. Curves are vertically\nshifted for the aim of clarity.\nforT.Tminshould be ascribed to di\u000berent contribu-\ntions from the magnetocrystalline anisotropy and, possi-\nbly, from dynamical e\u000bects preluding to the AF phase.\nC. ESR. High-frequency regime\nWe performed measurements in the high-frequency\nregime atTvalues selected in such a way that all the\nfour samples are properly tuned within the FM phase\n(see Fig. 3). A comparison of the observed FMR lines\nat comparable \u0017values is presented in Fig. 12. Here, we\nclearly observe that the asymmetric line broadening is\nsizeably increasing when substituting the R3+ion from\nLa3+to Pr3+, Nd3+and \fnally Sm3+. Accordingly, we\nmainly recognize a further indication of what we have\nalready argued above, namely that the Rsubstitution\ninRCoPO gradually induces an increasing magnetocrys-\ntalline anisotropy and, accordingly, an inhomogeneous\nbroadening of the powder-averaged FMR line. The line-\nshapes presented in Fig. 12 are highly reminiscent of\nhard-axis anisotropy limit,42as discussed in more detail\nin the next section.\nIV. DISCUSSION\nIn Fig. 13 we report Hrdata extracted from Fig. 12\nand from similar measurements performed at \fxed T\nand at several \u0017values. As enlightened in the inset of\nFig. 13, we observe a non-linear behaviour in the \u0017(Hr)\ntrends of PrCoPO, NdCoPO and SmCoPO for small \u0017\nvalues. We also recognize that the non-linearity of the \u0017\nvs.Hrdatasets is progressively increasing for PrCoPO,\nthen NdCoPO and \fnally SmCoPO. On the other hand,\nFIG. 13: (Color online) Hrdata for the four samples extracted\nfrom Fig. 12 and from similar measurements performed at\n\fxedTand at several \u0017values. Continuous lines are best \fts\nto experimental data according to Eq. (7). The inset shows\nan enlargement of data in the low \u0017regime.\nLaCoPO displays a linear behaviour over the whole ac-\ncessed experimental window. By referring to the theory\nof FMR,38,39this property of LaCoPO can be considered\nas an a posteriori con\frmation of our original assump-\ntion about the sample morphology, namely, the powder\nis composed of approximately spherical grains. Accord-\ningly, we can neglect the e\u000bect of demagnetization factors\n(shape anisotropy) on the actual \u0017(Hr) trend, assuming\nthat the same holds for the other compounds as well.\nIn the light of the observed phenomenology, we analyze\n\u0017(Hr) data by referring to a basic model for magnets with\nuniaxial symmetry.43Data in Fig. 13 are indeed highly\nreminiscent of the hard-axis (easy-plane) limit for the\nmagnetocrystalline anisotropy43\n\u0017?=\r\n2\u0019p\nH(H+jHAnj) (7)\nwhereHAnis an e\u000bective magnetic \feld quantifying the\nmagnetocrystalline anisotropy within the FM phase and\n\ris the gyromagnetic ratio. The expression\nHAn=2K\nM(8)\nrelates this latter parameter to the usual magne-\ntocrystalline anisotropy constant K < 0 (easy-plane\nanisotropy) via the sample magnetization.43,44Results\nof the \ftting procedure to experimental data are shown\nin Fig. 13, denoting an excellent agreement with the ex-\nperimental data upon properly setting \randHAnvalues.\nIt should be remarked that Eq. (7) is relative to one\nspeci\fc branch of the K < 0 limit and, in particular, to\nthe case of the external magnetic \feld lying within the\neasy plane ( H?z, wherezdenotes the hard axis). In10\nFIG. 14: (Color online) Qualitative illustration of the inho-\nmogeneous broadening of the FMR line of SmCoPO arising\nfrom grains with di\u000berent orientations with respect to the\nexternal magnetic \feld. The continuous black lines repre-\nsent the two branches described by Eqs. (7) and (9), with\n\r=2\u0019= 2:91\u000210\u00003GHz/Oe and HAn= 9 kOe obtained for\nSmCoPO from the \ftting procedure in Fig. 13.\nthe opposite case ( Hkz), one expects43\nH <jHAnj:\u0017k= 0 (9)\nH >jHAnj:\u0017k=\r\n2\u0019(H\u0000jHAnj):\nThe exemplary trends for both \u0017?and\u0017kare visualized\nin Fig. 14 as continuous lines, after selecting \r=2\u0019=\n2:91\u000210\u00003GHz/Oe and HAn= 9 kOe, i. e., the values\npreviously obtained from a \ftting procedure to SmCoPO\ndata. Fig. 14 also reports some selected experimental\ncurves for SmCoPO at di\u000berent \u0017values. By consid-\nering each curve, it is clear that the overall \u001f00vs.H\nbehaviour is the result of the contribution of grains with\ndi\u000berent orientations with respect to H, giving rise to\na powder-averaged, inhomogeneously-broadened absorp-\ntion line. We further stress that the well-de\fned maxi-\nmum observed in the experimental curves at Hrshould\nTABLE I: Summarizing results from previous studies of dc\nmagnetization on the currently investigated samples.17,32For\neach sample, we report the ordered value for the magnetic mo-\nment per Co ions, \u0016, and the corresponding saturation mag-\nnetizationMs. Estimates were performed at temperatures\ncomparable to the conditions of the FMR measurements (see\nFig. 12).\nCompound \u0016(\u0016B/Co)Ms(erg/Oe cm3)\nLaCoPO 0.295 \u00060.01 40.5 \u00061.5\nPrCoPO 0.27 \u00060.01 37.1 \u00061.5\nNdCoPO 0.24 \u00060.01 32.8 \u00061.5\nSmCoPO 0.225 \u00060.01 31 \u00061.5\nFIG. 15: (Color online) Summarizing results for the estimated\nanisotropy parameters Kaccording to the model described in\nthe text. The dashed line is a guide to the eye. The inset\nshows the\r=2\u0019values estimated from the \ftting procedure\nof Eq. (7) to data in Fig. 13.\nbe associated to grains where the H?zcondition holds.\nThe stronger intensity compared to the Hkzbranch is\neasily explained from geometrical considerations of the\npowder average. Remarkably, the low- \u0017line is less asym-\nmetric than the other ones. We attribute this e\u000bect to\nthe low typical values of Hin this limit, which may result\nin an undetectable signal from the \u0017kbranch.\nEstimates of HAnvalues from Fig. 13 enable us to di-\nrectly estimate the Kanisotropy constants for the four\nsamples via Eq. (8). We used magnetization values de-\nrived from independent measurements17,32and, in par-\nticular, we employed the saturation values Msestimated\nat temperatures close to the conditions of FMR mea-\nsurements (see Fig. 12). The results are displayed in\nFig. 15, showing a well de\fned trend for Kas a function\nof the unit cell volume V. These results suggest that\ndecreasing the Vvalue is the physical origin for trigger-\ning anisotropic magnetic properties for RCoPO. Since\npreviously reported data18evidence that smaller Rions\ninduce a reduction in aandcaxes such that the c=aratio\nis approximately constant (i. e., the lattice is contracting\nisotropically), one reasonable conclusion is that the pnic-\ntogen height hPis reducing faster, making the local tetra-\nhedral environment progressively more distorted. How-\never, we also expect a sizeable interaction between fand\ndelectronic degrees of freedom from rare-earth and pnic-\ntogen ions, respectively, which is typically measured by\nmeans of local-probe techniques for RMX O oxides.45{47\nWhile this e\u000bect may be well enhanced by the increas-\ning chemical pressure, we argue that it may introduce\nan indirect transfer of anisotropic properties from the f\norbitals of Rions to the dbands arising from Co or-\nbitals and ultimately in\ruencing the magnetocrystalline\nanisotropy. In this respect, extending our measurements\nto a more complete set of RCoPO samples with di\u000berent\nprolaticity properties for the Rorbitals would lead to a11\nimportant check on which of the two proposed mecha-\nnisms is indeed the dominant one.\nAnother scenario can be considered in order to under-\nstand the origin of the observed behaviour. One robust\noutput of our investigation is that LaCoPO shows almost\nfully-isotropic magnetic properties within the FM phase,\na fact which is surprising in the light of the typically\nanisotropic properties of uniaxial magnets.44These fea-\ntures may be only apparently isotropic if one assumes\nthat a strong magnetocrystalline anisotropy could be ef-\nfectively compensated by shape anisotropy e\u000bects, under\nthe hypothesis that the spherical grains composing the\ninvestigated samples are coupled among them. While it\nseems quite unlikely that this compensation e\u000bect leads\nto completely symmetric lineshapes as the experimental\nones measured for LaCoPO, the main conclusions out-\nlined above (i. e., the magnetocrystalline anisotropy is\nenhanced by Rsubstitution) are robustly preserved also\nwithin this scenario.\nV. CONCLUSIONS\nWe reported on ferromagnetic resonance measure-\nments inRCoPO for di\u000berent Rions. 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B 83, 094526 (2011)."    },    {        "title": "1409.2845v1.Micro_strip_ferromagnetic_resonance_study_of_strain_induced_anisotropy_in_amorphous_FeCuNbSiB_film_on_flexible_substrate.pdf",        "content": "Micro-strip ferromagnetic resonance study of strain-induced anisotropy in amorphous\nFeCuNbSiB \flm on \rexible substrate\nFatih Zighem1,\u0003Anouar El Bahoui1, Johan Moulin2, Damien Faurie1,y\nMohamed Belmeguenai1, Silvana Mercone1, and Halim Haddadi3\n1Laboratoire des Sciences des Proc\u0013 ed\u0013 es et des Mat\u0013 eriaux,\nCNRS-Universit\u0013 e Paris XIII, Sorbonne Paris Cit\u0013 e, Villetaneuse, France\n2Institut d'Electronique Fondamentale, UMR 8622, Universit\u0013 e Paris Sud-CNRS, Orsay, France and\n3Laboratoire MSMP - Carnot Arts, ENSAM ParisTech,\nrue Saint-Dominique, 51006 Ch^ alons-en-Champagne\n(Dated: September 1th2014)\nThe magnetic anisotropy of a FeCuNbSiB (Finemet ®) \flm deposited on Kapton ®has been\nstudied by micro-strip ferromagnetic resonance technique. We have shown that the \rexibility of\nthe substrate allows a good transmission of elastic strains generated by a piezoelectric actuator.\nFollowing the resonance \feld angular dependence, we also demonstrate the possibility of controlling\nthe magnetic anisotropy of the \flm by applying relatively small voltages to the actuator. Moreover, a\nsuitable model taking into account the e\u000bective elastic strains measured by digital image correlation\nand the e\u000bective elastic coe\u000ecients measured by Brillouin light scattering, allowed to deduce the\nmagnetostrictive coe\u000ecient. This latter was found to be positive ( \u0015= 16\u000210\u00006) and consistent\nwith the usually reported values for bulk amorphous FeCuNbSiB.\nPACS numbers:\nKeywords: Voltage induced anisotropy, magnetoelastic anisotropy, ferromagnetic resonance, digital image\ncorrelation\nI. INTRODUCTION\nThe strain control of magnetization orientation distri-\nbution in magnetic thin \flms viathe magnetoelastic or\nthe magnetostrictive properties, is increasingly studied to\nface new challenges in magnetoelectronics and spintron-\nics [1{6]. An easy way for mastering the magnetization\nis to make adhering the thin \flm on a piezoelectric ac-\ntuator, so that the strains can be applied to the \flm by\nvarying continuously a voltage on the piezoelectric actua-\ntor. This latter will succeed in transferring the strains in\na more or less e\u000ecient way, depending on the chosen sys-\ntem. Obviously, more the elastic strains are transferred\nat the interface, more the indirect magnetoelectric e\u000bect\nis optimized. Generally, at a \frst step, the magnetic thin\n\flms are deposited on a substrate of usually one hun-\ndred microns thick. At a second step the \flm/substrate\nis cemented on an actuator [7{10]. At this point two in-\nterfaces will play their role in the transmission of strains:\nthe actuator/substrate interface and the substrate/\flm\none. This double transfer limits the desired phenomenon,\nespecially when the substrate is sti\u000b such as for com-\nmonly used wafer (Si, GaAs, ...) [11]. Besides, in this\ncon\fguration, it is generally hard to predict perfectly\nthe amount of the induced strains inside the thin \flm\n(\"xxand\"yy) even if the piezoelectric coe\u000ecients dijk\nof the actuator are known. This is due to the partial\nstrains transmissions from the actuator to the substrate,\nespecially in the case of sti\u000b substrates. This well-known\n\u0003Electronic address: zighem@univ-paris13.fr\nyElectronic address: faurie@univ-paris13.frreported limitation (a few ten percents of losses in best\ncases) can be avoided by depositing the magnetic thin\n\flm on a compliant substrate such as polyimides that\nare more and more used in \rexible spintronics [12{14].\nMore precisely we reported in Ref. [9], by comparing\n\flm and actuator strains, that one advantage of study-\ning thin \flms on polymer substrates is the very good\nstrains transmissions (nearly 100%) resulting from the\nsubstrate compliance. However, once well-known strains\nare transferred to the \flm, the estimation of the stresses\n(\u001bxxand\u001byy) is straightforward only if the elastic coef-\n\fcients of this latter are known. Otherwise Hook's law\nis not applicable. Unfortunately, the elastic coe\u000ecients\nare not always well-known, this is especially not the case\nwhen studying thin \flm of new functional alloys.\nIn this paper, voltage induced strain e\u000bect on the in-\nplane magnetic anisotropy has been quantitatively stud-\nied in a ferromagnetic \flm deposited on a \rexible sub-\nstrate and glued onto a piezoelectric actuator. In partic-\nular, the structural (Young's modulus Eand Poisson's\nratio\u0017) and magnetoelastic (magnetostriction coe\u000ecient\nat saturation \u0015) properties have been completely deter-\nmined by combining di\u000berent techniques. Our general\nmethodology is based on the combination from one hand\nof i) the control of the applied stress state in a mag-\nnetic thin \flm and from another hand on ii) the magnetic\nuniform precession mode resonance \feld measurement\nas function of the stress. In order to control the stress\nstate in the magnetic \flm, the \flm/substrate system is\nglued onto a piezoelectric actuator that allows applying\nin-plane strains ( \"11and\"22) when applying voltage to\nthe actuator as it is shown on Figure 1. To properly es-\ntimate the in-plane strains by the double interfaces, we\nadopted the Digital Image Correlation (DIC) techniquearXiv:1409.2845v1  [cond-mat.mtrl-sci]  9 Sep 20142\nVPiezoelectric actuator\nKapton ® susbtrate\n(125µm)Finemet ® film \n(530nm)\nFigure 1: Sketch of the studied heterostructure showing the\n530 nm thick Finemet ®\flm deposited onto a 125 µm thick\nKapton ®substrate and glued onto the piezoelectric actuator.\nThe arrows qualitatively represent the in-plane strains of the\npiezoelectric actuator.\nbased on the optical observation of the \flm surface and\nits evolution during the \flm straining (section IV).\nA non-destructive method (Brillouin light scattering\n[17{19]) has been used to quantify the elastic coe\u000ecients\nof the amorphous magnetic thin \flm (BLS study in sec-\ntion III). The magnetic anisotropy dependence upon the\napplied stress has been followed by measuring the behav-\nior of the high frequency resonance \feld while applying\nan electric \feld inside the actuator. This resonance \feld\nis directly linked to the magnetic anisotropy and it can\nbe estimated by performing Micro-Strip FerroMagnetic\nResonance (MS-FMR )[20, 21] experiments in several ge-\nometrical con\fgurations (section IV). Moreover, by ad-\njusting an appropriate model (section II) to the exper-\nimental data, we have shown how to estimate with ac-\ncuracy the thin \flm e\u000bective magnetostriction coe\u000ecient\nat saturation ( \u0015).\nII. THEORETICAL BACKGROUND\nThe voltage-induced strain e\u000bect has been experimen-\ntally studied by ferromagnetic resonance through uni-\nform precession mode resonance \feld as a function of the\napplied voltage. Indeed, the resonance \feld (or resonance\nfrequency) of the uniform precession mode is in\ruenced\nby the magnetoelastic behavior of the thin \flm. All the\nexperimental spectra presented in this work have been\nperformed at room temperature. Moreover, they have\nbeen done at relatively \\high\" applied magnetic \feld in\norder to have a uniform magnetization inside the \flm.\nThe expression of the resonance \feld is derived taking\ninto account in-plane strains ( \"xxand\"yy). These in-\nplane strains will be induced by applying an electric \feld\ninside the piezoelectric actuator. When any magnetoelas-\ntic contribution is required, the magnetic energy density\nof a ferromagnetic thin \flm, using the coordinates system\nof II, can be written as:F0=Fzee+Fdip+Fexch+Fanis (1)\nWhere the three \frst terms stand for the Zeeman,\nthe dipolar and the exchange contributions, respectively.\nThe last term, which corresponds to the anisotropy con-\ntribution, will be written as an ad hoc in-plane uniaxial\nanisotropy characterized by the anisotropy constant Ku.\nThis is correct when no out-of-plane or surface contri-\nbutions to the anisotropy are observed in the \flm. This\nterm can then be written as:\nFanis=\u0000Ku\nM2s\u0010\n~M\u0001~ ux\u00112\n(2)\n~ uxis a unit vector along the \\easy axis\" (along xdi-\nrection) while ~MandMsare respectively the vector and\nthe module of the magnetization. The stress e\u000bect will\nbe modeled through a magnetoelastic density of energy\nFmewhich will be added to F0:\nFme=\u00003\n2\u0015\u0012\u0012\n\r2\nx\u00001\n3\u0013\n\u001bxx+\u0012\n\r2\ny\u00001\n3\u0013\n\u001byy\u0013\n(3)\n\u001bxxand\u001byybeing the in-plane principal stress tensor\ncomponents while \rxand\rycorrespond to the direction\ncosines of the in-plane magnetization. The presence of\na unique saturation magnetostriction coe\u000ecient \u0015is due\nto the amorphous structure of the thin \flm. The relation\nbetween the principal stress components ( \u001bxx,\u001byy) and\nstrains (\"xx,\"yy) tensors is thus given by an isotropic\nHook's law where Eis the Young's modulus and \u0017is the\nPoisson's ratio:\n\u001bxx=\u0012E\n1 +\u0017\u0013\u00121\n1\u0000\u0017\"xx+\u0017\n1\u0000\u0017\"yy\u0013\n(4)\n\u001byy=\u0012E\n1 +\u0017\u0013\u00121\n1\u0000\u0017\"yy+\u0017\n1\u0000\u0017\"xx\u0013\n(5)\nIn these conditions, minimizing the total volume mag-\nnetic energy density (i. e. F=F0+Fme), the reso-\nnance \feld of the uniform precession mode can be ob-\ntained thanks to the following relation [15, 16]:\n\u00122\u0019f\n\r\u00132\n=\u00121\nsin\u0012M\u00132 \n@2F\n@\u00122\nM@2F\n@'2\nM\u0000\u0012@2F\n@\u0012M'M\u00132!\n(6)\nWherefis the microwave driving frequency. The dif-\nferent energy derivatives are calculated at the equilib-\nrium direction of the magnetization. In the above ex-\npression,\ris the gyromagnetic factor \r=g\u00028:794\u0002106\ns\u00001.Oe\u00001while\u0012Mand'Mstand for the polar and the\nazimuthal angles of the magnetization. Note that for\nan in-plane applied magnetic \feld, the equilibrium polar\nangle is'M=\u0019\n2because of the large e\u000bective demagne-\ntizing \feld associated with the planar \flm geometry and3\nan explicit expression is obtained for f:\nf2=\u0010\r\n2\u0019\u00112\nH1H2 (7)where:\nH1= 4\u0019Ms+Hrescos ('\u0000'H) +2Ku\nMscos2'+3\u0015\nMs\u0000\n\u001bxxcos2'+\u001byysin2'\u0001\n(8)\nH2=\u00122Ku\nMs+3\u0015\nMs(\u001bxx\u0000\u001byy)\u0013\ncos 2'+Hrescos ('\u0000'H) (9)\nHere'His the angle between the in-plane applied mag-\nnetic \feld and the magnetic easy axis ( xdirection). The\nanalysis can be simpli\fed if the resonance \feld is larger\nthan the e\u000bective uniaxial anisotropy and magnetoelas-tic \feld:~Hu=\u0000~r~MFanis and~Hme=\u0000~r~MFme, re-\nspectively. Indeed, in this condition, the magnetization\ndirection will be almost parallel to the applied magnetic\n\feld ('M\u0018'H). The resonance \feld is thus given by:\nHres=1\n20\n@s\u0012\n4\u0019Ms+\u0012\nHu+3\u0015\nMs\u001bxx\u0013\nsin2'+3\u0015\nMs\u001byycos2'H\u00132\n+ 4\u00122\u0019f\n\r\u00132\n\u00004\u0019Ms1\nA\n\u0000Hu\u00121\n4+3\n4cos 2'H\u0013\n\u00003\u0015\nMs\u001bxx\u00121\n4+3\n4cos 2'H\u0013\n\u00003\u0015\nMs\u001byy\u00121\n4\u00003\n4cos 2'H\u0013\n(10)\nzM\ny\nxM\nH\nH\nxx\nyyye\nxeze\nM\nuH\nFigure 2: Schematic illustration showing angles, \felds and\ncoordinate systems used in the text.\nThe \frst term essentially represents a constant shift\nin the resonance \feld baseline because 4 \u0019Msand2\u0019f\n\r\nare found to be larger than the magnetoelastic and the\nuniaxial anisotropy \felds. The other terms correspond\nto the angular variation of the resonance \feld due to the\nuniaxial anisotropy \feld (second term) and to the voltage\ninduced magnetoelastic anisotropy \feld (third and fourth\nterms).III. AS-DEPOSITED THIN FILM\nCHARACTERIZATION AT ZERO STRESS\nAPPLIED\n1. Finemet thin \flm deposition and structure\nAn amorphous 530 nm thick Finemet ®\flm was de-\nposited onto a 125 \u0016m thick polyimide \rexible substrate\n(Kapton ®) by radio frequency sputtering. The deposi-\ntion residual pressure was of around 10\u00007mbar, while a\nworking Ar pressure was of 40 mbar and the RF power\nwas of 250 W. A 10 nm thick Ti bu\u000ber layer was de-\nposited on the substrate to ensure a proper adhesion of\nthe Finemet ®\flm. Finally, another 10 nm thick Ti\ncap layer was deposited on the top of the Finemet ®\n\flm in order to protect it from oxidation. The composi-\ntion of the \flm has been measured by EDS (Energy Dis-\npersive Spectroscopy) and is close to that of the target\n(Fe73:5Cu1Nb3Si15:5B7) while the thickness of the \flm\n(530 nm) has been measured by Scanning Electron Mi-\ncroscopy and mechanical pro\flometry.\nThe \flm/substrate system has been then glued onto\na piezoelectric actuator. Figure 1 presents a sketch of\nthe studied heterostructure for studying the so-called in-\nverse magnetoelectric e\u000bect. In this Figure \"11and\"22\ncorrespond to the in-plane strains at the surface of the\nactuator (top) and at the interface with the \rexible sub-4\n\t\r experimental\t\r simulated\t\r \n711151923Frequency (GHz)1501209060300Intensity (counts)R\t\r S1\t\r S2\t\r S3\t\r S4\t\r S5\t\r S6\t\r \nFigure 3: Brillouin light scattering spectrum of the amor-\nphous Finemet ®\flm on Kapton ®substrate. The trans-\nferred wave-vector is q= 2:14\u0002105cm\u00001.Rdenotes Rayleigh\nsurface wave while Sicorrespond to the Sesawa guided waves.\nThe red dashed line corresponds to the theoretical calculated\nspectrum while the continuous blue line corresponds to the\nexperimental data.\nstrate. This latter is again used here as it is the best pos-\nsible medium for e\u000eciently transfer the in-plane strain\nbetween the actuator and the ferromagnetic \flm. Max-\nimum values of in-plane strains at a \fxed voltage will\nbe obtained as compared to those obtained when using\na rigid substrate. Indeed, there is roughly two orders of\nmagnitude between the Young's modulus values of rigid\nsubstrates and the ones of a \rexible one ( \u00184 GPa for\nKapton ®) and\u0018180 GPa for Si). Nevertheless, even\nif the interacting vector is the voltage induced in-plane\nstrain transferred from the actuator to the ferromagnetic\n\flm, this is not su\u000ecient to quantitatively analyze the in-\ndirect magnetoelectric e\u000bect. As stressed out in the pre-\nvious section, in fact, the magnetoelastic anisotropy of\nthe Finemet ®material depends directly on the Young's\nmodulusEand the Poisson's ratio \u0017of the thin \flm. To\ndetermine them, we performed a Brillouin light scatter-\ning (BLS) study\n2. Elastic coe\u000ecients estimation by Brillouin light\nscattering (BLS)\nDuring the last twenty years, BLS has proved to be\nvery e\u000ecient for achieving a complete elastic charac-\nterization of thin \flms and multilayered structures [17{\n19]. In a BLS experiment, a monochromatic light beam\nprobes and reveals acoustic phonons characteristic of\nthe investigated medium. The power spectrum of these\nphonons is mapped out from the frequency analysis of\nthe light scattered within a solid angle. Because of the\nwave vector conservation in the phonon-photon interac-\ntion, the wavelength of the revealed elastic waves is of\nthe same order of magnitude as that of light. This meansthat the wavelength is much larger than the inter-atomic\ndistances, so that the material can be described as a\ncontinuum within an e\u000bective-medium approach. The\nBLS spectra were collected in air at room temperature\nwith typical acquisition times of a few hours. 30 mW p-\npolarized monochromatic light coming from a solid state\nlaser (\u0003 = 532 nm) was focused on the surface of the\nsample. We used back-scattering con\fguration and the\nscattered light was analyzed thanks to a (3+3)-pass tan-\ndem Fabry{P\u0013 erot interferometer, so that the value of the\nwave vector of the probed surface acoustic waves is exper-\nimentally \fxed to the value q=4\u0019\n\u0015sin (\u0018) = 2:14\u0002105\nrad.cm\u00001, where\u0018= 65°is the incidence angle of the\nlight beam.\nFor nearly opaque \flms with thicknesses around the\nacoustic wavelength (0.3 - 0.4 \u0016m), we can observe the\nsurface acoustic waves with a sagittal polarization. The\nRayleigh wave ( R), the so-called Sesawa guided waves\n(S1toSi) and the corresponding phase velocities are\nthen measured. In our case, the amorphous \flm can\nbe considered as isotropic. Thus, two isotropic elastic\ncoe\u000ecients (Young's modulus Eand Poisson's ratio \u0017\nof the thin amorphous \flm) in\ruence the Rayleigh and\nSezawa modes, so that they can be evaluated by a best\n\ft procedure of the experimental velocities to the cal-\nculated dispersion curves. By taking into account only\nthe ripple mechanism for the scattered intensity by the\nsurface acoustic waves, the experimental spectra are well\n\ftted, leading to the following values : E= 145\u00021010\ndyn.cm\u00002(\u0011145 GPa) and \u0017= 0:27. The estimated\nYoung's modulus is close to the one of amorphous bulk\nFeCuNbSiB rubbans [22].\n3. Magnetic parameters at zero applied voltage\nDuring this study, the magnetic properties have been\nprobed by using Micro-Strip FerroMagnetic Resonance\n(MS-FMR) which is now a common technique to scruti-\nnized the dynamic magnetic properties in the microwave\nregime. This setup allows the determination of the res-\nonance \feld Hresof the uniform precession mode by\nsweeping the applied magnetic \feld in presence of a \fxed\npumping radio frequency \feld ~hrf(i. e. microwave driv-\ning frequency f).\nIn order to enhance the signal to noise ratio, a weak\nmodulation of the static applied magnetic \feld (here \u00185\nOe at 175 Hz) is performed. Thus, this setup gives ac-\ncess to the \feld \frst derivative of the rf absorption as\na function of the applied magnetic \feld. We have \frst\nstudied the magnetic properties of the Finemet ®thin\n\flm in zero-applied voltage. The angular ( 'H) depen-\ndence of the resonance \feld has been studied for dif-\nferent microwave driving frequency f. This dependence\nis presented in Figure 4a) for three frequencies (8, 9.47\nand 12 GHz). Note that the measurements performed\nat 9.47 GHz have been performed by using a resonant\ncavity thanks to an Electron Paramagnetic Resonance5\n0 120 240 3605006007008001100115012001250\n81 2 1 6 2 01238GHz\n9.47GHz\n12GHzResonance field(Oe)\nIn‐planeangle(deg.)\nalonghardaxis:\n\nalongeasyaxis:\nResonance field(kOe)\nFrequency (GHz)a)\nb)\nFigure 4: a) In-plane angular ( 'H) dependence of the reso-\nnance \feld at three di\u000berent driven frequencies (8, 9.47 and\n12 GHz). b) Variation of the uniform precession resonance\n\feld variation as a function of the microwave driven frequency\nfat zero applied voltage. Open symbols are obtained with\n'H= 0 (easy axis: ~Hk~ ex) and the \flled ones are obtained\nwith'H=\u0019=2 (hard axis: ~Hk~ ey). The solid and dashed\nlines in a) and b) are best \fts to the experimental data using\nequation 10 with the following parameters: \r= 1:885\u0002107\nHz.Oe\u00001,MS= 965 emu.cm\u00003,Ku= 3:8\u0002104erg.cm\u00003and\n\u001bxx=\u001byy= 0.\n(EPR) set-up and are clearly consistent with the MS-\nFMR ones. All the measurements show that the angular\ndependencies are governed by a uniaxial anisotropy hav-\ning an easy axis along xdirection (the resonance \feld is\nindeed minimum at 'H= 0). Figure 4b) presents the\nresonance \feld variation as function of the microwave\ndriving frequency falong the easy axis ( 'H= 0, open\ncircles) and along the hard axis ( 'H=\u0019\n2, \flled circles).\nThe continuous lines in Figure 4 are calculated thanks\nto equation 10. The in-plane strains are equal to zero\n(\"xx(V= 0) =\"yy(V= 0) = 0) in absence of applied\nvoltage and therefore the resonance \feld depends only on\n\r,MsandKu[25]. The best \fts to experimental data are\nobtained using the following parameters: \r= 1:885\u0002107\ns\u00001.Oe\u00001,MS= 965 emu.cm\u00003(i. e. 4\u0019Ms\u001812100\nG) andKu= 3:8\u0002104erg.cm\u00003(i. e.Hu\u001880 Oe).\nThe obtained Land\u0013 e factor ( g\u00182:1) value is the typical\nfor metallic ferromagnets and the Msvalue is in a good\nagreement with previously determined values for simi-\nlar \flms. However, due to the amorphous structure of\nthe material, the in-plane magnetic anisotropy in equiv-\nalent Finemet ®\flm deposited onto rigid substrate (Si)\nis generally weaker (a few Oe) than the \u001880 Oe found\nhere. As previously suggested [26{29], the origin of this\nuniaxial anisotropy is certainly due to a non zero mag-\nnetoelastic anisotropy at zero-applied voltage. This \\ini-\ntial\" anisotropy can be due to a slight curvature along a\ngiven direction taking place during the elaboration pro-\ncess which could lead to a magnetoelastic anisotropy atzero-applied voltage. Thereafter, this residual anisotropy\nwill be modeled as an ad hoc uniaxial one.\nIV. MAGNETIC ANISOTROPY STRAINS\nEFFECT\nA. Measured in-plane strains\nThe piezoelectric actuator is characterized by a\nmain axis direction (direction 2 in Figure 1). The\n\flm/substrate system has been glued in order to have\nthe main deformation axis of the actuator perpendicu-\nlar to the easy axis, so in this condition, the direction\n1 (resp. 2) of the actuator refers to x(resp.y) direc-\ntion of the thin \flm. Digital image correlation (DIC)\nmethod enables accurate measurements of changes in dig-\nital images [30, 31]. This method uses tracking and image\nregistration to make full-\feld non-contact measurements\nof displacements and strains in a wide variety of engi-\nneering applications (mechanics of materials, micro- and\nnano-technology, ...). Here, two CCD cameras mounted\non a tripod have been positioned vertically in top of the\n\flm/substrate/actuator system, the \feld of view is \fxed\nto approximately 2 \u00022 cm2. Given the 2448 \u00022050\nnumber of pixels of the cameras, the area per pixel is\nabout 12:5 nm2. The respective positions of the cameras\nhas been calibrated by using a 10 mm \u00028 mm calibra-\ntion pattern. Figure 5a) presents a part speckle pattern\nwhich has been generated at the \\uniform\" top surface\nof the heterostructure by using a spray paint in order\nto generate a contrast which will serve to calculate the\ndeformation. Indeed, a \frst image (reference image) is\ntaken at zero applied voltage; then, a sequence of images\nare taken at di\u000berent applied voltages and are compared\nto the reference. The \feld strain at the surface of the\nsystem has been extracted for each applied voltage by\nperforming DIC calculations which are performed by us-\ning the reference image and the di\u000berent images coming\nfrom the sequence. The DIC calculations have been per-\nformed by using ARAMIS which is a commercially avail-\nable software package [32]. From the \felds strain, the\nmean in-plane strains are extracted as function of the\napplied voltage. Note that shear strains can also be ex-\ntracted because of the use of two cameras in the present\nsetup. However, the shear strains values are found to be\nnegligible in this study and will be neglected thereafter.\nImages were collected by performing applied voltage\nloops (from 0V to 100V and back to 0V) with a frame rate\nof about 0.1 FPS; the step of applied voltage was \fxed\nto around 5 V. Furthermore, prior to the measurements,\nseveral images have been taken in absence of voltage in\norder to estimate the DIC setup statistical errors, esti-\nmated to be\u00185\u000210\u00006. Moreover, di\u000berent images have\nbeen taken as a function of time at 0 V after saturating\nthe actuator at 100 V. After approximately 5 hours, a dif-\nference of about\f\f4\u000210\u00005\f\fin the in-plane strains values\nis found; this value rises to\f\f1\u000210\u00004\f\fafter several days6\n0 2 55 07 5 1 0 0‐0.50.00.51.0\nBlueregion:\n1122\nRedregion:\n1122Strain(1x10‐3)\nVoltage(V)\n0 2 55 07 5 1 0 0‐50050100\nxx\nyyStress(MPa)\nVolta ge(V)b) c)\nupdown\nup\ndownupdown\nup\ndown\na)\nV\napplied\nxxapplied\nyyuncoated \narea\nspeckled\npattern\nthinfilm\narea\n V\napplied\nxxapplied\nyyuncoated \narea\nspeckled\npattern\nthinfilm\narea\nFigure 5: a) Top view sketch of the system showing the spray-painted speckle pattern in order to generate a contrast at the\nsurface of the \flm. The blue and red areas correspond to the top surface of the \flm region and to an uncoated area of the\nactuator, respectively. b) Extracted mean in-plane strains from the blue (\flm surface) and red (uncoated actuator surface)\nareas by performing DIC calculations. Continuous lines correspond to \"xxand\"yywhile and symbols correspond to \"11and\n\"22. c) Calculated mean in-plane stresses \u001bxxand\u001byyusing equations 4 and 5 with E= 145\u00021010dyn.cm\u00002and\u0017= 0:27.\nIn b) and c), the applied external voltage was swept from 0 V to 100 V and back to 0 V with steps of around 5 V. The\n\frst measurement at 0 V has been performed after \\saturating\" the actuator at 100 V to avoid training e\u000bect of the actuator\npolarization.\n(which is relatively high). This di\u000berence is due to the\ntraining e\u000bect of the polarization. In addition, ARAMIS\nhas been used to calculate the DIC in two di\u000berent 3 \u00023\nmm2regions: an uncoated area of the actuator and an\narea located at the top of the thin \flm.\nFigures 5b) presents the extracted mean in-plane\nstrains\"11and\"22. Indeed, similar quasi-homogeneous\nstrain \felds as function of the applied voltage have been\ncalculated from the two regions. Thus a 100% in-plane\nstrain transmission in between the piezoelectric actuator\nand the \flm is observed, it can be conclude that \"11=\"xx\nand\"22=\"yy. Non linear and hysteretic variations are\nobserved for both \"11and\"22which is certainly due to\nthe intrinsic properties of the ferroelectric material used\nin the fabrication of the actuator [9]. One can note that\n\"22is found to be positive and \"11is found to be negative\nin the voltage range [0-100 V]. Moreover, it is interesting\nto note that a linear variation of \"22as a function of \"11\nis found\"22'\u00001:7\"11. The maximum achieved values\nof\"22and\"11(\u00181\u000210\u00003\u0018\u00000:5\u000210\u00003at 100 V, re-\nspectively) show that the \flm is not deteriorated by the\nplasticity regime because it is obtained for higher values:\nthis is experimentally con\frmed by the excellent repro-\nducibility of the experiments (even after several days). In\nthis condition and because of the amorphous structure of\nthe magnetic \flm, the in-plane stresses can be calculated\nusing equations 4 and 5 with Eand\u0017values previously\ndetermined by BLS. Obviously, \u001b11and\u001b22also present\nnon linear and hysteretic variations as function of the ap-\nplied voltage. This representation of the in-plane stresses\ncould be useful, especially in the case of uniaxial in-plane\nstress which is not the case here. Indeed, for \u001b22, a sim-\nilar linear variation is found with \u001b22'\u00003:6\u001b11.\nFrom the voltage dependence of \"11and\"22(or\u001b11.\nand\u001b22), the indirect magnetoelectric e\u000bect can be quan-\ntitatively studied in the the system by probing resonance\n\feld uniform mode.\n02 5 5 0 7 5 1 0 00255075100\n0 3 06 09 0 1 2 0 upsweep\ndownsweepHres(Oe)\nAppliedvoltage(V)upsweep\ndownsweep\n linearfit\nyy(MPa)updown800 1000 1200 1400‐15‐10‐505\n0V\n20V\n40V\n60V\n80V\n100VIntensity(a.u.)\nAppliedmagnetic field(Oe)a)11GHz\nHres\n11GHz 11GHzFigure 6: a) Experimental spectra recorded at 11 GHz with\n'H=\u0019\n2at di\u000berent applied voltages. (b) Resonance \feld\nvariations (at 11 GHz and 'H= 0) as a function of the ap-\nplied voltage (b) and of the applied stress (c). The open\nsymbols correspond to upsweep (0 to 100 V) while \flled sym-\nbols correspond to downsweep (100 to 0 V). Note that \u000eHres\nhysteresis disappears when the data are plotted as function of\n\u001byy(or\u001bxx). The solid lines in Figure 7b are guided for the\neyes while the solid line in Figure 7c corresponds to a linear\n\ft.7\n0 30 60 90 120115012001250\n\n\nResonance field(Oe)\n22(MPa)0 30 60 90 120950100010501100\n\n\n\nResonance field(Oe)\n22(MPa)0 30 60 90 120500550600650\n\n\n\nResonance field(Oe)\n22(MPa)\n0 2 04 06 08 0 1 0 0175018001850\n\n\n\nResonance field(Oe)\nAppliedvolta ge(V)0 2 04 06 08 0 1 0 01500155016001650\n\nResonance field(Oe)\nAppliedvoltage(V)0 2 04 06 08 0 1 0 01300135014001450\n\n\nResonance field(Oe)\nAppliedvoltage(V)8GHz 11GHz 12GHz\n13GHz 14GHz 15GHz0H\n4/\n\nH\n2/\n\nH0H\n4/\n\nH\n2/\n\nH0H\n4/\n\nH\n2/\n\nH\n0H\n4/\n\nH\n2/\n\nH0H\n4/\n\nH\n2/\n\nH0H\n4/\n\nH\n2/\n\nH\nFigure 7: Resonance \feld variation as a function of the induced in-plane stress \u001b22(top graphs) and the applied voltage (down\ngraphs) measured at di\u000berent microwave driven frequencies (8-15 GHz) and in-plane angular angles ( 'H=0,\u0019\n2and\u0019\n4). Note\nthat the observed hysteretic and non linear variations (see Figure 6 disappears). Open circles are obtained with 'H= 0\n(initial easy axis: ~Hk^x), the \flled circles are obtained with 'H=\u0019\n2(hard axis: ~Hk^y) and the open triangles are obtained\nwith'H=\u0019\n4. The di\u000berent lines are best \fts to the experimental data using equation 10 with the following parameters:\n\r= 1:885\u0002107s\u00001.Oe\u00001,MS= 965 emu.cm\u00003,Ku= 3:8\u0002104erg.cm\u00003,E= 145 GPa and \u0017= 0:27 and\u0015= 16\u000210\u00006.\n\u000bme(V.cm\u00001.Oe\u00001)\u000b\u001byyms(Oe.MPa\u00001)\nf(GHz) 'H\n0\u0019\n20\u0019\n2\n8 -1.62 1.4 -0.67 0.77\n10 -1.62 1.39 -0.62 0.77\n11 -1.76 1.41 -0.63 0.77\n12 -1.68 1.41 -0.64 0.76\n13 -1.7 1.4 -0.58 0.75\n14 -1.76 1.41 -0.62 0.76\n15 -1.85 1.44 -0.58 0.75\n16 -1.81 1.76 -0.59 0.62\n18 -1.81 1.7 -0.6 0.64\n20 -1.831 1.37 -0.6 0.79\nTable I:\u000bmeand\u000b\u001byy\nmsas function of the microwave driving\nfrequencyfextracted along the xandydirection.\nB. Strain induced anisotropy and magnetoelastic\nbehavior\nFigures 6a) shows typical MS-FMR experimental spec-\ntra recorded at 11 GHz with an applied magnetic \feld\nalongydirection (initial hard axis: 'H=\u0019\n2) at dif-\nferent applied voltages (0, 20, 40, 60, 80 and 100 V).\nA shift of the resonance \feld \u000eHresof around 100 Oe\nbetween spectra recorded at 0 and 100 V is clearly ob-\nvious. The corresponding variation of \u000eHres(de\fned as\n\u000eHres=Hres(0)\u0000Hres(V)) as a function of the appliedvoltage is reported in Figure 6: a non linear and hys-\nteretic variation is observed. Here, \flled circles represent\nthe upsweep (0 to 100 V) and open circles the downsweep\nof the applied voltage (100 to 0 V). The spectra presented\nin Figure 6a) show that the resonance \feld decreases\nwhen increasing the applied voltage. Since \"yy>0 and\n\"xx<0, it can be conclude that the magnetostriction\ncoe\u000ecient at saturation \u0015of the thin \flm is positive. In-\ndeed, a negative magnetostriction coe\u000ecient would have\nled to an increase of Hres, as it is the case in Ni poly-\ncrystalline thin \flm [9, 33].\nFurthermore, a linear variation of \u000eHresappears if the\nvoltage-stress dependence (see Figure 5) is used. Fig-\nure 6c) illustrates such a behavior where the continu-\nous line is the linear \ft of the slope which is found\naround\u000b\u001byyms= 0:76 Oe.MPa\u00001. This feature indicates\nthat the non linear and the hysteretic variations of \u000eHres\nas a function of Vis not related to the magnetoelastic\nanisotropy; it is completely due to the intrinsic properties\nof the ferroelectric material used for the actuator fabri-\ncation. In addition, in \frst approximation, if the non\nlinear and the hysteretic behavior of \u000eHresas a func-\ntion ofVis neglected and adjusted by a linear \ft, an ef-\nfective magnetoelectric coupling ( \u000bmein V.cm\u00001.Oe\u00001)\ncan be estimated (by considering the static electric \feld\ninside the actuator. Table I presents the di\u000berent val-\nues of\u000bmeand\u000b\u001byymsas function of the microwave driv-\ning frequency measured along the initial easy and hard\naxes ('H= 0 and\u0019\n2). The \frst observation is that the\nvoltage-induced magnetoelastic e\u000bect is frequency inde-\npendent for this system. However, a weak di\u000berence\nof\u000bmeextracted from the measurements performed at8\n1120119012600\n60\n120\n180240300\n1120\n119012600V\n80V\n100V\nResonance field(Oe)\n0VPiezoelectric actuator\neasyaxis\n0VPiezoelectric actuator\neasyaxis 0V\n80V\nPiezoelectric actuator\n80Visotropic\nPiezoelectric actuator\n80Visotropic\n100V\nPiezoelectric actuator\n100V\neasyaxis\nPiezoelectric actuator\n100V\neasyaxis\nFigure 8: In-plane angular ( 'H) dependence of the reso-\nnance \feld measured at 12 GHz for three di\u000berent applied\nvoltages: 0, 80 and 100 V. The symbols are experimental\ndata (triangles: 0 V, open circles: 80 V and \flled circles:\n100 V) while the solid lines are calculated thanks to equa-\ntion 10 with the following parameters: \r= 1:885\u0002107\ns\u00001.Oe\u00001,MS= 965 emu.cm\u00003,Ku= 3:8\u0002104erg.cm\u00003,\nE= 145\u00021010dyn.cm\u00002and\u0017= 0:27 and\u0015= 16\u000210\u00006.\nThe Sketches correspond to 3D view of the heterostructure\nshowing the voltage-switch of the easy axis from xtoydirec-\ntion.\n'H= 0 and'H=\u0019\n2is found. Such e\u000bect is most prob-\nably due to the misalignment (of a few degrees) of the x\ndirection and the direction 1 of the actuator which may\noccurred when gluing of the \flm/substrate system onto\nthe piezoelectric actuator and is predicted by the equa-\ntion 10.\nFigure 7 presents variations of Hresas a function of the\napplied voltage-induced stress (three top graphs) and as\na function of the applied voltage (three down graphs)\nfor di\u000berent 'Hangles (0,\u0019\n4and\u0019\n2) at various mi-\ncrowave driving frequencies. In \frst approximation, the\ninduced magnetoelastic anisotropy can be viewed as a\nuniaxial magnetoelastic anisotropy \feld ~Hmealongydi-\nrection, which is thus perpendicular to the \\initial\" uni-\naxial anisotropy \feld ~Hu. In absence of applied voltage,the resonance \feld along xdirection ('H= 0) is smaller\nthan the one measured along ydirection ('H=\u0019\n2), the\ndi\u000berence of these two last resonance \felds is roughly\nequal to\r\r\r2~Hu\r\r\r. When increasing the applied voltage (or\ninduced-stress), the xdirection will be less easy which\nleads to an increase (resp. decrease) of the resonance\n\feld along x(resp. along y) direction because of the\ncompetition between ~Hmeand~Hu. At 80 V ( \u001b11\u0018\u000030\nMPa and\u001b22\u0018105 MPa), the resonance \felds are equal\nfor the three studied angles which means that ~Huis to-\ntally compensated by ~Hme. A quantitative analysis of\nthe voltage-induced variation of the resonance \feld has\nbeen performed by using equation 10. By introducing\nthe stress-voltage dependence, the only undetermined pa-\nrameter is the saturation magnetostriction coe\u000ecient \u0015;\nthe best \fts of all the experiments gives \u0015= 16\u000210\u00006,\nwhich is slightly lower with the bulk material. A good\nagreement is found between the experimental variations\nand the calculated ones for the di\u000berent frequencies (see\n7). Note that the non linear and the hysteretic variations\nofHresas a function of Vare well reproduced. In ad-\ndition, at'H=\u0019\n4, an experimentally con\frmed almost\nconstant variation as predicted by equation 10.\nFigure 8 presents angular variations of the resonance\n\feld at di\u000berent applied voltage: 0, 80 and 100V which\ncorrespond to \u001b11\u00180, -30 and -45 MPa and \u001b22\u00180, 105\nand 130 MPa, respectively. At zero-voltage, the angular\nvariation of the resonance \feld found in Figure 4a) is re-\ntrieved, the horizontal peanut shape represented by blue\ntriangles is characteristic of a uniaxial anisotropy along x\ndirection ('H= 0). The 80 V value was chosen because\nof the \\exact\" compensation of this uniaxial anisotropy\nby the induced magnetoelastic one. At this voltage, the\n\flm is in-plane isotropic in the magnetic point of view. It\nis experimentally con\frmed by the circle shape formed by\nthe open circles. Finally, at 100 V, a uniaxial anisotropy\ncharacterized by a vertical peanut shape is observed along\nyaxis ('H=\u0019\n2). Thus, a voltage-switch of the e\u000bective\neasy axis from xdirection (at 0 V) to ydirection (at 100\nV) has been performed. The sketches of Figure 4b qual-\nitatively present such e\u000bect. Finally, the solid lines of\nFigure 4a are calculated thanks to equation 10 with the\nparameters previously determined.\nV. CONCLUSIONS\n•Magnetic anisotropy in Finemet ®thin \flms de-\nposited on Kapton ®substrate has been studied by\nMicro-Strip FerroMagnetic Resonance (MS-FMR).\n•We have shown that the \rexibility of Kapton ®\nsubstrate allowed tailoring the magnetic anisotropy\nof the \flm by applying small voltage-induced\nstrains.\n•The \rexibility of the Kapton ®substrate induced\nan initial uniaxial anisotropy that is generally not9\nfound in ferromagnetic \flms whose thickness is a\nfew hundred nanometers.\n•The knowledge of the applied elastic strains versus\napplied voltage measured by Digital Image Correla-\ntion and Finemet ®\flm elastic constants measured\nby Brillouin light Scattering allowed estimating the\ne\u000bective magnetostriction coe\u000ecient of the \flm.\nAcknowledgments\nThe authors gratefully acknowledge the CNRS for his\n\fnancial support through the \\PEPS INSIS\" program(FERROFLEX project) and the Renatech network sup-\nporting the IEF clean room facilities. This work has been\nalso partially supported by the French Research Agency\n(ANR) in the frame of the project ANR 2010 JCJC\n090601 entitled \\SpinStress\" and by the Universit\u0013 e Paris\n13 through a \\Bonus Qualit\u0013 e Recherche\" project. The\nauthors thank Pr. Dr. Philippe Djemia for discussion\nconcerning Brillouin Ligth Scattering measurements and\nanalysis. The authors are also grateful to Dr. Brigitte\nLeridon (LPEM-ESPCI, ParisTech) for putting at their\ndisposal the experimental EPR setup.\n[1] Jing Ma , Jiamian Hu , Zheng Li and Ce-Wen Nan, Adv.\nMater. 23, 1062 (2011)\n[2] Carlos A. F. Vaz , Jason Ho\u000bman , Charles H. Ahn and\nRamamoorthy Ramesh, Adv. Mater., 22, 2900 (2010)\n[3] Pedro Martins and Senentxu Lanceros-M\u0013 endez, Adv.\nFunct. Mater., 23, 3371 (2013)\n[4] J. Ma, J. Hu, Z. Li, C.-W. Nan, Adv. Mater. 23, 1062-\n1087 (2011)\n[5] T. H. E. Lahtinen, J. O. Tuomi, S. van Dijken, Adv.\nMater. 23, 3187-3191 (2011)\n[6] K. Roy, S. Bandyopadhyay, J. 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Yang* \nDepartment of Physics, The  Ohio State University, Columbus, OH, 43210, USA \n†These authors made equal contributions to this work  \n*Emails: hammel@physics.osu.edu; fyyang@physics.osu.edu \n \nAbstract  \nEpitaxial Y3Fe5O12 thin films  have been deposited by off -axis sputtering , which \nexhibit  excellent crystallin e quality , enabling observation of large  spin pumping signals in \nPt/Y 3Fe5O12 and W/Y 3Fe5O12 bilayers  driven by cavity ferromagnetic resonance. The inverse \nspin Hall voltages reach 2.10 mV and - 5.26 mV in 5-mm long Pt/Y 3Fe5O12 and W/Y 3Fe5O12 \nbilayers, respectively , excited by a radio -frequency magnetic field of 0.3 Oe . From the \nferromagnetic resonance linewidth broadening, the interfacial spin mixing conductance of \n4.56 × 1014 Ω-1m-2 and 2.30 × 1014 Ω-1m-2 are obtained for Pt/Y 3Fe5O12 and W/Y3Fe5O12 \nbilayers, respectively.   \n \nPACS:  76.50.+g, 75.47.Lx, 75.70.Ak, 61.05.cp \n  \n1 Ferromagnetic resonance (FMR)  driven spin pumping of pure spin currents  has \ngenerat ed intense interest  for its potential application in  next-generation spintronics  [1-17]. \nDue to  the exceptionally low magnetic damping, Y3Fe5O12 (YIG) has been regarded as one of \nthe best ferromagnets  (FM)  for microwave applications and FMR  spin pumping [1- 9]. The \ninverse spin Hall effect (ISHE)  is an effective tool for studying  spin pumping  from FM s into \nnonmagnetic materials  (NM) [1-4, 12, 14, 15] . In addition to Pt which is widely used as a \nNM due to its large ISHE, β-phase W and Ta are expected to generate large ISHE voltages \n(though of the opposite sign) , making them attractive in this role as well. T o date , no clear  \nISHE detection of  FMR spin pumping in W/FM structures  has been reported. Generating a \nhigh spin current density  with a modest radio -frequency ( rf) field, hrf, requires a  FM with low \ndamping  and YIG is highly attractive for this purpose  [18]. In this letter, we report \nobservation of ISHE voltages , VISHE, of 2.10 mV (0.420 mV/mm) and 5.26 mV (1.05 mV/mm) \nfor Pt/YIG  and W/YIG bilayers, respectively, excited by a rf field of 0.3 Oe in a FMR cavity.  \nThere are two common methods in generating magnetic resonance in FMs for spin \npumping, cavity FMR and microstrip waveguide s [3, 7, 8, 12, 19]. FMR cavities produce \nmodest -strength , uniform rf fields  over a relatively large space (cm -scale) ; while microstrip \nwaveguides produce rf fields typically in micro n to sub- mm scale, and when made very close \nto the FMs, can generate fairly large h rf [12, 19]. Since t he magnitude of rf field  determines \nthe excitation strength  for spin pumping  and only a few reports  on microstrip spin pumping \npresented values of h rf [12, 19], in this letter, we  mainly  compare our results with previous \nreports of spin pumping using cavity FMR.  \n2 Most YIG epitaxial films and single crystals are produced by liquid- phase epitaxy \n(LPE)  with thicknesses from  100 nm to  millimeters [20]. Pulsed laser deposition (PLD) has \nalso been used to grow epitaxial YIG thin films  [21-23], although no ISHE measurement of \nspin pumping is  reported . Using a new approach of ultrahigh vac uum off-axis sputtering \n[24-26], we deposit epitaxial YIG  thin films on (111) -oriented Gd 3Ga5O12 (GGG) substrate s \n(see s upplementary information for details) .  \nThe crystalline quality of the YIG films is de termined  by high-resolution x-ray \ndiffraction (XRD ). A representative θ-2θ scan of a 20 -nm YIG film in Fig. 1a indicat es a \nphase -pure epitaxial YIG film. Figure 1b shows θ-2θ scans near the YIG  (444) peak for four \nfilms with thickness es, t = 10, 20, 50 and 80 nm, from which the out -of-plane lattice constant \nof the YIG films are obtained: c  = 12.426 Å, 12.393 Å, 12.383 Å and 12.373 Å, respectively . \nExcept for the 10- nm film, all other YIG films have lattice constants very close (within \n0.14%) to the bulk value of 12.376 Å, indicating essentially strain- free films.  Pronounced \nLaue oscillations  are observed in all films, reflecting smooth surface s and sharp YIG/GGG \ninterfaces . The  XRD rocking curves ( insets to Fig. 1b)  exhibit  a full width  at half maximum \n(FWHM)  of 0.027°, 0.0092°, 0.0072°, and 0.0053 ° for the 10, 20, 50, and 80 nm thick films , \nrespectively , which reach the resolution limit of conventional high- resolution XRD systems , \ndemonstrating  excellent crystalline quality . In this letter, we focus on two 20- nm YIG films \n(YIG -1 and YIG -2) for FMR and spin pumping measurements. \nRoom -temperature FMR  measurements  of the YIG films are  carried out in a cavity at \na microwave frequency f = 9.65 GHz  and power  Prf = 0.2 mW. Fig ure 2 shows a n FMR \n3 derivative spectr um of a 20- nm YIG film (YIG -1) with an in -plane magnetic field  H along \nthe x-axis ( θH = 90 °, see top- right inset to Fig. 2  for FMR measurement geometry ), which \ngives a  peak -to-peak linewidth (∆H) of 7.4 Oe  (for YIG -2, ∆H = 11.7 Oe). The angular \ndepende nce of the resonance field ( Hres) of the YIG film is shown in the bottom -left inset to  \nFig. 2b, where  Hres is defined as the field at which the derivative of  the FMR absorption \ncrosses zero. We obtain the effective magnetization, 4 πMeff = 1794 Oe , from a fit to  Hres(θH) \nemploying q uantitative analysis [ 27, 28], which agrees  well with the values reported for  \nsingle crystal YIG [ 29].  \nOur s pin pumping measurement s are conducted at room temperature on three bilayer \nsamples: Pt( 5nm)/YIG -1, Pt(5nm)/YIG -2 and  β-W(5nm)/YIG -2, all made  by off -axis \nsputtering . The samples  with approximate dimensions of 1 mm ×  5 mm  are placed in the \ncenter of the FMR cavity  with H applied in the xz -plane  while the ISHE voltage  is measured \nacross the 5-mm long P t or W  layer along the y-axis, as illustrated in Fig. 3a . The transfer of \nangular momentum to the Pt  or W  conduction electrons [ 30, 31] resulting  from FMR \nexcitation of the YIG magnetization  (M) can be described as a spin current Js injected  along \nthe z-axis with its polarization (𝜎𝜎) parallel to M . This spin current  is converted by spin -orbit \ninteractions to a charge current J c ∝ θSHJs×𝜎𝜎, where θSH is the spin -Hall angle of Pt  or W [ 32]. \nFigure 3b shows the VISHE vs. H spectr a for Pt/YIG -1 and W/YIG -2 at θH = 90° (field \nin-plane)  and Prf = 200 mW , which  generates an rf field hrf ~0.3 Oe. At this moderate h rf \nexcitation,  VISHE reaches a large  value of 1.74 mV  (0.35 mV/mm)  in Pt/YIG -1, significantly  \nlarger than  previously reported spin pumping signal s using cavity FMR  [1, 6, 9-11, 13-16]. \n4 The W/YIG -2 bilayer exhibits an even larger V ISHE of -5.26 mV (-1.05 mV/mm) , where the \nnegative sign reflect s the opposite spin Hall angle s of W and Pt [ 33]. \nFigure 3c shows t he rf -power  dependence of V ISHE for Pt/YIG -1 and W/YIG -2 at θH = \n90°. The linear relationship between  VISHE and Prf indicates that the observed ISHE voltage is  \nnot near saturation and can potentially be further increased by larger h rf (~0.3 Oe in our \nmeasurements) since V ISHE ∝ (hrf)2 [19]. Figure 3d shows a series of  VISHE vs. H spectr a for \nvarying θH at Prf = 200 mW  for the two samples . VISHE vs. H is antisymmetric about H = 0 as \nexpected  from FMR spin pumping  since the reversal of  H switches  M (hence 𝜎𝜎) and, \nconsequently, changes the sign of J c. When  H is rotated from in -plane to out -of-plane, VISHE \ngradually vanishes. M approximately follow s H at all angles  since 2500 Oe < H res < 5000 Oe, \nall larger than 4 πMeff = 1794 Oe  of our YIG  film. Figure 3e shows the angula r dependence of \nVISHE for Pt/YIG -1 and W/YIG -2 normalized by the maximum  magnitude of  VISHE at θ H = 90°. \nThe clear sin usoidal  shape is characteristic of ISHE  since [ 15] \nVISHE ∝ Jc ∝ θSHJs×𝜎𝜎 ∝ θSHJs×M ∝ θSHJs×H ∝ θSHsinθH,     (1) \nthus confirm ing that the observed  ISHE voltage arises from  FMR  spin pumping . The s pin \npumping signals we observed in insulating YIG  cannot be explained by  artifacts due to \nthermoelectric  or magnetoelectric effect s, such as anisotropic magneto resistance (AMR)  or \nanomalous Hall  effect  (AHE) [ 13, 16, 32, 34, 35] . \nWhile a spin current is generated by transfer of angular momentum from YIG to metal, \nsimultaneously, the coupling between YIG and metal exerts an additional damping to the \nmagnetization precession in YIG, resulting in increased linewidths  [10, 12], as show n in Fig. \n5 4 for  the three samples before  (∆H0) and after  (∆H1) the deposition of Pt or W. A clear \nlinewidth broadening is observed for all three samples: ∆ H1 - ∆H0 = 19.9, 24.3 and 12.3 Oe \nfor Pt/YIG -1, Pt/YIG -2 and W/YIG -2, which give V ISHE of 1.74, 2.10 and 5.26 mV , \nrespectively. We note that the magnitude of V ISHE appears more correlated  to the linewidth \nchange than the original linewidth s of the YIG films : Pt/YIG -2 has  larger linewidth increase  \n(24.3 Oe) and V ISHE (2.10 mV) than Pt/YIG -1 (∆H1 - ∆H0 = 19.9 Oe,  VISHE = 1.74 mV) \nalthough YIG -2 (∆H0 = 11.7 Oe) has a larger linewidth than YIG -1 (∆H0 = 7.4 Oe). This can  \nbe understood that while narrower FMR linewidth leads to a larger  FMR  cone angle, the \nlinewidth change determines the  interfacial  spin mixing conductance which is critically \nimportant for  spin pumping efficiency  [10, 12],  \n𝐺𝐺𝑟𝑟=𝑒𝑒2\nℎ2√3𝜋𝜋𝑀𝑀s𝛾𝛾𝑡𝑡F\n𝑔𝑔𝜇𝜇B𝜔𝜔(Δ𝐻𝐻1−Δ𝐻𝐻0),        (2) \nwhere Gr, γ, 𝑔𝑔 and 𝜇𝜇B are the real part of spin mixing conductance, the  gyromagnetic ratio , \n𝑔𝑔 factor  and Bohr magnetron, respectively . Using Eq. (2), w e obtain 𝐺𝐺𝑟𝑟 = 4.56×  1014 and \n2.30×  1014 Ω-1m-2 for Pt/YIG -2 and W/YIG -2, respectively,  which agree with the theoretical \ncalculations [ 36] and are among the highest of  reported experimental values [ 3, 5, 8, 9].  \nPreviously, spin pumping of Pt/YIG excited by similar cavity FMR gave ISHE \nvoltages in the µV range [ 1, 9, 11, 16]. The large spin pumping signals and high spin mixing \nconductance observed in our YIG films may be att ributed to two possible rea sons. First, the \nsmall thickness (20 nm) of our films compared to LPE films (100 nm or larger) may play an \nimportant role, as suggested by a recent report [ 7] that a 200- nm YIG film shows much \nhigher spin pumping efficiency than 1- µm and 3- µm films excited by a microstrip waveguide.  \n6 Secondly, the YIG films made by our off -axis sputtering method may be different in \ncrystalline quality and FMR characteristics from those by other techniques.  Compared to \ncavity FMR, microstrip waveguides can pote ntially provide much stronger rf fields , e.g. 16 \nOe in Ref . 19 and 4.5 Oe in Ref . 12, which can significantly increase the magnitude of ISHE \nvoltages ( VISHE ∝ hrf2 in the linear regime) [ 7, 12 ]. Further investigation of spin pumping in \nthese thin YIG films  using microstrip waveguides will access larger dynamic range of  spin \npumping. In addition, t he mV -level ISHE voltages reported here  using a moderate h rf will \nallow miniaturization of spin pumping structures while maintaining  signals sufficiently large \nto explore opportunities  such as magnon -based electronics and other  next generation \ntechnologies  [18]. It also provides a material platform for probing the fundamental \nmechanisms in spin pumping  for quantitative characterization of coupling mechanisms and \ninterfacial  phenomena.  \nAcknowledgements  \nThis work is supported by the Center for Emergent Materials at the Ohio State \nUniversity, a NSF Materials Research Science and Engineering Center (DMR -0820414) \n(HLW, YP, and FYY) and by the Department of Energy through grant DE -FG02 -03ER46054 \n(RA, PCH). Partial support is provided by Lake Shore Cryogenics Inc. (CHD) and the \nNanoSystems Laboratory at the Ohio State University.  \n  \n7 Reference:  \n1. Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. \nKawai, K. Ando, K. Takanashi, S. Maekawa  and E. Saitoh, Nature  464, 262- 266 (2010).  \n2. C. W. Sandweg , Y . Kajiwara, A. V . Chumak, A. A. Serga , V . I. Vasyuchka , M. B. \nJungfleisch , E. Saitoh, a nd B. Hillebrands , Phys. Rev. Lett.  106, 216601 (2011) . \n3. V.  Castel,  V. 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B 84, 054423 (2011) . \n35. S. Y . Huang , X. Fan , D. Qu , Y . P. Chen, W. G. Wang , J. Wu , T. Y . Chen, J. Q. Xiao , and  C. \nL. Chien, Phys. Rev. Lett. 109, 107204 (2012) . \n10 36. Y . T. Chen, S. Takahashi , H. Nakayama, M. Althammer , S. T. B. Goennenwein, E. Saitoh, \nand G. E. W. Bauer , Phys. Rev. B  87, 144411 (2013). \n  \n11 Figure Captions:  \nFigure 1. (a) Wide angle  semi -log θ-2θ XRD scan of a 20- nm thick YIG film grown on GGG  \n(111) . (b) Semi -log θ-2θ scans of 10, 20, 50, and 80 nm thick YIG films near the YIG (444) \npeak, all of which exhibit  clear Laue oscillations corresponding to the film thickness . The \nvertical short  lines mark th e positions  of the YIG (444) peak. The scans are offset from each \nother for clarity.  The insets are the rocking curves of the four YIG films  taken  for the first \nsatellite peak to  the left of the man peak at the 2θ  angle marked by the up arrows.  The \nshoulde r in the rocking curve of the 80 -nm film is likely due to twinning in the film.  \nFigure 2 . Room -temperature FMR derivative spectrum  dIFMR/dH vs. H of a 20-nm YIG film \n(YIG -1) at θH = 90° (field in-plane) gives a linewidth of 7.4 Oe. Top-right inset: schematic of \nFMR experimental geometry. Bottom- left inset: angular dependenc e of Hres for the  YIG film  \nand t he fit  (solid green curve) agree s well with the experimental data, from which  4πMeff = \n1794 Oe and g = 2 .0 were obtained . \nFigure 3. (a) Schematic  diagram  of the ISHE voltage measurement  setup . (b) VISHE vs. H \nspectra at θH = 90° and Prf = 200 mW for two Pt(5nm)/YIG(20nm)  (Pt/YIG -1 and Pt/YIG -2) \nand a W(5nm)/YIG(20nm)  (W/YIG -2) bilayer s give an ISHE voltage of 1.74 mV , 2.10 mV  \nand -5.26 mV , respectively . (c) rf power dependence of  VISHE with a least -squares fit for the \nthree samples. (d) VISHE vs. H spectra at different  θH for Pt/YIG -1 and W/YIG -2. The curves \nare offset for clarity.  The non- zero ISHE voltage at θH = 0° and the difference in Hres between \nPt/YIG -1 and W/YIG -2 at the same θ H are due to slight misalignment of the sample with \n12 respect to H . (e) Angular dependence of  the normalized  VISHE for Pt/YIG -1 and W/YIG -2, \nwhere the red and blue  curve s show  sinθH and - sinθH, respectively .  \nFigure 4 . FMR derivative absorption spectr a of YIG thin films  at Prf = 0.2 mW  before ( ∆H0, \nblue) and after ( ∆H1, red) the deposition of (a) 5- nm Pt on YIG -1 (VISHE = 1.74 mV ), (b) \n5-nm Pt on YIG -2 (VISHE = 2.10 mV ), and (c) 5- nm W on YIG-2 (VISHE = 5.26 mV) , which \nshow linewidth increase from ∆ H0 = 7.4 Oe to ∆H1 = 27.3 Oe, from 11.7 Oe to 36.0 Oe, and \nfrom 11.7 Oe to 24.0 Oe, respectively . \n  \n13  \nFigure 1.    \n  101103105\n10 20 30 40 50 60 70 80 90Intensity (c/s)\n2θ (deg)(a)\nYIG(444)\nGGG(444)YIG(222)\nYIG(666)\nGGG(666)t = 20 nm\n1011031051071091011101310151017101910211023\n49 50 51 52Intensity (c/s)\n2θ (deg)(b)\nt = 80 nm\nt = 50 nm\nt = 20 nm\nt = 10 nmGGG(444)\nYIG(444)25.2 25.25 25.3\nω (deg)FWHM=\n0.027o25.6 25.65ω (deg)FWHM=\n0.0092o25 25.05\nω (deg)FWHM=\n0.0072o25.1 25.15 25.2\nω (deg)FWHM=\n0.0053o\n14  \nFigure 2 . \n  -2-1012\n2500 2550 2600 2650 2700 2750dIFMR/dH (a.u.)\nH (Oe)∆H = 7.4 OePrf = 0.2 mWYIG-1\nt = 20 nm\n300040005000\n0 30 60 90Hres (Oe)\nθH (deg)\n15  \nFigure 3.  \n  \n-101\n0 90 180 270 360\nθH (deg)VISHE /|VISHE(90o)|\n(e)W/YIG-2 Pt/YIG-1-6-5-4-3-2-1012\n0 100 200VISHE (mV)\nPrf (mW)Pt/YIG-2\nPt/YIG-1\nW/YIG-2(c)\n-5-4-3-2-1012\n-100 -50 050100VISHE (mV)\nH - Hres (Oe)(b)Pt/YIG-1Pt/YIG-2\nW/YIG-2 θH = 90o(a)\n-15-10-5051015\n-4000 -2000 0 2000 4000\nH (Oe)VISHE (mV)(d) θH = 0o\nθH = 30o\nθH = 60o\nθH = 90o\nθH = 120o\nθH = 150o\nW/YIG-2 Pt/YIG-1\nVISHE = 5.26 mV VISHE = 1.74 mV\n16  \n \nFigure 4 .  \n \n \n \n     \n \n  -2-1012(a)YIG-1, ∆H0 = 7.4 Oe\nPt/YIG-1, ∆H1 = 27.3 Oe\nVISHE = 1.74 mV∆H1 - ∆H0 = 19.9 Oe\n-2-1012\n-40 -20 0 20 40\nH - Hres (Oe)(c)\nVISHE = 5.26 mVYIG-2, ∆H0 = 11.7 Oe\nW/YIG-2, ∆H1 = 24.0 Oe\n∆H1 - ∆H0 = 12.3 Oe-2-1012(b)\nVISHE = 2.10 mVdIFMR/dH (a.u.)YIG-2, ∆H0 = 11.7 Oe\nPt/YIG-2, ∆H1 = 36.0 Oe\n∆H1 - ∆H0 = 24.3 Oe\n17 Supplementary  Information  for \nLarge Spin Pumping from Epitaxial Y 3Fe5O12 Thin Films to Pt and W Layers  \nH. L. Wang, C. H. Du, Y . Pu, R. Adur, P. C. Hammel, and F. Y . Yang  \n \n1. Growth of Y3Fe5O12 films \nSingle -crystal line Y3Fe5O12 (YIG) epitaxial thin films were grown on (111) -oriented \nGd3Ga5O12 (GGG) substrate s in an off -axis ultrahigh vacuum ( UHV ) sputtering system with \na base pressure below  5 ×10−9 Torr. Horizontal sputtering sources and 90° off-axis geometry \nwere used for film deposition [S1-S3]. The optimal growth conditions include:  a total Ar/O 2 \npressure of 11.5 mTorr with  an O2 concentration of 0.15%, a substrate temperature 750 °C, \nand a radio -frequency  sputtering power of 50 W . The deposition rate for YIG is 0.33 nm/min \nand t he film thickness ranges from 10 to 200 nm. F or FMR and spin pumping measurement s, \nthe film thickness is  typically 20  nm.  \n \n2. Deposition  of Pt and W  film s \nPt and W films of 5 -nm thick were deposited in the same off-axis UHV sputtering \nsystem  for YIG film growth. DC magnetron sputtering was used with a deposition rate of \nabout 1.7 nm/minute for Pt and W.  \n \n3. Magnetization characterization  by FMR  \nSaturation m agnetization and g factor of the YIG films  were determined from FMR \n18 resonance field as a function of θH. Resonant condition can be derived by  minimiz ing the \ntotal free energy F. For a material with tetragonal symmetry [ S4], F can be expressed by:  \n𝐹=−𝑯·𝑴+1\n2𝑀𝑀�4𝜋𝑀𝑀eff cos2𝜃−12𝐻𝐻4⊥cos4𝜃− 18𝐻𝐻4||(3+cos4𝜙)sin4𝜃−\n𝐻𝐻2||sin2𝜃sin2(𝜙−𝜋𝜋\n4 )�,            (S1) \nwhere θ and φ are angles of magnetization ( M) in the equilibrium position with \nrespect to the film normal and in -plane easy axes, respectively. The first term in Eq. ( S1) is \nthe Zeeman energy and the second term is the effective demagnetizing energy ( 4πMeff) which \nincludes both the shape anisotropy (4 πMs) and out -of-plane uniaxial anisotropy H2⊥, where \n4π𝑀𝑀eff=4π𝑀𝑀s−H2⊥.The remaining terms are out -of-plane cubic anisotropy ( H4⊥), \nin-plane cubic anisotropy ( H4||), and in- plane uniaxial anisotropy ( H2||).  \nThe equilibrium orientation ( θ, φ) of magnetization can be obtained by minimizing \nthe free energy, and t he FMR resonance frequency ω in equilibrium is given by [ S4]: \n�𝜔𝜔\n𝛾𝛾�2\n=1\n𝑀𝑀2sin2𝜃𝜃�𝜕𝜕2𝐹𝐹\n𝜕𝜕𝜃𝜃2 𝜕𝜕2𝐹𝐹\n𝜕𝜕𝜙𝜙2−�𝜕𝜕2𝐹𝐹\n𝜕𝜕𝜃𝜃𝜕𝜕𝜙𝜙�2\n�,      \n (S2) \nwhere 𝛾𝛾=𝑔𝑔𝜇𝜇𝐵𝐵/ℏ is the gyromagnetic ratio.  We used a numerical procedure to \nobtain the equilibrium angle s at resonance condition at different  θH [S5]. By fitting the data in \nthe bottom -left inset to  Fig. 2 , we obtained 4πMeff = 1794 Oe and g factor = 2.0 . Given that \nYIG is magnetically  soft, all the anisotropy terms should be small.  \n \n  \n19 Reference:  \nS1. A. J. Hauser,  R. E. A. Williams,  R. A. Ricciardo,  A. Genc,  M. Dixit,  J. M. Lucy,  P. M. \nWoodward, H.  L. Fraser, and F .  Y. Yang , Phys. Rev. B 83, 014407 (2011) . \nS2. A. J. Hauser, J. R. Soliz, M. Dixit,  R. E. A. Williams,  M. A. Susner, B. Peters,  L. M. Mier,  \nT. L.  Gustafson, M. D. Sumption, H. L. Fraser, P. M. Woodward and F. Y . Yang , Phys. Rev. \nB Rapid Comm. 85, 161201(R) (2012).  \nS3. C. H. Du , R. Adur, H. L. Wang,  A. J. Hauser,  F.  Y. Yang, and  P. C. Hammel , Phys. Rev. \nLett. 110 , 147204 (2013).  \nS4. X. Liu, W. L. Lim, L. V . Titova, M. Dobrowolska, J. K. Furdyna, M. Kutrowski, and T. \nWojtowicz , J. Appl. Phys. 98, 063904 (2005).  \nS5. Y . Q. H e, P. E. Wigen, J. Magn. Magn. Mater. 53, 115- 120 (1985).  \n \n \n20 "    },    {        "title": "1102.2185v2.Circularly_Polarized_Resonant_Rayleigh_Scattering_and_Skyrmions_in_the__ν____1_Quantum_Hall_Ferromagnet.pdf",        "content": "FIG.1:Energy ofthePLtransitions asafunction of\nthemagnetic field attemperature of~1.4K.Inthe\ninset isreported theintensity ofthePLpeaks .The\nschematic view ofthe spin split level L0isalso\nreported .\nFIG.2:The light scattering plus emission spectra\naroundn=1for (a) thes+s+and (b)s-s-\npolarizations atT~1.4K.FIG.3:The ratio between theRRS andnon resonant\nlight scattering intensities for the four crosses\npolarizations .The lines refers totheSkyrmion model\nofRef.[1]with thedifferent values oftheparameter\nS=A= 2,3and4."    },    {        "title": "1607.02985v2.Triple_resonant_Brillouin_light_scattering_in_magneto_optical_cavities.pdf",        "content": "arXiv:1607.02985v2  [physics.optics]  22 Sep 2016Triple-resonant Brillouinlightscattering inmagneto-op tical cavities\nJ. A. Haigh,1,∗A. Nunnenkamp,2A. J. Ramsay,1and A. J. Ferguson2\n1Hitachi Cambridge Laboratory, Cambridge, CB3 0HE, UK\n2Cavendish Laboratory, University of Cambridge, Cambridge , CB3 0HE, UK\n(Dated: November 16, 2021)\nAn enhancement in Brillouin light scattering (BLS) of optic al photons with magnons is demonstrated in\nmagneto-optical whispering gallery mode (WGM) resonators tuned to a triple-resonance point. This occurs\nwhenboththeinputandoutputopticalmodesareresonantwit hthoseofthewhisperinggalleryresonator,witha\nseparationgiven bythe ferromagnetic resonance (FMR)freq uency. The identificationandexcitationof specific\noptical modes allows us to gain a clear understanding of the m ode-matching conditions. A selection rule due\nto wavevector matching leads to an intrinsic single-sideba nd excitation. Strong suppression of one sideband is\nessential forone-to-one frequency mapping incoherent opt ical-to-microwave conversion.\nExtending microwave-optical transducers into a regime\nwhereinter-conversionbetweensingleopticalandmicrowa ve\nphotons is possible in a coherent manner [ 1] is an impor-\ntant technologicalaim, as it would open up many avenuesin,\nforexample,implementingexistingsuperconductingquant um\ndevices[2]inawiderquantumnetwork[ 3]. Furthermore,fre-\nquency shifting of single photons would enable quantum op-\ntical devices to take advantage of wavelength division mult i-\nplexing. Strongprogresstowardsthesegoalshasbeenmadei n\ncavityoptomechanics[ 4–7],andoptimizedelectro-opticmod-\nulators[8–10].\nRecently,microwave-opticalinter-conversionhasalsobe en\nexplored in a cavity opto-magnonic system [ 11,12], where\nmagnetic Brillouin light scattering [ 13] has been reported in\nhighQoptical WGMs of a transparent magnetic sphere [ 14].\nIn this system, the collective excitations of the magnetic m o-\nment, magnons, play a role analogous to the phonons in a\ncavity optomechanics system [ 15]. An important feature of\nthis opto-magnonicsystem is the non-reciprocityof the BLS ,\nwhere only one sideband has been observed [ 11,12]. A key\nrequirementforacoherenttransducerisaone-to-onemappi ng\nofthe frequencycomponents,andhencea strongsuppression\nofonesideband. Incontrasttoanoptomechanicssystem,due\nto conservationof angularmomentum,optically inducedcre -\nationandannihilationofmagnonsrequiresachangeinoptic al\npolarization [ 16]. When combined with the geometric bire-\nfringenceofaWGMresonator,thisresultsinanon-reciproc al\ntriple-resonance condition, where the optical pump and sig -\nnal of opposite polarizationare resonantwith differentca vity\nmodes, whose frequency splitting is equal to the driven fer-\nromagnetic resonance [ 17]. Hence, side-band suppression is\nenforcedbyaselection-rule,ratherthanbydetuningthepu mp\nlaserfromtheopticalcavity,asisusuallythecaseinacavi ty-\noptomechanicssystem.\nIn this Letter we show that the non-reciprocal triple-\nresonanceconditionbetweenopticalmodesforpumpandsig-\nnal of the inter-conversion can be achieved with the precise\nmode identification allowed by prism coupling to the mag-\nnetic sphere. This is in contrast to previous measurements\n[11,12], where, due to the waveguide coupling used, the ex-\nact identification of the optical modes involved has been dif -\nficult, with the resonance condition being met accidentallyFigure 1. (a) Top view of experimental setup. The scattered l ight,\nwithpolarizationorthogonal totheinputbeam, emittedata different\nangleduetothebirefringenceoftherutilecouplingprism, isspectro-\nscopically analyzed with a scanning Fabry-P´ erot etalon. A dc mag-\nnetic field Hdcis applied along the z-axis. (b) Microwave antenna\ntodrive ferromagnetic resonance inthe YIGsphere (side vie w). The\nmicrowave drive is provided by a vector network analyzer (VN A).\nThe FMR modes are identified by measuring the microwave reflec -\ntioncoefficient |S11|asafunctionoffrequency,afterwhichtheVNA\nis configured in continuous-wave mode to drive the resonance . (c)\nCoordinate systems usedinthe analysis.\n[12]. For microwave driving of the uniform Kittel magneti-\nzationmodein the plane ofthe WGM, the polarizationof the\npumplasercanbeusedtoselectthescatteringdirectionvia the\nfixed change in the azimuthal mode index. We identify that\nthis selectivity arises from wavevector matching around th e\noptical path of the pump and signal light-fields and the geo-\nmetricaldependenceofthemagneto-opticalcoupling. Fina lly,\nmeasurements of the BLS intensity as a function of detuning\nfromthetriple-resonanceconditionshowexcellentagreem ent\nwith a simple analytical model. Our experiments allow us\ntopreciselycharacterizetheresonantsingle-photonmagn eto-\nopticalcouplingstrength[ 18,19].\nThe experimental setup is shown in Fig. 1(a). A prism\ncoupler [20] is used to match the input angle, and therefore\nthe wavevector, to the low order WGMs. The mode struc-\nture is probed by measuring the reflected intensity with same\npolarization as the input using a photodiode (PD I), as the\ninput laser wavelength is tuned. The light emitted from the2\nFigure 2. Optical mode identification for an r≈250µm YIG\nsphere. (a) Schematic of mode families for radial indices q= 1,2,\nand forh- andv-polarization. The free spectral range λFSRis indi-\ncated by the black solid line. The h-vsplittingλh-vis shown by\nthe black dotted line, while the dashed line indicates the sp litting\nbetween adjacent modes with ∆q= 1. The azimuthal index mde-\ntermines the number of wavelengths around the circumferenc e and\nradial index qdetermines the number of radial nodes. (b) Represen-\ntative plots of real part of the electric field(i) in the WGM pl ane for\nm= 20,q= 1,and(ii)incross-sectionfor q= 1,2. (c)Reflectance\nspectrum for v-polarized input. (d) The dispersion of the FSR λFSR\nis used toidentify the strongest mode family as q= 1. The splitting\nbetweenthedifferentmodes λq,1↔2isusedtoidentifythesecondas\nq= 2. Solid and dashed lines show the calculated dispersions [ 24]\nfittedwithsmalladjustments of the sphere radius.\ncavity with opposite linear polarization to the input is emi t-\nted at a different angle due to the birefringence of the ru-\ntile prism. Thispolarization-scatteredcomponentis anal yzed\nwith a scanning Fabry-P´ erot etalon on an avalanche photodi -\node (PD II). A microwave antenna (Fig. 1(b)) is placed close\nto the YIG sphere to drive ferromagnetic resonance and the\nmagnetic field from a permanent magnet (NdFeB) mounted\non a stage is used to tune the FMR frequency. The setup can\nbeswitchedtomeasurethesamequantitiesforbothlinearpo -\nlarizationsoftheinputbeam.\nFirst,weidentifytheopticalWGMs. Thedcmagneticfield\nis fixed in the out-of-plane direction. Since there is no stat ic\ncomponent of the magnetization along the direction of prop-\nagation, mixing between linear polarized modes due to the\nFaraday effect is negligible [ 14]. We therefore use the stan-\ndard analytical forms of the WGM electric field distribution s\nandresonantwavelengths[ 21,22],withtwolinearlypolarized\ncomponents perpendicular and parallel to the sphere surfac e.\nThesemodes,whichwe label horizontal handvertical v[23]\n(seeFig.1(c)),aresplit inenergyduetothegeometricalbire-\nfringence from the different surface boundary conditions f or\ntwoelectric fieldcomponents.\nThe basic mode structure is shown schematically in\nFig.2(a). The expected reflectance spectra for h(pink) and\nv(green) polarized modes are shown including modes with\nradial index q= 1,2for sets of modes with a difference∆m= 1in the azimuthal index m. The mode indices\nare defined in Fig. 2(b). The free spectral range is given\nbyλFSR=λ2\n0/2πrnYIGto a good approximation in the rel-\natively large spheres ( r∼100µm,m∼1000) which\nwe study. In the same limit, the h-vsplitting is given by\nλh-v=λFSR/radicalbig\nn2\nYIG−1/nYIG[21]. ForYIG,with nYIG≈2.2,\nλh-v≈0.9λFSR. Therefore the closest adjacent modesof op-\nposite polarization are for different mindices, separated by\nmv−mh= 1andλeff\nh-v= 0.1λFSR.\nFig.2(c) presents a reflection spectrum for a h-polarized\ninput. Two familiesof modesare observed. These are identi-\nfied asq= 1andq= 2from comparison of the wavelength\ndispersion, shown in Fig. 2(d), to the expectedsplitting. This\ndemonstratesthe highly selective excitation of the WGM, al -\nlowing clear identification of the matching conditionsfor e n-\nhancedwavelengthconversion.\nWith the dc magnetic field in the out-of-plane direction z,\nwe now introduce the microwave drive field in the in-plane\nxdirection. This drives ferromagnetic resonance (FMR), the\nprecession of the magnetization about the static field. The\nmagneto-staticmodes[ 25]oftheYIGspherecanbeidentified\nby measuring the microwave reflection coefficient S11of the\nmicrowaveantenna. The FMR spectrumas a functionof per-\nmanent magnet position is shown in Fig. 3(a) along with the\nexpected Kittel mode frequency calculated from the positio n\ndependentmagnetic field (blue line) [ 26]. From this field de-\npendence and the relative strength of the absorption, the un i-\nformKittelmodecanbeidentified. Duringdatacollectionth e\nmicrowavedrivetrackstheFMRfrequencytocompensatefor\nfluctuationsinthe dcmagneticfield.\nTo achieve the triple-resonancecondition, we use a sphere\nofradius500µm,whichhas λeff\nh-vcorrespondingto ωv−ωh≈\n7GHz, and drive the FMR of the uniform Kittel mode close\nto that frequency. The cross-polarized emission of the cavi ty\nis spectrally analyzed using the etalon, and example data ar e\nshowninFig. 3(b). Thetoppanelshowsameasuredspectrum\nforhpolarizedinput. Therearetwosetsofpeaks,eachmatch-\ning the 10 GHz FSR of the etalon. The largest is the elastic\nscattered light at the same wavelength as the input laser. Th e\nanti-Stokes signal is marked with a blue arrow and is higher\ninfrequencyby ≈7GHz. ThereisnomeasurableStokespeak\nfor this input polarizationfor any input wavelength. The bo t-\ntom panel shows a measured spectrum for vpolarized input.\nHerethereis onlya Stokespeak(orangearrow),lower infre-\nquencyby the microwavedrive. In the following,we demon-\nstratethatthisasymmetrybetweentheStokes/anti-Stokes sig-\nnal [11,12], different for the two input polarization, follows\nfromaselectionruleintheBLSprocess. Thelinewidthofthe\nBLS peakis limited bythe 200MHz resolutionofthe etalon.\nWe further note that when the magnetic field is reversed, the\nBLS issubstantiallyreduced.\nFigure3(c,d) comparesthe BLS peak amplitudeas a func-\ntion of detuning of the input laser from the resonance to the\nreflectivity spectra. The BLS is enhanced when the h(v) po-\nlarized input laser is resonant with the h(v) polarized, q= 1,\nWGM.3\nFigure 3. (a) FMR of YIG sphere measured through microwave re -\nflection coefficient |S11|of the antenna as a function of permanent\nmagnet position. The blue dashed line shows the expected dep en-\ndence of the uniform Kittel mode from the known dependence of\nmagnetic field on distance from a cuboid magnet [ 26] and the gy-\nromagnetic ratio γ= 28GHz T−1. The magnetic field range is\nHdc≈100-320mT. TheQ-factor of the magnetic mode is QFMR≈\n400(as this is due to Gilbert damping, the rate ��FMR≈10-20MHz\ndepends linearly on FMR frequency [ 27]). (b) Measured spectra\nof emitted signal (PD II) for h(upper) an vinput polarization for\nωFMR/2π≈7GHz. Orange and blue arrows label Stokes and anti-\nStokespeaks,respectively. Wecanexcludethesuppresseds ide-band\ndown to the signal-to-noise ratio, maximum ≈20(slightly different\nfor the two input polarizations due to different experiment al condi-\ntions). (c,d) Lower panels: maximum of BLSintensity as a fun ction\nof input laser wavelength, for handvinput, respectively. Upper\npanels: Reflected optical intensity (PD I), shown for compar ison.\nThex-axes is detuning from the resonant wavelength of the hpo-\nlarized mode. For vinput measurements, this is set by the mea-\nsuredh-vsplittingλeff\nh-v. For the optical modes Qv≈2×105and\nQh≈1×105(dissipationrates κv≈1GHz,κh≈2GHz).\nTo explore the triple-resonance condition, the wavelength\ndependenceof the BLS peak is measuredas a functionof the\nFMR frequency ωFMR. This is shown in Fig. 4for (b)hand\n(c)vinput polarization. For h(v) input, we only observe the\nStokes (anti-Stokes) signal, and the color corresponds to t he\nintensityofthatsignal. AstheWGMsaresensitivetochange s\nin sphere temperature with dissipated microwave power, the\nwavelength scans are aligned at the dip in reflected intensit y\n(PDI),andarenormalizedtothepeakvalueforthatFMRfre-\nquency in order to highlight the mode structure. An example\nof the reflected intensities (PD I) for both input polarizati ons\nare shown for comparison in Fig. 4(a), these are independent\noftheFMRfrequency.\nIn Fig.4(b, c) there are two maxima in the efficiency of\nthe BLS process. The first peak is independent of the FMR\nfrequencyandisalignedwiththeWGMoftheinputpolariza-\ntion. This corresponds to a cavity enhancement of the input\nFigure 4. BLS scattering amplitudes for different input lin ear polar-\nizations (h: pink,v: green). (a) Reflected intensity (PD I) for com-\nparison. Azimuthal mode indices are labeled for clarity. Th e two\ncurves are plotted on separate scales. (b) Color-plot of BLS inten-\nsity forh-input polarization as a function of input laser wavelength\nand FMR frequency. Each scan for fixed FMR frequency has been\nnormalized tothe peak amplitude for that scan. (c) As in (b), but for\nv-inputpolarization. Dashedlinesin(b)and(c)indicateth eresonant\nwavelengths for the two polarizations. The x-axis in all panels is in\ndetuning fromtheresonant wavelengthof the hpolarizedmode. For\nvinput measurements, thisis setby the measured h-vsplitting.\nlight field. For small FMR frequenciesthereis a secondpeak\nwhose wavelength is linear in the FMR frequency. For h(v)\npolarized input, the black lines in Fig. 4(b, c) corresponding\ntoωv−ωFMRandωh+ωFMRrespectively are in reasonable\nagreementwiththedata. Hence,thesecondpeakcorresponds\nto a cavity enhancement of output light field, shifted by the\nFMRfrequency.\nBytuningtheFMRfrequencytomatchthe h-vsplitting,we\nachieve the triple-resonance condition. This scattering i s be-\ntweenmodesofdifferentazimuthalmodeindices, ∆m=±1.\nInfact,thisisconsistentwithourexpectation,as,inthef rame\nof the light propagatingaroundthe mode, the in-planedrive n\nmagnetization rotates with respect to the direction of prop -\nagation. This means that the magnetic mode has an effec-\ntive wavevector,and azimuthalintegrationof the electrom ag-\nnetic energy leads to a selection rule mv−mh= 1[28]. It\nis this required change in mode index that allows the triple-\nresonance condition to be achieved for reasonable magnetic\nfield strengths, as the FSR is approximately equal to the h-\nvsplitting so that the two modes with mv−mh= 1are\nclosely spaced in frequency. This is in contrast to previous\nwork [11,12], which has suggested ∆m= 0, requiring sub-\nstantially highermagneticfields. We also note that in scatt er-\ningtheradialindex qis unchanged, ∆q= 0.\nFurthermore,we can see that the Stokes/anti-Stokesasym-\nmetry persists even detuned from the triple-resonance cond i-\ntion. This indicates that the asymmetry is not governed sim-\nply by the optical density of states. In fact, the selection r ule\nmv−mh= 1means that interaction Hamiltonian for the\nmagnonmode ˆbandtwoopticalmodes ˆah,ˆav,reducestotwo4\nFigure5. Comparisonbetween(a)experimentand(b)theoryo fBLS\nintensity as a function of input wavelength detuning and FMR fre-\nquency for hinput polarization. The blacklines are the wavelengths\ncorresponding to ωhandωv−ωFMR. Both experimental data and\nmodel are normalized to the peak value at each FMR frequency t o\nallow better comparison of the mode structure. (c) Peak BLS e ffi-\nciency as a function of FMR frequency. The red line is the expe cted\ntrendgiven bythe maximum of Eq.( 2)for fixed ωFMR.\nterms [28], correspondingto the observed Stokes/anti-Stokes\nasymmetry,selectedbytheinputpolarization:\nˆHint= ¯hG(ˆbˆa†\nvˆah+ˆb†ˆa†\nhˆav). (1)\nHence, the scattering process is non-reciprocal due to the\nwavevector matching around the WGM and azimuthal de-\npendence of the magneto-optical coupling. From the known\nstrengthoftheFaradayeffectinYIG,wecalculatethesingl e-\nphotoncouplingrate G= 1Hz [28].\nWe can compare the measured data to a simple analytical\nmodelbasedonthese threemodes[ 28]. Theamplitudeof the\nscattered field as a function of the detuning from the triple-\nresonance condition ωFMR−ωv+ωLand of the h-polarized\ninputfrequency ωh−ωLis\n|/angbracketleftˆav,out/angbracketright|2= (2)\n4G2|¯ah,in|2|¯bin|2κvκh/κFMR/bracketleftBigκ2\nh\n4+(ωh−ωL)2/bracketrightBig/bracketleftBig\nκ2v\n4+(ωFMR−ωv+ωL)2/bracketrightBig.\nThis is the product of two Lorentzians, correspondingto res -\nonantenhancementoftheinputandoutputfieldsrespectivel y.\nAll the parameters are known from independent measure-\nments,sothatwecanplotthisexpressioninFig. 5(b),withex-\ncellentagreementwiththedataplottedalongside(Fig. 5(a)).\nFinally, we plot the maximum BLS amplitude for each\nFMR frequency in Fig. 5(c). The variation in the data is due\nto changes in the microwave power transmitted to the YIG\nsphere at different frequencies. The red line is the expecte dvalue given by Eq. ( 2), vertically scaled to match the data,\nwithgoodagreementin thegeneraltrend.\nTosummarize,wehavedemonstratedthetuningofacavity\nmagneto-opticalsystemtoatriple-resonanceconditionfo ren-\nhancedBrillouin light scattering. A selection rule ∆m=±1\nin the azimuthal index of the optical mode arises due to\nwavevector matching around the optical path of the WGM.\nDue to conservation of total angular momentum, a change in\nthe optical orbital angular momentum of ∆m=±1, results\nin the annihilation/creation of one magnon, and up/down-\nconversion of the light, respectively. The modes closest to\nenergy-matching conditions have mv−mh= 1, and hence\nthe polarization of the input laser selects either a Stokes o r\nAnti-Stokes frequency conversion. Since the asymmetry of\nthe BLS arises from a selection rule, a strong asymmetry can\nalso be observed away from cavity resonance. This mecha-\nnismhassimilaritiestoBLSbetweentwoopticalmodesinop-\ntomechanics[ 29]. Wefurthernotethatnon-transversecompo-\nnentsoftheopticalmodes[ 30]arenotincludedinourmodel,\nand are therefore not needed to explain the asymmetry in the\nBLS [11,12].\nAsymmetries in magnon BLS have been reported previ-\nously due to other mechanisms. Localization of surface\nmagnon modes with a given chirality [ 31] is not relevant\nhere, as we study the uniform magnetic mode, and spin-spin\ncorrelations between different components introduced by t he\ndemagnetizing field are only relevant in a thin film geome-\ntry [32]. It is possible that interference between the first-\n(Faraday) and second-order (Voigt) magneto-optical effec ts\n[16,33] may result in minor corrections to the differing am-\nplitudes.\nAlthough the single-photon coupling rate is significantly\nsmallerthanthelinewidthsoftheopticalandmagneticmode s,\nthe scaling of the coupling with the magnetic mode vol-\nume suggests that interesting regimescould be achieved wit h\nsmaller devices. Optimizing the Q-factor mode volume ra-\ntio [34] could be achieved with photonic-crystal defect cavi-\nties[35],orplasmonicresonances,whichhavebeenshownto\nenhance static magneto-optical parameters [ 36–38]. In addi-\ntion, for the current system, cooling to low temperatures[ 39]\nshouldimprovethe Q-factorsof all modessignificantly. This\nmay open up the possibility of spontaneousBrillouin coolin g\n[29] orlasing[ 40]ofthemagneticmodein thissystem.\nAcknowledgments. This project was partly funded by EP-\nSRC under EP/M50693X/1. A. J. F. is supported by ERC\ngrant648613anda HitachiResearchFellowship. A.N.holds\naUniversityResearchFellowshipfromtheRoyalSocietyand\nacknowledges support from the Winton Programme for the\nPhysicsofSustainability.\n∗jh877@cam.ac.uk\n[1] C. A. Regal and K. W. Lehnert, “From cav-\nity electromechanics to cavity optomechanics,”5\nJ.Phys.: Conf.Ser. 264, 012025 (2011) .\n[2] M. H. Devoret and R. J. 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TheelectricfieldsoftheWGMmodesindielectricspheresha veapprox-\nimate analytical forms [ 21,24]. Here, we are interested in a single pair of linearly polari zed modes with polarization vector /vector v\nperpendiculartothe WGM planeand /vectorhperpendicularto thespheresurface,see Fig.1(c). In a sphe ricalcoordinatesystem with\naxisalong /vectorh,/vector v, andthe directionofpropagation /vectork= (/vectorh×/vector v), wecanwrite\n/vectorE=Eh/vectorh+Ev/vector v. (S2)\nThe componentof the magnetization which enters the energy i s that along /vectork, i.e.Mk=−Mxsinφ+Mycosφfor a counter-\nclockwisedirectionofpropagation. Substituting M±=Mx±iMywe obtain Mk=−i1\n2/parenleftbig\nM+e−iφ−M−eiφ/parenrightbig\n. The coupling\nconstantcanbe foundfromthemagneto-opticalcouplingpar tofu,\nuint=−i\n4ε0f/integraldisplay\ndV(MkE∗\nhEv−MkE∗\nvEh) (S3)\n=−1\n8ε0f/integraldisplay\ndV/parenleftbig\ne−iφM+E∗\nhEv−eiφM−E∗\nhEv\n−e−iφM+E∗\nvEh+eiφM−E∗\nvEh/parenrightbig\n.\nTheelectricfieldisquantized[ 19]bysubstituting\nEh,v→ˆE+\nh,v=i/radicalBigg\n¯hωh,v\n2ε0n2\nYIGVWGMF(r,θ)1√\n2πe−imh,vφˆah,v (S4)\nwheremh,vis the azimuthal mode index, defined positive for counter-cl ockwise propagation, and F(r,θ)is the mode func-\ntion, normalized such that/integraltext\nFdV=VWGMapproximately equal for the two modes. We leave the azimutha l part outside the\nmode function to emphasize a selection rule which will becom e apparent, and replace the magnetization components with t he\nraising and lowering operators for the total spin M+→(M0/S0)ˆS+, whereM0andS0are the total magnetization and spin,\nrespectively. Thisgives\nˆHint=−ε0f\n8¯hω\n2ε0n2\nYIGM0\nS01\n2π/integraldisplay\ndφ (S5)\n×/bracketleftBig\ne−iφ(1−mh+mv)ˆS+ˆa†\nhˆav−eiφ(1+mh−mv)ˆS−ˆa†\nhˆav\n−e−iφ(1+mh−mv)ˆS+ˆa†\nvˆah+eiφ(1−mh+mv)ˆS−ˆa†\nvˆah/bracketrightBig\n,\nwherewehaveusedtheapproximation ω=ωv≈ωh.\nIn the experiment, the optical cavity modes of interest are s eparated by ≈7GHz, of the order of the FMR frequency, and\nhavemv−mh= 1. Thissatisfiestheselectionrulegivenbythe azimuthalint egrationforsecondandthirdterms. Thefirst and\nfourth terms would be satisfied by mv−mh=−1, but the frequency separation of these modes is ≈90GHz, such that the\ntriple-resonance condition is far from being met. This resu lts in the measured Stokes/anti-Stokes asymmetry. We note t hat for\ntheoppositedirectionofopticalpropagationthemodeindi ceswillchangesign mh→ −mh,suchthattheoppositetermswould\nsurvive. Forthe situationofinterest,we canwritedownthe interactionHamiltonian\nˆHint= ¯hg(ˆS−ˆa†\nhˆav+ˆS+ˆa†\nvˆah), (S6)\nwithcouplingconstant\ng=ε0fM0\n8¯hS0¯hω\n2ε0n2\nYIG. (S7)2\nThe expression can be simplified by putting Faraday constant in terms of Verdet constant f= (2cnYIG/ωM0)V, usingS0=\nNspins/2,\ng=1\n4Nspinsc\nnYIGV, (S8)\nwhereNspinsis the numberofspins in the sphere. Using the Holstein-Prim akoffapproximationthe interactionHamiltoniancan\nalsobewrittenintermsofmagnoncreationandannihilation operators ˆb†,ˆb,wheretheassociationwiththeraisingandlowering\noperators ˆS±will depend on the equilibrium direction of the magnetizati on. For dc magnetic field in the +zdirection, the\nsubstitutionis ˆS−=√2S0ˆb†,whichresultsinEq.(1)ofthemaintext\nˆHint= ¯hG(ˆbˆa†\nvˆah+ˆb†ˆa†\nhˆav), (S9)\nwiththe single-photoncouplingstrength\nG=/radicalbig\nNspinsg=Vc′\n4/radicalBigg\n1\nNspins, (S10)\nwherec′=c/nYIGis the speed of light in YIG [ 11]. The terms in ˆHintthen satisfy energy conservation, as ωv< ωh, e.g. for\nb†ˆa†\nhˆavwe haveωFMR=ωh−ωv. This would not be the case for the opposite magnetic field dir ection,ˆb†→ˆb. We observe\nthisin experiment;foroppositedirectionofthe magneticfi eld the Stokes/anti-Stokessymmetryisreversedbut theamp litudeis\nseverely reduced. Note that equivalentlyfor the opposite d irectionof optical propagation,the opposite magnetic fiel d direction\nwouldbe required.\nFor a 1 mm sphere we estimate G/2π≈1Hz, from the number density of spins nspin= 2.1×1028m−3,Vsphere=\n4.2×10−10m3(Nspins= 1×1019),V= 3.77radcm−1[11],andnYIG= 2.2.\nDYNAMICALMODEL\nThederivationofEq.(2)in themaintextisasfollows. Oursy stem isdescribedbythe Hamiltonian\nˆH= ¯hωhˆa†\nhˆah+¯hωvˆa†\nvˆav+¯hωFMRˆb†ˆb+¯hG(ˆbˆa†\nvˆah+ˆb†ˆa†\nhˆav) (S11)\nIn the frame of the laser drive, separating mean and fluctuati ons, we have ˆah=e−iωLt(¯ah+ˆdh)andˆav=e−iωLt(¯av+ˆdv).\nDrivingthe h-polarizedmodewithdrivestrength ¯ah,inleadsforsmallcoupling Gto¯av= 0and\n¯ah=√κh¯ah,in\nκh\n2−i(ωh−ωL). (S12)\nForsmallcoupling GwecanlinearizetheHamiltonian\nˆH′= ¯h(ωh−ωL)ˆd†\nhˆdh+¯h(ωv−ωL)ˆd†\nvˆdv+¯hωFMRˆb†ˆb+¯hG|¯ah|(ˆb†ˆdv+ˆbˆd†\nv). (S13)\nFrom this we derive quantum Langevin equations assuming sta ndard linear damping for both optical modes κh,κvas well\nas the magnon mode κFMR(this damping rate is linear in ωFMR, corresponding to the Gilbert damping in the Landau-Lifshi tz\nequation[ 27])\n˙ˆb=−iωFMRˆb−κFMR\n2ˆb+√κFMRˆbin−i¯Gˆdv (S14)\n˙ˆdv=−i(ωv−ωL)ˆdv−κv\n2ˆdv+√κvˆdv,in−i¯Gˆb (S15)\nwiththe driving-enhancedcouplingconstant ¯G=G|¯ah|.\nIntheFourierdomainweobtain\nˆb(ω) =+√κFMRˆbin−i¯Gˆdv\nκFMR\n2−i(ω−ωFMR)(S16)\nˆdv(ω) =+√κvˆdv,in−i¯Gˆb\nκv\n2−i[ω−(ωv−ωL)]. (S17)3\nIntheexperimentwe driveFMRresonantly, /angbracketleftˆbin(ω)/angbracketright=¯bin2πδ(ω−ωFMR), so thecoherentresponsetosecondorderin Gis\n|/angbracketleftˆdv/angbracketright|2=4G2|¯ah,in|2|¯bin|2κh/κFMR/bracketleftBig\nκ2\nh\n4+(ωh−ωL)2/bracketrightBig/bracketleftBig\nκ2v\n4+(ωFMR−ωv+ωL)2/bracketrightBig (S18)\nwhere we integrated over the frequency ωas the resolution of the spectrometer is of the order of the ca vity linewidth κhand\nmuchlowerthantheFMRlinewidth κFMR.\nWiththe input-outputrelation ˆav,out= ˆav,in−√κvˆavwefinallyobtainEq.(2)ofthemaintextas\n|/angbracketleftˆav,out/angbracketright|2=4G2|¯ah,in|2|¯bin|2κvκh/κFMR/bracketleftBigκ2\nh\n4+(ωh−ωL)2/bracketrightBig/bracketleftBig\nκ2v\n4+(ωFMR−ωv+ωL)2/bracketrightBig. (S19)"    },    {        "title": "1710.10534v1.High_frequency_dynamics_modulated_by_collective_magnetization_reversal_in_artificial_spin_ice.pdf",        "content": "High frequency dynamics modulated by collective magnetization reversal in arti\fcial\nspin ice\nMatthias B. Jung\reisch,1,\u0003Joseph Sklenar,2Junjia Ding,1Jungsik Park,2\nJohn E. Pearson,1Valentine Novosad,1Peter Schi\u000ber,2and Axel Ho\u000bmann1\n1Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA\n2Department of Physics and Frederick Seitz Materials Research Laboratory,\nUniversity of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA\n(Dated: July 22, 2018)\nSpin-torque ferromagnetic resonance (ST-FMR) arises in heavy metal/ferromagnet heterostruc-\ntures when an alternating charge current is passed through the bilayer stack. The methodology\nto detect the resonance is based on the anisotropic magnetoresistance, which is the change in the\nelectrical resistance due to di\u000berent orientations of the magnetization. In connected networks of\nferromagnetic nanowires, known as arti\fcial spin ice, the magnetoresistance is rather complex ow-\ning to the underlying collective behavior of the geometrically frustrated magnetic domain structure.\nHere, we demonstrate ST-FMR investigations in a square arti\fcial spin-ice system and correlate our\nobservations to magnetotransport measurements. The experimental \fndings are described using\na simulation approach that highlights the importance of the correlated dynamics response of the\nmagnetic system. Our results open the possibility of designing recon\fgurable microwave oscillators\nand magnetoresistive devices based on connected networks of nanomagnets.\nState-of-the-art nano-patterning enables the fabrica-\ntion of networks of connected ferromagnetic nanowires,\nwhich serve as magnetic metamaterials. The class of\nthese structures known as arti\fcial spin ice (ASI) is de-\nsigned to mimic the frustrated behavior of crystalline spin\nice. ASI systems show degenerate ground states, com-\nplex magnetic ordering and collective behavior [1{3]. ASI\nstructures are not only model systems in which to study\ngeometrical frustration, but have also signi\fcant tech-\nnological potential as recon\fgurable metamaterials and\nmagnetic storage media [4, 5]. On the other hand, ASI\ncan be considered as a recon\fgurable magnonic crystal,\nwhich can serve as an integral part in magnonic circuits\nto tailor spin-wave properties by design [6{8].\nMagnetoresistance (MR) and magnetization dynam-\nics in nanostructures are two of the most important\ntopics in contemporary studies of magnetism [9]. In\nthis regard, various aspects of MR in nanostructures\nand ASI structures in particular have been reported.\nThe \frst MR measurements in ASI systems indicated\nthat anisotropic magnetoresistance (AMR) [10], com-\nbined with nanowire domain reversal events, were re-\nsponsible for \feld-dependent MR [11{13]. Recently,\nangular- and \feld dependent magnetoresistive measure-\nments and simulations of networks of connected Ni 81Fe19\n(Py) nanowires quantitatively revealed the importance\nof the vertex regions to the AMR [14, 15]. Furthermore,\ncomplex dynamics in the high-frequency range have been\nobserved in ASI. It was shown that the speci\fc behav-\nior of individual modes in the spectra can be correlated\nwith the con\fguration of the magnetic moments in the\nnanowire legs [16{25].\nA big step forward towards the integration of such\nhigh-frequency devices in (spin-) electronics is the uti-\nlization of the spin-Hall e\u000bect in a heavy metal layer al-lowing for the conversion of electric charge currents into\nspin-polarized electron currents. In a bilayer of a ferro-\nmagnet and a heavy metal such as Pt or Pd, this spin\ncurrent di\u000buses from the heavy metal to the ferromagnet\nwhere it exerts a torque on the magnetization leading to\nmagnetization precession. If the ferromagnet is metal-\nlic, this precession results in a time-varying resistance\nin the ferromagnet on account of the AMR. Spin-torque\nferromagnetic resonance (ST-FMR) combines these two\ne\u000bects to electrically drive and detect spin dynamics [26].\nThe main advantage of ST-FMR over conventional FMR\nis that it does not rely on an inductive detection tech-\nnique which simpli\fes detecting resonant excitations in\nferromagnets. Furthermore, the spin torque can assist\nthe excitation of spin dynamics to induce unidirectional\ne\u000bects in the dynamic motion of the magnetization [27].\nST-FMR was studied in a wide range of materials, com-\npositions and structures, [26, 28{33]. However, despite\ntheir interesting unconventional collective behavior and\nmagnetotransport properties, magnonic crystals and ASI\nhave not been explored by ST-FMR yet.\nIn this Letter, we study networks of permal-\nloy/platinum (Ni 80Fe20/Pt) nanowires arranged on a\nsquare lattice by ST-FMR. Complementary to these dy-\nnamic characterizations in the rf-frequency range, we\ncarry out magnetotransport measurements and \fnd dis-\ntinct features in the longitudinal MR data, which cor-\nrespond closely to features observed in the ST-FMR\nmeasurements. Correlating the experimental angular-\ndependent transport data with a micromagnetic-based\ntransport model reveals that the sharp features in both\nthe MR and the ST-FMR can be explained through the\ninteraction of collective magnetization switching of the\nspin-ice lattice and AMR. We \fnd evidence that the dy-\nnamic ST-FMR measurements are more sensitive to thearXiv:1710.10534v1  [cond-mat.other]  28 Oct 20172\nFIG. 1. (Color online) Experimental setup consisting of a\nshortened CPW made of Ti/Au with the square ASI made of\nPy/Pt integrated into the signal arm (S), see inset. A bias-T\nallows for simultaneous rf-signal transmission and voltage de-\ntection by lock-in technique (ST-FMR). A multimeter is used\nfor the MR measurements (two-wire). The lattice constant is\ngiven bya, the hole width by bandc=a\u0000bis the bar width.\nA magni\fed SEM image is shown in the supplementary ma-\nterial, Fig. S1.\noccurrence of the switching of the nanowire legs trig-\ngered by external \feld changes than the magnetoresis-\ntance measurements. Our results have direct implica-\ntions for magnetoresistive and spin-torque devices asso-\nciated with complex magnetic nanostructures, where re-\ncon\fgurable magnonic and/or spin-ice states can be pre-\npared by applying a speci\fc charge current magnitude,\nmagnetic \feld value and/or microwave power and fre-\nquency.\nThe samples were fabricated in the following fashion:\nFirst, the square spin-ice structures of various dimensions\nwere de\fned by electron beam lithography. 15 nm-thick\npermalloy and 5 nm-thick Pt layers were deposited using\nmagnetron sputtering at rates <0:7\u0017A/s without break-\ning the vacuum. The ASI lattices cover an area of approx-\nimately 75\u000210\u0016m2in total, and the lateral dimensions\nof each investigated lattice are summarized in Tab. I.In a\nsubsequent step, a shortened coplanar waveguide (CPW)\nmade of Ti/Au (3 nm/120 nm) was fabricated by electron\nbeam evaporation and photolithography. Figure 1 shows\na typical scanning electron microscopy image (SEM); the\ninset shows a zoomed-in view of the ASI structure and\nthe dimensions of the lattice (magni\fed SEM image in\nthe supplementary materials [34], Fig. S1).\nFigure 1 illustrates the experimental setup. It con-\nsists of a shortened CPW with the square ASI structure\nintegrated into the signal arm. For the ST-FMR mea-\nsurements a bias-T is used to allow for a transmission of\na high-frequency signal from a rfsource and simultane-\nous voltage detection by a lock-in ampli\fer (modulation\nFIG. 2. (Color online) Frequency-dependent ST-FMR spec-\ntrum (microwave power: 6 dBm), here: lattice A. The blue-\nshaded area highlights the anomaly observed in the ST-FMR\ndata. The inset shows the low \feld regime around the blue-\nshaded area on a magni\fed scale. The resonance shifts to\nhigher magnetic \felds as the frequency is increased (black\narrows). Figure S2 shows the corresponding frequency-\feld\ndependence.\nfrequency 3 kHz). The geometry of our ST-FMR exper-\nimental setup is limited to a \fxed orientation (in-plane\nmagnetic \feld parallel to one of the main axes of the lat-\ntice,\u0012\u001945\u000e, see Fig. 1). For the MR measurements, a\ntwo-wire method was used by connecting the CPW to a\nKeithley Nanovoltmeter (2182A). A current of 5 \u0016A is\nprovided by a Keithley current source (6221) The trans-\nport measurements were performed as a function of the\nin-plane angle \u0012, see Fig. 1.\nFigure 2 shows an example of a ST-FMR spectrum\ntaken on lattice A. The results are consistent between\nthe di\u000berent lattices A,B,C. For the investigated ASI\nsamples, a typical FMR spectrum is more complex than\nthose observed from simpler micro- and nanostructures\n[26, 28, 29, 35] and di\u000ber from high-frequency spectra\nof ASI structures [17, 20]. We also note that passing a\nrfsignal through the nanostructure results in inhomo-\ngeneous phase variation in the magnetization dynamics;\nthis precludes a lineshape analysis and, thus, the deter-\nmination of the spin-Hall angle. However, it is yet possi-\nble to clearly identify the various modes using spatially-\nresolved dynamic micromagnetic simulations as will be\ndiscussed below. As is apparent from Fig. 2, the reso-\nnance shifts to higher magnetic \felds as the frequency is\nTABLE I. Studied spin-ice lattices. A;B;C : Py thickness:\n15 nm, Pt thickness: 5 nm. Parameters illustrated in Fig. 1.\nLattice hole width b(nm) bar width c(nm)\nA 620 252\nB 751 257\nC 497 2543\nFIG. 3. (Color online) (a) Experimental MR data for in-\nplane angles \u0012= 35\u000eand\u0012= 55\u000e, latticeA. Measurement\ncon\fguration as shown in Fig. 1 and sketched in the inset of\n(b);Iis the charge current. The negative \feld sweeps and\npositive \feld sweeps (small arrows) are symmetric under \feld\nreversal. The \feld range (positive sweep direction) labeled (I)\nand (II) indicate \felds before and after a collective change in\nthe magnetization (` avalanche ') is triggered, respectively. (b)\nCorresponding simulated MR (only positive sweep shown).\nincreased (black arrows). Independent of the lattice pa-\nrameters, the highest-lying mode (indicated by black ar-\nrows in Fig. 2) sti\u000bens with increasing magnetic \feld, see\nFig. S2 in the supplementary material. Power-dependent\nST-FMR measurements reveal a linear dependence of the\nresonance signal at all tested frequencies, Fig. S3 in the\nsupplementary material shows exemplarily the results for\nf= 4 GHz.\nBesides the expected resonances in the ST-FMR data,\na surprising feature is observed in the spectra: Indepen-\ndent of the excitation power, we \fnd an anomaly in the\nvoltage spectra of all lattices. This anomaly is character-\nized by a distinct change in the detected voltage, see blue\nshaded area in Fig. 2 and Fig. S3(a) in the supplemen-\ntary material. The inset in Fig. 2 shows the sharp feature\nat low magnetic \felds on a magni\fed scale. The nega-\ntive \feld sweeps and positive \feld sweeps are symmetric\nunder \feld reversal (not shown here). Dependent on the\nlattice parameter and excitation frequency, this jump in\nthe voltage is more or less pronounced. Furthermore, the\n\feld at which the jump occurs, varies slightly for di\u000ber-\nent driving frequencies. Naturally, one might ask if our\nobservation is due to the complex collective behavior of\nthe studied magnetic nanostructure? We address this\nquestion by exploring magnetotransport measurements\nin the studied nanostructures and correlating them with\na micromagnetic-based transport model that gives us a\nmicroscopic understanding of our results. In addition, we\ncarry out detailed micromagnetic simulations to under-\nstand the spatially-resolved dynamics.\nThe measured MR at two di\u000berent in-plane angles \u0012=\n35\u000eand\u0012= 55\u000eis depicted in Fig. 3(a) (up and down\nsweep indicated by arrows next to the traces). The traces\nfor the two measurement angles are signi\fcantly di\u000berent.\nThe 35\u000ecurve starts o\u000b at a higher voltage than the\n55\u000ecurve (at given magnetic \feld), has a maximum at\nzero-\feld and drops signi\fcantly at a small magnetic \feld\nFIG. 4. (Color online) Simulated magnetization con\fgura-\ntion (a,b) and y-component of the electric \feld (c,d), respec-\ntively. Results of the micromagnetic simulations at \u0012= 35\u000e\nbefore (a) and after (b) the collective switching occurs; de-\nnoted as (I) and (II) in Fig. 3(a). (c) and (d) show the elec-\ntric \feld maps of the respective states. The sample is \frst\nmagnetized in the negative \feld direction and then swept in\nthe positive direction. The sharp features in the MR, Fig. 3,\nis mainly determined by the vertex region: they realign from\nhorizontal to vertical direction when the avalanche is trig-\ngered. This switching in the vertex region causes changes in\nthe AMR as is apparent from the change in the polarity of\nthey-component of the electric \feld in (c) and (d). We note\nthat this e\u000bect is also seen in the x-component of the electric\n\feld as shown in the supplementary material, Fig. S5.\nat 200 Oe before it recovers and then slowly decreases\nwith increasing \feld. In contrast, the 55\u000etrace slowly\nincreases while the \feld is swept up, saturates around\nzero-\feld and then jumps up at 200 Oe before it decreases\nagain. The negative \feld and positive \feld sweeps (small\narrows) are symmetric under \feld reversal.\nTo better understand the underlying mechanisms in\nthe observed MR behavior of our nanostructures, we use\na combination of micromagnetic simulations and the phe-\nnomenology of AMR, following Ref. [14]. The magnetiza-\ntion pro\fles were obtained using Mumax3 [36]. In order\nto simulate the experimentally acquired \feld-dependent\nresistance traces the micromagnetic \feld maps [37] are\nconverted to electric \feld maps that take into account\n\frst order changes in the electric \feld due to AMR [10].\nThe electric \feld associated with anisotropic magnetore-\nsistance is given by:\nE=\u001a0J+ \u0001\u001a(^m\u0001J)^m; (1)\nwhere Jis the electric current vector, ^mis the unit vector\nof the magnetization, \u001a0is the isotropic resistivity, and\n\u0001\u001ais the anisotropic magnetoresistivity. The \frst order\nelectric \feld can be integrated along paths that mimic the\nexperimental con\fguration to then simulate the observed\nMR [14]. We assume that half of the current I=2 \rows\nin either of the nanowire legs and the full current I\rows4\nin the vertex, see Fig. S4 (supplementary material). A\nremarkably good agreement between experimentally ob-\nserved [Fig. 3(a)] and modeled MR [Fig. 3(b)] is found.\nAlthough the model only requires the lattice parameters\nas input, it captures the main experimental \fndings. Ad-\nditional MR measurements on square networks made of a\nsingle layer permalloy con\frm our \fndings and highlight\nthe generality of our model, see Fig. S6 in supplementary\nmaterial.\nSnapshots of our simulations at characteristic \feld val-\nues at an in-plane angle of 35\u000eare shown in Fig. 4\n[denoted as (I) and (II) in Fig. 3(a)]. Figures 4(a,c)\nshow the magnetization and respective electric \feld maps\nright before the sharp feature in the magentoresistance\nis observed [ H < 200 Oe, labeled as (I)]. Respectively,\nFig. 4(b) and (d) illustrate the maps right after the sharp\nfeatures in the MR have been observed [ H\u0019200 Oe, la-\nbeled as (II)]. The system is initially magnetized at a\nnegative \feld and then swept in positive \feld direction.\nThe micromagnetic simulations reveal that sharp features\nin the MR data occur when a rapidly evolving collective\nchange in the magnetization (` avalanche ') happens [(I)\nand (II) in Fig. 3(a)]. The avalanches in our arti\fcial\nspin ice are triggered by the external magnetic \feld. The\ndistinct signature in the MR is mainly determined by\nthe vertex region: The moments in the vertex are ini-\ntially aligned horizontally in the negative \feld direction\nand reorient to the vertical direction after the avalanche\nis triggered. This switching from a horizontal to a ver-\ntical alignment causes a change in the AMR as is ob-\nvious from the corresponding electric \feld maps shown\nin Fig. 4(c) and (d) [14]. At the vertices, the polarity\nof they-component of the electric \feld switches states,\nwhile the electric \feld in the remaining portion of the\nnetwork remains unchanged. We note that this e\u000bect is\nalso seen in the x-component of the electric \feld as shown\nin supplementary material, Fig. S5. As the magnetic \feld\nincreases, the moments in the vertex region re-align hor-\nizontally in the positive \feld direction and the MR goes\nback to approximately the same value as before.\nThe simulation results indicate that the MR data for\nan in-plane measurement angle of 35\u000edi\u000bers strongly\nfrom the results obtained at 55\u000esince the magnetic mo-\nments in the vertex region are orthogonal to each other.\nThis is why the MR drops when the islands switch at 35\u000e,\nand increases at 55\u000e: The electric current and the mag-\nnetization are perpendicular to each other at 35\u000e, and\nthey are parallel to each other in the 55\u000econ\fguration\n(see Fig. S7 in supplementary material).\nOur results clearly reproduce the previous experimen-\ntal MR results [14, 15] that the vertex regions in spin-\nice structures strongly in\ruence experimentally measured\nMR. Strikingly, the occurrence of the sharp features in\nthe transport correlates well with the appearance of cor-\nresponding signatures in the dynamic ST-FMR data at\nthe same magnetic \felds. This leads to the question:Do the observed features in the dynamic spectra stem\nfrom an AMR mixing mechanism related to a dynamic\nmode of the system that appears at/during the onset of\nan avalanche or from the collective switching itself?\nTo address this question, we carried out dynamic mi-\ncromagnetic simulations [38]. The results are summa-\nrized in Fig. 5. Figure 5(a) shows a simulated M=H loop\n(top panel) and the predicted resonance modes and their\ncomparison to the experimental observations. Exemplar-\nily the results for a driving frequency f= 6 GHz is shown\n(bottom panel). The in-plane magnetic \feld angle in the\nsimulations was chosen so as to resemble the experimen-\ntal con\fguration, but at the same time to break the lat-\ntice symmetry (50\u000e). They-scale of the simulated spectra\nis the dynamic response of the mz-component (intensity\nof precession/m2\nz) to a constant sinusoidal driving fre-\nquency of 6 GHz as a function of the magnetic \feld. A\nlow intensity means that there is no resonance at 6 GHz\nat a given magnetic \feld. A strong intensity means a\nlarge precession amplitude at 6 GHz at a given magnetic\n\feld. A good agreement between simulation and experi-\nment is found. In particular, the \feld asymmetry in the\nrange between +100 Oe and +170 Oe found in the ex-\nperiment is also observed in the dynamic micromagnetic\nsimulations. This \feld range also agrees very well with\nthe reversal regime found in the magnetoresistance mea-\nsurements (Fig. 3). Meanwhile the resonances detected\nat +600 Oe and +930 Oe in the experiment, are found\nat +540 Oe and +770 Oe in the simulations. The dif-\nference between micromagnetics and experiment may be\nattributed to details of imperfections in our simulations;\ne.g., simulations occur at 0 K and do not account for any\nlithographic imperfections. Furthermore, the simulation\nonly considers the response of a limited number of unit\ncells with periodic boundaries due to computational limi-\ntations, and assumes that the roughness is uniform across\nthe entire sample. The predicted mode at +1610 Oe was\nnot detected in the experiment; as we will discuss below,\nthis mode is an edge mode. Typically edge modes occur\nin simulations, in which a perfect structure without any\nimperfections is assumed. Clearly, this assumption is not\nful\flled in real spin-ice networks which may be the rea-\nson why we do not observe this mode in the experiment.\nPlease note that the polarity of the simulated intensity\nwas manually mirrored for negative \felds to better match\nthe experimental \fndings (which are detected based on\nST-FMR). The reason is that Mumax3 simulates the res-\nonances, not ST-FMR directly which changes sign due to\nthe spin-Hall e\u000bect.\nIn the following, we discuss the two-dimensional maps\nof the dynamic magnetization at the resonances indicated\nin Fig. 5(a). These maps illustrate where in the spin-ice\nlattice the intensity is the largest (small intensity: red,\nlarge intensity: blue). We also show the corresponding\nstatic magnetization con\fguration of the vertex regions\nas insets. Figure 5(b) shows the low-\feld range at +1005\nFIG. 5. (Color online) Results of dynamic micromagnetic simulations. (a) Top panel: Simulated magnetization in x-direction\nas a function of the external \feld, which is applied at 50\u000ein the simulations. The coercive \feld is about 190 Oe. Bottom\npanel: Predicted resonance modes (black, straight line) and their comparison to experimental spectrum (blue, dashed line) at 6\nGHz. A reasonable agreement between experiment and simulation is found. Please note that the mz-component of the dynamic\nmagnetization has been used to represent the magnetization precession. (b-e) Spatially-resolved maps of the dynamic response\nat a given magnetic \feld. The color scale represents the dynamic mz-component of the precession at 6 GHz and illustrates\nwhere the particular modes at a given magnetic \feld are located. The insets show the corresponding static magnetization\ncon\fguration for comparison. 2D maps of the precession are shown for the \felds indicated in (a): (b) Reversal region, (c)\nbulk mode, (d) higher order mode and (e) edge mode. The static magnetization con\fguration in the reversal regime, (b), only\nchanges slightly, while the dynamic maps show a big change when the \feld is swept.\nOe, +140 Oe and +170 Oe, as well as the total magnetiza-\ntion dynamics in the \feld range, which is the sum of +100\nOe, +140 Oe and +170 Oe. The dynamics in this \feld\nrange is quite uniform. From a comparison of the static\nand dynamic maps in the low \feld range we conclude that\nthe dynamic signal is much more sensitive to external\n\feld changes, which implies that the dynamic measure-\nments are more sensitive to the occurrence of avalanches\nthan the magnetoresistance measurements since they rely\nonly on the static magnetization. The sharp features due\nto avalanche formation could not be observed in simi-\nlar spin-pumping experiments [20]. This might indicate\nthat an inhomogeneous phase of the microwave signal,\nout-of-plane Oersted \felds and/or spin torques are re-\nquired to couple to this low-\feld mode. This mode is\nmost pronounced at 6 GHz in the experiment as well as\nin the simulations. It is non-existent in the experiment\nat higher frequencies and hardly observable in the corre-\nsponding simulations. At lower frequencies this low-\feld\nmode is narrower in \feld range and less intense than at 6\nGHz. This experimental observation is con\frmed in the\nsimulations; not shown here.\nFigure 5(c-e) show the simulated two-dimensionalmaps of the dynamic response at +540 Oe, +770 Oe\nand +1610 Oe. The mode at +540 Oe is identi\fed as\na bulk-like mode, Fig. 5(c), while the resonance at +770\nOe corresponds to a higher order mode, Fig. 5(d), and\nthe resonance at +1610 Oe is an edge mode, Fig. 5(e).\nWe emphasize that the ST-FMR recti\fcation is given\nby a time-averaged mixing of the microwave current with\nthe resistance, V=hRIi. Therefore, it is likely that\ntwo components contribute to the ST-FMR signal: (1)\na static (heterodyne) magnetoresistance change and (2)\na dynamic (homodyne) anisotropic resistance due to the\nformation of a change of the magnetization con\fguration,\noccurring due to the switching, that is susceptible to the\nmicrowave/spin-torque drive.\nIn summary, we demonstrated that the collective mag-\nnetization behavior in an ASI strongly a\u000bects the dy-\nnamic ST-FMR spectra, as well as the magnetoresistive\nbehavior in those structures. We provide a microscopic\npicture of this unexpected response by means of micro-\nmagnetic simulations. The sharp features observed in\nthe experimental data occur due to a sudden change in\nthe magnetization con\fguration when the \feld is swept.\nWe show that the angular-dependent alignment of the6\nvertex region strongly a\u000bects the resistance and even\nmore importantly the resonance condition leading to a\nsigni\fcantly di\u000berent ST-FMR response. Our \fndings\nclearly demonstrate the possibility to read out collec-\ntive switching processes in ASI by transport, as well as\nhigh-frequency dynamics. The observation of a spatially-\ncon\fned magnonic modes in ASI is a \frst step towards\nthe realization of microwave oscillators in connected net-\nworks of ferromagnetic nanowires. Given the geometric\nfreedom enabled by modern lithography techniques, our\nresults open the possibility of designing innovative recon-\n\fgurable microwave oscillators and magnetoresistive de-\nvices based on connected ferromagnetic networks which\nmight also provide desirable functionalities, e.g., for neu-\nromorphic computing [39].\nWork at Argonne including experiment design, sample\nfabrication and characterization, ST-FMR and magne-\ntoresistance measurements, mircomagnetic simulations,\ndata analysis, and manuscript preparation, was sup-\nported by the U.S. Department of Energy, O\u000ece of\nScience, Materials Science and Engineering Division.\nLithography was carried out at the Center for Nanoscale\nMaterials, an O\u000ece of Science user facility, which is sup-\nported by DOE, O\u000ece of Science, Basic Energy Sci-\nence under Contract No. DE-AC02-06CH11357. 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IEEE\n104, 2024 (2016)."    },    {        "title": "1208.2802v1.Disorder_effects_on_resonant_tunneling_transport_in_GaAs__Ga_Mn_As_heterostructures.pdf",        "content": "arXiv:1208.2802v1  [cond-mat.mes-hall]  14 Aug 2012Disorder effects on resonant tunneling transport in GaAs/(G a,Mn)As heterostructures\nChristian Ertler∗1and Walter P¨ otz1\n1Institute of Theoretical Physics, Karl-Franzens Universi ty Graz, Universit¨ atsplatz 5, 8010 Graz, Austria\nRecent experiments on resonant tunnelingstructures compr ising (Ga,Mn)As quantumwells [Ohya\net al.,Nature Physics 7, 342 (2011)] have evoked a strong debate regarding their int erpretation as\nresonant tunneling features and the near absences of ferrom agnetic order observed in these struc-\ntures. Here, we present a related theoretical study of a GaAs /(Ga,Mn)As double barrier structure\nbased on a Green’s function approach, studying the self-con sistent interplay between ferromagnetic\norder, structural defects (disorder), and the hole tunnel c urrent under conditions similar to those in\nexperiment. We show that disorder has a strong influence on th e current-voltage characteristics in\nefficiently reducing or even washing out negative differentia l conductance, offering an explanation\nfor the experimental results. We find that for the Be lead dopi ng levels used in experiment the\nresulting spin density polarization in the quantum well is t oo small to produce a sizable exchange\nsplitting.\nPACS numbers: 85.75.Mm, 73.23.Ad, 73.63.-b, 72.25.Dc\nI. INTRODUCTION\nDilute magnetic semiconductors (DMS) are produced\nby doping of semiconductors with transition metal el-\nements, which provide local magnetic moments aris-\ning from open electronic dorfshells.[1, 2] Bulk\nGa1−xMnxAs may be regarded as the prototype: Mn re-\nsiding on the Ga site (Mn Ga) donates a hole, associated\nwith valence band p-orbitals, and provides a local mag-\nnetic moment associated with partly filled Mn d-orbitals.\nMnGais a moderately deep acceptor with the energy lev-\nels lying about 100 meV above the valence band edge.[3]\nBy increasing the Mn density the acceptor levels become\nmore and more broadened, developing into an impurity\nband which allows hole propagation and, for sufficiently\nhigh doping level, is believed to merge with the valence\nband.[4] At the same time, Mn more and more takes un-\nwanted lattice positions, such as the antisite and inter-\nstitial position in the fcc lattice, or may even form Mn\nclusters, all leading to strong electron-hole compensation\nwhich eventually destroys ferromagnetic ordering. There\nis somedebateas tothe orderin whichthese eventsoccur\nas the Mn concentration is increased. Probably due to\nthe presence of unintentional defects in (Ga,Mn)As sam-\nples, depending on growth conditions, experimental evi-\ndence has led to somewhat conflicting conclusions about\nthe precise position of the Fermi level in ferromagnetic\nbulk(Ga,Mn)As.[2]Someexperimentscanbe interpreted\nby placing it into the top of a GaAs–like valence band\nedge which is broadened by disorder.[1] Others suggest\nthe existence of an isolated impurity band in the ferro-\nmagnetic state.[5–7]\nRecently a systematic series of experiments in form\nof non-equilibrium tunneling spectroscopy on double-\nbarrier resonant tunneling structures with a (Ga,Mn)As\nquantum well [7–9] were conducted to provide a deeper\n∗email:christian.ertler@uni-graz.atinsight into this question. The group reported a near ab-\nsence of ferromagnetic order in the well under bias and\nobtained weak signatures of resonant tunneling, observ-\nable only in the second derivative of the current-voltage\n(IV) characteristic. Their conclusion that the Fermi en-\nergy lies in the impurity band has evoked strong debates\nand an alternative explanation has been given, which\nproposes that the resonant–tunneling signature is caused\nmerelybythe confined statesin apotential pouch formed\nat the contact/barrier heterointerface.[10] In this expla-\nnation the observed dependence of the peak positions\non the quantum well width is completely attributed to\nthe increased series resistance which, however, seems to\nbe insufficient to account for all well–width-dependent\ntrends in the experimental results, as discussed in detail\nin a reply by Tanaka et al. which again emphasizes the\nexistence of quantized levels in the (Ga,Mn)As quantum\nwells.[11]\nIndeed, quantization effects can be expected in\n(Ga,Mn)As for a layer thickness of about ≤3 nm since in\na recent scanning tunneling microscopy experiment the\nradius of the Mn acceptor wave function has been de-\ntermined to be about 2 nm [5] and one can expect that\nnearthe band edge Bloch-likeand delocalizedeigenstates\nwill coexist in the picture of merging impurity and va-\nlence bands.[12] Tunneling spectroscopic experiments of\n(Ga,Mn)As quantum well structures have indicated such\neffects.[7] However, the signatures in the current–voltage\ncharacteristicsappeartoberatherweakandnoregionsof\nnegative differential resistance due to resonances associ-\natedwith (Ga,Mn)Aswelllayershavebeenobservedasof\nyet, with the notable exception of an asymmetric magne-\ntoresistance resonant–tunneling structure.[13] This sug-\ngests that a significantconcentration ofunwanted defects\nand/or disorder may be present, depending on growth\nconditions, as it is known to be the case in thin layers of\namorphous Si, in which similarly weak signatures have\nbeen found.[14, 15] The density of imperfections due to\nthe presence of Mn interstitial or antisite defects can be\nas high as 20% of the nominal Mn doping, which makes2\n(Ga,Mn)As a heavily compensated system.[16, 17] Even\nlowerstructural quality for (Ga,Mn)As must be expected\nat heterointerfacessince the need oflow-temperatureepi-\ntaxy for growing the (Ga,Mn)As layers is harmful to\nforming clean interfaces with other materials. Moreover,\nthe interstitial defects may be trapped near the inter-\nfaces in post-growth annealing procedures which have\nbeen found successful for bulk(Ga,Mn)As. This suggests\nthat transport through thin layers of (Ga,Mn)As is in-\nfluenced by disorder and defects more severely than in\nannealed bulk structures.\nThe growth of heterostructures, on the other hand,\nprovides the appealing opportunity to drive (Ga,Mn)As\nlayers into a genuine non-equilibrium situation by means\nof an external bias which modifies their local hole den-\nsity, possibly leading to bias-dependent ferromagnetic\nbehavior.[18, 19] However, drawing conclusions from the\nphysics of a thin (Ga,Mn)As layer regarding the Fermi\nenergy position in the bulk is an intricate problem, since\na Fermi energy in a (Ga,Mn)As quantum well under bias\nconditions is not well defined.\nIn a recent series of studies we have investigated the\nferromagnetic bias anomaly in (Ga,Mn)As–based het-\nerostructures and reached the conclusion that, for suf-\nficiently high hole densities in the thin (Ga,Mn)As quan-\ntum wells, ferromagnetic ordering becomes bias depen-\ndent leading to variable spin-polarized currents.[18, 19]\nHere, we study the low doping regime (relative to the\nMnconcentration)andusearefinedmodelforthevalence\nband stateswhich accountsforboth heavyandlight–hole\nstates. This allows us a direct comparison to recent ex-\nperiments and, as will be shown, enhances the effect of\ndisorder on suppressing a resonant–tunneling signature\nin the IV characteristic. Based on a four band Kohn-\nLuttingerHamiltonianthetransportpropertiesareinves-\ntigated within a self–consistent non-equilibrium Green’s\nfunction method which accounts for space charge effects\nand a hole-density-dependent exchange splitting. We\nshow that disorder reduces or even completely washes\nout regions of negative differential conductance in the IV\ncurve. We find that, for the Be lead doping levels as\nused in experiment, the resulting spin density polariza-\ntion in the quantum well is low and thus leads to almost\nvanishing ferromagnetic order. Our theoretical model is\npresented in Sect. II and the results and relevance to\nexperiment are discussed in Sect. III. Summary and con-\nclusions are given in Sect. IV.\nII. PHYSICAL MODEL\nHere we describe our transport model for heterostruc-\ntures composed of layers of GaAs, GaAlAs, and\n(Ga,Mn)As grown along the z-axis. In this study the\nband structure of the top ofthe valence bands is modeled\nby the Kohn-Luttinger Hamiltonian [20], which allows us\nto take into account the mixing of heavy hole (HH) and\nlight hole (LH) bands, which is of crucial importance forgetting a realistic transmission function for holes tun-\nneling through a double-barrier structure as shown in\nRef. 21. Ordering the four spin-3/2 basis vectors at\nthe Γ-point as uσ=|3\n2,mσ/an}bracketri}htwithmσ={3\n2,1\n2,−1\n2,−3\n2},\nthe wave-vector-dependent Kohn-Luttinger Hamiltonian\nreads\nH(k) =\nP+Q−S R 0\n−S†P−Q0R\nR†0P−Q S\n0R†S†P+Q\n.(1)\nThe matrix elements can be expressed in terms of the\ndimensionless Luttinger parameters γ1,γ2andγ3:\nP(k) =/planckover2pi12\n2mγ1k2,\nQ(k) =/planckover2pi12\n2mγ2(k2\nx+k2\ny−2k2\nz) (2)\nS(k) =/planckover2pi12\n2m2√\n3γ3(kx−iky)kz,\nR(k) =/planckover2pi12\n2m√\n3[−γ2(k2\nx−k2\ny)+2iγ3kxky],\nwheremis the free electron mass. In order to consider-\nably simplify the numerical demands for the calculation\nof macroscopic quantities, such as the current density,\nwhich require the summation over the in-plane momen-\ntum, we apply the axial approximation in which the con-\nstant energy surface in the k-space becomes cylindrically\nsymmetric but for which HH-LH band mixing is still in-\ncluded. Within the axial approximation the transmis-\nsion function only depends on the absolute value of the\nin-plane momentum k2\nρ=k2\nx+k2\ny. Space-dependent (in\nz-direction) potentials are taken into account within the\nenvelope function approximation, which effectively leads\nto replacing kz→by−i∂z. By approximating the intro-\nduced spatial derivatives on a finite grid of spacing aone\nends up with an effective nearest-neighbor tight-binding\nHamiltonian of tridiagonal form\nH=/summationdisplay\nl,σσ′ε(l)\nσσ′c†\nl,σcl,σ′+/summationdisplay\nl,σσ′tσσ′c†\nl+1,σcl,σ′+h.c.,(3)\nwithc†\nl,σdenoting the creation operator for site land\norbitalσ. Theon-siteandhoppingmatrices,respectively,\ntake the form\nεσσ′=\nC10−B0\n0C20−B\n−B0C20\n0−B0C1\n, (4)\ntσσ′=\nD1−iE0 0\n−iE D 20 0\n0 0 D2iE\n0 0 iE D 1\n. (5)3\nHere, the matrix elements are given by\nC1=/planckover2pi12\n2m[k2\nρ(γ1+γ2)+2(γ1−2γ2)/a2]\nC2=/planckover2pi12\n2m[k2\nρ(γ1−γ2)+2(γ1+2γ2)/a2]\nB=/planckover2pi12\n2m√\n3γk2\nρ (6)\nD1=/planckover2pi12\n2m[−(γ1−2γ2)/a2]\nD2=/planckover2pi12\n2m[−(γ1+2γ2)/a2]\nE=−/planckover2pi12\n2mγ3kρ√\n3/a\nwithγ= (γ2+γ3)/2. This effective tight-binding model\nhas the advantage that space-dependent potentials, ex-\nchange splittings in magnetic layers, and structural im-\nperfections can be readily included in the orbital onsite\nenergies of the model, i.e., the diagonal elements of the\nonsite matrix using\nε(l)\nσσ=εσσ+Ul−eφ−σ\n2∆l+εrand (7)\nwithUldenoting the intrinsic hole band profile due to\nthe band offset between different materials, φis the the\nelectrostatic potential, eis the elementary charge, ∆ lde-\nnotes the local exchange splitting in the magnetic mate-\nrials with σ=±1, andεrandlabels a random shift due to\ndisorder, as will be detailed below.\nWith the ferromagnetic order being mediated by the\nitinerant carriersthe exchangesplitting of the hole bands\nself-consistently depends on the local spin density of the\nholes. It can be derived within an effective mean-field\nmodel taking into account two correlated mean magnetic\nfields stemming from the ions’ d–electrons spin polariza-\ntion/an}bracketle{tSz/an}bracketri}htandtheholespindensity /an}bracketle{tsz/an}bracketri}ht= (n↑−n↓)/2.[22–\n24] The exchange splitting ofthe hole bands is then given\nby\n∆(z) =−Jpdnimp(z)/an}bracketle{tSz/an}bracketri}ht(z), (8)\nwithzbeing the longitudinal (growth) direction of the\nstructure, Jpd>0 is the exchange coupling between the\np-likeholesandthed-likeimpurityelectrons,and nimp(z)\nis the impurity density profileofmagneticallyactiveions.\nThe effective impurity spin polarization /an}bracketle{tSz/an}bracketri}htis induced\nby the magnetic field caused by the mean hole spin po-\nlarization, yielding\n/an}bracketle{tSz/an}bracketri}ht=−SBS/parenleftbiggSJpd/an}bracketle{tsz/an}bracketri}ht\nkBT/parenrightbigg\n, (9)\nwhere,kBis the Boltzmann constant, Tis the lattice\ntemperature, and BSis the Brillouin function of order S,\nhere with S= 5/2 for the Mn impurity spin. Combining\nthe last two expressions gives the desired result\n∆(z) =Jpdnimp(z)SBS/braceleftbiggSJpd[n↑(z)−n↓(z)]\n2kBT/bracerightbigg\n.(10)Since the hole spin density /an}bracketle{tsz/an}bracketri}htis changed by the in- and\nout-tunneling holes, the magnetic and transport proper-\nties of the system are coupled self-consistently.\nTo obtain realistic potential drops between the two\nleads space-charge effects have to be taken into account.\nIn the Hartree approximation the electric potential is de-\ntermined by the Poisson equation,\nd\ndzǫd\ndzφ=e[Na(z)−n(z)], (11)\nwhereǫandNa, respectively, denote the dielectric con-\nstant and the acceptor density. The local hole density at\nsite|l/an}bracketri}htcan be obtained from the non-equilibrium “lesser”\nGreen’s function G<:\nn(l) =−i\nAa/summationdisplay\nk/bardbl,σ/integraldisplaydE\n2πG<(E;lσ,lσ),(12)\nwithAandk/bardbl, respectively, being the in-plane cross sec-\ntional area of the structure and the in-plane momentum.\nThelesserGreen’sfunctionisdeterminedbytheequation\nof motion\nG<=GRΣ<GA(13)\nwhereGRandGA= [GR]+denotes the retarded and\nadvanced Green’s function, respectively. The scattering\nfunctionΣ<= Σ<\nl+Σ<\nrdescribestheparticleinflowfrom\nthe left ( l) and right ( r) reservoirs [25] with\nΣ<\nl,r=f0(E−µl,r)(ΣA\nl,r−ΣR\nl,r),(14)\nwheref0(x) = [1 + exp( x/kBT)]−1is the Fermi dis-\ntribution function and µlandµr, respectively, denote\nthe quasi–Fermi energies in the contacts. The retarded\nand advanced self-energy terms ΣR= ΣR\nl+ ΣR\nrand\nΣA= [ΣR]+, respectively, couple the simulated system\nregion to the left and right contacts. The surface Green’s\nfunction of the leads is needed to obtain the contact self-\nenergy ΣRand is calculated by using the algorithm of\nL´ opez-Sancho et al.[26] The retarded Green’s function of\nthe system, finally, is given by\nGR=/bracketleftbig\nE+iη−Hs−ΣR/bracketrightbig−1. (15)\nwhich we calculate by consecutively adding one layer of\nthe system after another which, in our case, solely re-\nquires the inversion of a 4x4 matrix for each additional\nlayer.\nThe transport equations Eqs. (13) and (15) couple via\nthe spin-resolved hole density to the exchange splitting\nof the hole bands Eq. (10), and the Poisson equation\nEq. (11). For a given applied voltagethis system of equa-\ntions is solved in a self-consistent loop until convergence\nof the electrostatic potential and the exchange field is\nreached. A small external magnetic field is applied ini-\ntially to aid spontaneous symmetry breaking. For the\nnext bias iteration, the self–consistent solution from the\nprevious bias value is used for an initial guess. Having4\nobtained the self-consistent potential profile the trans-\nmission probability T(E,k/bardbl) from the left to the right\nreservoir is calculated by\nT(E,k/bardbl) = Tr/bracketleftbig\nΓlGRΓrGA/bracketrightbig\n(16)\nwith Tr[·] denoting the trace operation and the lead cou-\npling functions being defined by Γ (l,r)=i[ΣR\n(l,r)−ΣA\n(l,r)].\nThe steady–state current density is obtained by an in-\ntegrationoverall incoming k/bardblstates and the total energy\nE:\nj=e\n2π/planckover2pi1/summationdisplay\nk/bardbl/integraldisplay\ndE T(E,k/bardbl)[fl(E,µl)−fr(E,µr)] (17)\nwith the applied bias V= (µl−µr)/ebeing defined as\nthe difference in quasi-Fermi levels of the contacts.\nWe model disorder effects in the (Ga,Mn)As layers by\nperformingaconfigurationalaverageoverstructureswith\nrandomly chosen diagonal elements of the onsite matrix\nresulting in an ensemble of tight-binding Hamiltonians.\nThe diagonal onsite matrix elements are sampled accord-\ningtoaGaussiandistributionaround εrand= 0in Eq.(7)\nfor increasing standard deviations ( σdis= 20, 40, 60, and\n80 meV). For each specific structure (and Hamiltonian)\nthe current-voltage (IV) curve is solved self–consistently\nfor an upsweep ofthe applied voltage. This effective one–\ndimensional modeling of disorder must be viewed as a\nlimited estimate since it corresponds to a cross–sectional\naverage of transport though uncorrelated effective lin-\near chains. As such, any disorder correlations parallel to\nthe interface are neglected. Such correlations in disor-\nder will play a role in the establishing of ferromagnetic\norder in real structures relative to the idealized homoge-\nneous mean–field model adopted here, since both ferro-\nmagnetic order and disorder effects are highly dependent\nupon spatial dimensionality.[27, 28] Also, this type of av-\neraging cannot model the effects of Mn clusters. Addi-\ntional types of disorder from Mn clustering, Mn intersti-\ntials, etc. however may be present in real structures, but\ntheir type and concentration may differ from sample to\nsample.\nIII. RESULTS\nWe investigate a double barrier structure consisting of\nBe-doped GaAs leads and a (Ga,Mn)As quantum well,\nsimilar to the experimental setups presented in Refs.8, 9.\nThe structural similarity of the layer materials allows\none to use the same valence band model for the whole\nstructure, which considerably simplifies the theoretical\ndescription. In recent resonant tunneling spectroscopic\nexperiments ofthe groupofTanaka[7], however, aSchot-\ntky barrier contact with Au at one side and a GaAs:Be\ncontact with an AlAs barrier on the other side of the\n(Ga,Mn)As quantum well was used. If the transport is\nprimarily determined by the resonant levels in the well,z Energy (eV)log(LDOS) at V= 0\n  \n1204060800.10.20.3\n−5−4−3−2−10\nFIG. 1: (Color online) Logarithm of the local density of stat es\n(LDOS) at zero bias for the perfect structure (no disorder)\nusing an inverted energy scale. The self–consistent potent ial\nprofile is indicated by the (red) solid line, whereas the Ferm i\nenergy position of the leads is plotted as (red) dashed line.\nThe position of the impurity band in the well is indicated\nschematically by the (white) bold solid line below the valen ce\nband edge.\nour results obtained for a completely zincblende struc-\nture will be relevant also for the Schottky-barrier system\ninvestigated in experiment.\nFor the simulations we assume following parameters,\ncomparable to the experimental values of Ref. 8: barrier\nthickness dbar= 4 layers ( ≈2 nm), quantum well width\ndw= 10 layers ( ≈5 nm), barrier height Vbar= 300 meV,\nrelative permittivity of GaAs ǫr= 12.9, exchange cou-\npling constant Jpd= 0.06 eV nm3, 5% Mn doping, and\nthe lead temperature T= 4.2 K. The Fermi level in the\nGaAs leads is chosen to µl= 10 meV, which corresponds\nto a Be-doping of about 1018cm−3as used in experiment.\nThe dimensionless Luttinger parameters are set to stan-\ndard values for GaAs: γ1= 6.85,γ2= 2.1, andγ3= 2.9.\nWhile (Ga,Mn)As inevitably is a disordered system,\nour starting point is an idealized system for which the\nvalence band edge is identical to that of GaAs. Sub-\nsequently, valence band disorder is added in increasing\nsteps allowing for a systematic qualitative account of its\nconsequencesonthe IV-curve. Forconveniencewe usean\ninverted hole energy scale but retain positively charged\nholes. In order to simulate the presence of Mn Gaimpu-\nrity band levels, partially populated by holes, we use a\npositive (repulsive) charge background of 1018cm−3in\nthe(Ga,Mn)Aslayerleadingtoanupwardshiftofthepo-\ntential profile in the well region, as shown in Fig. 1. The\nlocal density of states of the ideal double-barrier struc-\nture (in absence of disorder) at zero bias and V= 0.1 V,\ncorresponding to the dominant current peak of the IV\ncurve (see Fig. 5), is shown in Fig. 1 and Fig. 2, respec-\ntively. The resonant levels in the well are clearly visible\nand the solid line indicates the self-consistent potential\nprofile of the structure. At zero bias the valence band\nedge of (Ga,Mn)As is lifted relativeto that of the contact\nGaAs layers by about 30 meV thus partially aligning the5\nz Energy (eV)log(LDOS) at V = 0.1 V\n  \n120406080−0.100.10.20.3\n−12−10−8−6−4−2\nFIG. 2: (Color online) Logarithm of the local density of stat es\n(LDOS) at the bias V= 0.1 V for the perfect structure (no\ndisorder) using an inverted energy scale. The self–consist ent\npotential profile is indicated bythe(red) solid line. The Fe rmi\nenergy position of the left and right lead is plotted as (red)\ndashed and dotted line, respectively. Holes of the impurity\nband (indicated schematically by the bold (white) solid lin e\nbelow the valence band edge) can tunnel out to the collector\nreservoir.\n00.05 0.1 0.15 0.210−610−3100T (a.u.)\nEnergy (eV)  \nσdis = 0\nσdis = 20 meV\nσdis = 60 meV\nFIG. 3: (Color online) The transmission function at zero bia s\nfor the perfect structure (solid line), for moderate disord er of\nσdis= 20 meV (dashed line), and stronger disorder σdis=\n60 meV (dotted line) by sampling over 3 ×104configurations.\nMnGalevels (indicated schematically by the bold (white)\nsolid line) with the chemical potential which, in turn, lies\nabout 10 meV above the valence band edge of the GaAs\nleads. Therefore, when a bias greater than about 10 mV\nis applied the Mn Galevels can no longer be filled reso-\nnantly from the emitter side and, beyond about 30 mV,\nno longer from either emitter or the collector. The latter\nsituation is shown schematically in Fig. 2. This loss of\nholes from the Mn Galevels in the well region and the\ninsufficient resupply of holes from the GaAs emitter into\nthe resonant valence band levels lead to a breakdown of\nferromagnetic order in the (Ga,Mn)As well under small\nbias,i.e.a zero bias anomaly.\nThe transmission function and local density of states\nin the center of the Ga 1−xMnxAs quantum well region00.05 0.1 0.15 0.210−2100102LDOS in the well (a.u.)\nEnergy (eV)  \nσdis = 0\nσdis = 20 meV\nσdis = 60 meV\nFIG. 4: (Color online) Local density of states (LDOS) in\nthe middle of the quantum well at zero bias for the perfect\nstructure (solid line), for moderate disorder of σdis= 20 meV\n(dashed line), and stronger disorder σdis= 60 meV (dotted\nline) by sampling over 3 ×104configurations.\n00.05 0.1 0.15 0.200.511.522.5x 104\nV (V)j (A/cm2)\n  σdis = 0\nσdis = 20 meV\nσdis = 40 meV\nσdis = 60 meV\nσdis = 80 meV\nFIG. 5: (Color online) IV-characteristic of a magnetic doub le\nbarrier structure for increasing degree of disorder measur ed\nby the standard deviation σdisof the Gaussian distribution of\nthe randomly chosen diagonal onsite matrix elements εrand.\nis plotted for increasing degree of disorder in Fig. 3 and\nFig. 4, respectively. Disorder leads to defect level in the\nband gap and a significant spectral broadening of the\nresonant levels associated with the lowest valence band\nresonances. Note also the strong increase of transmission\nprobability in the low–bias regime from disorder, opened\nby resonances for tunneling.\nAs a key difference compared to the bulk (Ga,Mn)As\nsituation, the hole density inside a (Ga,Mn)As quantum\nwell is established by forming a steady state situation\nwith the external leads. Since the Be doping level in the\nGaAs contacts is significantly lower than the Mn con-\ncentration this leads to a strongly reduced hole concen-\ntration (over bulk) in the (Ga,Mn)As wells under bias.\nEven under favorable bias conditions, in which resonant\nlevels in the quantum well can be populated from the ex-\nternal leads, the quantum well hole density remains on\nthe order of the hole density in the GaAs leads (in our6\ncase≈1018cm−3), which is at least two orders of magni-\ntude smaller than in bulk (Ga,Mn)As. Therefore, in our\nsimulations we find practically vanishing ferromagnetism\n(exchange splitting) for all bias values. In previous stud-\nies we have shown that for higher hole doping of the\nleads a bias-dependent ferromagnetic state with a maxi-\nmum exchange splitting of the order of tens of meV can\nbe expected, being detectable by a significant spin po-\nlarization of the collector current.[18, 19] In this earlier\nstudy we focused on the transport through the first HH-\nsubband using a simpler effective mass band model. The\npicture that the quantum well hole density is determined\nby the coupling to the leads surely applies to the case\nwhen the impurity band merges with the valence band,\nleading at most, to a broadening and shift of the valence\nband edge. In the presence of an isolated impurity band\nlooselybound holesmayexist (at leastatlowvoltages)in\nthe confined impuritybands lyingenergeticallybelowthe\nfirst valence 2D-subband (on the inverted energy scale).\nThe main result of the paper is given in Fig. 5, which\nshows the IV-characteristics for increasing degree of dis-\norder. Using a parallelized code typically 240 configura-\ntions are used for each characteristic, which needs about\n4 days of computation on a 12-node Opteron server for\na single IV-curve. For increasing disorder the resonances\nin the IV-curve become more and more broadened and\nstart to overlap until they are almost washed out. Here\na model which takes into account the HH-LH band mix-\ning in the well is essential in order to see this effect.\nIf ignored, the LH resonances would dominate the cur-\nrent magnitude comparedto the HH peaks and regionsof\npronounced negative differential resistance would persist\neven for unphysically high degrees of disorder. For the\nlowdopingregimeoftheleads(relativetotheMnconcen-\ntration in the quantum well), as considered here, we find\nan almost vanishing ferromagnetism (∆ <10−2meV) in\nthe well leading to a vanishing current spin polarization.\nThis results also suggest that, for a given (small) degreeof disorder, enhancing the confinement, e.g. by using\na thinner quantum well and/or higher barriers, should\nlead to smaller overlap between the subbands, thus en-\nhancing resonant–tunneling features the IV-curve. Al-\nternative external pressure may be used to enhance the\nsplitting between the lowest HH and LH subbands.\nIV. CONCLUSIONS\nIn summary, we have shown that two factors can be\nrelevant to understand the experimental findings of weak\nresonant–tunneling features and an absence of ferromag-\nnetic order arising from thin (Ga,Mn)As quantum–well\nlayers:\n(i) The presence of considerable disorder in thin layers\nof (Ga,Mn)As which conspires with HH-LH band mixing\nin the quantum well to efficiently weaken any signatures\nof resonant tunneling in the IV characteristics.\n(ii) The observed (almost) vanishing of ferromagnetic\norderin the (Ga,Mn)As quantum well can be understood\nby finding orders of magnitude lower hole densities in the\nwell as compared to the bulk (Ga,Mn)As case of equal\nMn concentration.\nWe thus find the original interpretation of observing\nquantum size effects in a (Ga,Mn)As quantum well by\nthe group of Tanaka [7–9] to be plausible and consistent\nwith our numerical results.\nV. ACKNOWLEDGMENT\nWe acknowledge helpful discussions with Prof.\nMasaaki Tanaka. 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Mermin, Solid State Physics\n(Saunders, Philadelphia, 1976)."    },    {        "title": "1302.5908v2.Electron_spin_resonance_in_a_dilute_magnon_gas_as_a_probe_of_magnon_scattering_resonances.pdf",        "content": "arXiv:1302.5908v2  [cond-mat.str-el]  3 Dec 2013LA-UR-13-20791\nElectron spin resonance in a dilute magnon gas\nas a probe of magnon scattering resonances\nYusuke Nishida\nTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and\nDepartment of Physics, Tokyo Institute of Technology, Ooka yama, Meguro, Tokyo 152-8551, Japan\n(Dated: February 2013)\nWestudytheelectronspinresonance inadilutemagnon gas th atisrealized inaferromagnetic spin\nsystem at low temperature. A quantum cluster expansion is de veloped to show that the frequency\nshift of the single-magnon peak changes its sign and the line width reaches its maximum across a\nscattering resonance between magnons. Such characteristi c behaviors are universal and can be used\nto experimentally locate the two-magnon resonance when an e xternal parameter such as pressure\nis varied. Future achievement of the two-magnon resonance m ay have an impact comparable to\nthe Feshbach resonance in ultracold atoms and will open up a r ich variety of strongly correlated\nphysics such as the recently proposed Efimov effect in quantum magnets. We also suggest how the\nemergence of an Efimov state of three magnons and its binding e nergy may be observed with the\nelectron spin resonance.\nPACS numbers: 75.10.Jm, 75.30.Ds, 75.30.Gw, 76.30.-v\nI. INTRODUCTION\nScattering resonances play an important role in\nphysics. In the field of ultracold atoms, Feshbach res-\nonances induced with magnetic field are used to control\nthe interaction between atoms.1This unparalleled tun-\nability of the interaction has been essential to the recent\nremarkable advance of the field and led to experimen-\ntal realization of a rich variety of physics, such as the\nBCS-BEC crossover in Fermi gases2,3and the Efimov ef-\nfect in Bosegases.4Thesestronglycorrelatedphenomena\nin the vicinity of the scattering resonance are universal,\ni.e., independent of microscopic details. Also, an atom\nloss peak caused by the three-atom or atom-dimer reso-\nnance has been used as a signature of the emergence of\nan Efimov trimer.4,5\nTurningtothefieldofcondensedmatter,thescattering\nresonance can be induced between collective excitations\nin ferromagnetic spin systems (magnons), for example,\nby tuning the exchange coupling.6Because the ferromag-\nnetic coupling is sensitive to the structure of ions,7it is\nexperimentally possible to vary the exchange coupling\nsignificantly with pressure8and bring the system close\nto the two-magnon resonance. Future achievement of\nthe two-magnon resonance may have an impact compa-\nrable to the Feshbach resonance in ultracold atoms and\nwill open up a rich variety of strongly correlated physics\nsuch as the recently proposed Efimov effect in quantum\nmagnets.6But, what is an experimental signature of the\ntwo-magnon resonance?\nIn this paper, we study the electron spin resonance\n(ESR) in a dilute magnon gas as a probe of magnon\nscattering resonances. ESR is a powerful experimental\ntechnique to investigate the dynamics of interacting spin\nsystems,9,10while theoretical calculationsof its spectrum\narein generalchallengingandthusoften sufferfromsome\nlimitations. For example, the previous approaches devel-\noped by Kubo-Tomita11and Mori-Kawasaki12are basedon perturbations in terms of small spin anisotropies,13\nwhich are actually of no use for our purpose because\nmagnon scattering resonances take place at large spin\nanisotropies.6Therefore, a new approach must be devel-\nopedwherespinanisotropiescanbetreatednonperturba-\ntively. One suchapproachis aquantum cluster (orvirial)\nexpansion which uses a fugacity z≡e−β∆as a small pa-\nrameter to perform a systematic expansion.14Here ∆ is\na single-particle excitation energy, β≡1/(kBT) the in-\nverse temperature, and the system is assumed to be so\ndilute that z≪1, but importantly, no assumptions are\nneeded on the form of the interaction.\nWe first demonstrate this quantum cluster expansion\nin Sec. II by computing a single-particlespectral function\ninadiluteBosegas, whichservesasthe universalformula\nfor ESR at low temperature and can be used to extract\nthe scattering length between magnons. Then in Sec. III,\nweapplythesamemethod toESRinadilutemagnongas\nand show that the frequency shift of the single-magnon\npeak changes its sign and the linewidth reaches its max-\nimum across a scattering resonance between magnons.\nSuch characteristic behaviors are universal and can be\nused to experimentally locate the two-magnon resonance\nwhen an external parameter such as pressure is varied.\nFinally, Sec. IV is devoted to the conclusion of this paper\nand discussion where we also suggest how the emergence\nof an Efimov state of three magnons and its binding en-\nergymaybeobservedwithESR.Belowweset¯ h=kB= 1\nand the unspecified range of integration is assumed to be\nfrom−∞to∞.\nII. SINGLE-PARTICLE SPECTRAL FUNCTION\nIN A DILUTE BOSE GAS\nBeforewe workon ESR in a dilute magnon gas, it is in-\nstructive to demonstrate the quantum cluster expansion2\nin a dilute Bose gas, which is described by a Hamiltonian\nH=/integraldisplay\ndrψ†\nr(ε−i∇+∆)ψr−g\n2/integraldisplay\ndrψ†\nrψ†\nrψrψr.(1)\nHereεp=p2/(2m) is the single-particle dispersion rela-\ntion,µ=−∆<0 is the chemical potential, and the bare\ncoupling constant gis related to the s-wave scattering\nlengthasby 1/g=mΛ/(2π2)−m/(4πas) with Λ being\na momentum cutoff. Our purpose in this section is to\ncompute the single-particle spectral function\nA(ω,p) =−2ImG(ω+i0+,p), (2)\nwhereG(iωn,p) withωn≡2πn/βbeing the Matsubara\nfrequency is the Fourier transform of the imaginary-time\npropagator\nG(iωn,p) =−/integraldisplayβ\n0dτ/integraldisplay\ndreiωnτ−ip·r∝angbracketleftψr(τ)ψ†\n0(0)∝angbracketright.(3)\nAs we will show in the next section, A(ω,0) serves as the\nuniversalformula forESR at low temperature and can be\nused to extract the scattering length between magnons.\nThe imaginary-time propagator is defined by\n−∝angbracketleftψr(τ)ψ†\n0(0)∝angbracketright=−1\nZTr[e−βHψr(τ)ψ†\n0(0)],(4)\nwhereZis the grand canonical partition function: Z=\nTr[e−βH]. The systematic expansion over the fugacity\nz=e−β∆≪1 can be developed by writing the grand\ncanonical trace as a sum over canonical traces with fixed\nparticle number N: Tr[·] =/summationtext∞\nN=0trN[·]. Because\ntrN[e−βH]∝zN, we obtain\nZ= 1+Vz\nλ3+O(z2), (5)\nwhereVis the system volume and λ≡/radicalbig\n2πβ/mis the\nthermal de Broglie wavelength. Accordingly, the particle\nnumber density is found to be\nn=1\nVβ∂lnZ\n∂µ=z\nλ3+O(z2). (6)\nTherefore, the quantum cluster expansion is valid when\nthe system is so dilute that the mean interparticle dis-\ntance is much larger than the thermal de Broglie wave-\nlength:n−1/3≫λ.\nThe numerator in Eq. (4) can be expanded over zin\nthe same way. By denoting the Fourier transform of each\nterm as\nGN(iωn,p)≡ −/integraldisplayβ\n0dτ/integraldisplay\ndreiωnτ−ip·r\n×trN[e−βHψr(τ)ψ†\n0(0)]∼O(zN),(7)\nthe leading term is easily evaluated as\nG0(iωn,p) =1−e−βεpz\niωn−εp−∆. (8)/MinuΣ6/MinuΣ4/MinuΣ2 2 4 6Λ\nas\n/MinuΣ1123Σ/Star/LParen10/RParen1/Slash1/LParen1zT/RParen1\nFIG. 1. Real and imaginary parts of σ∗(0)/(zT) in Eq. (14)\nas functions of λ/as, represented by solid and dashed curves,\nrespectively.\nOn the other hand, after a straightforward calculation,\nthe next-to-leading term is evaluated as\nG1(iωn,p) =V/λ3+e−βεp\niωn−εp−∆z\n+/integraldisplaydq\n(2π)3F(iωn,p;q)\n(iωn−εp−∆)2e−βεqz+O(z2),(9)\nwhere\nF(iωn,p;q)≡8π/m\n1/as−/radicalbig\n−m(iωn−∆+εq−εp+q/2)\n(10)\nis the forward scattering amplitude between a particle\nwith energy-momentum ( iωn−∆,p) and an on-shell par-\nticle with momentum q. Then, by writing the Fourier\ntransform of the imaginary-time propagator (3) in the\nstandard form\nG(iωn,p) =1\niωn−εp−∆−Σ(iωn,p)(11)\nand comparing it with its systematic expansion obtained\nin Eqs. (5), (8), and (9), we find that the self-energy\nΣ(iωn,p) must have the following quantum cluster ex-\npansion:\nΣ(iωn,p) =z/integraldisplaydq\n(2π)3F(iωn,p;q)e−βεq+O(z2).(12)\nTherefore, the self-energy at O(z) is determined only by\nthe two-particle physics, i.e., binary collisions with ther-\nmally excited particles.\nThe resulting single-particle spectral function (2) at\nO(z0) is simply a delta function located at ω=εp+∆\ncorresponding to the single-particle energy. By including\nthe self-energy correction Σ( iωn,p)∼O(z)≪1, it be-\ncomes a sharp peak whose line shape within the accuracy\nup toO(z) is described by the Lorentzian\nApeak(ω,p)≈−2Imσ(p)\n[ω−εp−∆−Reσ(p)]2+[Imσ(p)]2,\n(13)3\n/MinuΣ6/MinuΣ4/MinuΣ2 0 2 4∆Ω\nT0.10.20.30.40.50.6TA/LParen1Ω,0/RParen1\nFIG. 2. Single-particle spectral function A(ω,0) forz= 0.1\nandλ/as= 4 as a function of δω≡ω−∆. The single-\nparticle peak goes up to TA(ω,0)≃51 and there is a thresh-\nold singularity at the two-particle binding energy E2/T=\n−(λ/as)2/(2π).\nwhere we introduced the on-shell self-energy\nσ(p)≡Σ(εp+∆+i0+,p)\n=z/integraldisplaydq\n(2π)38π/m\n1/as+i|p−q|/2e−βεq.(14)\nTherefore, the energy shift and the decay width of a par-\nticle with momentum pin a dilute Bose gas are given by\nthe real and imaginary parts of σ∗(p), respectively. The\ncorresponding normalized quantity σ∗(p)/(zT)∼O(1)\natp=0isplottedinFig.1asafunctionof λ/as. Wefind\nthat the energyshift changesits signandthe decaywidth\nreaches its maximum across the scattering resonance at\nas→ ∞. Such characteristic behaviors are universal and\nshared by ESR at low temperature.\nBesides the single-particle peak at ω=εp+ ∆ +\nReσ(p), there exists an additional structure in A(ω,p)\natω < ε p+ ∆ +E2when two particles form a bound\nstate with binding energy E2=−1/(ma2\ns)<0. This\nstructure is due to the pole of the two-particle scattering\namplitude (10) and shows a square-root threshold singu-\nlarity atω=εp+∆+E2in the single-particle spectral\nfunction as A(ω,p)∼z/radicalbig\nεp+∆+E2−ω. This feature\natp=0is demonstrated in Fig. 2 by choosing z= 0.1\nandλ/as= 4 as an example. We also note that the\nsingle-particle spectrum function of bosons is universal\nup toO(z2), i.e., independent of an ultraviolet cutoff.\nTheO(z2) term involves the three-particle physics and\nthus depends on an additional parameter, so-called Efi-\nmovparameter κ∗.15See the discussionin Sec. IVonhow\nthe emergence of an Efimov state and its binding energy\nmayappearin the expected behaviorof A(ω,0) atO(z2).III. ELECTRON SPIN RESONANCE IN A\nDILUTE MAGNON GAS\nA. Quantum cluster expansion\nWe now apply the same method to ESR in a di-\nlute magnon gas. For definiteness, we consider a spin-\nSHeisenberg model with exchange ( Jz∝negationslash=J >0) and\nsingle-ion (D∝negationslash= 0) anisotropies, which is described by\nH=−1\n2/summationdisplay\nr/summationdisplay\nˆe(JS+\nrS−\nr+ˆe+JzSz\nrSz\nr+ˆe)\n−D/summationdisplay\nr(Sz\nr)2���B/summationdisplay\nrSz\nr. (15)\nHere/summationtext\nˆe=/summationtext\nˆe=±ˆx,±ˆy,±ˆzis a sum over six unit vec-\ntors of a simple cubic lattice and spin operators S±\nr≡\nSx\nr±iSy\nrandSz\nrobey the usual commutation relations:\n[S+\nr,S−\nr′] = 2Sz\nrδr,r′and [Sz\nr,S±\nr′] =±S±\nrδr,r′. ESR ex-\nperimentsmeasureanabsorptionintensityofelectromag-\nnetic radiation polarized perpendicular to the magnetic\nfieldaxis. Within the linearresponsetheoryforacircular\npolarization, the absorption intensity normalized by the\nsystem volume and the intensity of the incident radiation\nis given by13,16\nI(ω) =ω\n2Imχ(ω+i0+,0), (16)\nwhereχ(iωn,p) is the Fourier transform of the\nimaginary-time susceptibility\nχ(iωn,p) =/integraldisplayβ\n0dτ/summationdisplay\nreiωnτ−ip·r∝angbracketleftS−\nr(τ)S+\n0(0)∝angbracketright.(17)\nWhile the ESR experiments can measure the spectrum\nonly at zero momentum, we develop the formulation for\ngeneralpand setp=0at the end.\nThe ground state for a sufficiently large magnetic field\nB <0 is a fully polarized state with all spins pointing\ndownwards: Sz\nr|0∝angbracketright=−S|0∝angbracketrightandS−\nr|0∝angbracketright= 0. Accordingly,\nwe redefine the Hamiltonian to absorb the ground state\nenergyE0=−/summationtext\nr(3JzS2+DS2−BS) so thatH|0∝angbracketright=\n0. Because of the U(1) symmetry under rotation S±\nr→\ne±iθS±\nr, the magnetization relative to the ground state\nδM≡ ∝angbracketleftSz\nr∝angbracketright+Sis aconservedquantitywhichcorresponds\nto a particle number density of magnons. Then at low\ntemperature,magnonsarethermallyexcitedandthusthe\nsystem becomes a dilute magnon gas. A single magnon\nhas the dispersion relation εp=SJ/summationtext\nˆe[1−cos(p·ˆe)]\nwith the excitation energy ∆ = −6SJ+6SJz+2SD−\nD−B. As long as the fugacity is small, z=e−β∆≪1,\nthe quantum cluster expansion can be developed for the\ndilute magnon gas similarly to the previous dilute Bose\ngas. The imaginary-time susceptibility is defined by\n∝angbracketleftS−\nr(τ)S+\n0(0)∝angbracketright=1\nZTr[e−βHS−\nr(τ)S+\n0(0)],(18)4\nwhereZis the grand canonical partition function: Z=\nTr[e−βH]. By writing the grand canonical trace as a sum\nover canonical traces with fixed magnon number N, we\nobtain\nZ=∞/summationdisplay\nN=0trN[e−βH] = 1+Vz\nρ3+O(z2),(19)\nwhere we used tr N[e−βH]∝zNand introduced the\nanalog of the thermal de Broglie wavelength by ρ≡\nae2βSJ/I0(2βSJ) withabeing the lattice constant. Ac-\ncordingly, the relative magnetization is found to be\nδM=1\nVβ∂lnZ\n∂B=z\nρ3+O(z2). (20)\nThe numerator in Eq. (18) can be expanded over zin\nthe same way. By denoting the Fourier transform of each\nterm as\nχN(iωn,p)≡/integraldisplayβ\n0dτ/summationdisplay\nreiωnτ−ip·r\n×trN[e−βHS−\nr(τ)S+\n0(0)]∼O(zN),(21)\nthe leading term is easily evaluated as\nχ0(iωn,p) =−2S1−e−βεpz\niωn−εp−∆. (22)\nOn the other hand, after a straightforward calculation,\nthe next-to-leading term is evaluated as\nχ1(iωn,p) =−2SV/ρ3+e−βεp\niωn−εp−∆z\n−2S/integraldisplayπ/a\n−π/adq\n(2π/a)3Γ(iωn,p;q)\n(iωn−εp−∆)2e−βεqz+O(z2),\n(23)\nwhere\nΓ(iωn,p;q)≡/summationdisplay\nˆe[Jcos(p+q\n2·ˆe)−Jzcos(p−q\n2·ˆe)]γ(ˆe)\n−2Dγ(0) (24)\nis the forward scattering amplitude between a magnon\nwith energy-momentum ( iωn−∆,p) and an on-shell\nmagnon with momentum q. Here the unknown func-\ntionγ(r) =γ(−r) implicitly depends on ( iωn,p;q) and\nsatisfies the Lippmann-Schwinger equation\nγ(r) = 2cos(p−q\n2·r)+/integraldisplayπ/a\n−π/adk\n(2π/a)3cos(k·r)\n×/summationtext\nˆe[Jcos(p+q\n2·ˆe)−Jzcos(k·ˆe)]γ(ˆe)−2Dγ(0)\niωn−∆+εq−ε(p+q)/2+k−ε(p+q)/2−k.\n(25)\nBy setting r=ˆx,ˆy,ˆzandr=0, we obtain four cou-\npled equations to determine γ(ˆe) andγ(0) appearing in\nEq. (24).Then, by writing the Fourier transform of the\nimaginary-time susceptibility (17) in the standard form\nχ(iωn,p) =−2S\niωn−εp−∆−Ξ(iωn,p)(26)\nand comparing it with its systematic expansion obtained\nin Eqs. (19), (22), and (23), we find that the self-energy\nΞ(iωn,p) must have the following quantum cluster ex-\npansion:\nΞ(iωn,p) =z/integraldisplayπ/a\n−π/adq\n(2π/a)3Γ(iωn,p;q)e−βεq+O(z2).\n(27)\nTherefore, the self-energy at O(z) is determined only by\nthe two-magnon physics, i.e., binary collisions with ther-\nmally excited magnons. The resulting ESR spectrum\n(16) atO(z0) is simply a delta function located at ω= ∆\ncorresponding to the single-magnon energy at p=0. By\nincluding the self-energy correction Ξ( iωn,0)∼O(z)≪\n1, it becomes a sharp peak whose line shape within the\naccuracy up to O(z) is described by the Lorentzian\nIpeak(ω)≈ω\n2−2SImξ(0)\n[ω−∆−Reξ(0)]2+[Imξ(0)]2,(28)\nwhere we introduced the on-shell self-energy: ξ(p)≡\nΞ(εp+∆+i0+,p). Therefore, the frequencyshift and the\nlinewidth of the single-magnonpeak are given by the real\nand imaginary parts of ξ∗(0), respectively.17Also, when\ntwo magnons form a bound state with binding energy\nE2<0, the ESR spectrum shows an additional struc-\nture atω<∆+E2similarly to Fig. 2, while we will not\ninvestigate it further.\nB. Solution and results\nOur remaining task is to solve the Lippmann-\nSchwinger equation (25) with iωn= ∆+i0+andp=0:\nγ(r) = 2cos(q\n2·r)+/integraldisplayπ/a\n−π/adk\n(2π/a)3cos(k·r)\n×/summationtext\nˆe[Jcos(q\n2·ˆe)−Jzcos(k·ˆe)]γ(ˆe)−2Dγ(0)\nεq−εq/2+k−εq/2−k+i0+(29)\nto determine γ(ˆe) andγ(0) in the two-magnonscattering\namplitude (24). In the low-temperature limit T→0, the\nintegration over qin the self-energy (27) is dominated by\nthe region q≃0because of the Boltzmann factor. In\nthis case, by expanding the right-hand side of Eq. (29)\nup toO(q), we find γ(ˆx) =γ(ˆy) =γ(ˆz) and the in-\ntegration over kcan be performed analytically. Then,\nby substituting the obtained analytical solutions for γ(ˆe)\nandγ(0)intoEq.(24), wefindthatthetwo-magnonscat-\ntering amplitude takes the same form as that of bosons5\nS= 1/2 S= 1 (D= 0) S= 1 (Jz=J)as/a\n2.0 2.5 3.0 3.5 4.0 4.5\n/MinuΣ15/MinuΣ10/MinuΣ5051015\n4.0 4.5 5.0 5.5 6.0 6.5\n/MinuΣ15/MinuΣ10/MinuΣ5051015\n4.4 4.6 4.8 5.0 5.2 5.4\n/MinuΣ15/MinuΣ10/MinuΣ5051015Reξ∗(0)/(zT)\n2.0 2.5 3.0 3.5 4.0 4.5\n/MinuΣ1.5/MinuΣ1.0/MinuΣ0.50.00.51.01.5\n4.0 4.5 5.0 5.5 6.0 6.5\n/MinuΣ1.5/MinuΣ1.0/MinuΣ0.50.00.51.01.5\n4.4 4.6 4.8 5.0 5.2 5.4\n/MinuΣ1.5/MinuΣ1.0/MinuΣ0.50.00.51.01.5Imξ∗(0)/(zT)\n1.5 2.0 2.5 3.0 3.5 4.0 4.50.00.51.01.52.02.53.0\n3.5 4.0 4.5 5.0 5.5 6.0 6.50.00.51.01.52.02.53.0\n4.2 4.4 4.6 4.8 5.0 5.2 5.40.00.51.01.52.02.53.0\nJz/J Jz/J D/J\nFIG. 3. Top panels show scattering lengths in Eq. (31) for S= 1/2 (left), S= 1 with D= 0 (middle) as functions of Jz/J,\nandS= 1 with Jz=J(right) as a function of D/J. The vertical lines indicate the locations of two-magnon re sonances where\nas→ ∞. Middle and bottom panels show real and imaginary parts of ξ∗(0)/(zT) in Eq. (28) at T/J= 0.01,0.1,1, and 10,\nrepresented by solid, dashed, dash-dotted, and dotted curv es, respectively.\nin Eq. (10):\nlim\nq→0Γ(∆+i0+,0;q) =1\na38π/m\n1/as+i|q|/2\n=1\na3F(∆+i0+,0;q),(30)\nwhere 1/m= 2SJa2is the inverse effective mass of\nmagnons and\nas\na=3\n2π[1−D\n3J−Jz\nJ(1−D\n6SJ)]\n2S−1+Jz\nJ(1−D\n6SJ)+3W[1−D\n3J−Jz\nJ(1−D\n6SJ)]\n(31)\nis the scattering length between magnons with\nW≡/integraldisplayπ/a\n−π/adk\n(2π/a)32/summationtext\nˆe[1−cos(k·ˆe)]\n=√\n6\n96π3Γ/parenleftbigg1\n24/parenrightbigg\nΓ/parenleftbigg5\n24/parenrightbigg\nΓ/parenleftbigg7\n24/parenrightbigg\nΓ/parenleftbigg11\n24/parenrightbigg\n(32)being the Watson’s triple integral for a simple cubic\nlattice.18We note that the same scattering length was\nobtained in Ref. 6 with a different approach.\nAccordingly, the low-temperature limit of the on-shell\nself-energy at p=0appearing in Eq. (28) reduces to\nthat in Eq. (14),\nlim\nT→0ξ(0) =σ(0), (33a)\nand thus the line shape of the single-magnon peak is de-\nscribed by the universal formula\nlim\nT→0Ipeak(ω) =ω\n2SApeak(ω,0),(33b)\nwhereApeak(ω,p) is the single-particle spectral function\nof bosons obtained in the previous section as Eq. (13).\nThis result is actually expected because the magnon gas\nat low temperature is so dilute that the system becomes\nindependent of microscopic details and thus described\nby only a few low-energy parameters such as mandas\n[additionally κ∗atO(z2)]. Therefore, it is possible to\nextract the scattering length between magnons by fitting6\nthe universal formula (33) to the experimentally mea-\nsured temperature dependence of the line shape of the\nsingle-magnon peak.\nAway from the low-temperature limit, the line shape\n(28) is model dependent and has to be evaluated nu-\nmerically by solving Eq. (29) for general q. The fre-\nquency shift and the linewidth of the single-magnonpeak\nare given by the real and imaginary parts of ξ∗(0), re-\nspectively, and the corresponding normalized quantity\nξ∗(0)/(zT)∼O(1) is plotted in Fig. 3. For demon-\nstration, we choose three distinct cases where S= 1/2,\nS= 1 withD= 0 as functions of Jz/J, andS= 1\nwithJz=Jas a function of D/Jat four different tem-\nperatures,T/J= 0.01,0.1,1,10. Figure 3 also displays\nthe corresponding scattering length (31) which indicates\nthat the two-magnon resonances as→ ∞are located at\nJz/J= 2.94,Jz/J= 4.87, andD/J= 4.77, respectively,\nwhere magnons interact strongly. We find that the line\nshape of the single-magnon peak is well described by the\nuniversal formula (33) at low temperature T < Jand\nthusthefrequencyshiftchangesitssignandthelinewidth\nreaches its maximum across the two-magnon resonance.\nSuch characteristic behaviors become sharper with de-\ncreasing temperature and can be seen moderately even\natintermediatetemperature T≃J, whiletheydisappear\nat higher temperature T >J.\nIV. CONCLUSION AND DISCUSSION\nIn this paper, we studied ESR in a dilute magnon\ngas that is realized in a ferromagnetic spin system at\nlow temperature. We developed the quantum cluster\nexpansion up to O(z) which is determined by the two-\nmagnon physics and showed that the frequency shift\nof the single-magnon peak changes its sign and the\nlinewidth reaches its maximum across a scattering res-\nonance between magnons. Such characteristic behaviors\nare universal and can be used to experimentally locate\nthe two-magnon resonance when an external parameter\nsuchaspressureisvaried. Futureachievementofthetwo-\nmagnonresonancemayhaveanimpactcomparabletothe\nFeshbach resonance in ultracold atoms and will open up\na rich variety of strongly correlated physics such as the\nrecently proposed Efimov effect in quantum magnets.6\nIt is straightforward in principle to continue this sys-\ntematic expansion to include higher-ordercorrections. In\nparticular, the O(z2) term involves the three-magnon\nphysics and thus it is possible to probe the Efimov ef-fect with ESR. When the system comes across the crit-\nical coupling where an Efimov state of three magnons\nemerges from the scattering threshold, the linewidth of\nthe single-magnon peak as a function of the external pa-\nrameter is expected to show an additional peak struc-\nture caused by the three-magnon resonance on either\nside of the two-magnon resonance. This feature is in\nanalogy with ultracold atom experiments where an atom\nloss peak caused by the three-atom or atom-dimer reso-\nnance has been used as a signature of the emergence of\nan Efimov trimer.4,5Similarly, a pair of universal four-\nmagnon states associated with every Efimov state19,20\nmay be observed with ESR through an additional peak\nstructure in the linewidth at O(z3) caused by the four-\nmagnon resonance.21,22Besides such characteristic be-\nhaviors in the single-magnon peak, we expect an addi-\ntional structure in the ESR spectrum at ω <∆ +EN\nwhenNmagnonsformaboundstatewithbindingenergy\nEN<0. This structure shows a threshold singularity at\nω= ∆+ENasI(ω)∼zN−1(∆+EN−ω)(3N−5)/2, which\nmay be used to measure binding energies of magnon Efi-\nmov states. Therefore, ESR is a powerful experimental\ntechnique to investigate the interaction among magnons\nand their spectrum.\nSo far we considered ferromagnetic spin systems where\nthe single-magnon dispersion relation has a minimum at\nzero momentum. On the other hand, it is also possi-\nble to induce the two-magnon resonance and thus the\nmagnon Efimov effect in spin systems with antiferromag-\nnetic or frustrated exchange couplings where the single-\nmagnon dispersion relation has a minimum at nonzero\nmomentum.6,23,24In these cases, however, the sharp sig-\nnature of magnon scattering resonances discussed in this\npaper will not appear in ESR because it measures the\nspectrum only at zero momentum. Therefore, different\nexperimental techniques such as inelastic neutron scat-\ntering that can scan momentum space should be used\nto probe magnon scattering resonances. The quantum\ncluster expansion developed in this paper will be useful\nto compute any other physical observables in a dilute\nmagnon gas.\nACKNOWLEDGMENTS\nThe author thanks C. D. Batista and Y. Kato and\nacknowledges many valuable discussions during his visit\nto RIKEN and YITP in the fall of 2012. This work was\nsupportedbyaLANLOppenheimerFellowshipandJSPS\nKAKENHI Grant Number 25887020.\n1C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.\nMod. Phys. 82, 1225 (2010).\n2C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett.\n92, 040403 (2004).3M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Rau-\npach, A. J. Kerman, and W. Ketterle, Phys. Rev. Lett. 92,\n120403 (2004).7\n4T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl,\nC. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola,\nH.-C. N¨ agerl, and R. Grimm, Nature (London) 440, 315\n(2006).\n5S. Knoop, F. Ferlaino, M. Mark, M. Berninger, H. Sch¨ obel,\nH.-C. N¨ agerl, and R. Grimm, Nat. Phys. 5, 227 (2009).\n6Y. Nishida, Y. Kato, and C. D. Batista, Nat. Phys. 9, 93\n(2013).\n7J. B. Goodenough, Magnetism and the Chemical Bond\n(Wiley, New York, 1963).\n8T. Kawamoto, M. Tokumoto, H. Sakamoto, and K. Mi-\nzoguchi, J. Phys. Soc. Jpn. 70, 1892 (2001).\n9K. Katsumata, J. Phys.: Condens. Matter 12, R589\n(2000).\n10Y. Ajiro, J. Phys. Soc. Jpn. 72, 12 (2003).\n11R. Kubo and K. Tomita, J. Phys. Soc. Jpn. 9, 888 (1954).\n12H. Mori and K. Kawasaki, Prog. Theor. Phys. 27, 529\n(1962); Prog. Theor. Phys. 28, 971 (1962).\n13M. Oshikawa and I. Affleck, Phys. Rev. B 65, 134410\n(2002).\n14L. D. Landau and E. M. Lifshitz, Statistical Physics\n(Butterworth-Heinemann, Oxford, 1980).\n15E. Braaten and H.-W. Hammer, Phys. Rep. 428, 259\n(2006).16M. Brockmann, F. G¨ ohmann, M. Karbach, A. Kl¨ umper,\nand A. Weiße, Phys. Rev. B 85, 134438 (2012).\n17In the absence of the spin anisotropies Jz=JandD=\n0, the two-magnon scattering amplitude (24) at q=0\nvanishes so that the ESR spectrum shows no frequency\nshift and linewidth, which is consistent with the general\nargument given in Ref. 13. Also, by expanding Eqs. (24)\nand (25) in terms of the small spin anisotropies ǫ∼Jz/J−\n1,D/J≪1, it is easy to find that the frequency shift and\nthe linewidth are Re ξ∗(0)∼O(ǫ) and Im ξ∗(0)∼O(ǫ2),\nrespectively, which are again consistent with the previous\nresults (Ref. 11–13).\n18G. N. Watson, Q. J. Math. 10, 266 (1939).\n19H.-W. Hammer and L. Platter, Eur. Phys. J. A 32, 113\n(2007).\n20J. von Stecher, J. P. D’Incao, and C. H. Greene, Nat. Phys.\n5, 417 (2009).\n21F. Ferlaino, S. Knoop, M. Berninger, W. Harm,\nJ. P. D’Incao, H.-C. N¨ agerl, and R. Grimm, Phys. Rev.\nLett.102, 140401 (2009).\n22F. Ferlaino and R. Grimm, Physics 3, 9 (2010).\n23H. T. Ueda and K. Totsuka, Phys. Rev. B 80, 014417\n(2009).\n24H.T. UedaandT.Momoi, Phys.Rev.B 87, 144417(2013)."    },    {        "title": "1504.02319v2.Spin_transfer_torque_effects_in_the_dynamic_forced_response_of_the_magnetization_of_nanoscale_ferromagnets_in_superimposed_ac_and_dc_bias_fields_in_the_presence_of_thermal_agitation.pdf",        "content": "1 Spin -transfer torque effects in the d ynamic forced response of the magnetization of \nnanoscale ferromagnets  in superimposed ac and dc bias fields in the presence of thermal \nagitation  \n \nD. J. Byrne,1 W. T. Coffey,2 Y. P. Kalmykov,3 S. V. Titov , 4 and J. E. Wegrowe5 \n1School of Physics, University College Dublin, Belfield, Dublin 4, Ireland  \n2Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland  \n3Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, F -66860, \nPerpignan, France  \n4Kotel’nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, \nVvedenskii Square 1, Fryazino, Moscow Region, 141120, Russia  \n5Laboratoire des Solides Irradiés, Ecole Polytechnique, 91128 Palaiseau Cedex, France  \n \n \nAbstract  \nSpin-transfer torque ( STT) effects on the stationary  forced response of nanoscale \nferromagnets  subject to thermal fluctuations and  driven by an ac  magnetic field  of arbitrary strength  \nand direction  are investigated  via a generic nanopillar model of a spin -torque device comprising two \nferromagnetic strata representing the free and fixed layers and a nonmagnetic conducting spacer all \nsandwiched betwee n two ohmic contacts . The S TT effects a re treated via  Brown’s  magnetic \nLangevin equation generalized to include the Slonczewski STT term thereby  extending t he \nstatistical moment method  [Y. P. Kalmykov et al., Phys. Rev. B 88, 144406  (2013) ] to the forced \nresponse  of the  most general version of the  nanopillar  model . The dynamic susceptibility , nonlinear \nfrequency -dependent dc magnetization, dynamic magnetic hysteresis loops, etc. are  then evaluated \nhighlighting STT effects on both the low -frequency thermal relaxation processes and the high -\nfrequency ferromagnetic resonance , etc., demonstrating a  pronounced dependence of these  on the \nspin polarization current and  facilitating  interpretation  of STT experiments .  \n  2 I. INTRODUCTION  \nOne of the most significant developments in magnetization reversal by thermal agitation in \nnanoscale ferromagnets since the seminal treatment  of Néel [1] and Brown [2] has been the spin-\ntransfer torque (STT) effect [3-5] existing  because an electric current with spin polarization in a \nferromagnet has an associated flow of angular momentum [3 -7] thereby exerting a  macroscopic \nspin-torque. Consequently , the magnetization \nM  of the ferromagnet may be altered by spin-\npolarized  currents , which underpin  the novel subject of spintronics [8], i.e., current -induced control \nover magnetic nanostructures. Applications include  (a) very high speed current -induced \nmagnetization switching by reversing the orie ntation of magnetic bits [5,9]  and (b) using spin-\npolarized currents to manipulate steady state microwave oscillations [9] via the steady state \nmagnetization precession due to STT representing the conversion of dc input current into an ac \noutput voltage [5 ]. Now due to  thermal fluctuations [5,9], STT devices invariably represent an open \nsystem on the nanoscale  in an out -of-equilibrium steady state quite unlike conventional \nnanostructures characterized by the Boltzmann equilibrium distribution. T herefore , the thermal  \nfluctuations cannot be ignored . They lead  to mainly noise -induced switching at currents far less than \nthe critical switching current without noise as well as introducing randomness into the precessional \norbits, which  now exhibit energy -controlle d diffusion [10]. Thus , the effect of the noise is generally \nto reduce the current -induced switching time. This phenomenon has been corroborated by many \nexperiments (e.g., [11]) demonstrat ing that STT near room temperature alters thermally activated \nswitch ing processes , which then  exhibit a pronounced dependence on both material and geometrical \nparameters.  However , in marked contrast to the well -developed zero temperature limit, T = 0, and \nto nanomagnets  at finite  temperature without STT, various treatments  of the thermally activated \nmagnetization reversal in STT systems  (e.g., escape rates  [12-15] and stochastic dynamic \nsimulations  [16-19]) are still in a state of flux  [20]. Therefore , accurate solutions  of generic STT \nmodels at finite temperatures are necessary both to properly assess such theories and to achieve \nfurther improvements in the design and interpretation of experiments, particularly due to the \nmanifold practical applications in spintronics, ra ndom access memory technology, and so on.  \nThe archetypal model (Fig. 1) of a STT device is a nanostructure comprising two magnetic \nstrata labelled the free and fixed layers and a nonmagnetic conducting spacer. The fixed layer is \nmuch more strongly pinned a long its orientation than the free one. On passing an electric current \nthrough the fixed layer it  becomes spin-polarized which , as it encounters  the free layer, induces a \nSTT that alters the  magnetization  \nM of that layer . Both ferr omagnetic layers are assumed to be \nuniformly magnetized [8] . Although the single -domain or macrospin  approximation cannot explain  \nall observations of the magnetization dynamics in spin -torque systems,  nevertheless  many \nqualitative features needed to interpret  experimental dat a are satisfactorily reproduced . Thus the \ncurrent -induced magnetization dynamics in the free layer  including thermal fluctuations  may be 3 described by the Landau -Lifshitz -Gilbert -Slonczewski  equation [3], i.e., the Landau -Lifshitz -\nGilbert  equation [ 21] including the STT augmented by a Gaussian white noise field \n()th  so \nbecoming a Langevin equation [ 5,7,20], viz.,  \n \nX e   u Z \nY M  \n \neasy axis  H0+ Hcos  t  \nfixed layer  free layer   Je ep \n(a) (b) \nFIG. 1.  (Color on line)  (a) Geometry of the problem:  A STT device consists of two ferromagnetic \nstrata labelled the free and fixed layers, respectively, and a normal conducting spacer all sandwiched \non a pillar between two ohmic contacts [12]. The fixed layer has a fixed magnetization along the \ndirection \nPe . \neJ is the spin -polarized current density, \nM  is the magnetization of the free layer, \n0H  \nis the dc bias magnetic field  and \ncos t H  is the applied ac field . (b) Free energy potential  presented \nin the standard form of superimposed easy -plane and in -plane easy -axis anisotropies.  \n \n  eff ST     u u H u h u u\n . (1) \nHere \n1\nSMuM  is a unit vector along  \nM, \nSM  is the saturation magnetization,  is the \ngyromagnetic -type constant, \n  is a dimensionless phenomenological damping parameter, \nrepresenting the  combined  effect of all the microscopic degrees of freedom ,  \n \neff\n0S1 V\nMHu  (2) \nis the effective magnetic field comprising the anisotropy and external  applied  fields, while  \n/u  \ndenotes  the gradient  operator  on the surface of the unit sphere , \n7 2 1\n04 10 JA m    in SI units,  \nV\n is the free energy density of the free layer , and the STT term \nSTu\n  in Eq. (1) is defined as  \n \nST\n0S1\nM uuu\n , \nwhere \n  is the non -conservative potential due to the spin -polarized current [ 3,4,20]. \nAlmost invariably,  the effect s of thermal fl uctuations combined with STT have  been \ninvestigated via the magnetic Langevin equation (1) or its associated  Fokker -Planck equation [20] . \nThe magnetic  Langevin  equation without STT was originally proposed by Brown [ 2] for theor etical \ntreatment of the magnetization  reversal  in magnetic nano particle s. His primary objective was to  \nsecurely  anchor Néel’s conjectures [ 1] concerning the nature of the superparamagnetic  relaxation \nof a single domain ferromagnetic particle within the framework of the  theory of stochastic processes  \n4 (in essence,  Brown’s theory  of the magnetization relaxation in magnetic nanoparticles [2] is an \nanalog of the Debye theory [22,23 ] of dielectri c relaxation of polar liquids ). During the last decade , \nvarious analytical and numerical approaches to the calculation of the measurable parameters of STT \ndevices  via the magnetic Langevin equation in cluding  STT have been developed . These include  \ngeneraliz ations (e. g., Refs. [12-14, 20]) of the Kramers escape rate theory [24 -27] and stochastic \ndynamics simulations (e.g., Refs. 12, 16 -19). For example, the pronounced time separation between \nfast precessional and slow energy changes in lightly  damped ( << 1) closed phase space trajectories \n(called Stoner -Wohlfarth orbits) at energies near the barrier energy has bee n exploited in Refs. [7, \n12, 13] to formulate a one-dimensional Fokker -Planck equation  for the energy distribution  function  \nessentially similar to that derived by  Kramers [24] for point particles. These generalizations yield  \nSTT effects in the thermally assisted magnetization reversal via the Langevin and/or Fokker -Planck \nequations as a function  of temperature, damping, external magnetic field, and s pin-polarized current . \nIn particular , varying the spin -polarized current  may alter the reversal time by several orders of \nmagnitude concurring with experimental results [11].  \nNow we  have previously treated  [28] STT effects on certain out -of-equilibrium time - and \nfrequency -independent stationary observables in the presence of a dc bias field alone  via the generic \nnanopillar model (Fig. 1) by solving  the magnetic Langevin equation (1) using the statistical \nmoment method  [27]. The se observables comprise  the stationary distribution of the magnetization \norientations, the effective potential, the in -plane component of the m agnetization of the free layer , \nand the static susceptibility . In particular , these  time- and frequency -independent  observables have  \nbeen  studied [28] for wide ranges  of the spin-polarized current, the dissipati ve coupling to the \nthermal bath, the anisotropy parameters  and the magnitude and orientation of the applied external \ndc field , which wa s supposed constant  in time. Besides  the calculation of these stationary \nobservables, the reversal time of the in -plane com ponent of the magnetization of the free layer has \nalso been evaluated  [28] via the smallest nonvanishing eigenvalue of the corresponding Fokker -\nPlanck operator [29] again as a function of the parameters  mentioned . Now in Ref.  28, the external \napplied  (bias)  field wa s supposed  time-independent , i.e., it represents a dc  field applied in the infinite \npast. Hence , the results  of Ref. 28 cannot be applied to virtually all dynamical aspects of the time -\ndependent magnetization response . These include  magneti zation switching  of STT devices and line \nshape of STT nano -oscillators driven by ac external magnetic fields and currents [30-35], stochastic \nresonance  [36-39], etc. In particular , as shown  experimentally, the magnetization reversal in STT \ndevices driven b y superimposed dc and ac currents or by a direct spin -polarized current combined \nwith an ac magnetic field may allow one  a more efficient STT magnetization reversal in comparison \nto that achievable by pu rely dc currents alone. There the problem is of  technological interest  in the \ncontext of improving switching characteristics of magnetic random access memories  [35]. Despite \nthe potential applications,  an accurate theoretical description of STT effects in the response  of a \nnanomagnet  to an ac force  of arbitrary  strength in the presence of thermal agitation has not yet been 5 fully developed due to the inherent  difficulties generally associated with  modelling  a nonlinear \nresponse . As a result , most  of the theoretical methods , which were  developed for  STT ef fects (see, \ne.g., [30 -39]), concern the ac response over limited  ranges of the frequency and amplitude . Hence , \nthey do not cover  many  other dynamical characteristics  of nanomagnets including  the nonlinear \ncomplex magnetic susceptibility and dynamic magnetic hysteresis (DMH) loops , which  require the \nresponse to a strong ac magnetic field over a wide  frequency range.  Therefore , to comprehensively  \ninvestigate the influence of STT on the dynamical characteristics  of the generic nanopillar model \n(Fig. 1) due to an ac magnetic field of arbitrary strength  and frequency , we generalize  the approach \nof Titov et al.  [40] developed originally for zero STT . The advantage  of this approach over all others \nis that one can obtain the nonlinear  response characteristics for all frequencies of interest  ranging  \nfrom the very low ones corresponding to overbarrier relaxation processes up to the very high \nfrequencies appropriate to  the ferromagnetic resonance (GHz) range  using a single model . Now  a \npriori  STT effects in the ac stationary  response of a n anomagnet  inherently  pose a more complicated \nproblem than the time -independent  out-of-equilibrium case of Ref. 28 because the observables are \nnow both time- and frequency -dependent . However , these difficulties may  be overcome us ing the \nmatrix continued fraction method  [27,29 ] just as with the nonli near ac response  without STT \n[40,41]. \nThe paper is arranged as follows. In Sec. II, the basic equations for the calculation of the ac \nstationary response  are given . In Sec. III, the spectra of the linear dynamic susceptibility in all \nfrequency range s characterizing the magnetization dynamics are given demonstrating a strong \ndependence on STT. In Sec. IV, STT effects on spectra of the nonlinear dynamic susceptibility  and \nthe stationary time-independent  but frequency -dependent  magnetization are illustrated, while  STT \neffects on DMH loops and specific absorption rate are studied in Sec. V. Appendixes A and B \ncontain a detailed account  of the matrix continued fraction solution  for the stationary response of a \nnanoscale ferromagnet to an ac magnetic field of arbitrary strength . \nII. STATISTICAL MOME NT EQUATIONS  \nNow  the main thrust of our investigation is the study of  STT effects on  the complex magnetic \nsusceptibilit y and DMH  loops  of a nanoscale ferromagnet subjected to superimposed ac and  dc bias \nfields \n0 cos tHH  of arbitrary strengths and orientations using the generic nanopillar model  \nillustrated  by Fig. 1 . Here the normalized free energy per unit volume \n( , , )Vt    of the free layer \nmay conveniently be written as  (\n0H and \nH  are assumed parallel ) \n \n \n 2 2 2\n0( , , ) cos sin cos\ncos cos ( , ),Vt\nt       \n    \n    (3) \nwhere  and  are the angular coordinates specifying the orientation of the magnetization  \nM in \nspherical polar coordinates  (see Fig. 1b ), \n02\nSMD  \n  and \n/DD\n  are the dimensionless 6 anisotropy  and biaxiality parameters  respectively , \nD\n and \nD  account for both demagnetizing and \nmagnetocrystalline anisotropy effects [20], \n0 0 S 0 MH   and \n0SMH   are the dc and ac \nexternal field parameters, respectively,  \n/ ( )v kT , v is the volume of the free lay er, k is \nBoltzmann’s constant, T is the absolute temperature,  while  \n is the angle between \nH  and \nM  so \nthat \n \n\n1 2 3cos ( , )\nsin cos sin sin cos .H \n         \n  uH  (4) \nHere \n1cos sin   , \n2sin sin   , and \n3cos  are the direction cosines of the applied dc \nand ac fields. The first term on the right hand side of Eq. (3), namely,  \n 2 2 2cos sin cos     \nconstitutes a conservative potential taken in the standard form of superimposed easy -plane and in -\nplane easy -axis anisotropies  (see Fig. 1b) . This potential , in general , represents  an energyscape with \ntwo minima and two saddle points compelling  the magnetization to align in a given direction in \neither  of the energy minima in the equatorial or XY plane  [28]. As in Ref. [ 28], Z is taken as the hard \naxis while the X-axis is the easy one. Furthermore, t he non-conservative potential \n  due to the \nspin-polarized current may sensibly be approximated [28] for all polar angles \n , \n and arbitrary  \norientation of the unit vector \nPe  (identifying the magnetization direction in the fixed layer ) by a \nfinite series of spherical harmonics  \n( , )lmY  [42], viz.,  \n \n2\n0( , )r\nrs rs\nr s rBY   \n  , (5) \nwhere the expansion coefficients \nrsB  are listed explicitly in Ref. [28 ].  \nNow  the task of calculating the ac stationary response  from the Langevin equation (1) can \nalways be reduced  to the solution of an inf inite hierarchy of differential -recurrence relations  for the \nstatistical moments (averaged spherical harmonics  \n( ),lmYt  where the angular brackets \n  mean \nstatistical averaging ) as with zero STT [40, 41]. Such a hierarchy has  been derived in Ref. [28]  for \nthe non-conservative  potential due to spin -polarized current given by Eq. (5) and the biaxial \nanisotropy plus the Zeeman term due to a spatially uniform dc  bias field \n0H . In like manner, we \ncan generalize  this derivation to our case representing the response to  a dc bias field tempora lly \nmodulated by a n ac field \ncos t H . Here  the total free energy  density V is given by  Eq. (3) above  \nand the infinite hierarchy of 25-term differential -recurrence relation s for \n()lmYt  becomes  \n \n22\n;\n22( ) ( ) ( )N lm lm l rm s l rm s\nrsdY t e t Y tdt   \n  , (6) \nwhere the coefficients \n; ()lm l rm set  are now time -dependent and  are given explicitly in Appendix A , \n1\n0()N    \n is the fr ee rotational diffusion time of  the magnetization , and \n00(2 )SMD \n  7 is a normalizing  time. By using  Eq. (4) and the definition of the spherical harmonics of first rank, \nviz., [ 42] \n \n10\n113( , ) cos ,4\n3( , ) sin ,8iY\nYe  \n  \n\n\n  \nthe magnetization \nS ( ) cos ( )HM t M t  in the direction of the ac  driving field \ncos t H  may be  \nformally expressed via  the statistical moments \n10()Yt  and \n11()Yt  as \n \n  S 3 10 1 2 114( ) ( ) 2 Re ( ) ( )3HM t M Y t i Y t        . (7) \nHowever , due to the sinusoidal term in the applied field \n0 cos tHH , the stationary response  of \n()HMt\n must , in general , be developed in  a Four ier series because with the notable exception of  the \nlinear response all harmonics of th e ac field will now be involved , viz., \n \nS1 ( ) ( )k ik t\nH\nkM t M m e\n , (8) \nwhere  the Fourier coefficients \n1()km  of the kth harmonic of \n()HMt  are given by  \n \n1 3 10 1 2 1 1 1 2 112( ) 2 ( ) ( ) ( )3k k k km c i c i c              (9) \nand \n()k\nlmc  are themselves the Fourier coefficients in a Fourier series development in the time of \nthe average spherical harmonics  \n \n ()k ik t\nnm nm\nkY t c e\n . (10) \nThe coefficients \nc ( )k\nlm  can then be evaluated using matrix continued fractions as described in \nAppendix B. Equation  (8) includes the linear response as a special case, \n0 , whereupon all \nhigher harmonics may be discarded in Eq. (8) and only the term \n1\n1()m  linear in \n  remains.  \nHaving  determined the  Fourier amplitudes \n1()km , we have  \n()HMt  and other rel ated \nparameters  such as the dynamic susceptibilities , etc. This procedure  will also yield  the DMH  loop \nrepresenting a parametric plot of the stationary  time-dependent magnetizatio n as a function of the \nac field, i.e., \n()HMt  vs. \n( ) cosH t H t  , and the area enclosed by the loop , viz., \n \n0 ( ) ( )H A v M t dH t\n . (11) \nEquation (11) represents  the energy loss per nanomagnet in one  cycle of the ac field . The physical \nmeaning of \nA  is that it determines the so -called specific absorption rate (SAR) defined as \n/ (2 ) SAR A\n. Here we shall calculate (because of its direct relation to the complex \nsusceptibility ) the normalized area of the DMH loop \n0 / (4 )nSA A v M H   given by  [43] 8  \n1\n11( ) ( ) Im( )42nH\nSA M t dH t mMH \n . (12) \nThe DMH phenomenon (originally predicted in nanomagnets by Ignachenko and Gekht [ 44]) is of \nmuch  practical  interest since it occurs in magnetic information storage and magnetodynamic \nhyperthermia occasioned by induction heating of nanomagnets . \nThe vectors  \n0H, \nH, and \nPe  (as defined in spherical polar coordinates  in Fig. 1 ) are assumed  \nthroughout to lie in the equatorial or XY plane  with colatitudes  \n/2  and \n/ 2,P  \nrespectively , so that  the orientations of \n0H , \nH, and \nPe  are entirely specified  by the azimuthal \nangles of the applied field s \n and spin polarization \nP , respectively. The values \n0P  \ncorrespond to the particular configuration whereby the vectors \n0H , \nH, and \nPe  are all directed along \nthe easy ( X-)axis. The spin polarization azimuthal angle  \nP, biaxiallity parameter  \n, spin-\npolarization factor  \nP, and damping  \n selected are  \n0P , \n20 , \n0.3P  (\n0.3 0.4P  are \ntypical values for ferromagnetic metals [20]) , and \n0.01  (for high damping \n1 , the STT \neffects become very small [28] ). For \n0.034 D\n , \n5 1 12.2 10 mA s  , \n61101.4 AmSM  \n(cobalt) , we have  \n11\n04.8 10 s. Furthermore, for \n24~10v\n3m  and \n~ 293 KT , the dc and ac field \nparameter s \n0 and \n  are of the order of unity for \n31\n00, ~ / ( 2 A0 m 31 ).S H H kT v M  .  \nIII. LINEAR DYNAMIC SUSCE PTIBILITY  \nFor a weak  ac field, \n0 , all nonlinear effects in the response may be  ignored,  so that the \nmagnetization \n()HMt  is simply given by  the linear response  \n \n 0 ( ) Re ( )it\nHM t M e    , (13) \nwhere  \n2\n0\n0S\n0/\nS1 cos ( ) ( )2H stM M M m M t dt\n\n    \n \nis the stationary time - and frequency -independent magnetization, \nst  is the statistical average, and \n1\n1 ( ) 2 ( ) / m   \n is the linear dynamic susceptibility which is independent of the ac field strength. \nThe corresponding plots  of the real and imaginary parts of the normalized linear susceptibility \n( ) /  \n vs. \nN  are shown in Fig. 2 , where \n(0)  is the static susceptibility . Just as with the \nzero STT case, analysis and subsequent interpretation of the linear response radically simplifies at \nlow frequencies because the low-frequency behavior of \n( ) ( ) ( ) i         can then be \naccurately described by a single Lorentzian, viz.,  \n \n( ) 1\n1i\n   . (14) 9 \n\n5 4 3 2 1\n'()/(a)\n = 0\n5\n54\n3\n21\nN43\n2\n''()/1\n\n554 3\n21'()/\n1: J = 6     \n2: J = 0\n3: J = 3   \n4: J = 6\n5: J = 12 =  /4\n5\n543\n21\nN432\n''()/\n1(b) \nFIG. 2. (Color on line) Real and imaginary parts of the normalized linear susceptibility \n( ) /    vs. \nthe normalized frequency \nN  for various spin-polarized current parameter s J = 6, 0, 3, 6, 12 \nand for various orientations of the appl ied fields \n0  (a) and \n/4  (b) with  the anisotropy \nparameter  \n20  and the dc field parameter \n02 . Solid lines: matrix continued fraction solution. \nAsterisks: Approximate Eq. (14) with the reversal time \n1\n1  calculated using the inde pendent \nmethod of Ref. [28].  \nIn Eq. (14), \n is the longest (overbarrier) relaxation time  without the ac external field, and \n  is a \nparameter accounting for the mid - and high -frequency relaxation processes. Now \n  is related to the \nfrequency \nmax  of the low -frequency peak in the loss spectrum \nIm[ ( )] , where it attains a  \nmaximum, and the half -width \n  of the spectrum of the real part of the susceptibility \nRe[ ( )]  \nvia \n \n11\nmax    . (15) \nSince \n  is the magnetization reversal time (effectively the inverse escape rate) , it can be associated \nwith the inverse of the smallest nonvanishing  eigenvalue \n1  of the Fokker -Planck operator as \ncomprehensively described in Ref. [28] . Comparison of \n  as extracted from the spectra \n()  via \nEq. (14) with \n1\n1  calculated independently via \n1  of the Fokker -Planck operator [28] shows \nthat both methods yield identical results.  Also  varying of the material and geometrical model \nparameters may alter the reversal time  \n by orders of magnitude concurring with experimental \nresults [11].  The dependence of \n  on the model parameters (damping  \n, spin-polarized current \nparameter J, the external field strength and orientation, etc.) has been given in Ref. 28. \nThe main features of the normalized linear susceptibility plots are as follows. For J of \nintermediate magnitudes, the overall picture is more or less similar to that for J = 0, i.e., we have \nthe usual low -frequency overbarrier (interwell) relaxation, mid -frequency intrawell relaxation, and 10 high-frequency ferromagnetic resonance (FMR)  beha vior in a biaxial potential . Thus , we have, in \ngeneral, three dispersion regions  in \n Re ( ) /    and three corresponding absorption bands in the \nmagnetic loss spectrum \n ( Im )/    (see Fig. 2). The broad low -frequency peak in \n ( Im )/  \n corresponds to slow reversal of the magnetization vector over the potential barriers  \nand is accurately described by the approximate Eq . (14). The most pronounced STT effect is that \nthe decrease of J from large positive values initially  shifts the low -frequency relaxation peak to \nlower  frequencies until the peak frequency \nmax  reaches a minimum at some intermediate value of \nJ above which the peak is shifted to higher  frequencies (only the shift to higher frequencies is shown \nin Fig. 2) . This minimum frequency peak corresponds to the particular situation , where the STT has \nannul led the effect of the dc bias field so that the effective potential has equal well depths. This \ncorresponds to the maximum relaxation time at a definite value of \nmaxJ , which  has been depicted \ngraphically in Fig.  10 of Ref. [2 8]. For h igh positive or negative J, the magnitude  of the  low-\nfrequency peak in \n ( Im )/    decreases until it merges with the mid -frequency peak, signifying \nthat the overbarrier relaxation process has been completely extinguished  due to the action of STT. \nThus , high magnitude spin-polarized current  seems to have virtually  the same effect on the \nmagnetization reversal as that of a strong dc bias field in single domain ferromagnetic particles  at \nzero STT  [45,46] (see also [27], Chap . 9). Here at a certain cri tical value of that field [ 45,46] which \nis much less than the nucleation field the integral relaxation time (area under the curve of the \nmagnetization decay) diverges exponentially from the overbarrier  relaxation time due to the \ndepletion of the population of the shallowest well of the potential  by the dc field . This event is also \nsignified by the virtual disappearance of the low -frequency peak in the magnetic loss spectrum. \nThus, all that remains are t he dynamical processes within the wells. The explanation appears to be \nthat the high positive or negative J reduc es the effective potential barrier  so that the overbarrier time \ndecreases.  Regarding the second peak at intermediate frequencies, this is due t o fast near -degenerate \nexponential decays in the wells of the effective potential and comprises the usual longitudinal \nintrawell  relaxation. Lastly, we see a third  FMR  peak at the Larmor frequency \npr . The origin of \nthis peak lies in  the magnetization precession in an effective field due to both the anisotropy and \napplied dc  field. Notice that the susceptibility is strongly influenced by the azimuthal angle \n  of \nthe applied ac field. For \n0 , high -frequency resonant harmonic modes are discernible in the \nFMR band generating a comb -like structure with characteristic frequencies \n, 2,3,...prn n  \nreminiscent of that which occurs in inertia -corrected dielectric relaxation of polar mo lecules at THz \nfrequencies under the influence of a mean field potential [27,47]. This comb -like structure virtually \ndisappears, however, for \n0 . The STT has no effect on the  mid-frequency and FMR regions \nwith any apparent changes being purely an arti fact of the normalization .  11 IV. NONLINEAR RESPON SE \nIn strong applied ac fields,  > 1, pronounced frequency -dependent nonlinear effects occur \n(see Figs. 3 and 4 illustrating the dependence of the nonlinear response on the ac field stren gth \nparameter \n ). In contrast to the linear response,  the stationary time-independent  but now frequency -\ndependent  magnetization \n0\n0 S 1( ) ( )M M m  and the nonlinear dynamic susceptibility  \n1\n1 ( ) 2 ( ) / m   \n as well as all other higher harmonics \n1()km  with k > 1 now strongly depend on \nthe magnitude \n  of the ac field . Moreover , for given \n , all \n1()km  also markedly  depend  on the \nazimuthal angle \n , dc bias field \n0 , anisotropy parameters \n  and \n , damping , and spin-\npolarized current parameter J, while the time -independent component of the magnetization \n0M  \nalters profoundly  leading to new non linear effects. In particular , that component  in typical nonlinear \nfashion  becomes dependent on both the amplitude and frequency of the ac field  (see Fig. 3). Such \nbehavio r is in sharp  contrast to that of nanomagnets in an ac field omitting  STT, where  one must \nalso apply a dc bias in combination with a strong  ac field in order to observe the frequency \ndependence of the dc response . This effect being  due to entanglement  of the  nonlinear  ac and dc \nresponses  [48]. However , with STT included the ac field amplitude and frequen cy dependence of \n0M\n always exists even for zero dc bias field , i.e., \n00 . Hence , the spin -polarized current seems \nto have the same effect on \n0()M  as that of a dc bias field at zero STT . Furthermore  the dc response \nin Fig.  3 is not an odd function  of J due to the nature of the non -conservative potential  \n.  \n  \n\n123\n41\n4J\n3 21: = 0.01\n2:  = 1\n3: = 2\n4:  = 31 = 0.01\n = 5\n0 = 0\n =0M / MS a\n \n  \n\n = 23\n1\nJ\n321:0.1\n2:10\n3:10001M / MSb\n \nFIG. 3. (Color on line) (a) Time -independent (dc) component of the magnetization \n0/S MM  vs. the \nspin-polarized current parameter J for (a) various ac field amplitudes \n  = 0.01, 1, 2, and 3 (\n0.01  \nrepresents linear response) and \n1N  and for (b) various frequencies \n110 ,10,N  and \n310  at \n2\n at \n5 , \n0 , and \n00 . Solid lines: the matrix continued fraction solution . 12 \n \n\n\nN'') =  /4 0= 2   J = 3     = 0.01  \n(a)\n\n')\n\npr  pr  pr / 2 3\n44\n412\nN'') =  /4 h0= 0.05   J = 3    = 0.01  \n32\n4 ')1(b) 1:   =0.01\n2:   =2.0\n3:   =4.0\n4:   =6.0 \nFIG. 4. (Color on line) (a) Real and imaginary parts of the nonlinear susceptibility \n Re ( )  and \n Im ( )\n vs. \nN  for various ac field amplitudes \n  and J = 3, \n/4  \n20 , and \n02 . \nSolid and d ashed lines: linear and nonlinear response , respectively, using the matrix continued \nfraction solution. (b) The high -frequency parts of the spectra alone.  \n0.000.05\n55 =  /4 0= 2     =4\n = 0.01   = 20\n'')1: J = 6\n2: J = 0\n3: J = 3\n4: J =6\n5: J =12\nN'()\n432 1432\n1\n \nFIG. 5. (Color on line) Real and imaginary parts of the nonlinear susceptibility \n Re ( )  and \n Im ( )\n vs \nN  for various spin-polarized current parameter J = 6, 0, 3, 6, 12 and \n20\n, \n/4 , \n02 , and \n4 . Solid lines: the matrix continued fraction solution . \nNow  in strong ac fields, it appears that the low -frequency band of \n Im ( )  deviates \nsubstantially from the Lorentzian shape so that it can no longer be approximated by the single \nLorentzian Eq. (14). Nevertheless, Eq. (15) may still be used in order to estimate an effective \nmagnetization reversal time \n  as \n1 . Furthermore , as the ac field strength \n  increases, the \nmagnitude of the low -frequency peak in \n Im ( )  is enhanced (Fig. 4)  and also with  increas ing \n\n the overbarrier peak on initially shift ing to lower frequencies, attains  a minimum frequency, \nthereafter  shifting to higher frequencies. Omitting  STT, this minimum frequency peak will occur  at 13 \n0\n. However, this is not true when STT is included as the STT acts in combination with the \napplied field (Fig. 4). We remark that the reversal time \n  may also be evaluated from the spectra of \nboth the dc component \n0()M  and the higher -order harmonics \n1()km  with k >1 because the low -\nfrequency parts of these spectra are themselves dominated by overbarrier relaxation processes with \nthe characteristic time \n . Now  as seen i n Fig. 4 (b), again with increasing , the magnitude of the \nmain FMR peak at the precession frequency \npr  decreases and also broadens showing pronounced \nsaturation effects. Moreover , a new high -frequency dispersion of resonant charac ter near the \nfrequency \n~ / 2pr  due to parametric resonance  appears just as  that commonly occurring in \nnonlinear oscillators driven by an ac external force. Nevertheless , the high -frequency (\npr ) \nbehavior  of the spectrum  remains virtually unchanged (see Fig. 4). Parametric excitations of a \ncurrent -biased nanomagnet by a microwave magnetic field were observed recently by Urazhdin et \nal. [34] amply demonstrating that this phenomenon can be used to determine dynamical properties \nof nanomagnets.   \nThe nonlinear susceptibility for various J (Fig. 5) exhibits many of the same characteristics \nas the corresponding linear susceptibility (Fig. 2), i.e., three dispersion regions  in \n Re ( ) /    and \nthree corresponding absorption bands in the magnetic loss spectrum \n ( Im )/    for J of \nintermediate magnitude, merging of the overbarrier peak with the mid -frequency peak for high \npositive or negative J, and virtually no STT effect in the mid -frequency and FMR regions. For \n0J  \nthe magnitude of the low-frequency overbarrier peak decreases and the peak  is shifted to higher \nfrequencies, while for \n0J  the magnitude of this peak increases and the peak  also shifts to higher \nfrequencies.  \nV. D YNAMIC MAGNETIC HYSTERESIS  \nFor a weak  ac field, \n0 , the DMH loops [\n( ) ( ) /HS m t M t M  vs. reduced ac field \n( ) cosh t t\n] are ellipses with normalized area \nnA  given by Eq. (12); the behavior of \n1\n1 ~ Im(~ )nAm\n being similar [cf. Eq. (12)] to that of the magnetic loss \n()  (see Figs. 2,4,  \nand 5). Now f or moderate  ac fields, \n1 , the DMH loops still have an ellipsoidal shape implying  \nthat only a few harmonics actually contribute to the nonlinear response. However , in strong  ac fields, \n1\n, the loop shape alters substantially (see Figs. 6 -9). In Fig. 6 , the DMH loops are plotted  for \nvarious values of \nJ  and ac amplitude \n  showing that both the ir shape s and their areas alter as these \nparameters vary. The pronounced frequency dependence of the DMH is highlighted in Fig s. 8 and \n9 for \n0  and \n/4 , respectively, which also illustrates their azimuthal angle dependence . \nIn the  low frequency band  (Figs. 8a-8c and 9a -9c), the negative and positive J shifts the DMH loops \nto the left and right respectively. Moreover , at low frequencies, the field changes are quasi -14 adiabatic , so that the magnetization reverse s due to the cooperative  shuttling action of thermal \nagitation, STT , and ac field. In contrast a t high frequencies  (see Figs. 8e, 8f, 9e  and 9f), the origin \nof the  DMH lies in  the resonant  dispersion and absorption in the FMR band . Here , the phase \ndifference  \n between \n()HMt  and \n( ),Ht  governing  loop orientation , may undergo a pronounced \nvariation in the very high frequency FMR band  as is typical of a resonant process . In particular, the \nphase difference  may exceed \n/2  (see, e.g., [ 49]). Obviously, this large resonant effect does not \nexist at low and intermediate frequencies, where \n  is always less than \n/2 . At FMR frequencies, \nDMH occurs  due to the resonant beha vior of the nonlinear response ( see Fig. 8f) and under such \nconditions  the switching  may be termed “resonant”, leading  naturally  to the concept of resonant \nswitching of the magnetization . Since the resonant DMH occurs at very high (GHz) frequencies, the \nmagnetization switching is , therefore , for the most part  gover ned by the frequency of the external \ndriving field, or equivalently, the rate of change of the amplitude of the latter. Hence , the \nmagnetization may be advantageously switched  in this situation , because the field needed  to reverse \nit is then much smaller t han the quasi -static coercive force  [49]. \nBy plotting the normalized area \nnA  vs. the spin-polarized current (Fig. 10), \nnA can \ninvariably be represented as a bell curve with the height, width, and center of the peak determined \nby the various parameters. This is similar to a plot of \nnA  vs. the dc bias field strength \n0  except tha t \nthe latter will always have the center of the peak along \n00 . In Fig. 10 a, on increasing the ac field \nstrength \n , \nnA also increases and the range of J, for which  a significant DMH lo op area  exists, \nbroadens. In strong  ac fields, \n1 , the normalized area alters substantially (see Fig . 10a). \nNevertheless , \nnA is still determined by \n1\n1 Im( )m  [cf. Eq. (12)]. Thus  \nnA strongly depends on the \nfrequency \n , the angles \n  and \nP, ac and dc bias field amplitudes \n  and \n0 as well as the \nanisotropy parameters \n  and \n, damping , and the spin -polarized current p arameter \nJ . In Fig. \n10b, on increasing the driving frequency,  the normalized area initially increases, reaches a \nmaximum, and then decreases.  \n \n \n \n 15 \n  \n1: J =1\n2: J = 0\n3: J = 1\n(a)\n2h(t)\n321m(t) \n  \n1\n = 0.01\n = 5\n0 = 0\n =0\n(b)\n1 3 2h(t)3 21m(t)\n \nFIG. 6. (Color on line) DMH loops for various spin-polarized current parameter J = 1, 0, 1  and (a)\n2\n and (b) \n5  with \n1,N  \n0 , \n0.01 , \n5 , and \n00  (calculated using the matrix \ncontinued fraction solution ). \n  \n  =0(a)\nh(t)32\n1: J = 1\n2: J = 0\n3: J = 111\n = 0.01\n = 5\n = 3\n0 = 0m(t)\n \n  \n (b)\n2 3\nh(t)m(t)\n1: J = 2\n2: J = 0\n3: J = 23 21\n \nFIG. 7. (Color on line) DMH loops for various spin polarized current parameter J = 1, 0, 1 (a) and \nJ = 2, 0, 2 (b) with \n1N , \n0 , \n0.01 , \n00 , \n3 , and \n5 .  \n 16 \n  \n\n0.01\n1: J =1\n2: J = 0\n3: J = 1\nh(t)321m(t)\n = 0.01\n = 5\n = 5\n0 = 0\n =0(a)\n  \n\nb1: J =1\n2: J = 0\n3: J = 1\nh(t)321m(t)\n0.1\n  \n\n\nh(t)3 21: J =1\n2: J = 0\n3: J = 11m(t)\nc\n  \n\n(d)1: J = 1\n2: J = 0\n3: J = 1\nh(t)\n32110m(t)\n  \n\nh(t)\n321: J = 1\n2: J = 0\n3: J = 11 100 m(t)\ne\n  \n\nh(t)\n32\n1: J = 1\n2: J = 0\n3: J = 11m(t)3981\n(f) \nFIG. 8. (Color on line) DMH loops for various spin polarized current parameters J = 1, 0, 1 and \nfrequencies \n210N  (a), \n110  (b), 1 (c), 10 (d), \n210  (e), and \n3981  (f) with \n0 , \n0.01 , \n00,\n \n5 , and \n5 . 17 \n  \n\n(a) = 0.01\n = 5\n = 5\n0 = 0\n = /4h(t)321: J = 1\n2: J = 0\n3: J = 11m(t)0.01\n  \n\nh(t)\n321: J = 1\n2: J = 0\n3: J = 11m(t)\n =/4\n p=00.1\n(b)\n  \n\nh(t)321: J = 1\n2: J = 0\n3: J = 11m(t)\n = /4\n p=01\n(c)\n  \n\nh(t)\n321: J = 1\n2: J = 0\n3: J = 1\n1m(t)10\n(c)\n  \n\nh(t)\n321: J = 1\n2: J = 0\n3: J = 1\n1m(t)100\n(e)\n  \n\n(f)h(t)\n32\n1: J = 1\n2: J = 0\n3: J = 11m(t)3981 \n \nFIG. 9. (Color on line) DMH loops for various spin-polarized current parameter s J = 1, 0, 1 and \nfrequencies  \n210N  (a), \n110  (b), 1 (c), 10 (d), \n210  (e), and 3981  (f) with \n/4 , \n0.01 , \n00,\n \n5 , and \n5  (calculated using the matrix continued fraction solution ). 18 \n    0.10.20.3\n14\nJ3 21: = 0.01\n2: = 1\n3: = 2\n4: = 3\n1An a \n    0.20.4\n = 2b\n4\n5 J3\n21:0.1\n2:1\n3: 10\n4:100\n5:1000\n1 = 0.01\n = 5\n = 20\n0 = 0An\n \nFIG. 10. (Color on line) Normalized area of the DMH loop \nnA , Eq. (12), vs. spin-polarized current \nparameter J (a) for various ac field amplitudes \n  and \n1N  and (b) for v arious frequencies \nN  \nand \n2  with \n0 , \n0.01 , \n00,  and \n5  (calculated using the matrix continued fraction \nsolution ). \n \nVI. CONCLUSIONS  \n We have treated STT effect s on the ac stationary forced response  of nanoscale ferromagnets \ndriven by  an ac magnetic field of arbitrary strength using a nonperturbative approach originally \ndevelo ped [27] for nanomagnet s omitting STT . Our method , based on the solution of the \ndifferential -recurrence relation for the infinite hierarchy of statistical moments generated by either \nthe Langevin or Fokker -Planck equations as augmented by STT  terms , indicates  that STT \nprofoundly  alters the nonlinear response of a nanomagnet  leading to new effects. Furthermore , the \nstatistical  moment  approach  holds  for the most comprehensive  formulation of the generic nanopillar \nmodel (Fig. 1), i.e., for arbitrary directions  of the dc and ac external fields allo wing us to treat  STT \neffects on frequency -dependent characteristics under conditions  which are  otherwise inaccessible. \nClearly,  at low damping, the stationary response  to an ac driving field  is very sensitive to both the \nintensity of the spin -polarized cur rent and the frequency and amplitude of that field owing to the \nintrinsic  coupling between the magnetization precession and its thermally activated reversal. \nFurthermore, our calculations, since they are valid for ac fields of arbitrary  strength and orientation, \nquantify the role played by STT in nonlinear phenomena in nanoscale ferromagnets such as \nnonlinear stochastic resonance and dynamic  magnetic  hysteresis, nonlinear ac field effects on the \nswitching field curves, etc., where pertur bation theory is no longer valid. In addition , the moment  \nmethod  yields  the response for all frequencies of interest  including very high frequencies covering \nthe ferromagnetic resonance (GHz) range exemplifying  various nonlinear phenomena such as 19 parametri c resonance and higher harmonic generation  (which we hope will stimulate new \nexperiments) . Hence , the high -frequency linear and nonlinear FMR spectra (see Figs. 2 and 4) may \nbe suitable for the purpose of  explaining  the line shape of STT nano -oscillators d riven by ac external \nmagnetic fields and currents . Likewise , the DMH loops and their area (yielding the  Joule heating \nduring the switching process ) as well as the calculations  of the effective magnetization reversal time \nvia the low -frequency band of the magnetic loss spectra may be useful  for the  prediction, modeling, \nand interpretation of switching processes in recording techniques. Furthermore , the DMH arising \nfrom a high-frequency  periodic signal may be exactly evaluated permit ting quantitative analysis of \nultrafast switching  of the magnetization.  In particular , accurate solutions in the manner outlined of \nthe hierarchy of the statistical moment equations  for a generic model are essential for the future \ndevelopment of both escape rate theory and stochastic dynamics simulations of the magnetization \nreversal process  in STT systems  just as they were in single domain particles . For the limit of zero \nSTT, our results concur with established solutions for nanomagnets with biaxial anisotropy [41]  \nwhile, for nonzero STT, they constitute  rigorous benchmark solutions with which calculations of \nnonlinear response characteristics vi a any other approach  must comply . Finally, t he statistical \nmoment method may be similarly generalized  to the forced response of a nanoscale ferromagnet \ndriven by an alternating current.  \nACKNOWLEDGMENTS  \nWe would like to thank FP7 ‐PEOPLE‐Marie Curie Actions ‐ International Research Staff Exchange \nScheme (Project No. 295196 DMH) for financial support. Moreove r, W. T. Coffey thanks \nAmbassade de France in Ireland  for research visits to Perpignan. D. Byrne, acknowledges the \nSimSci Structured Ph.D. Program at University College Dublin for financial support. This research \nis supported by the Programme for Research In Third Level Institutions (PRTLI) Cycle 5 and co -\nfunded by the European Regional Development Fund . We also thank P. M. Déjardin for helpful \nconversations.  \nAPPENDIX A : EXPLICIT FORM OF THE COEFFICIENTS  \n;()nm n met  \nBy apply ing the general approach [ 27,28,40] for the derivation of differential -recurrence \nrelations from the magnetic Langevin equation (1) as specialized to the potential s Eqs. (3) and (5), \nwe have the  25 term differential -recurrence Eq. (6) for the statistical moments \n( ) ( )lm lmc t Y t , viz.,  \n \nN 2 2 2 1 2 2 1 2 2\n1 2 1 1 1 1 1 1 2\n21( ) ( ) ( ) ( ) ( ) ( )\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n( ) ( ) ( )nm nm n m nm n m nm n m nm n m nm n m\nnm n m nm n m nm n m nm n m nm n m\nnm nm nm nmdc t v c t v c t v c t v c t v c tdt\nw c t w t c t w t c t w t c t w c t\nx c t x t c t x   \n        \n   \n        \n \n    \n    \n  12\n1 2 1 1 1 1 1 1 2\n2 2 2 1 2 2 1 2 2( ) ( ) ( ) ( ) ( )\n( ) ( ) ( ) ( ) ( ) ( ) ( )\n( ) ( ) ( ) ( ) ( ).nm nm nm nm nm nm\nnm n m nm n m nm n m nm n m nm n m\nnm n m nm n m nm n m nm n m nm n mt c t x t c t x c t\ny c y t c t y t c t y t c t y c t\nz c t z c t z c t z c t z c t \n\n   \n        \n   \n        \n    \n      (16) 20 Here the coefficients \n()nmxt , \n()nmyt , etc. corresponding to the matrix elements \n;()nm n met  in Eq. (6) \nhave the same form as those in Eq. (C1) of Ref. [2 8] save that they are now time-dependent . To \nkeep in st ep with the notation of Ref. [28 ], we define reduced fields  \n \n00 / (2 )S h H M D\n  and \n/ (2 )S h H M D\n  (17) \nThus  in our particular case of a single ac forcing term, we have  \n \n00 ( ) 2 cos ( )i t i th t h h t h h e e      (18) \nEquation  (18) then implies  that for a constant fie ld superimposed on a periodic ac  one, the time-\ndependent coefficients \n()nmxt , etc. in Eq. (16) may be written as  the sum of a dc  term and one \noscillating  at the fundamental frequency, viz. ,  \n \n01( ) ( )i t i t\nnm nm nmw t w w e e   , \n \n01( ) ( )i t i t\nnm nm nmx t x x e e   , \n \n01( ) ( )i t i t\nnm nm nmy t y y e e   , \netc. The various coefficients are then given by  \n 0* 03\n10\n2\n*2 * *\n10 11 1 1( 1)( , )23\n2 ( 1) 3 1( , ) ( , ) ( , ) ,2 1 2 3 2 3nm P P P\nPP\nP P P P P Pmh nnx i i mb JY\nc b J n n mY Y Ynn \n         \n          \n \n \n1 3,nmmhxi\n  \n \n\n0 0\n12\n* * *\n1 1 10 1 112\n2 (1 2 )( , ) ( , ) ( , ) ,6 3 2 1 2 3nm\nPP\nP P P P P P Pihx n m n m i\nc b J mi b JY Y Ynn\n      \n   \n    \n  \n \n1\n12 1,2nmihx n m n m i  \n   \n \n0 *2\n113 1 2 1 4( , ) ,4(2 1)(2 3) 3PP\nnm P Pn m n m n m n m c b JxYnn  \n        \n  \n \n\n 22\n0*\n0 3 10\n*2 * *\n10 11 1 1( 1) 1( , )2 1 2 3 2 3\n4( , ) ( , ) ( , ) ,3P\nnm P P\nPP\nP P P P P Pb Jn n m imy n h Ynn\nim c b JY Y Y    \n             \n   \n \n22\n1\n3( 1)\n2 1 2 3nmnmy n hnn  21  \n\n\n0*\n11\n**\n0 1 2 10 1 112( , )1 2 3 2 6\n2 ( 2 )( , ) ( , ) ,23P\nnm P P\nPP\nP P P Pn m n m b JnyYnn\ni c b J n m nh i Y Y\n      \n\n       \n \n  \n \n\n1\n1212\n2 1 2 3 2nmn m n m ny h inn      \n  \n \n\n0 *2\n111 2 3( , ) ,4 3 1 2 3 2PP\nnm P Pn m n m n m n m c b Jy i Ynn\n       \n  \n \n 22\n0\n03 2\n* *2 * *\n10 10 11 1 11( 1)4 1 2\n( 1) 2( , ) ( , ) ( , ) ( , ) ,33nm\nP P P\nP P P P P P P Pn m imw n hn\nb J n im c b JY Y Y Y  \n               \n       \n \n22\n1\n3 2( 1)41nmnmw n hn  \n \n\n0*\n11 2\n**\n0 1 2 10 1 11 ( 1)( , )4 1 6\n2 ( 1 2 ) 1( , ) ( , ) ,23P\nnm P P\nPP\nP P P Pn m n m b J nwYn\ni c b J n m nh i Y Y\n      \n\n  \n    \n  \n \n1\n12 21 1\n2 4 1nmn m n m nw h in    \n  \n \n0 *2\n11 22 1 1 4( , ) ,4 3 4 1PP\nnm P Pn m n m n m n m c b J iwYn\n      \n  \n \n\n 2 2 2 2\n2\n*2 * *\n10 11 1 1( 1) ( 2)\n2 3 2 1 2 5\n2 1( , ) ( , ) ( , ) ,23nm\nPP\nP P P P P Pn m n m nzn n n\nc b JY Y Y                \n     \n \n\n22\n**\n10 1 1( 1) 2 ( 3) 22( , ) ( , ) ,3 2 3 2 1 2 5PP\nnm P P P Pn m n m n m c b J nz Y Yn n n   \n         \n \n\n*2\n111 2 (3 )(4 )( , ) ,4 3 2 3 2 1 2 5PP\nnm P Pn m n m n m n m c b J nzYn n n\n              \n \n\n 2 2 2 2\n*2 * *\n10 11 1 1( 1) ( ) 11\n2 1 2 1 2 3 2\n2( , ) ( , ) ( , ) ,3nm\nPP\nP P P P P Pn m n m nvn n n\nc b JY Y Y\n               \n   22  \n\n22\n**\n10 1 12 ( 1)( ) 22 1( , ) ( , )3 2 1 2 1 2 3PP\nnm P P P Pn m n m n m c b J nv Y Yn n n   \n     \n , \n \n\n*2\n113 2 1 ( ) 1( , )4 3 2 1 2 1 2 3PP\nnm P Pn m n m n m n m c b J nvYn n n\n       \n . \nHere  \n \n3/2\n3 3/24\n3(1 ) 16PPbPP , \n3\n3 3/2(1 )\n3(1 ) 16PPcPP  \nare model -dependent coefficients determined by the spin polarization factor \n(0 1)PP , the \ndimensionless spin -polarized current parameter J is defined as  \n \n2\n0S e\npv M JJkTJ  (19) \nwhere \neJ  is the current density, taken as positive when the electrons flow from the free into the \nfixed layer, and \n2\n0  /pSJ M e d\n  (\ne is the electronic charge, \n  is Planck’s reduced constant, and \nd\n is the thickness of the free layer). A typical value of \n pJ  for a 3 nanometer thick layer of cobalt \nis \n91.1 10pJ  A/cm2 while the largest current densit y reported in experiments is \n7810 10eJ A \ncm2 (cf. Ref. [20], p. 237). However, for weak  damping \n1 , the ratio \n/PPc b J  appearing in \nthe coefficients \n0\nnmx , etc. listed above may be of the same order of magnitude as the anisotropy \nparameters \n  and \n  so explain ing the strong STT effects  on the magnetization dynamics . In \ncontrast , for high damping \n1 , the STT effects become very small [28].  \nAPPENDIX B  : CALCULATION OF THE STATISTICAL MOMENTS VIA MATRIX \nCONTINUED FRACTIONS   \nThe differential -recurrence relations Eq. (16) can be solved by matrix continued fraction \nmethods just as in the case of  zero STT term [41] with some modifications of the algorithm. By \nintroducing vec tors \n()k\nnc  \n  0,1,2,...n  with elements composed of the Fourier amplitudes \n()k\nnmc  \nin Eq. (10), viz.,  \n \n22\n22 0\n0 00\n2 1 2 1\n2 12 1()\n()( ) , ( ) ,\n()\n()k\nnn\nk\nnn k\nnk\nnn\nk\nnnc\ncc\nc\nc\n\n\n\n  \n\n\n\n\n \n\n\n\ncc\n   \nwe have from Eq. (16) a matrix  differential -recurrence relation for the \n()k\nnc , viz.,  23  \n  1 N 1\n1 1 1 1\n11\n11\n11( ) ( ) ( )\n( ) ( ) ( ) ( )\n( ) ( ) 0,k k k\nn n n n n n\nk k k k\nn n n n n n\nkk\nn n nik     \n   \n\n\n    \n\n  \n  \n         \n  q c q I c q c\np c c p c c\np c c  (20) \nwhere the supermatri xes \n,,n n nq q q , \n,,n n np p p , \n,,n n nr r r  are (cf. Eq. (C3) of [ 28]) \n \n0 0 0\n2 2 2 2 2\n0 0 0\n2 1 2 1 2 1 2 1 2 1,,n n n n n\nn n n\nn n n n n\n         \n                 X W Z Y V 0q q q\nY X 0 Z W V , \n \n11 1\n22 2\n1 11\n21 2 1 2 1,,nn n\nn n n\nn nn\n          00 XW 0Yp p p\nW0 00 YX . \nHere  the submatrices  \nnV, \nnZ , \ni\nnW , \ni\nnX, and \ni\nnY  (i = 0, 1)  have virtually the same  form and \nthe same nonzero elements as  the submatrices  \nnV, \nnW , \nnX , \nnY , and \nnZ  from Ref. [28] defined \nin terms of the time-independent  elements \nnmv , \nnmw , etc. The only difference s which occur  are \nhighlighted by the superscript i = 0, 1  in the submatrices  \ni\nnW , \ni\nnX , and \ni\nnY , indicating that the \nelements  \nnmw , \nnmx , \nnmy , etc. appearing in these submatrices  must now be replaced by \ni\nnmw , \ni\nnmx , \ni\nnmy\n, etc., respectively .  \nNext, we introduce  super column vectors via  \n \n0\n0 00\n0\n0\n0\n\n\n\n\n\n\n\nC c\n ,   \n2\n1\n0\n1\n2()\n()\n()\n()\n()n\nn\nn n\nn\nn\n\n\n\n\n\n\n\n\n\n\n\n\nc\nc\nC c\nc\nc\n    n = 1,2,3,… , (21) \nthen we have from Eq.  (20) the tridiagonal matrix recurrence relations  \n \n1 1 1 2 1 0  Q C Q C Q C , (22) \n \n11 0n n n n n n\n   Q C Q C Q C . (23) \nHere  n = 1, 2, 3, … and  the tridiagonal supermatrices \nnQ  and \nnQ , and column vectors \n10QC  and \nnC\n  are defined as  \n \n11 n l m n lm n l m nlm     \n  Q p q p , \n \n  1 N 1 n l m n lm n l m nlmim         Q p q I p , 24  \n1\n10 1\n11\n4\n \n\n\n\n\n\n\n\n\n\n\n\n0\n0\np\nQC q\np\n0\n0\n ,   \n1\n1\n11\n1\n10\n1\n110\n0\n0\n0\n0\nw\nw\nw\n\n\n\n\n\n\n\n\n\n\n\np ,   \n0\n22\n0\n21\n0\n20\n0\n21\n1 0\n22\n0\n11\n0\n10\n0\n11v\nv\nv\nv\nv\nw\nw\nw\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nq , \nThe exact solution of Eqs. (22) and (23) is then rendered by the matrix continued fraction  \n \n1 1 1 0C S Q C , (24) \nwhere \n1S  is defined by the recurrence equation  \n \n1\n11 n n n n n\n S Q Q S Q . \nThe vector \n1C  in Eq. (24) contains all the Fourier amplitudes needed for  both the linear and  \nnonlinear ac stationary response s. These results are valid for arbitrary field strength meaning that  \ncalculating  the \n()k\nnmc  and thus the forced response may be reduced to computing matrix continued \nfractions.  When  the spin-polarized current parameter J = 0, i.e., omitting  STT, the above solution \nagrees in all respects with that given  in Ref. [41] for the ac response  of a nanomagnet with biaxial \nanisotropy subjected to superimposed  external ac and dc fields of arbitrary strength and orientation.  \n The solution Eq. (24) is easily computed on a standard PC  as the matrix continued fraction \ninvolved converge rapidly  in most cases . In our calculation, the infinite matrix continued fraction \n1S\n was approximated by (i ) a matrix -continued fraction of finite order (by putting \n1nS0  at some \nn = N) and by (ii) a finite dimension of the vector \n1C  (by choosing the number of harmonics k = K) \nin such way that a further increase of N and K did not change the significant digits  in calculated \nobservables . The values of N and K depend on the model parameters  values and must be chosen \ntaking into account the desired degree of accuracy of the calculation. The calculations have shown \nthat t he results are stable for the ranges of parameters \n0 25,  \n00 25 , \n0 10 , and \n0.005\n yielding  an accuracy of not less than six significant digits in most cases. 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Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland\n2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia\n3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia\n(Dated: February 22, 2022)\nWe show that the Higgs mode of superconductors can couple with spin dynamics in the presence\nof a static spin-splitting \feld and Rashba spin-orbit coupling. The Higgs-spin coupling dramatically\nmodi\fes the spin susceptibility near the superconducting critical temperature and consequently\nenhances the spin pumping e\u000bect in a ferromagnetic insulator/superconductor bilayer system. We\nshow that this e\u000bect can be detected by measuring the magnon transmission rate and the magnon-\ninduced voltage generated by the inverse spin Hall e\u000bect.\nSuperconductors (SC) with broken U(1) symmetry\nhost two kinds of collective modes associated with the\norder parameter \ructuations: the phase mode and the\namplitude mode. Coupled to a dynamical gauge \feld,\nthe phase mode is lifted up to the plasma frequency [1]\ndue to the Anderson{Higgs mechanism [2, 3]. The other\ncollective mode in SC is the amplitude mode [4, 5] with\nan energy gap of 2\u0001, called the Higgs mode by anal-\nogy with the Higgs boson [3] in particle physics. It was\ncommonly believed that unlike the phase mode the Higgs\nmode usually does not couple linearly to any experimen-\ntal probe. That is why in earlier experiments, the Higgs\nmode was only observed in charge-density-wave (CDW)\ncoexisting systems [6{11]. With the advance of terahertz\nspectroscopy technique [12] it became possible to inves-\ntigate the Higgs mode through the nonlinear light{Higgs\ncoupling [13{17]. In these experiments, the perturba-\ntion of the order parameter is proportional to the square\nof the external electromagnetic \feld \u000e\u0001/E2, so very\nstrong laser pulses are required.\nRecently, it has been shown that in the presence of a\nsupercurrent the Higgs resonance can actually contribute\nto the total admittance Y\ndue to the linear coupling of\nthe Higgs mode and the external electromagnetic \feld\n[18{22]. This can be understood from a symmetry ar-\ngument. Suppose the external electric \feld is linearly\npolarized in the xdirectionE= ^xExei\nt. The linear\ncoupling of the Higgs mode and the external \feld is rep-\nresented by the susceptibility \u001f\u0001E=\u0000@2S\n@\u0001@Eobtained\nfrom the action Sdescribing the electron system con-\ntaining the pair potential \feld \u0001. Without a supercur-\nrent, the system preserves the inversion symmetry ( ^I)\nand the mirror symmetry in the xdirection ( ^Mx). On\nthe other hand \u001f\u0001Eis odd under both these operations\nbecauseEchanges sign under ^Iand ^Mxwhereas \u0001 re-\nmains the same. Therefore \u001f\u0001Ehas to vanish. In the\npresence of a supercurrent, the inversion symmetry and\nthe mirror symmetry are both broken and there is no re-\nstriction for \u001f\u0001Exfrom these symmetries, so \u001f\u0001Ecan be\nFIG. 1. System under consideration. A superconductor thin\n\flm is placed on the top of a FI with in-plane magnetization.\nThe SC and FI are coupled via spin exchange interaction. The\nmagnon in FI can be injected into SC in a process known as\nthe spin pumping e\u000bect. For magnon frequency \n = 2\u0001 0the\nSC Higgs mode greatly increases the spin pumping.\nnonzero. This symmetry argument also explains why the\nHiggs mode does not couple with an external \feld in the\ndirection perpendicular to the supercurrent.\nNow a natural question arises: without a supercurrent\ndoes the Higgs mode couple linearly with other exter-\nnal probes, such as spin exchange \felds? As we show in\nthis Letter it does. The above discussion indicates that\nthe decoupling of the Higgs mode is protected by cer-\ntain symmetries. In order to couple the Higgs mode to\nan external \feld one needs to break these symmetries.\nHere we show how it happens in a ferromagnetic insu-\nlator (FI)/superconductor (SC) bilayer system (Fig. 1).\nMagnons with momentum qand frequency \n in the FI\ncan be injected into the SC in a process known as spin\npumping [23{29]. We predict that the Higgs mode in\nthe SC couples linearly with the magnon mode in the\nFI in the presence of Rashba spin-orbit coupling and the\nmagnetic proximity e\u000bect into the SC. In this system\nthe symmetries protecting Higgs-spin decoupling are bro-\nken: in particular, the (spin) rotation symmetry and thearXiv:2108.06202v2  [cond-mat.supr-con]  21 Feb 20222\ntime-reversal symmetry. Near the critical temperature,\nsuperconductivity is suppressed and \u0001 0becomes compa-\nrable with the magnon frequency \n. When the magnon\nfrequency matches the Higgs frequency \n M= 2\u0001 0, the\nHiggs mode is activated and the magnon absorption is\nhugely enhanced which can be detected through the in-\nverse spin Hall e\u000bect (iSHE) [30{32]. This e\u000bect can pos-\nsibly explain the voltage peak observed in the experiment\n[33].\nWe consider a SC/FI bilayer in which the FI and the\nSC are coupled via the exchange interaction as shown in\nFig. 1. For simplicity, we assume that the thickness d\nof the SC \flm is much smaller than the spin relaxation\nlength and the coherence length so that we consider it as\na 2D system. The magnetization of the FI can be written\nasm=m0+m\n, wherem0is the static manetization\npolarized in the zdirection and m\nis the dynamical\ncomponent perpendicular to m0. When magnons (spin\nwaves) are excited in the FI, they can be injected into\nthe SC in a process known as the spin pumping e\u000bect.\nThe DC interface spin current \rowing from the FI into\nthe SC is polarized in the zdirection and given by [34]\nIz=X\n\n;q\u00002JsdIm[~\u001fss(\n;q)]m2\n\n;q; (1)\nwhereJsdis the exchange coupling strength and\nm\nis the Fourier amplitude of m\n. ~\u001fss(\n;q) is\nthe total dynamical spin susceptibility ~ \u001fss(\n;q) =\nS+(\n;q)=h+(\n;q), whereSis the dynamical spin of\nthe SC,his the proximity induced exchange \feld h=\nJsdm=d[35] and for a vector A= (Ax;Ay;Az) the\u0006\ncomponent is de\fned as A\u0006=Ax\u0006iAy. One can see\nthat for a \fxed Jsd, the e\u000eciency of the magnon injection\nis soley determined by ~ \u001fss(\n;q). The spin susceptibility\nof superconductors has been extensively studied [29, 36].\nHowever the previous theories, based on the static mean-\n\feld description, failed to explain the peak of the iSHE\nsignal observed in the spin Seebeck experiment [33]. In\nthis work, we start with the general partition function of\nthe SC,Z=R\nD[\u0016\t;\t;\u0016\u0001;\u0001]e\u0000Sobtained by performing\nthe Hubbard-Stratonovich transformation. The action S\nis given by\nS=\fX\nK;Q\u0016\tK(\u0000i!+\u000fk\u0000h\u0001\u001b) \tK+ \u0001Q\tK+Q\t\u0000K\n+\u0016\u0001\u0000Q\u0016\tK\u0016\t\u0000K\u0000Q+\u0016\u0001\u0000Q\u0001Q\nU;(2)\nHereK= (!;k) andQ= (\n;q) are the four-momenta\nof the electrons and magnons, respectively. != (2n+\n1)\u0019Tand \n = 2n\u0019T are the Matsubara frequencies with\nn2Zand\f= 1=T.\u000fkis the energy dispersion of the\nelectron in the normal state, his the proximity induced\nexchange \feld, and Uis the BCS interaction. In the\nmean-\feld theory, one can ignore the path integral over\u0001 and replace it by its saddle point value \u0001 0which is\ndetermined by the minimization of the action@S\n@\u0001j\u0001=\u0001 0=\n0 after integrating out the fermion \felds.\nTo include the Higgs mode, we go beyond the mean-\n\feld theory and write the order parameter as \u0001 = \u0001 0+\u0011,\nwhere\u0011is the deviation of \u0001 from its saddle point value\n\u00010. Here we only consider the amplitude \ructuation of\n\u0001, so\u0011is real. Expanding the action to the second order\nin\u0011and the strength of the external Zeeman \feld h\u0006\ngivesS=S0\u0000S2with [37]\nS2=\fX\nQ\u0002\u0011(\u0000Q)h\u0000(\u0000Q)\u0003\u0014\n\u0000\u001f\u00001\n\u0001\u0001\u001f\u0001s\n\u001fs\u0001\u001fss\u0015\u0014\u0011(Q)\nh+(Q)\u0015\n:\n(3)\nHere, all the susceptibilities are functions of Q.S0is the\nmean-\feld action without the external \feld. In usual su-\nperconductors the o\u000b-diagonal susceptibilities \u001f\u0001sand\n\u001fs\u0001vanish as required by the time-reversal symmetry\nand the (spin) rotation symmetry because these oper-\nations change the sign of h+but have no e\u000bect on \u0011\n[38, 39]. In the system under consideration, the proxim-\nity induced static exchange \feld breaks the time-reversal\nsymmetry and RSOC breaks the (spin) rotation symme-\ntry. Thus the pair-spin susceptibility does not have to\nvanish, allowing for a nonzero Higgs{spin coupling.\nThen it is straightforward to calculate the total spin\nsusceptibility ~ \u001fssby integrating out the \u0011\feld\n~\u001fss=\u001fss\u0000\u001fs\u0001\u001f\u0001\u0001\u001f\u0001s: (4)\nThe imaginary part of \u001f\u0001\u0001is sharply peak at the Higgs\nfrequency \n = 2\u0001 dramatically modifying the total spin\nsusceptibility.\nPhenomenological theory . Before we go to the detailed\ncalculations, we use a simple phenomenological theory\nto illustrate the e\u000bect of RSOC. It has been shown that\nRSOC can induce a Dzyaloshinskii-Moriya (DM) interac-\ntion in superconductors described by the DM free energy\n[40]\nFDM=X\niZ\ndrj\u0001j2d\u000b;i\u0001(h\u0002rih); (5)\nwhere both \u0001 = \u0001( r) andh=h(r) are position depen-\ndent.d\u000b;iis the DM vector proportional to the strength\nof spin-orbit coupling \u000b. For RSOC d\u000b/\u000b[\u001bx;\u0000\u001bz],\nwhere\u000bis the spin-orbit coupling strength and \u001bis\nthe Pauli matrix acting on the spin space. To \fnd\nthe pair spin susceptibility we write \u0001 = \u0001 0+\u0011(t),\nh=h0^z+h+(t)(^x+i^y), where ^nis the unit vector\nin thendirection with n=x;y;z , and generalize the\nDM free energy to the time dependent DM action. Here\nwe consider the case where the spin wave is propagating\nin thezdirectionh+(t;r) =P\n\n;qzh+(^x+i^y)ei(\nt\u0000qzz).\nFocusing on the \frst order terms in \u0011(t) andh+(t) and3\nFourier transforming them to momentum and frequency\nspace, the DM action can be written as\nSDM1=\fX\n\n;qziqz\u00010h0h+(\n;qz)\u0011(\n;qz)~d\u000b;z(\n;qz)\n\u0001(i^x\u0000^y);(6)\nwhere ~d\u000b;iis the dynamical DM vector, which has the\nsame \fniteness and spin structure as d\u000b;ifrom symmetry\nanalysis. From the above expression, one can see that\nthe Higgs mode couples linearly with the spin degree of\nfreedom in the presence of RSOC.\nSpin susceptibility . We adopt the quasiclassical ap-\nproximation to systematically evaluate the susceptibili-\nties. In the di\u000busive limit, this system can be described\nby the Usadel equation [18, 36, 41{45]\nFIG. 2. Imaginary part of the pair susceptibility. This can\nbe interpreted as the spectral weight of the Higgs mode. A\nsigni\fcant peak emerges when the driving frequency matches\nthe Higgs frequency \n = 2\u0001 0. The inset shows the height\nof the Higgs peak PHas a function of the inverse of the mo-\nmentum q. Parameters: \u0001 0= 0:8\u0001T0,h0= 0:5\u0001T0with\n\u0001T0\u0011\u00010(T= 0).\n\u0000if\u001c3@t;^gg=D~r\u0010\n^g~r^g\u0011\n\u0000i[H0;^g] +h\nXei(\nt\u0000qzz);^gi\n:\n(7)\nHere ^gis the quasiclassical Green function, D=vF\u001c2=3\nis the di\u000busion constant and \u001cis the disorder scat-\ntering time. H0=\u0000ih0\u001b3+ \u0001 0\u001c1, whereh0is the\nproximity induced e\u000bective static exchange \feld and \u001ci\nis the Pauli matrix acting on the particle-hole space.\n~r= (~rz;~rx) is the covariant derivative de\fned by\n~rz\u0001=rz+i\u000b[\u001bx;\u0001],~rx\u0001=rx\u0000i\u000b[\u001bz;\u0001]. The Usadel\nequation is supplemented by the normalization condition\n^g2= 1. In the quasiclassical approximation the approxi-\nmate PH symmetry of the full Hamiltonian becomes ex-act. In the linear response theory, the external oscillat-\ning \feldXis small and can be treated as a perturbation.\nThus we can write the quasiclassical Green function as\n^g= ^g0ei!(t1\u0000t2)+ ^gXei(!+\n)t1\u0000i!t2\u0000iqzz, where ^g0is the\nstatic Green function and ^ gXis the perturbation of the\nGreen function describing the response to the external\n\feld. Solving the Usadel equation we obtain the quasi-\nclassical Green function, the anomalous Green function\nF=NeTr [\u001c1^g]=4iand the\u001b+component of spin in the\nSChsi=NeTr [\u001b\u0000\u001c3^g]=4i, whereNeis the electron den-\nsity of states at the Fermi surface and Tr is the trace.\nThe susceptibilities can be evaluated as\n^\u001f=\u0014\n\u001f\u00001\n\u0001\u0001\u001f\u0001s\n\u001fs\u0001\u001fss\u0015\n=\"@F\n@\u0011+1\nU@F\n@h+\n@hsi\n@\u0011@hsi\n@h+#\n: (8)\nLet us \frst set X= \u00010\u001c1and consider the pair suscep-\ntibility. We assume the RSOC is weak and treat \u000bas a\nperturbation. At q= 0 and 0th order in \u000b, we have\n\u001f\u0001\u0001(i\n) =\"\nNeT\n2X\n!;\u001b4\u00012+ \n2\ns\u001b(!)(4!2\u0000\n2)#\u00001\n;(9)\nwheres\u001b(!) =p\n(!+i\u001bh)2+ \u00012, with\u001b=\u00061. To get\nthe pair susceptibility as a function of real frequency, we\nneed to perform an analytical continuation [38]. Thus\ni\n is replaced by \n + i0+. One can see that the \u001f\u0001\u0001is\npeaked at the Higgs frequency \n = 2\u0001.\nWe numerically calculate \u001f\u0001\u0001with \fnite momentum\nand show the results in Fig. 2 [38, 46]. One can see that\nthe imaginary part of the inverse of the pair suscepti-\nbility exhibits a sharp peak when the driving frequency\nequals 2\u0001 0. With a \fnite momentum, the Higgs mode is\ndamped in the sense that the peak in the Higgs spectrum\nhas a \fnite height and width.\nFIG. 3. Real part (a) and imaginary part (b) of pair-spin\nsusceptibility. The solid line is the approximate result calcu-\nlated from Eq. (12) and the circles show the numerical solu-\ntion from Eq. (7). Parameters used here are: \n = 0 :8\u0001T0\nfor the blue lines, \n = \u0001 T0for the red lines, h0= 0:5\u0001T0,\nDq2\nz=D\u000b2= 0:01\u0001 T0.\nTo study the response of this system to the external\nexchange \feld we set X=h+\u001b+\u001c3. Again we treat \u000bas\na perturbation and write the Green function as\n^g= ^g0ei!(t1\u0000t2)+ (^gh0+ ^gh\u000b)ei(!+\n)t1\u0000i!t2\u0000iqzz;(10)4\nwhere ^gh0is 0th order in \u000band ^gh\u000bis \frst order in \u000b.\nThe 0th order solution in \u000bis given by [38]\n^gh0= ^gh00\n\u001b+=i[\u001c3\u0000^g\"(1)\u001c3^g#(2)]h\n\u001b+\ns\"(1) +s#(2);(11)\nwhere ^g\"=# =(!\u0006ih0)\u001c3+\u0001\u001c1\ns\"=#ands\"=# =p\n(!\u0006ih0)2+ \u00012. ^gh00is a 2\u00022 matrix in the\nparticle-hole space. Without doing detailed calculations,\none can immediately see that \u001f\u0001shas to vanish without\nRSOC because ^ ghhas no\u001b0component. In this case\nthe external exchange \feld cannot activate the Higgs\nmode. To get a \fnite pair-spin susceptibility we need to\nconsider the \frst order terms in \u000bwhich break the spin\nrotation symmetry. The \frst order solution in \u000byields\n^gh\u000b= diag(^gh\u000b\";^gh\u000b#) with\n^gh\u000b\"=#= 2iD\u000b^g0\"=#\u0002\n^gh00;^g0\"=#\u0003\ns\"=#(!1) +s\"=#(!2): (12)\nFIG. 4. (a) Total spin susceptibility as a function of tem-\nperature with a \fxed frequency. (b) Total spin susceptibility\nas a function of frequency with a \fxed temperature. The\ntwo temperatures have been chosen so that \u0001( T1) = 0:2\u0001T0\nand \u0001(T2) = 0:1\u0001T0. The Higgs peak thus shows up when\n\n = 2\u0001(T). The parameters used here are: h0= 0:5\u0001T0,\nDq2\nz=D\u000b2= 0:01\u0001 T0.\nSince the 0th order term does not contribute to the\npair-spin susceptibility, we have \u001f\u0001s= Tr[\u001c1^gh\u000b]=4ih+.\nWe compare this analytical result with the non-\nperturbative numerical solution of the Usadel equation\nin Fig. 3. It shows that the perturbative approach is ac-\ncurate at high temperatures when D\u000b2\u001c\u00010;T, and\ncaptures the qualitative behavior of \u001f\u0001salso at the low\ntemperatures. Another feature of this pair spin suscepti-\nbility is that at a lower frequency (\n = 0 :8\u0001T0),\u001f\u0001sissuppressed at low temperatures because the spin excita-\ntion is frozen by the pair gap at low temperatures. On\nthe other hand, at higher frequency (\n = \u0001 T0),\u001f\u0001sis\nslightly enhanced at low temperatures.\nWe can also get the bare spin susceptibility from ^ gh0,\n\u001fss= Tr[\u001b\u0000\u001c3^gh0]=4ih+. Then it is straightforward\nto calculate the total spin susceptibility according to\nEq. (4). The results are shown in Fig. 4. The total\nspin susceptibility exhibits a signi\fcant peak near criti-\ncal temperature. This is a signature of the Higgs mode\nwith the frequency \n = 2\u0001 0. The dependence of the to-\ntal susceptibility on the strength of RSOC is studied in\nthe supplementary information [38]. The details depend\nsensitively on the amount of disorder, as in the disordered\ncase increasing RSOC leads to a stronger spin relaxation.\nWe note that even though the pair-spin susceptibility is\nlinear in momentum qz, the magnon momentum need\nnot be large for the detection of the Higgs mode. This is\nbecause the spectral weight of the Higgs mode is propor-\ntional to 1=q2\nzat the Higgs frequency, so that the height\nof the peak in the total spin susceptibility is independent\nof the magnon momentum.\nExperimental detection . We propose that the Higgs\nmode in Rashba superconductors can be detected in the\nspin pumping experiment as shown in Fig. 1. Magnons\nin the FI with momentum qand frequency \n are injected\nfrom one side of FI and propagate in the zdirection to-\nwards the other end. Due to the spin pumping e\u000bect,\npart of the magnons can be absorbed by the SC on top\nof it and converted to quasiparticles. This spin injection\ncauses a spin current Is\rowing in the out-of-plane di-\nrection. In the presence of RSOC, Isis converted into a\ncharge current Ievia the iSHE Ie=\u0012z\nxzIs, where\u0012is the\nspin Hall angle [47]. When the width of the SC is smaller\nthan the charge imbalance length the non-equilibrium\ncharge accumulation cannot be totally relaxed resulting\ninto a \fnite resistance \u001aof the SC. Therefore a voltage\ncan be measured across the SC, given by\nV=\u0012z\nxz\u001aX\n\n;q\u00002JsdIm[~\u001fss(\n;q)]m2\n\n: (13)\nThus by tuning the temperature or the frequency of\nmagnon, one can observe a peak in the voltage [33].\nMeanwhile we can also obtain the magnon absorption\nrate de\fned as the energy of the absorbed magnons di-\nvided by time\nW= 2\nX\n\n;q\u00002JsdIm[~\u001fss(\n;q)]m2\n\n: (14)\nThis magnon absorption rate results in a dip in the\nmagnon transmission rate which is experimentally mea-\nsurable.\nConclusion . In this Letter, we consider a FI/SC bi-\nlayer with RSOC in the bulk of the SC. Using symme-5\ntry arguments and microscopic theory, we show that the\nHiggs mode in the SC couples linearly with an exter-\nnal exchange \feld. This Higgs{spin coupling hugely en-\nhances the total spin susceptibility near a critical phase\ntransition point, which can be detected using iSHE or\nvia strong frequency dependent changes in the magnon\ntransmission. Note that in this work, we consider the dif-\nfusive limit where the disorder strength is stronger than\nthe RSOC and exchange \feld. However, our conclusion\non Higgs{spin coupling should still be valid in the case\nof strong RSOC. In fact, we expect that the coupling is\nmuch stronger with strong SOC in the clean limit. In\nthe di\u000busive limit, the RSOC together with disorder ef-\nfectively generate spin relaxation which reduces the prox-\nimity induced exchange \feld suppressing the Higgs{spin\ncoupling. On the other hand, in the clean case without\ndisorder this e\u000bect is absent and hence the Higgs{spin\ncoupling can be stronger. We also compare the Higgs\nmode in the di\u000busive limit and ballistic limit in the sup-\nplemental material [38].\nThis work was supported by Jenny and Antti Wi-\nhuri Foundation, and the Academy of Finland Project\n317118 and the European Union's Horizon 2020 Research\nand Innovation Framework Programme under Grant No.\n800923 (SUPERTED).\n[1] N. Bittner, D. Einzel, L. Klam, and D. Manske, Phys.\nRev. Lett. 115, 227002 (2015).\n[2] P. W. Anderson, Phys. Rev. 130, 439 (1963).\n[3] P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).\n[4] I. Kulik, O. Entin-Wohlman, and R. Orbach, J. Low\nTemp. Phys. 43, 591 (1981).\n[5] A. Pashkin and A. Leitenstorfer, Science 345, 1121\n(2014).\n[6] P. B. Littlewood and C. M. Varma, Phys. Rev. B 26,\n4883 (1982).\n[7] P. B. Littlewood and C. M. Varma, Phys. Rev. Lett. 47,\n811 (1981).\n[8] M.-A. M\u0013 easson, Y. Gallais, M. Cazayous, B. Clair,\nP. Rodiere, L. Cario, and A. Sacuto, Phys. Rev. B 89,\n060503(R) (2014).\n[9] R. Grasset, Y. Gallais, A. 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Benfatto, Phys-\nical review letters 115, 157002 (2015).\n[38] Supplementary material includes details of the symmetry\noperators and derivation of the susceptibilities.\n[39] In general h+is inhomogeneous and the gradient of h+\nbreaks the mirror symmetry locally. Thus we do not need\nto consider the mirror symmetry of the SC.\n[40] M. A. Silaev, D. Rabinovich, and I. Bobkova, arXiv\npreprint arXiv:2108.08862 (2021).\n[41] K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n[42] W. Belzig, F. K. Wilhelm, C. Bruder, G. Sch on, and\nA. D. Zaikin, Superlatt. and Microstruc. 25, 1251 (1999).\n[43] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys. 77, 1321 (2005).\n[44] F. S. Bergeret and I. V. Tokatly, Phys. Rev. B 89, 134517\n(2014).\n[45] F. S. Bergeret and I. V. Tokatly, Phys. Rev. Lett. 110,\n117003 (2013).\n[46] The code used to obtain the results in this manuscript6\ncan be found at https://gitlab.jyu.\f/jyucmt/sssc-higgs- rashba .\n[47] I. V. Tokatly, Phys. Rev. B 96, 060502(R) (2017)."    },    {        "title": "2403.12745v1.Acoustoelectric_non_local_spin_wave_power_detector_for_studying_magnon_phonon_coupling.pdf",        "content": "Acoustoelectric non-local spin wave power detector for studying magnon-phonon\ncoupling\nHiroki Matsumoto,1, 2Yasuhiro Todaka,1Takuya Kawada,1, 3Masashi Kawaguchi,1and Masamitsu Hayashi1, 4\n1Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan\n2Institute for Chemical Research, Kyoto University, Kyoto 611-0011, Japan\n3Department of Physics, Osaka University, Osaka 560-0043, Japan\n4Trans-scale quantum science institute, The University of Tokyo, Tokyo 113-0033, Japan\n(Dated: March 20, 2024)\nWe have developed a simple detection scheme to study spin waves excited by surface acoustic\nwave (SAW) in ferromagnetic thin films. Metallic antennas made of Ta and a ferromagnetic element\nare placed along the SAW propagation path. The SAW excites spin waves in the ferromagnetic\nelement and induces acoustoelectric current in the antennas, which are detected as a DC voltage.\nThe DC voltage takes an extremum at the spin wave resonance condition, which demonstrates that\nthe antenna acts as a non-local spin wave detector. The antennas placed before and after the fer-\nromagnetic element along the SAW propagation path can probe spin wave excitation from reflected\nand transmitted SAWs, respectively. Interestingly, we find characteristics of spin wave excitations\nthat are different for the reflected and transmitted SAWs: the former excites spin waves with larger\nfrequency with broader linewidth compared to the latter. The results show that the antennas act as\na non-local spin wave power detector and can be used to map out the spin wave spectra in a unique\nway, providing insights into the magnon-phonon coupling in magnetic nanostructures fabricated on\nphononic SAW devices.\nSurface acoustic wave (SAW)[1, 2] is an acoustic\nphonon that can be readily excited using piezoelectric\nsubstrates. A widely used approach to generating SAW\nis to pattern a comb-shaped electrode, often referred to\nas an interdigital transducer (IDT), on a piezoelectric\nsubstrate and apply a microwave electrical signal to the\nelectrode[3]. The electric field applied to the IDT is con-\nverted to an oscillating strain due to the substrate piezo-\nelectricity. The strain is launched as an acoustic wave\nalong the substrate surface, propagating a distance of\nhundreds of micrometers to millimeters[4]. One may de-\nposit a thin film on the path to which the SAW travels,\nthus allowing excitation of acoustic phonons in the film.\nOwing to its high coherence, SAW has been used\nto study the interaction between acoustic phonons and\nother degrees of freedom in solids[5, 6]. Among them,\ncoupling with spin waves[7] in magnetic thin films are\ngaining attraction since their eigenenergy and wavelength\ncan be close to those of SAW[8–12]. Recent studies have\nexamined the strength of the coupling between SAWs\nand spin waves[13–16]. The coupling often manifests it-\nself as extra absorption of SAW power that travels across\na magnetic element. Studies have also shown that trans-\nmission of SAW becomes non-reciprocal when it couples\nto spin waves[17–23].\nThe spin wave-SAW coupling, or magnon-phonon cou-\npling in general, may cause non-linear effects in the SAW\nand spin waves characteristics[24]. In analogy to non-\nlinear optics, higher harmonic generation, parametric\namplification and parametric down conversion are known\nto arise due to non-linearity. It is, however, difficult to\nprobe such non-linear effects unless one relies on sophisti-\ncated time- and space-resolved magnetic or atomic imag-\ning techniques (e.g. magneto-optical Kerr effect[11, 25],\nBrillouin light scattering[26–28], photoemission electronmicroscopy[4] and pulse induced microwave impedance\nmicroscopy[29]).\nHere we show a simple approach to detecting SAW-\ninduced spin waves via voltage measurements. The con-\ncept of the detection scheme is illustrated in Fig. 1(a).\nA metallic antenna, made of a Ta thin film, and a fer-\nromagnetic island are patterned on a piezoelectric sub-\nstrate along the path where the SAW travels. The\nSAW excites spin waves in the ferromagnetic island un-\nder the resonance condition. Simultaneously, the SAW\ninduces acoustoelectric current jAE[30, 31] in the anten-\nnas, which can be detected as a DC voltage. Previous\nstudies have shown that jAEis proportional to the SAW\npower PSAW[32–34]. The DC voltage takes a minimum\nat the spin wave resonance condition, which can be ac-\ncounted for by SAW power absorption to the ferromag-\nnetic island via spin wave excitation. We show that the\nantennas placed before and after the ferromagnetic island\nalong the SAW propagation path can probe, respectively,\nspin waves excited from SAWs reflected and transmitted\nat the island.\nFigure 1(b) shows a schematic illustration of the ex-\nperimental setup. We fabricate a pair of IDTs (IDT1\nand IDT2) composed of Ta(5)/Cu(50)/Pt(3) (units in\nnanometers) on a Y+128o-cut lithium niobate (LiNbO 3)\nsubstrate by radio frequency (rf) magnetron sputtering\nand a liftoff process. The width and the gap of the IDT\nfingers are set to 2 µm. The delay line (i.e. the path\nalong IDT1 to IDT2) is patterned along the crystalline\nX-axis of LiNbO 3, which coincides with the x-axis shown\nin Fig. 1(b). + k(−k) is defined as the propagation direc-\ntion of SAW from IDT1 to IDT2 (IDT2 to IDT1) along\nthex-axis. Upon forming the IDTs, a ferromagnetic Ni\nisland made of Ni(10)/Cu(3)/MgO(2)/Ta(1) is placed at\nthe center of the delay line. The MgO(2)/Ta(1) servesarXiv:2403.12745v1  [cond-mat.mes-hall]  19 Mar 20242\nFIG. 1. (a) Schematic illustration of the detection of SAW-driven spin waves by voltage measurement. SAW is excited at the\nIDT and propagates to the ferromagnetic thin film. The SAW generates acoustoelectric current in the metallic antenna, which\nis detected as DC voltage. When the SAW frequency matches the spin wave resonance condition, SAW power is absorbed\nby excitation of spin waves, which results in the reduction of the acoustoelectric current. (b) Schematic illustration of the\nexperimental setup. The island including 10 nm-thick Ni layer is located at the center of the delay line and a pair of metallic\nantenna (composed of Ta/MgO/Ta) are placed on both sides of the island. We apply rf power to IDT1 (IDT2) to excite SAW\nalong + k(−k). The device geometry are p= 8µm,l= 400 µm,l′= 450 µm,L1= 450 µm and L2= 200 µm.\nas a capping layer. To avoid oxidation of Ni via sput-\ntering of MgO, the Cu(3) layer is inserted. The size\nof the island is 450 ×400µm2. Subsequently, a pair\nof metal antennas composed of Ta(3)/MgO(2)/Ta(1) is\nformed on both sides of the Ni island. Electrodes made\nof Ta(5)/Cu(50)/Pt(3) are attached to the antennas. All\nfilms (IDTs, Ni island, antennas and the electrodes) are\nmade by rf magnetron sputtering. Details of the device\ngeometry are described in Fig. 1(b). An external in-plane\nmagnetic field with magnitude µ0Hextis applied during\nthe measurements. The field angle φHis defined as the\nrelative angle between the magnetic field and + x. All\nmeasurements are performed at room temperature.\nFirst, we characterize the device without exciting any\nspin waves in the ferromagnetic island. A vector network\nanalyzer (VNA) is used to evaluate the S-parameters.\nPort i(i= 1,2) of the VNA is connected to IDT i. An\nrf electrical signal is supplied to port iand the transmit-\nted electrical signal is measured at port jto obtain the\nscattering matrix Sji. We find the fundamental mode (at\n∼0.48 GHz) and the 5th harmonic mode ( ∼2.35 GHz) as-\nsociated with the Reyleigh SAW of the LiNbO 3substrate\nappear in the S12andS21spectra measurements. In the\nexperiments described below, we use the 5th harmonic\nmode since its frequency is sufficiently large to excite spin\nwaves in Ni[8, 9]. fSAW (∼2.35 GHz) is defined as the\nSAW resonant frequency hereafter. During the spectrum\nmeasurements, a magnetic field of µ0Hext= 110 mT ( φH\n= 90o) is applied to the device. The magnitude of the\nfield is chosen such that spin wave excitation in the Ni\nisland is suppressed in the frequency range studied.\nFigure 2(a) shows the input frequency dependence of\n|S21|2around fSAW. Here we show the square of the\ntransmission amplitude, which is proportional to the\ntransmitted power. |S21|2takes a maximum at fSAW.\nMultiple sub-peaks found in the spectra near fSAWresult\nfrom multiple reflection echo between the IDTs[35]. The\nFIG. 2. (a, b) Input rf frequency dependence of |S21|2(a) and\nthe voltages VLandVRof the antennas placed after the Ni\nisland (b). The rf power applied to the IDT is 10 mW. (c)\nInput rf power dependence of VLandVRatf=fSAW∼2.35\nGHz. A magnetic field of µ0Hext= 110 mT is applied along\nthey-axis ( φH= 90o) during the measurements.\nresponse of the DC voltage across the metallic antennas\nis shown in Fig. 2(b). An rf signal source is connected to\none of the IDTs: SAW travels along + k(−k) when the\nsource is connected to IDT1 (IDT2). From the source,\nan amplitude modulated rf signal is applied to the IDT,\nwhich excites an amplitude modulated SAW. The volt-\nage across the antenna is measured using a lock-in am-\nplifier. The frequency and phase of the lock-in amplifier\nare locked to those of the rf signal source. The voltage\nmeasured at the left and right antennas are defined as VL\nandVR, respectively: see Fig. 1(b) for the details of the\nconfiguration. In Fig. 2(b), the solid (dotted) lines show\nVL(VR) when the power of the rf signal source is set to\n10 mW. The top and bottom panels show measurement\nresults when the SAW travels along + kand−k, respec-\ntively.3\nAs evident, the magnitude of the voltage is enhanced\nwhen the rf signal source frequency is matched to fSAW.\nWhen the SAW travels along + k, we find a positive volt-\nage both for VLandVRwhereas the sign of the voltages\nchanges when the SAW propagation direction is reversed.\nThis is consistent with the notion that the observed volt-\nage is induced by the acoustoelectric current[32, 34, 36].\nNote that the magnitude of the peaks differs depend-\ning on the SAW propagation direction and the probe\nposition. For example, the peak height of VLis larger\nthan that of VRwhen SAW propagates along + kand\nvice versa for −k. The results show that the peak volt-\nage of the antenna placed in the SAW propagation path\nbefore the Ni island is larger than that placed after the\nisland. We consider this is simply caused by mechanical\ndamping/electromagnetic absorption of the SAW power\nat the Ni island. Note that there is a slight difference\nin the extremum position (in frequency) of the voltage\npeaks when the SAW moves along + kand−k. We infer\nthis is caused by unintended difference in the geometry\n(e.g. pitch size) of IDT1 and IDT2. The rf power depen-\ndence of VLandVRare shown in Fig. 2(c). Clearly, VL\nandVRboth linearly scale with the power, which corrob-\norates the assumption that the detected voltages are due\nto the acoustoelectric current[32–34].\nNext, we vary the magnitude and direction of the in-\nplane magnetic field so as to excite spin waves in the\nNi island. Here we compare the S-parameter measure-\nments using the VNA and the voltages at the antennas.\nIn Figs. 3(a) and 3(b), we show the normalized SAW\ntransmission power, |S21|2\nnorm and|S12|2\nnorm, plotted as\na function of µ0HextandφH. The normalized power\n|Sij|2\nnorm (i, j= 1,2) represents |Sij|2obtained under a\ngiven field ( µ0Hext,φH) normalized by |Sij|2measured at\n(110 mT, φH). For a fixed field angle, both |S21|2\nnorm and\n|S12|2\nnorm take a minimum at small |µ0Hext|. Using the\nKittel formula[37] with negligible magnetic anisotropy,\nwe estimate the ferromagnetic resonance field µ0Hresat\nfSAW to be ∼10 mT. The field at which the transmit-\nted power takes a minimum is roughly equal to µ0Hresat\nthe frequency studied, indicating that the SAW power is\nabsorbed by spin waves excited in the Ni island.\nThe corresponding normalized DC voltages ( Vnorm\nL and\nVnorm\nR) obtained at the antennas are shown in Figs. 3(c)\nand 3(d), respectively. Similar to |Sij|2\nnorm,Vnorm\nL(R)is\ndefined as the voltage VL(R) measured at a given field\n(µ0Hext,φH) normalized by VL(R)measured at (110 mT,\nφH). Here we show the normalized voltage of the an-\ntennas placed on a SAW propagation path after the Ni\nisland ( Vnorm\nR for + kandVnorm\nL for−k). Thus the SAWs\nthat travel and reach the antennas carry information of\nthe spin waves excited at the Ni island. As evident, the\nresults resemble that of scattering matrix measurements\n(Fig. 3(a,b)). These results show that the antenna serves\nas a non-local probe of SAW induced spin wave excitation\nin ferromagnetic elements.\nIn Figs. 3(e) and 3(f), we show the normalized voltage\nof the antennas placed on the SAW propagation path be-\nFIG. 3. (a-d) Normalized transmitted power |S21|2\nnorm (a)\nand|S12|2\nnorm (b) and the corresponding normalized antenna\nvoltages Vnorm\nR (c) and Vnorm\nL (d), obtained from the antennas\nplaced after the Ni island, plotted as a function of the external\nmagnetic field µ0Hextand its orientation φH. (e,f) Normal-\nized antenna voltages Vnorm\nL andVnorm\nR, obtained from the\nantennas placed before after the Ni island, vs. µ0Hextand\nφH. An rf signal is applied to IDT1 (IDT2) and the SAW\npropagates along + k(−k) for (a,c,e) ((b,d,f)). Input rf power\nis fixed to 10 mW.\nfore the Ni island ( Vnorm\nL for +kandVnorm\nR for−k). Here\nwe expect no apparent signal from the spin waves at the\nNi island if one considers SAWs that had traveled from\nthe IDT to the antenna. However, we find clear contrast\nin the contour plots in Fig. 3(e,f). Albeit the difference\nin the signal amplitude, the overall shape of the color\ncontrast is similar to that of Fig. 3(a-d), which suggests\nthat the voltage reflects SAW induced spin waves at the\nNi island.\nTo elaborate on this effect in more detail, we plot the\nµ0Hextdependence of VLandVRwith a fixed field angle\n(φH= 60o) in Figs. 4(a) and 4(b), respectively. The blue\n(red) lines show data for + k(−k). Note that here we\nshow voltages that are not normalized (the offset voltage\nis also shown). Let us compare the voltages from the two\nantennas placed before ( VL) and after ( VR) the Ni island\nwhen the SAW propagates along + k(i.e. the blue lines of\nFig. 4(a,b)). First, the voltage at large |µ0Hext|is larger\nforVLthat VR. This is simply caused by the mechani-\ncal/electrical absorption of the SAW power at the Ni is-\nland; see the discussion pertaining to Fig. 2(b). Second,\nin contrast to the voltage dips found at µ0HresforVR, we\nfind a peak structure in VL. These features can be under-\nstood if one considers the following scenario. The SAW\nlaunched at the IDT travels toward the Ni island. A frac-\ntion of the SAW that reach the island are reflected due to4\nFIG. 4. (a,b) µ0Hextdependence of VL(a) and VR(b). Blue\n(red) lines: Measured voltage when an rf signal is applied to\nIDT1 (IDT2) and the SAW propagates along + k(−k). (c-\nf) Expanded view of the data shown in (a,b). The vertical\ndashed lines are aligned with the spin wave resonance field\nobtained from the transmitted SAWs, i.e. the positions of\nthe peaks in VLfor−kand the dips in VRfor + k,. The\nexternal magnetic field is applied along φH= 60o.\nthe difference in the mechanical/electrical properties of\nthe substrate/air and substrate/Ni interfaces. Since we\napply a continuous wave rf signal to the IDT, incident\nand reflected SAWs coexist in the path along the IDT-Ni\nisland and thus in the antenna. The acoustoelectric cur-\nrent associated with the incident SAW (traveling toward\nthe Ni island) in the antenna is the source of the large\noffset voltage found in Fig. 2(a) and does not depend\non the magnetic field since the incident SAW predomi-\nnantly represents states before the SAW arrives at the\nNi island. In contrast, the reflected SAW carries infor-\nmation on SAW-induced spin wave excitation at the Ni\nisland, similar to the transmitted SAW. Note that the di-\nrection to which the acoustic current flows in the antenna\nis defined by the SAW propagation direction. Thus, the\nsignal associated with SAW-induced spin wave excitation\nfor the reflected SAW becomes opposite to that for the\ntransmitted SAW (see e.g. Fig. 2(b)). The voltage mea-sured at the antenna is the net sum of contributions from\nthe incident and reflected SAWs, which can account for\nthe signal observed experimentally. The same argument\napplies to the dip structure found in VRfor−k(Fig. 4(b),\nred line).\nInterestingly, we find the position and width of the\npeak/dip in the measured voltages for the transmitted\nand reflected SAWs are different. To show this more\nclearly, an expanded view around the peak or dip is\nshown in Fig. 4(c-f). The vertical dashed lines are aligned\nto the spin wave resonance field obtained from the trans-\nmitted SAWs: i.e. the positions of the peaks in VLfor\n−kand those of the dips in VRfor + k. The correspond-\ning spin wave resonance field obtained from the reflected\nSAWs, i.e. the positions of the peaks in VRfor−kand\nthe dips in VLfor + k, are shifted to a larger |µ0Hext|,\nsuggesting that the excited spin waves have a different\nresonance frequency. The width of the peak or dip is also\nlarger for the spectra obtained with the reflected SAWs.\nThese results indicate that the reflected SAWs carry\ninformation of spin waves which is different from that\nof SAWs traversing the Ni island. Given that the spin\nwave resonance frequency increases with increasing spin\nwave resonance field for the condition under study, the\nresults in Fig. 4(c-f) show that spin waves with larger\neigenfrequency are excited when the SAW is reflected\nat the Ni island. The larger peak/dip width for the\nreflected SAWs suggests excitation of spin waves with\nlarger damping and/or excitation of multiple modes with\ndifferent frequencies. At the moment, it is not clear why\nthe peak/dip structure is different for the reflected and\ntransmitted SAWs. Note that the antenna used here al-\nlows one to probe SAW with frequency different from the\nSAW resonant (excitation) frequency. This is in stark\ncontrast to the IDTs, which can only detect, in principle,\ninteger multiples of the resonant frequency. Thus the an-\ntenna serves as a broadband power detector of excited\nSAWs, similar to the power detectors in microwaves and\noptics.\nIn conclusion, we have demonstrated non-local detec-\ntion of surface acoustic wave (SAW) driven spin waves\nusing metallic antennas placed near a Ni island. SAW\ninduced voltage that develops at the antennas, which is\nproportional to the power of the traversing SAW, is mea-\nsured. The voltage shows a magnetic field dependence\nthat is almost identical to that of SAW transmission co-\nefficient determined using a vector network analyzer. The\nresults suggest that the voltage measured at the antenna\nrepresents SAW power absorption due to SAW induced\nspin wave excitation. The DC voltage detected at the\nantennas placed before and after the Ni island along the\nSAW propagation path suggests that the frequency and\nthe linewidth of the spin waves excited by the reflected\nand transmitted SAWs at the island are different. Fur-\nther study is required to identify the exact mechanism\nthat causes such difference. These results show the po-\ntential of metallic antennas as a simple power detector to\nstudy magnon-phonon coupling in magnetic nanostruc-5\ntures.\nACKNOWLEDGREMENTS\nWe are grateful to T. Funato, M. Matsuo, H.\nKomiyama, K. Taga, R. Hisatomi, Y. Shiota and S.Nakatsuji for fruitful discussion. This work was sup-\nported by JSPS KAKENHI (Grant Number 20J20952,\n20J21915, 23KJ1159, 23KJ1419, 23H05463) from JSPS,\nMEXT Initiative to Establish Next-generation Novel\nIntegrated Circuits Centers (X-NICS) (Grant Number\nJPJ011438), and JSR Fellowship, the University of\nTokyo.\n[1] L. Rayleigh, On waves propagated along the plane surface\nof an elastic solid, Proceedings of the London Mathemat-\nical Society s1-17 , 4 (1885).\n[2] P. Delsing, A. N. Cleland, M. J. A. Schuetz, J. Knoerzer,\nG. Giedke, J. I. Cirac, K. Srinivasan, M. Wu, K. C. Bal-\nram, C. Bauerle, T. Meunier, C. J. B. Ford, P. V. Santos,\nE. Cerda-Mendez, H. Wang, H. J. Krenner, E. D. S. Nys-\nten, M. Weiss, G. R. Nash, L. Thevenard, C. Gourdon,\nP. Rovillain, M. Marangolo, J.-Y. Duquesne, G. Fischer-\nauer, W. Ruile, A. Reiner, B. Paschke, D. Denysenko,\nD. Volkmer, A. Wixforth, H. Bruus, M. Wiklund, J. Re-\nboud, J. M. Cooper, Y. Fu, M. S. Brugger, F. Rehfeldt,\nand C. 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Wang,4A.W.\nRushforth,4and A.J.Ferguson1,c)\n1)Cavendish Laboratory, University of Cambridge, CB3 0HE, Un ited Kingdom\n2)PRESTO, Japan Science and Technology Agency, Kawaguchi 332 -0012, Japan\n3)Hitachi Cambridge Laboratory, Cambridge, CB3 0HE, United K ingdom\n4)School of Physics and Astronomy, University of Nottingham, NG7 2RD, United Kingdom\n(Dated: 11 September 2018)\nWe measure electrically detected ferromagnetic resonance in micro devices patterned from ultra-thin Co/Pt\nbilayers. Spinpumpingandrectificationvoltagesareobservedandd istinguishedviatheirangulardependence.\nThe spin-pumping voltage shows an unexpected increase as the cob alt thickness is reduced below 2 nm. This\nenhancement allows more efficient conversion of spin to charge curr ent and motivates a theory modelling the\ndependence of impurity scattering on surface roughness.\nFerromagnetic/heavy metal (e.g. Co/Pt) bilayers pro-\nvide a model system in which to study spin transfer phe-\nnomena. A charge current in the heavy metal causes dif-\nfusion of a spin current into the ferromagnet via the spin\nhall effect1,2. The resulting angular momentum transfer\ncan either change the amplitude of magnetisation preces-\nsion induced by conventional ferromagnetic resonance3\nor directly drive magnetisation precession4. In addi-\ntion, switching of a nanoscale magnetic element has been\nachieved, indicating that the spin-Hall effect maybe used\nto control memory elements5. Conversely, the precessing\nmagnetisation in the ferromagnetic layer drives a spin\ncurrent into the heavy metal layer6,7, where the inverse\nspin-Hall effect8,9converts it into a measureable charge\ncurrent. This process, known as spin-pumping, has be-\ncome a common laboratory technique to create spin cur-\nrents in diverse materials10–15. A chargecurrent in ultra-\nthin Co/Pt bilayers has also been reported to act on\nthe magnetisation via a ‘Rashba’ spin-orbit torque16,17,\ndue to a relativistic magnetic field existing at the heavy\nmetal interface. In this Letter, in contrast to previous re-\nsearchonthickerlayers18,19, weinvestigatespin-pumping\nin ultra-thin Co/Pt bilayers in which the interface region\nis a significantproportionof the bulk ferromagnetand Pt\nlayers. By keeping the platinum layerthickness constant,\nwe eliminate any variation in the bulk inverse spin-Hall\ndetection. We examine the strength of the spin-pumping\nvoltage in the platinum layer as we vary the thickness of\nthe ferromagnet.\nIn our study the samples are thin bars of Co/Pt with\nnominal cobalt thickness dCo= 1, 1.25, 1.5, 1.75 and 2\nnm capped with a 3 nm Pt layer. From x-ray reflectiv-\nity (XRR) measurements we estimate the uncertainty in\nthe thickness of these layers to be 10%. An out-of-plane\na)Current address: Department of Electrical Engineering, Te chnis-\nche Universiteit Eindhoven, 5600 MB Eindhoven, The Netherl ands\nb)Current address: Centre for Materials Physics, Durham Univ er-\nsity, DH1 3LE, United Kingdom\nc)Electronic mail: ajf1006@cam.ac.uk\nVhz\nImw(a) (b)\ndCodPtw\n\t\n θ\nwImwb\nFIG. 1. (Colour online) (a) Measurement schematic showing\ncoplanar stripline on left with microwave current, Iw\nmw, gen-\nerating a perpendicular microwave field over the bar area. A\nmicrowave current, Ib\nmw, is coupled into the bar. The voltage\nis measured across the bar contacts with a lock-in amplifier.\n(b) The bar consists of a Pt layer deposited on top of a cobalt\nlayer. The in-plane angle θis defined as the angle between\nthe bar direction and the magnetisation.\nmicrowave magnetic field ( hzeiωt), for ferromagnetic res-\nonance(FMR), wasgeneratedoverthe sampleareabyan\non-chip coplanar stripline, shorted 1 µm away from the\nsample. The devices werefabricated from films sputtered\non thermally oxidised silicon. Electron beam lithography\nwas used for patterning, and then 1 x 10 µm bars and\nadjacent striplines were defined with Ar ion-milling. The\nbars are contacted by 200 nm thick gold contacts which\nwere deposited by evaporation at the same time as the\ngold striplines. A schematic of the device and measure-\nment is shown in figure 1.\nThe sample was mounted on a low loss printed circuit\nboard (PCB). A 15 GHz microwave signal was sent via\na coaxial cable into the PCB waveguide and then into\nthe shorted stripline to ground. The signal power in the\ncoaxial cable directly before the PCB was 14.5 dBm. As\nthe PCB waveguides and on-chip striplines are identical\nfor each device, we expect similar microwave currents,\nIw\nmw, in every stripline. In this measurement we assume\nthemicrowavefieldgeneratedisidenticalforeachsample.\nThe microwave signal was pulse modulated at low fre-\nquency (23.45 Hz) allowing a lock-in amplifier to detect\nthe DC voltage ( Vdc) across the sample. The sample2\nwas positioned in a 3-axis vector magnet at a temper-\nature of 250K. For a particular direction, the external\nmagnetic field was swept from high to low field, and the\nferromagnetic resonance was observed as a combination\nofsymmetricand antisymmetricLorentzianpeaksin Vdc.\nVdcis thought to be generated through two effects: the\ninverse-spin-Hall effect (ISHE) and rectification. During\nsteady-stateprecession, the drivingtorque is balanced by\nadampingtorque. ThePtlayeradjacenttotheferromag-\nnet is an efficient spin-current sink and contributes to the\ndamping by transferringangularmomentum between the\nCo and Pt layers via a spin-current. The spin-current,\njs, injected into the Pt layer through the ISHE generates\na transverse charge current given by19\njc=θISHE/parenleftbigg2e\n¯h/parenrightbigg\njs×σ (1)\nAn initial spin current j0\nsat the interface decays due\nto spin relaxation as it penetrates the Pt layer. This\nyields a total charge current of Icwhich creates a voltage\nVISHE=IcRacross the bar. θISHE,e, ¯handσrepresent\nthe spin-Hall angle, the elementary charge, the reduced\nPlanck constant and the spin-polarisation vector of the\nspin-current respectively.\nThemicrowavecurrentintheshortedstriplinecancou-\nple into the sample, to give another microwave current,\nIb\nmw. At resonance the magnetisation will precess at the\nsame frequency as this current. Precession of the mag-\nnetisation causes an oscillating component to the resis-\ntance, due to the anisotropic magnetoresistance (AMR)\nR=R0+∆Rcos2θ. This multiplies with the microwave\ncurrent to give a measurable Vdc. Combining this with\nVISHE, the real part of the voltage is given by the sum of\nsymmetric and antisymmetric parts19–21\nVdc=(VAMRcosφ+VISHE)∆H2\n(H−H0)+∆H2\n+VAMRsinφ∆H(H−H0)\n(H−H0)+∆H2(2)\nwithVAMRandVISHEgiven by\nVAMR=1\n2Ib\nmw∆RAxxsin(2θ)hz (3)\nVISHE=IcR=θSHw/parenleftbigg2e\n¯h/parenrightbigg\nλsdtanh/parenleftbiggdPt\nλsd/parenrightbigg\nj0\nsRsinθ\n(4)\nIntheseexpressions, Histheexternallyappliedmagnetic\nfield,H0is the resonant field and ∆ His the linewidth\nof the resonance. φis the phase difference between the\ncoupled current and the magnetisation precession. dPt\nandware the thickness of the platinum layer and the\nwidthofthebar. ∆ R,RandλsdaretheAMR coefficient,\nthe sample resistance and the spin diffusion length in Pt\nrespectively. Axxis related to the diagonal term of the\nAC magnetic susceptibility by χxx/MS, whereMSis the0 90 180 270 360-0.3-0.2-0.10.00.10.20.3 \n θ (degrees)0.0 0.1 0.2 0.3 0.4 0.5-0.2-0.10.00.10.20.3\nμ0H (T)Vdc (μV) Vdc (μV)(a)\n(b)\nFIG. 2. (Colour online) (a) Detectedvoltage for a 2nmdevice\nfor a single field sweep. The FMR peak is fitted (solid green\nline) by a combination of symmetric (dotted red line) and\nantisymmetric (dashed blue line) Lorentzian curves. (b) Th e\nangular dependences of the symmetric (full red circles) and\nantisymmetric (open blue circles) voltages are each fitted b y\na linear combination of sin θand sin2 θterms.\nsaturation magnetisation22. The magnetisation always\nliesin the plane ofthe sample due tothe demagnetisation\nfield and the negligible in-plane magnetic anisotropy.\nOnly rectification can produce an antisymmetric\nLorentzian, as the phase information needed to produce\nthe asymmetry is held in the relative phase of the resis-\ntance and microwave current. Also observe that the two\ndetection mechanisms have different angular dependen-\ncies, which allows us to distinguish them. The rectifica-\ntion voltage is proportional to sin2 θdue to the symme-\ntry of the AMR, whereas the angular dependence of the\nISHE, given by the cross product in equation 1, makes\nthe spin-pumping signal proportional to sin θ.\nWe measured FMR resonances for a series of in-\nplane angles and fitted the symmetric and antisymmetric\nLorentzian peaks (see figure 2a), defining VsymandVasy\nas the coefficients of the symmetric and antisymmetric\npeaks in equation 2. The angular dependencies of both\nthe symmetric and antisymmetric terms are fitted well\nby a combination of sin θand sin2 θcomponents. Figure\n2b shows the fitting for a sample with a 2 nm Co layer.\nNeither of the detection methods proposed explains the\nantisymmetric sin θcomponent. This component is only\nsignificant in the 1 nm Co layer.\nWe repeated the measurements for the five cobalt\nthicknesses, using identical device structures, and the\nsame experimental parameters. We also repeated mea-\nsurements in a second device for all cobalt layer thick-3\n-0.20.00.20.4Vsym (μV) sin(θ) \n  sin(2θ)\n sin(θ) \n  sin(2θ)(a)\n(b)\n1.0 1.5 2.0-0.2-0.10.00.10.2\ndCo (nm)Vasy (μV)\nFIG. 3. (Colour online) (a),(b) Cobalt thickness dependenc e\nof the fitted symmetric and antisymmetric sin θ(blue dia-\nmonds) and sin2 θ(red circles) voltage components.\nnesses except 1.75 nm to show the variation between de-\nvices. Figure3showsthedetectedvoltagesagainstcobalt\nthickness. Whilst there is a clear trend in the sin θcom-\nponents of both voltage parts, the sin2 θcomponents are\nnot consistent in magnitude or sign even between de-\nvices from the same layer structure. We attribute this\nto variation in the relative phase of the microwave cur-\nrent coupled into each device bilayer, Ib\nmw, and the mi-\ncrowave current in the coplanar stripline generating the\nmagnetic field, Iw\nmw. As the device and coplanar stripline\nmicrostructuresarenearlyidentical, weexpecttheampli-\ntude and phase of Ib\nmwto be dominated by the milli-scale\narrangement of bond wires and pads, which do vary be-\ntween devices. The bond-wire lengths ( ∼2 mm) are close\nto the free-space wavelength (20 mm) and could act as\nan antenna, coupling microwave current into the device\nbilayer. Unlike the rectification signal, the spin-pumping\nsignal is insensitive to Ib\nmwand consequently is repro-\nducible between devices.\nThe spin-current injected into the platinum layeris de-\npendent on both the Gilbert damping and effective mag-\nnetisation which are themselves dependent on the cobalt\nthickness:19\nj0\ns=g↑↓\neffγh2\nz¯h\n8πµ0Meff+/radicalBig\n(µ0Meff)2+4ω2\nγ2\nα2\neff/parenleftBig\n(µ0Meff)2+4ω2\nγ2/parenrightBig(5)\nHere,g↑↓\neffis the spin-mixing conductance, γis the gyro-\nmagnetic ratio and Meffis the effective magnetisation.\nThe effective Gilbert damping constant, αeff, has a con-\ntribution, not only from the volume of the ferromagnet,but also from the spin pumping at the interface19\nαeff=α0+gµBg↑↓\neff\nMSdCo(6)\nLikewise, the effective magnetisation has a bulk con-\ntribution from the demagnetisation field, but also from\na perpendicular uniaxial anisotropy originating from the\ninterface23\nMeff=MS−Hint\nU\ndCo(7)\nWe measured both αeffandMeffwith FMR. Values\nare shown in figure 4a and are fitted well by equations 6\nand 7 when g↑↓\neffis constant for all the cobalt thicknesses,\nshowing that there is no enhancement in the size of j0\ns\nwithdCo.\nThe symmetric sin θvoltage with the ISHE symme-\ntry was converted to a DC current by dividing, for each\ndevice, by the individual resistance measured. Figure\n4b shows both the charge current for the different layers\nand the relative size of the spin current calculated from\nequation 5.\nThe charge current generated in the device has a min-\nimum at around 1.75 nm, whereas the spin-current de-\ncreasesto zero asthe ferromagneticlayeris reduced. The\nreproducibility of the results for each repeated measure-\nment demonstrates that the increase in current in the\nthinnest layers cannot be attributed to variation in hz\nbetween devices. The conversion of the interfacial spin-\ncurrent to the charge current must therefore be depen-\ndent on the cobalt thickness. Our result is surprising as\nprevious studies of thicker Py/Pt bilayers have shown\nremarkable agreement with the theoretical model18,19.\nHowever, the minimum thickness of the ferromagnetic\nlayer measured in those studies was 5 nm, significantly\nthicker than the range we have measured.\nAsdPtis the same for each device, the enhancement\nin the charge current observed must originate from a dif-\nference in the ISHE at the interface, and not the bulk\nISHE in the Pt layer. The relative size of θISHE, calcu-\nlated from equation 4is plotted in figure4b and showsan\nenhancement of 2.4 times between the 2 nm and 1 nm Co\nlayer. XRR measurements on a thicker reference bilayer\nshow the substrate has a surface roughness of 0.42 ±0.07\nnm. We expect thinner films grown on the substrate to\nretain more of this roughness, and consequently to have\na larger Co/Pt interface region. This agrees with the\ninhomogeneous part of the linewidth shown in figure 4c\nwhich indicates that the Co layer becomes increasingly\nless uniform in the thinnest films.\nRecentstudies haveshownthatsurfaceandbulk impu-\nrities can greatly enhance the extrinsic SHE25,26due to\nskew scattering. We expect there could be an enhance-\nment tothe ISHEin oursamplesat, ornear, the interface\ncaused by Co impurities in the Pt layerand vice versa. A\nrougherinterface would lead to both a largersurface area\nwith a greater number of surface impurities, and also a\nwider interface region containing impurities.4\nMeff\neffα\neffα μ0ΔHin (T)ΔH inθISHE(a)\nMeff (T) μ0\nISHE(dCo)/θISHE(2nm) θ(b)\n(c)0.00.20.40.60.81.0\n0.0000.0020.004\n \n050100  \n0.00.20.4 \n1.0 1.5 2.00123 \n \n0.000.040.08\n \ndCo (nm)IC (pA)\n js0(dCo)/ js0(∞)\nIjs\nc0\nFIG. 4. (Colour online) (a) By measuring FMR out-of-\nplane of the sample and self-consistently fitting the mag-\nnetisation angle and resonant field to the Kittel and energy\nequations, we determine the effective magnetisation in each\nsample24. Measured values of Meff(red circles) are fitted well\nby equation 7 (dotted line). We calculate the Gilbert damp-\ning by measuring the frequency dependence of the linewidth,\n∆H= ∆Hin+ωαeff/γ, where ∆ Hinis the inhomogeneous\ncontribution to the linewidth. αeff(blue diamonds) is fitted\nwell by equation 6 (dashed line). (b) Cobalt thickness de-\npendence of the spin-pumping charge current is plotted (red\ncircles). The relative size of the spin-current (solid blue line),\nwhich is calculated using the fits to the measured values of\nMeffandαeff, decreases in the thinner layers. In contrast,\nthe charge current increases in the thinner layers. (c) The\nrelative size of θISHE(red circles) is enhanced in the 1nm Co\nlayer. The error bars show the standard error from fitting the\nsinθparameter to the angular-dependent symmetric voltage\ndata. The small variance between the data points of the same\nthickness could also be from a small difference in the size of\nthe microwave field in each device. The inhomogeneous part\nof the linewidth (blue diamonds) also shows an increase in\nthinner Co layers.\nOur experimental observation of the increase in the\nISHE in ultra-thin layers motivates development of a\ntheory modelling impurity scattering at rough interfaces.\nThe enhancement observed allows more efficient conver-\nsion of spin to charge current in ultra-thin layers.\nThe authors would like to acknowledge useful dis-\ncussions with Joerg Wunderlich, Gerrit Bauer and JanZemen. A.J.F acknowledges support from the Hitachi\nresearch fellowship and a Royal society research grant\n(RG110616). This work was partially funded by EPSRC\ngrant EP/H003487/1.\n1Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. 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Lett.\n109, 156602 (2012)."    },    {        "title": "0807.4711v1.Dose_dependence_of_ferromagnetism_in_Co_implanted_ZnO.pdf",        "content": "Dose dependence of ferromagnetism in Co-implanted ZnO\nNuman Akdogan,\u0003Hartmut Zabel, Alexei Nefedov,yand Kurt Westerholt\nInstitut f ur Experimentalphysik/Festk orperphysik, Ruhr-Universit at Bochum, D-44780 Bochum, Germany\nHans-Werner Becker\nInstitut f ur Physik mit Ionenstrahlen, Ruhr-Universit at Bochum, D-44780 Bochum, Germany\nS \u0018afak G ok\nLehrstuhl f ur Angewandte Festk orperphysik, Ruhr-Universit at Bochum, D-44780 Bochum, Germany\nRustam Khaibullin and Lenar Tagirov\nKazan Physical-Technical Institute of RAS, 420029 Kazan, Russia and\nKazan State University, 420008 Kazan, Russia\n(Dated: November 1, 2018)\nWe have studied the structural, magnetic and electronic properties of Co-implanted ZnO(0001)\n\flms grown on Al 2O3(1120) substrates for di\u000berent implantation doses and over a wide temperature\nrange. Strong room temperature ferromagnetism is observed with magnetic parameters depending\non the cobalt implantation dose. A detailed analysis of the structural and magnetic properties indi-\ncates that there are two magnetic phases in Co-implanted ZnO \flms. One is a ferromagnetic phase\ndue to the formation of long range ferromagnetic ordering between implanted magnetic cobalt ions\nin the ZnO layer, the second one is a superparamagnetic phase, which occurs due to the formation\nof metallic cobalt clusters in the Al 2O3substrate. Using x-ray resonant magnetic scattering, the\nelement speci\fc magnetization of cobalt, oxygen and Zn was investigated. Magnetic dichroism was\nobserved at the Co L2;3edges as well as at the O Kedge. In addition, the anomalous Hall e\u000bect is\nalso observed, supporting the intrinsic nature of ferromagnetism in Co-implanted ZnO \flms.\nPACS numbers: 85.75.-d, 75.50.Pp, 61.72.U-\nI. INTRODUCTION\nZnO is a II-VI semiconductor with a wide band gap\nof about 3.4 eV. The stable crystal structure of ZnO is\nthe wurtzite structure (hexagonal, with a= 3:25\u0017Aand\nc= 5:12\u0017A) [1], in which each atom of zinc is surrounded\nby four oxygen atoms in tetrahedral coordination. The\nmagnetic transition metal doped ZnO is interesting from\nthe view point of forming a transparent ferromagnetic\nmaterial, and it has the potential to be a highly multi-\nfunctional material with coexisting ferromagnetic, semi-\nconducting, and magneto-optical properties. Theoreti-\ncal predictions of room temperature ferromagnetism in\ntransition metal (TM)-doped ZnO [2, 3, 4] have initi-\nated a number of experimental works on these systems\nas a potential oxide-based diluted magnetic semiconduc-\ntor (DMS) material. The \frst observation of ferromag-\nnetism in Co-doped ZnO was reported by Ueda et al.\n[5]. They prepared Zn 1\u0000xCoxO thin \flms on sapphire\nsubstrates using PLD technique with xvarying between\n0.05 and 0.25. Following these initial theoretical and ex-\n\u0003Author to whom correspondence should be addressed. E-mail ad-\ndress: numan.akdogan@ruhr-uni-bochum.de\nPresent address: Department of Physics, Gebze Institute of Tech-\nnology, Gebze, 41400 Kocaeli, Turkey\nyPresent address: Lehrstuhl f ur Physikalische Chemie I, Ruhr-\nUniversit at Bochum, D-44780 Bochum, Germanyperimental reports, di\u000berent growth methods have been\nused to deposit Co:ZnO \flms, including radio-frequency\n(RF) magnetron co-sputtering [6], pulsed laser deposition\n(PLD) using a KrF laser [7, 8, 9, 10, 11, 12, 13, 14], com-\nbinatorial laser molecular beam epitaxy (LMBE) [15, 16],\nsol-gel method [17], as well as ion implantation [18]. Sap-\nphire has been widely used as substrate due to the small\nmismatch (2%) between (0001) oriented ZnO and Al 2O3\n(1120) substrates. In addition to cobalt, other 3 dtran-\nsition elements have also been used for doping, including\nMn [5, 15, 19, 20, 21], Ni [5, 11, 15], V [11, 15, 22], Cr\n[5, 11], and also Fe [5, 11, 15, 23].\nVarious solubility limits for Co in ZnO were reported\nby di\u000berent groups. Prellier et al. [10] have determined\na solubility limit of about 10 at.% in PLD-grown \flms.\nPark et al. [24] reported that cobalt nanoclusters start\nto form for x\u001512 at.% in samples grown by sol-gel and\nRF sputtering techniques. Lee et al. [17] observed some\nunde\fned Bragg peaks for a cobalt concentrations higher\nthan 25 at.%. Kim et al. [8] showed that the solubility\nlimit is less than 40 at.% in PLD-grown \flms. Ueda et\nal.[5] claimed that the solubility limit is lower than 50\nat.% and they clearly observed a phase separation into\nZnO- and CoO-rich phases in the \flm prepared using\nZn0:5Co0:5O targets. These controversial results from dif-\nferent research groups are likely due to di\u000berent growth\ntechniques used and/or due to di\u000berent growth condi-\ntions such as oxygen pressure and deposition tempera-\nture. Recently, we have reported that using ion implan-arXiv:0807.4711v1  [cond-mat.mtrl-sci]  29 Jul 20082\ntation cobalt concentrations of up to 50 at.% in ZnO\nare possible without cobalt cluster formation [18]. This\nhigh concentration is attributed to the properties of ion\nimplantation, which allows doping of transition metals\nbeyond their equilibrium solubility limits [25].\nRegarding the magnetic properties of Co-doped ZnO\n\flms, while several groups including ourself have ob-\nserved room temperature ferromagnetism for 50 at.%\n[18], 25-30 at.% [14, 17] and lower [10, 11, 13, 26] Co\nconcentrations, others reported the absence of ferromag-\nnetism at room temperature [8, 15, 24].\nIn this paper we report detailed studies using various\nexperimental techniques for the investigation of the struc-\ntural, magnetic and electronic properties of Co-implanted\nZnO \flms grown on sapphire substrates and for di\u000berent\ncobalt concentrations. Rutherford backscattering spec-\ntroscopy (RBS) and X-ray di\u000braction (XRD) were used\nto determine the depth pro\fle of implanted cobalt ions\nand to detect the formation of possible secondary phases\nsuch as metallic cobalt clusters. The magnetic properties\nof the \flms were characterized by the magneto-optical\nKerr e\u000bect (MOKE), a superconducting quantum inter-\nference device (SQUID) magnetometer, as well as x-ray\nresonant magnetic scattering (XRMS) techniques. In or-\nder to determine the type and concentration of carriers in\nCo-implanted ZnO \flms, Hall e\u000bect measurements were\nalso performed.\nII. SAMPLE PREPARATION\nAbout 35 nm thick ZnO(0001) \flms were grown on\n10\u000210 mm2epi-polished single-crystalline Al 2O3(1120)\nsubstrates by RF (13.56 MHz) sputtering of a ZnO target\n[27]. The sputtering was carried out in an atmosphere\nof 5\u000210\u00003mbar pure Ar (99 :999%) with a substrate\ntemperature of 500\u000eC. In order to increase the quality\nof ZnO \flms, the samples were annealed in an oxygen\natmosphere with a partial pressure of up to 2000 mbar\nand a temperature of 800\u000eC. After annealing, the ZnO\nsamples were implanted with 40 keV Co+ions with an\nion current density of 8 \u0016A\u0001cm\u00002using the ILU-3 ion\naccelerator (Kazan Physical-Technical Institute of Rus-\nsian Academy of Science). The sample holder was cooled\nby \rowing water during the implantation to prevent the\nsamples from overheating. The implantation dose varied\nin the range of 0 :25\u00002:00\u00021017ions\u0001cm\u00002. After implan-\ntation, the samples were cut into square pieces and gold\ncontacts were evaporated on the corners of the samples\nfor Hall e\u000bect studies (Fig. 1). A list of the Co-implanted\nZnO \flms used for the present study is given in Table I.\nFIG. 1: Sample preparation stages for Co-implanted\nZnO/Al 2O3\flms.\nTABLE I: List of the ZnO \flms implanted with 40 keV Co+\nfor varying Co ion dose.\nSample Dose ( \u00021017ion\u0001cm\u00002)\n1 0.25\n2 0.50\n3 0.75\n4 1.00\n5 1.25\n6 1.50\n7 2.00\nIII. EXPERIMENTAL RESULTS\nA. Structural Properties\nThe depth dependence of the cobalt concentration in\nCo-implanted ZnO/Al 2O3\flms was investigated using\nthe RBS technique at the Dynamic Tandem Laboratory\n(DTL) at the Ruhr-Universit at Bochum. The RBS data\nshow both a maximum of cobalt concentration located\nclose to the ZnO/Al 2O3interface and an extended in-\nward tail due to cobalt di\u000busion into the volume of the\nAl2O3substrate (Fig. 2). We also noticed that after ion\nimplantation the thickness of the ZnO layer has shrunk\n(e.g., from originally 35 nm to 28 nm for sample 6) due\nto sputtering e\u000bects. According to the SRIM algorithm\n[28], the average implanted depth of 40 keV Co ions in\nZnO/Al 2O3is about 20.4 nm with a straggling of 9.6\nnm in the Gaussian-like depth distribution (solid line in\nFig. 2). However, because of the surface sputtering, ion\nmixing and heating of the implanted region by the ion\nbeam, a redistribution of the implanted cobalt ions com-\npared to the calculated pro\fle has to be taken into ac-\ncount.\nHigh-angle XRD experiments provide information on\nthe structural coherence of the \flms and in our case\nalso of possible additional phases in the sample after\nion implantation. Fig. 3 shows high angle Bragg scans\nof the ZnO \flms before and after cobalt implantation.\nThe data were taken using synchrotron radiation at the\n\"Hamburg Synchrotron Radiation Laboratory\" (HASY-\nLAB) (for pure ZnO \flm) and at the \"Dortmund Electron\nAccelerator\" (DELTA) (for cobalt implanted ZnO \flms)\nwith an energy of E=11000 eV. Before implantation the\nx-ray di\u000braction pattern consists of a very strong Al 2O3\n(1120) peak and a ZnO(0001) re\rection to the left side.\nThe ZnO peak is surround by thin \flm Laue oscillations,3\nFIG. 2: Depth dependence of the cobalt concentration in\nZnO/Al 2O3implanted with Co ions with a dose of 0 :25\u0002\n1017ions\u0001cm\u00002(open symbols) and 1 :50\u00021017ions\u0001cm\u00002\n(black symbols), respectively. Solid line presents the calcu-\nlated SRIM pro\fle. The inset shows the experimentally ob-\nserved (symbols) and simulated (solid line) RBS spectra for\nsample 6.\nwhich are indicative for the high quality of the ZnO \flm.\nAfter implantation, the XRD di\u000braction pattern shows\na (10 10) re\rection of the Co hcp structure on the right\nside of the sapphire substrate peak. The ion bombard-\nment also causes an intensity reduction of the ZnO(0001)\npeak proportional to the implantation dose, indicating an\nincreasing amount of lattice defects. Furthermore, after\nimplantation we observe a shift of the ZnO (0001) peak to\nhigher angles. We attribute the ZnO lattice contraction\nto the substitution of Zn ions by Co cobalt ions, which\nhas a smaller ion radius. In addition, after implanta-\ntion a tail (shown by an arrow in Fig. 3) appears on the\nlow angle side of the main Al 2O3(1120) peak which is\nnot observed before implantation. This tail likely re\rects\nthe lattice expansion of the sapphire substrate upon Co\nimplantation.\nIn addition to the XRD experiments, we have also per-\nformed high resolution cross sectional transmission elec-\ntron microscopy (TEM) measurements for sample 6 [18].\nThe TEM results reveal the presence of metallic cobalt\nclusters in the Al 2O3sapphire substrate, but not in the\nZnO \flm. Co clusters with a diameter of about 5-6\nnm form a Co rich layer in the substrate close to the\nZnO/Al 2O3interface [18].\nB. Magnetic Properties\n1. Room temperature magnetization measurements\nFor the investigation of the magnetic properties of the\nCo implanted samples we used a high-resolution MOKE\nFIG. 3: High angle Bragg scans of the ZnO(0001) \flms on\nAl2O3(1120) before and after cobalt ion implantation.\nsetup in the longitudinal con\fguration with s-polarized\nlight [29, 30, 31]. Fig. 4 shows the hysteresis loops of\nCo-implanted ZnO \flms which were recorded at room\ntemperature. The MOKE data in Fig. 4 clearly indicate\nthat after cobalt implantation, non-magnetic ZnO be-\ncomes ferromagnetic at room temperature with a large\nremanent magnetization. With increasing cobalt con-\ncentration the implanted ZnO \flms exhibit sequentially\nparamagnetic, weak ferromagnetic and, \fnally, ferromag-\nnetic response with a square-like hysteresis at room tem-\nperature for the dose of 1 :50\u00021017ions\u0001cm\u00002. For\nthe highest dose (2 :00\u00021017ions\u0001cm\u00002) the square-like\nshape of the hysteresis loop disappears and the coercive\n\feld increases drastically. From this we infer that for the\nhighest dose level the cobalt atoms start to form clusters\nin the ZnO \flm. Moreover, although no in-plane mag-\nnetic anisotropy was observed by MOKE in Co-implanted\nZnO \flms, we observed a clear six-fold in-plane magnetic\nanisotropy by ferromagnetic resonance (FMR) technique\n[32]. The corresponding FMR data show that the easy\nand hard axes have a periodicity of 60 degree in the \flm\nplane, in agreement with the hexagonal structure of the\nZnO \flm.\nIn order to study in detail the observed ferromag-\nnetic behavior, the magnetic properties of Co-implanted\nZnO \flms were investigated using the XRMS technique.\nXRMS has proven to be a highly e\u000bective method for\nthe analysis of the magnetic properties of buried layers\nand interfaces, including their depth dependence [33, 34].\nMoreover, if the photon energy is \fxed close to the energy\nof the corresponding x-ray absorption edges, element spe-\nci\fc hysteresis loops can be measured [35]. Since there\nare three elements in the Co-doped ZnO \flm, the analysis\ncan be carried out separately for Co, O and Zn.\nThe XRMS experiments were performed using the4\nFIG. 4: Room temperature MOKE hysteresis curves of Co-\nimplanted ZnO \flms measured for varying implantation dose.\nALICE di\u000bractometer [36] at the undulator beamline\nUE56/1-PGM at BESSY II (Berlin, Germany). The\ndi\u000bractometer comprises a two-circle goniometer and\nworks in horizontal scattering geometry. A magnetic \feld\ncan be applied in the scattering plane and along the sam-\nple surface either parallel or antiparallel to the photon\nhelicity, which corresponds to the longitudinal magneto-\noptical Kerr e\u000bect (L-MOKE) geometry. The maximum\n\feld of \u00062700 Oe was high enough to fully saturate the\nmagnetization of the sample. The magnetic contribution\nto the scattered intensity was always measured by re-\nversing the magnetic \feld at \fxed photon helicity. As a\ncompromise between high scattering intensity and high\nmagnetic sensitivity for the investigation of the magnetic\nproperties at the Co Ledges, the scattering angle was\n\fxed at the position of 2 \u0012= 8:2\u000e(the angle of incidence\nis\u0012= 4:1\u000e) [18].\nThe magnetic contribution to the resonant scattering\ncan best be visualized by plotting the asymmetry ratio,\nAr= (I+\u0000I\u0000)=(I++I\u0000). In Fig. 5 we show the asym-\nmetry ratio taken at the Co Ledges for samples doped\nwith di\u000berent doses. The asymmetry ratio shows a strong\nferromagnetic signal for sample 6 (up to 30 %), and it\ndecreases with decreasing cobalt implantation dose. For\nsample 2, we observe only a very small magnetic signal at\n4.2 K. In addition to XRMS, we have also employed x-ray\nabsorption spectroscopy (XAS) experiments for sample 6.\nThe XAS spectrum clearly exhibits a multiplet structure\nof the Co L3peak, which is typical for oxidized cobalt\nshowing the presence of Co2+state in the ZnO \flm [18].\nThe magnetic signal at the Zn L3- (E=1021.8 eV) and\nthe O K- (526.8 eV) edges were also investigated. Within\nthe sensitivity limit no magnetic signal could be resolved\nfor Zn. However, a clear magnetic signal was observed at\nthe O Kedge for sample 6 [18]. In addition to sample 6,\na very small magnetic signal at the O Kedge was also\nFIG. 5: The asymmetry ratios taken at the Co Ledges\nfor sample 6 (1 :50\u00021017ions\u0001cm\u00002) and sample 4 (1 :00\u0002\n1017ions \u0001cm\u00002) shown by black and open symbols, re-\nspectively. Inset presents the asymmetry ratio of sample 2\n(0:50\u00021017ions\u0001cm\u00002) measured at 4.2 K.\nobserved for the samples 4 and 7 presented in Fig. 6.\nFIG. 6: The magnetic signal at the O Kedge for samples 4\n(black symbols) and 7 (solid line). Insets a and b show the\nhysteresis curves taken at the O Kedge for samples 4 and 7,\nrespectively.5\nFIG. 7: Temperature dependent magnetization curves of Co-\nimplanted ZnO \flms recorded by SQUID magnetometry for\nvarying implantation dose. FC and ZFC curves refer to \feld\ncooled and zero-\feld cooled protocols and are presented by\nclosed and open symbols, respectively. In both cases the data\nwere taken in a \feld of 100 Oe during the heating up cycle.\n2. Temperature dependent magnetization measurements\nIn order to check the temperature dependence of the\nmagnetization for ZnO \flms doped with di\u000berent doses,\nwe carried out \feld cooled (FC) and zero \feld cooled\n(ZFC)M\u0000Tmeasurements using a SQUID magnetome-\nter. For ZFC measurements, the samples are \frst cooled\nin zero \feld to 5 K and the magnetization is recorded\nduring warming up to 390 K with an applied \feld of 100\nOe, parallel to the \flm surface. For FC measurements,\nthe applied \feld of 100 Oe is kept constant during cooling\nto 5 K and the magnetization is recorded during warm-\ning at the same \feld value. Due to the clustering of\ncobalt in the Al 2O3substrate ([18]), the FC (closed sym-\nbols) and ZFC (open symbols) curves presented in Fig. 7\nFIG. 8: SQUID M\u0000Hloops of Co-implanted ZnO \flms\nmeasured for di\u000berent implantation doses at 5 K.\nFIG. 9: The dose dependence of the normalized remanent\nmagnetization (a), the coercive \feld (b) and the saturation\nmagnetization (c), respectively. The data taken at 5 K using\nSQUID magnetometry.\nalways show evidence for the presence of a superpara-\nmagnetic phase. There is a small peak at about 20 K\nin ZFC curve of sample 1 (0 :25\u00021017ions\u0001cm\u00002) and\nthis peak shifts to higher temperatures with increasing\ncobalt concentration. The trend in the M\u0000Tcurve of\nsample 1 (0 :25\u00021017ions\u0001cm\u00002) can be attributed to\nthe coexistence of a ferromagnetic phase originating from\nsubstituted Co2+ions in ZnO and the superparamag-\nnetic phase due to cluster formation in Al 2O3. Hystere-\nsis curves measured at 5 K (Fig. 8) indicate that the su-\nperparamagnetic phase in this sample is more dominant\nthan the ferromagnetic phase. The M\u0000Tmeasurements\nfor the samples implanted with higher doses exhibits su-\nperparamagnetism with a blocking temperature of about\n100 K and 250 K for sample 2 (0 :50\u00021017ions\u0001cm\u00002)6\nand sample 3 (0 :75\u00021017ions\u0001cm\u00002), respectively. The\nhysteresis curves of these \flms (Fig. 8) also show that\nthe superparamagnetic phase is still dominating over the\nferromagnetic phase. But the steep part of the hys-\nteresis curve of sample 3 and the increased coercivity\n(0:75\u00021017ions\u0001cm\u00002) are indicative for the onset of\na clear ferromagnetism phase at this dose. The tem-\nperature dependent magnetization curves of sample 4\n(1:00\u00021017ions\u0001cm\u00002), sample 5 (1 :25\u00021017ions\u0001cm\u00002)\nand sample 6 (1 :50\u00021017ions\u0001cm\u00002) show that these\nsamples have a blocking temperature of about 390 K or\neven higher. The magnetic hysteresis of these samples\nmeasured by SQUID (Fig. 8) clearly show a ferromag-\nnetic phase superimposed by a superparamagnetic com-\nponent. The ferromagnetic component is present even\nabove room temperature as seen in the MOKE exper-\niments in Fig. 4. Since MOKE probes only \flms near\ntheir surface, the superparamagnetic component in these\nsamples, which is deeper in the substrate, is not seen by\nMOKE experiments.\nIn the SQUID hysteresis curves there is another re-\nmarkable e\u000bect of the ferromagnetic phase as a function\nof dose. The coercivity HCdecreases systematically with\nincreasing Co dose up until a dose of 1 :50\u00021017ions\u0001\ncm\u00002, as seen in Figs. 8 and 9 (b). This behavior may\nbe explained as follows: with increasing Co dose the mag-\nnetization becomes more homogeneous and, since mag-\nnetic inhomogeneities are the main source of pinning for\nthe domain walls, HCdecreases with increasing Co dose.\nBetween 1:25\u00021017ions\u0001cm\u00002and 1:50\u00021017ions\u0001cm\u00002\nthe potential barrier for reversal of the ferromagnetic\ncomponent becomes smaller. Up to this level all inho-\nmogeneities are \flled. Any higher dose is counterproduc-\ntive, it decreases the saturation magnetization and en-\nhances the coercivity (see Figs. 9 (b) and (c)), indicating\nthat Co goes into antisites with eventually antiferromag-\nnetic (AF) coupling, loss of magnetization, and increase\nof the coercivity. CoO clusters are formed in the ZnO\nmatrix with AF spin structure and AF coupling to the\nremaining ferromagnetic Zn(Co)O \flm. The M\u0000Tdata\n(Fig. 7) and the room temperature (Fig. 4) and low tem-\nperature (Fig. 8) hysteresis measurements of sample 7\n(2:00\u00021017ions\u0001cm\u00002) clearly indicate that the cobalt\natoms start to cluster also within the ZnO layer at the\nhighest dose.\nC. Hall e\u000bect measurements\nIn ferromagnetic materials the Hall voltage consists\nof the ordinary term and an additional term that con-\ntributes to the Hall voltage due to their spontaneous\nmagnetization. This additional contribution, called\nanomalous Hall e\u000bect, is proportional to the sample mag-\nnetization [37]. Hence, the Hall voltage can be written\nas [37],VH=\u0010R0I\nt\u0011\nHcos\u000b +\u0010RA\u00160I\nt\u0011\nMcos\u0012; (1)\nwheretis the \flm thickness and Iis the current. R0\nandRAare the ordinary and anomalous Hall e\u000bect coe\u000e-\ncients, respectively. \u00160is the permeability of free space.\n\u000bis the angle between the applied magnetic \feld ( H)\nand sample normal. \u0012is the angle between the sample\nmagnetization ( M) and the sample normal. The \frst\nterm in Eq. 1 is the ordinary Hall e\u000bect and arises from\nthe Lorentz force acting on conduction electrons. This\nestablishes an electric \feld perpendicular to the applied\nmagnetic \feld and to the current. The anomalous Hall\ne\u000bect term is conventionally attributed to spin dependent\nscattering involving a spin-orbit interaction between the\nconduction electrons and the magnetic moments of the\nmaterial. At low applied magnetic \felds, the Hall volt-\nage (VH) is dominated by the magnetic \feld dependence\nof the sample magnetization M. When the applied mag-\nnetic \feld is high enough to saturate the sample magneti-\nzation, the magnetic \feld dependence of the Hall voltage\nbecomes linear due to the ordinary Hall e\u000bect.\nIn order to check whether this behavior is present in\nCo-implanted ZnO \flms and to determine the charac-\nter of the majority carriers, we have carried out Hall\ne\u000bect experiments. The Hall e\u000bect measurements were\nperformed at 4.2 K using a van der Pauw con\fguration\npresented in Fig. 10 as an inset.\nFIG. 10: AHE data of sample 6 (1 :50\u00021017ions\u0001cm\u00002) taken\nat 4.2 K. Inset shows the geometry of the AHE measurements.\nHis the external magnetic \feld applied perpendicular to the\n\flm surface.\nThe Hall e\u000bect data of sample 6 (1 :50\u00021017ions\u0001cm\u00002)\nare shown in Fig. 10. A sharp rise in the Hall voltage\nat low \feld, i.e., AHE, is followed by a slow decrease\ncorresponding to the ordinary Hall e\u000bect. It is important\nto note that the negative slope at high \felds indicates n-\ntype carriers in Co-implanted ZnO \flm with a 3D carrier\nconcentration of n3D= 1:931\u00021019\u0001cm\u00003. The Hall\nmobility measured at 4.2 K is about 90 cm2\u0001V\u00001s\u00001for\nsample 6. We have also observed similar behavior for7\nthe samples 3, 4, 5 and 7. However, for the lowest two\ndoses (samples 1 and 2), the measurements cannot be\ndone because of a too small signal-to-noise ratio of the\nHall voltage.\nIV. DISCUSSION\nFor the dose dependence of magnetic phases in ZnO\n\flms at room temperature we propose the following sce-\nnario : At low doses (0 :25\u00000:50\u00021017ions\u0001cm\u00002) the\nnumber of substituted cobalt ions in the ZnO layer is very\nsmall, which results in a paramagnetic signal at room\ntemperature. Increasing of cobalt implantation dose\nleads to an increasing number of substituted cobalt ions\nand after certain cobalt concentration they start to inter-\nact ferromagnetically. For this reason at the cobalt dose\nof 0:75\u00021017ions\u0001cm\u00002a weak ferromagnetic behavior\nis observed with a Tcbelow room temperature. At higher\ncobalt concentrations (1 :00\u00001:50\u00021017ions\u0001cm\u00002) the\nsubstituted cobalt ions in ZnO interact strongly and sta-\nbilize room temperature ferromagnetism. At the highest\ndose of 2:00\u00021017ions\u0001cm\u00002, in addition to the substi-\ntuted cobalt ions, metallic cobalt clusters are also present\nin the ZnO layer.\nAs discussed in detail in Ref. [18], the di\u000berence in\nthe shape of the hysteresis loops obtained by MOKE\nand SQUID is attributed to the surface sensitivity of the\nMOKE technique with a maximum penetration depth of\nabout 20-30 nm. The ZnO \flms have a thickness of 35\nnm before implantation. Because of surface sputtering,\nthe ZnO thickness decreases (e.g., decreased to about 28\nnm for sample 6) after implantation. Thus, MOKE pro-\nvides information only from the ZnO layer, not from the\nsapphire substrate, i.e. MOKE is only sensitive to the\nferromagnetic contribution from the ZnO layer. In this\nlayer a small fraction of nonmagnetic Zn atoms are sub-\nstituted by magnetic Co ions, giving raise to the MOKE\nhysteresis. However, SQUID measurements collect mag-\nnetic contributions from both the Co-implanted ZnO \flm\nand from the cobalt clusters in Al 2O3(Fig. 11). There-\nfore, the di\u000berence between the MOKE and SQUID data\nappear as a result of the depth-dependent Co content in\nthe implanted layer.\nFIG. 11: The cluster formation in Al 2O3substrate after\ncobalt ion implantation.\nAnother important result of this study is the obser-\nvation of oxygen spin polarization in Co-implanted ZnO\n\flms. This shows that the oxygen atoms are polarized\ndue to the spontaneous ferromagnetic order in ZnO \flms.\nThe main question that arises here is the mechanismwhich leads to the observed long range ferromagnetic or-\ndering in Co-doped ZnO. Recently, Patterson [38] calcu-\nlated the electronic band structure of Co substituted for\nZn in ZnO, for Zn and O vacancies, and for interstitial\nZn in ZnO using the B3LYP hybrid density functional\ntheory. He reported that the singly-positively charged\nO vacancy is the only defect in Co-doped ZnO which\ncan mediate ferromagnetic exchange coupling between\nCo ions at intermediate range (just beyond near neighbor\ndistances). In the ground state con\fguration the major-\nity Co spins are parallel, whereas the minority spins are\nparallel to each other and to the oxygen vacancy spin,\nso that there are exchange couplings between these three\nspins leading to an overall ferromagnetic ground state of\nthe Co ions. No substantial exchange coupling was found\nfor the positively charged interstitial Zn defect which has\nalso spin 1/2. The exchange coupling mechanism pro-\nposed by Patterson is essentially the same as the impu-\nrity band model of Coey et al. [39], where the polarons\nbound to the oxygen vacancies mediate ferromagnetic\ncoupling between Co ions. In order to have the mag-\nnetic moments of the Co ions aligned ferromagnetically,\none mediating electron is required with an oppositely di-\nrected spin. This is in line with a recent comparison of\nband structure calculations by Walsh et al. showing that\nthe electronic structure of Co-doped ZnO is consistent\nwith carrier mediated ferromagnetism [40]. The oxygen\nspin polarization has not explicitly been considered in the\naforementioned band structure calculations and may be\ndue to ferromagnetic splitting of nearest neighbor oxygen\np-levels. This has already been speculated by Methfessel\nand Mattis in their seminal review article on magnetic\nsemiconductors [41].\nThe reason for the observation of AHE and n-type\ncarriers in Co-implanted ZnO \flms can be explained by\nelectron doping via Zn interstitials. Normally, isovalent\nTM2+doping of ZnO does not introduce charge carriers\nitself, they need to be produced by additional doping [42].\nHowever, using ion implantation not only cobalt ions are\nintroduced in ZnO, but simultaneously many other de-\nfects are also be produced in the implanted region, such\nas Zn interstitials which are reported to form shallow\ndonors in ZnO [17, 43, 44]. This can be thought of as an\nadded advantage of ion implantation that it not only in-\ntroduces transition metal ions to induce ferromagnetism\nbut also introduces the required charge carriers into the\nZnO.\nV. SUMMARY\nIn conclusion, the structural, magnetic and electronic\nproperties of Co-implanted ZnO \flms, deposited by RF-\nsputtering methods on a (11 20) oriented sapphire sub-\nstrate, have been investigated. The structural data in-\ndicate a Co cluster formation in the sapphire substrate\nclose to the ZnO/Al 2O3interface but well separated from\nthe ZnO \flm. No indication of clustering in the ZnO layer8\nhas been found. The previously reported XAS data with\na multiplet \fne structure around the Co L3edge clearly\nshows that the implanted cobalt ions are in the Co2+ox-\nidation state, most likely substituting part of the Zn2+\nions in the host matrix. The combination of room tem-\nperature and low temperature magnetization measure-\nments indicates that there are two magnetic phases in\nthe Co-implanted ZnO/Al 2O3\flms. One is the ferromag-\nnetic phase due to the Co substitution on Zn sites in the\nZnO \flm, the second magnetic phase originates from Co\nclusters in the sapphire substrate. Furthermore, a clear\nferromagnetic signal at the O Kedge is observed which\nshows that the oxygen spin polarization is an important\nindicator for the observed long range ferromagnetic or-\ndering in the ZnO layer. In conclusion, implantation of\ncobalt ions into the nonmagnetic ZnO \flm causes intrin-\nsic ferromagnetism at room temperature and simultane-\nously creates n-type charge carriers without additionaldoping.\nAcknowledgments\nWe would like to acknowledge S. Erdt-B ohm and P.\nStauche for sample preparation and technical support.\nWe also would like to thank also Dr. C. Sternemann\nand Dr. M. Paulus for their assistance with the beam-\nline operation at DELTA, and G. Nowak for his help\nto perform XRD experiments at HASYLAB. This work\nwas partially supported by BMBF through Contracts\nNos. 05KS4PCA (ALICE Chamber) and 05ES3XBA/5\n(Travel to BESSY), by DFG through SFB 491, and by\nRFBR through the grant Nos 07-02-00559-a and 04-02-\n97505-r. N. Akdogan acknowledges a fellowship through\nthe International Max Planck Research School-SurMat.\n[1] R. W. G. Wycko\u000b, Crystal Structures, 2nd Edition (Wi-\nley, New York, 2001).\n[2] T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. 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Rev.\nLett. 82, 2552 (1999)."    },    {        "title": "2105.06606v1.An_energy_harvesting_technology_controlled_by_ferromagnetic_resonance.pdf",        "content": " \n1 \n An energy harvesting technology controlled by ferromagnetic resonance \n \nYuta Nogi,1 Yoshio Teki,2 and Eiji Shikoh1,a) \n \n1 Graduate School of Engineering, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka \n558-8585, Japan \n2 Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-\n8585, Japan \n \nWe have successfully demonstrated electrical charging using the electromotive force (EMF) generated \nin a ferromagnetic metal (FM) film under ferromagnetic resonance (FMR). In the case of Ni 80Fe20 \nfilms, electrical charge due to the EMF generated under FMR can be accumulated in a capacitor ; \nhowever, the amount of charge is saturated well below the charging limit of the capacitor. Meanwhile \nin the case of Co 50Fe50, electrical charge generated under FMR can be accumulated in a capacitor and \nthe amount of charge increases linearly with the FMR duration time. The difference between the \nNi80Fe20 and Co 50Fe50 films is due to the respective magnetic field ranges for the FMR excitation. \nWhen the FM films were in equivalent thermal states during FMR experiments, Co 50Fe50 films could \nmaintain FMR in a detuned condition, while Ni 80Fe20 films were outside the FMR excitation range.  \n2 \n The EMF generation phenomenon in an FM film under FMR can be used an energy harvesting \ntechnology by appropriately controlling the thermal conditions of the FM film. \n \na) E-mail: shikoh@eng.osaka-cu.ac.jp \n   \n3 \n Energy harvesting is an important technology to efficiently utilize the earth’s natural resources.1 \nThis technology harvests the existing micro-energy in an environment, and is different from \nconventional electric generation technologies such as electric power plants. So far the harvesting of \nsuch micro-energies has focused on the use of light, heat, vibration, electromagnetic fields, and their \nrelated phenomena.1 The energy obtained per system due to such harvesting methods is not very large ; \nhowever, the harvested electric power has the potential to be used to operate electronic devices. \nFerromagnetic resonance (FMR) is a magnetic phenomenon in which the magnetization \ndynamics in a magnetic material is controlled using both a static magnetic field ( H) and a high \nfrequency magnetic field in the GHz band.2 In research on spintronics, it has been discovered that an \nelectromotive force (EMF) is generated in the ferromagnetic metal (FM) film itself under FMR.3,4 The \nEMF originates from various physical phenomena such as the inverse Hall effect (ISHE),3-5 the \nanomalous Hall effect (AHE),3-5 and so on. In conventional devices, the EMF generated in the FM \nfilm under FMR must be carefully removed, for example, in spin injection and spin transport \nexperiments by the spin-pumping driven by the FMR.5-14 Meanwhile, the EMF generation \nphenomenon itself in an FM film under FMR is the focus of this study, independent of the EMF origins. \nWe have conceived an energy harvesting technology which uses this EMF generation phenomenon \nunder FMR. We have successfully demonstrated electrical charging using the EMF generated in an \nFM film itself under the FMR, and show that the EMF generation phenomenon under the FMR is  \n4 \n usable as an energy harvesting technology with appropriate control of the thermal conditions of the \nFM film. \nFigure 1(a) shows a schematic illustration of our sample structure and the experimental set-up \nto detect the EMF generated in the sample under FMR. An FM film was formed on a thermally-\noxidized silicon substrate using an electron beam deposition system at pressure <10-6 Pa. Ni 80Fe20 \n(Kojundo Chemical Lab. Co., Ltd., 99.99% purity) or Co 50Fe50 (Kojundo Chemical Lab. Co., Ltd., \n99.99%) was used as the FM film. The deposition rate and the substrate temperature during FM \ndeposition were set to 0.03 nm/s and room temperature (RT), respectively. No cover layer to prevent \nthe FM films from oxidizing was formed because only the EMF phenomenon generated in the FM \nfilms under the FMR was considered in this study, not the individual origins of EMFs. After the FM \ndeposition, the sample substrates were cut to the designed size as shown in Fig. 1(a). \nTo confirm the EMF properties generated in the FM films under FMR, a microwave TE 011-mode \ncavity in an electron spin resonance system (JEOL, JES-TE300) was used to excite the FMR in an FM \nfilm, and a nanovoltmeter (Keithley Instruments, 2182A) was used to measure the EMF. Lead wires \nto detect the output voltage properties were directly attached at both ends of the FM film with silver \npaste. \nTo evaluate the electric charging properties under the FMR, the electrical circuit shown in Fig. \n1(b) was connected to the FM film sample, in place of the nanovoltmeter used for the above EMF  \n5 \n confirmation. First, all of the switches S 1, S2 and S 3 were opened, and the capacitor (the capacitance \nis C) was completely discharged. In the case of charging experiments, S 1 and S 2 are closed, and the \nFMR of the FM film was excited by the same ESR system as described above. The electrical current \nderived from the EMF generated in the FM film under the FMR flowed and the electric charges were \naccumulated in the capacitor for the FMR duration time. The FMR condition in an in-plane field was \nset according to Kittel’s formula15 \nఠ\nఊ=ඥ𝐻ிெோ(𝐻ிெோ+4𝜋𝑀ௌ),   (1) \nwhere , , HFMR and MS are the angular frequency (2 f), the gyromagnetic ratios of 1.86 ×107 G-1s-1 \nfor Ni80Fe20,2,8 and 1.84×107 G-1s-1 for Co 50Fe50 calculated from the g-factor,16 the FMR field and the \nsaturation magnetization of the FM film, respectively. After the charging processes, both S 1 and S 2 \nwere opened. \nNext, the accumulated charges in the capacitor were discharged using a so-called RC-series \ncircuit, where the accumulated charges in the capacitor are consumed in a resistor (the resistance is R) \nas heat. Before starting discharge experiments for evaluation of the amount of charge accumulated in \nthe charging processes, S 3 was closed, that is, the same nanovoltmeter as above was connected. The \ntrigger for discharge is S 2. When S 2 is closed, an electric current due to the charge accumulated in the \nC starts to flow and is consumed at the resistor. In the RC-series circuit shown in Fig. 1(b), the \nelectrical voltage between the terminals of the resistor is defined as V(t), which is described by the  \n6 \n following equation:  \n𝑉(𝑡)= 𝑉𝑒𝑥𝑝ቀ−௧\nఛቁ,  (2) \nwhere V0, t, and  are the initial voltage corresponding to the accumulated charge during the FMR \nduration time, the duration time from the trigger of the discharge process, and the time constant of the \ndischarge circuit, which is defined as RC. In this study,  was always set to be 1 s for ease of \nmeasurement (for example, R : C = 1 : 1 F). All evaluations were performed at RT.  \n \n \nFIG. 1. (a) A schematic illustration of our sample structure and the experimental set-up to detect the \nEMF generated in the sample under FMR., (b) Electrical circuit to evaluate the electric charging \nproperties.  \n \nFigure 2(a) and (b) show the FMR spectrum of an Ni 80Fe20 film and the EMF generated in the \n \n7 \n same Ni 80Fe20 film itself under the FMR at the microwave frequency and power ( P) of 9.45 GHz and \n200 mW, respectively. In Fig. 2(b), the circles represent experimental data and the solid line is the \nfitted curve obtained using the following equation3-5,8-14 \n𝑉(𝐻) =𝑉ௌ௬௰మ\n(ுିுಷಾೃ)మା௰మ+𝑉௦௬ିଶ௰(ுିுಷಾೃ)\n(ுିுಷಾೃ)మା௰మ,  (3) \nwhere  denotes the damping constant (26 Oe for Ni 80Fe20 in this study). The first and second terms \nin Eq. (3) correspond to the symmetry term for H due to the ISHE, and the asymmetry term for H due \nto the AHE and/or other effects showing the same asymmetric voltage behavior relative to H, \nrespectively.3-5,8-14 VSym and VAsym correspond to the coefficients of the first and second terms in the \nEq. (3). The HFMR of the Ni 80Fe20 film was 1,100 Oe and the MS of the Ni 80Fe20 film was estimated to \nbe 646 emu/cc with Eq. (1). The output voltages from the Ni 80Fe20 film under the FMR are observed \nat HFMR. The observed EMF is mainly due to the self-induced ISHE in the Ni 80Fe20 film under FMR.3 \nFigure 2(c) and (d) show the FMR spectrum of a Co 50Fe50 film and the EMF generated in the \nsame Co 50Fe50 film itself under the FMR at the microwave frequency and P of 9.45 GHz and 200 mW, \nrespectively. In Fig. 2(d), the circles represent experimental data and the solid line is the fitted curve \nobtained using Eq. (3) with the  of 110 Oe for Co 50Fe50 in this study. The HFMR of the Co 50Fe50 film \nwas 572 Oe and the MS of the Co 50Fe50 film was estimated to be 1410 emu/cc with Eq. (1). The FMR \nspectrum of the Co 50Fe50 film is wider than that of the Ni 80Fe20 film, corresponding to the difference \nin magnetic anisotropy. Output voltages from the Co 50Fe50 film under the FMR are observed at HFMR.  \n8 \n The origins of the EMF generated in a Co 50Fe50 film under FMR are currently under investigation,17 \nwhile these might come from the ISHE, the AHE and so on, similarly to other FM films, like Ni 80Fe20, \nFe and Co films.3,4 While the apparent EMF generated in the Co 50Fe50 film at the HFMR is smaller than \nthat of the Ni 80Fe20 film, it is sufficient for the energy-harvesting experiments in this study.  \n \n \nFIG. 2. (a) FMR spectrum of a Ni 80Fe20 film and (b) the EMF generated in the same Ni 80Fe20 film \nunder FMR. (c) FMR spectrum of a Co 50Fe50 film and (d) the EMF generated in the same Co 50Fe50 \nfilm under FMR. The microwave frequency and power are 9.4 GHz and 200 mW, respectively.  \n \nThe energy-harvesting experiments are described below. First, using the charging circuit, the \nelectric current generated in the FM films under the respective FMR condition to satisfy Eq. (1) flows, \n \n9 \n and the capacitor is charged. The microwave power was 200 mW in all experiments except for the \nevaluation of P-dependence. Each FMR excitation was maintained for 30 min. with the FMR condition \nto satisfy the Eq. (1), and then, the capacitor was discharged. Figure 3(a) shows typical discharge \nproperties of the capacitor evaluated using the discharge circuit. Circles and triangles are experimental \ndata for a Ni 80Fe20 film and a Co 50Fe50 film, respectively. The solid lines are fitted curves obtained \nusing Eq. (2), and the respective data showed a good fit. The V0 is about 78 V for the Ni 80Fe20 film \nand 56 V for the Co 50Fe50 film. Figure 3(b) shows the P dependences of the discharge properties for \na Ni80Fe20 film. Each FMR duration time (the charging time) was 30 min. The solid lines are fitted \ncurves obtained using Eq. (2). Figure 3(c) shows the P dependence of the V0 analyzed from the Fig. \n3(b). The value of V0 increases linearly with the increase in P, that is, the charging is clearly due to the \nFMR phenomena. \n  \n10 \n  \nFIG. 3. (a) Typical discharge properties of capacitors evaluated using the discharge circuit. Circles and \ntriangles are experimental data for a Ni 80Fe20 film and a Co 50Fe50 film, respectively. (b) Microwave \npower (P) dependences of discharge properties for a Ni 80Fe20 film. Each FMR duration time (the \ncharging time) was 30 min. The solid lines are fitted curves obtained using Eq. (2). (c) The P \ndependence of the V0 obtained by analysis of FIG. 3 (b).  \n \nFigure 4 shows the FMR duration time dependence of the discharge properties of a Ni 80Fe20 \nfilm. The solid lines are fitted curves obtained using Eq. (2). Figure 4(b) shows the FMR duration time \ndependence of the V0 generated in the Ni 80Fe20 film, from analysis of Fig. 4(a). The amount of charge \nfrom the Ni 80Fe20 film tends to be saturated well below the voltage limit of the capacitor when the \nFMR duration time is over 15 min. To investigate the reason why the charge is saturated, we changed \n \n11 \n the capacitors and resistors while keeping  at 1s. Also, other Ni 80Fe20 films were tested. However, in \nall harvesting experiments with Ni 80Fe20 films, the charge was saturated against the FMR duration \ntime. Therefore, we changed the FM film from Ni 80Fe20 to another FM.  \n \n \nFIG. 4. (a) FMR duration time dependence of the discharge properties of a Ni 80Fe20 film. The solid \nlines are fitted curves obtained using Eq. (2). (b) The FMR duration time dependence of the V0 \ngenerated in the Ni 80Fe20 film from analysis of FIG. 4(a).  \n \nFigure 5 shows the FMR duration time dependence of the discharge properties of a Co 50Fe50 \nfilm. The solid lines are fitted curves obtained using Eq. (2). Figure 5(b) shows the FMR duration time \ndependence of the V0 generated in the Co 50Fe50 film obtained by analysis of Fig. 5(a). The amount of \ncharge from the Co 50Fe50 film increases almost linearly with increasing FMR duration time and is not \nsaturated. This behavior is different from the case of the Ni 80Fe20 film shown in the Fig. 4(b), and this \ncharacteristic shows good reproducibility with other Co 50Fe50 films. It is noticed that while the \n \n12 \n apparent EMF generated in the Co 50Fe50 film at the HFMR is smaller than that in the Ni 80Fe20 film, and \nthe FMR spectrum of a Co 50Fe50 film is much wider than that of a Ni 80Fe20 film. The FM film under \nFMR is basically heated. Notably, the FMR duration time in this study is very long compared with \ngeneral FMR experiments. The MS of an FM film becomes smaller at high temperature than at low \ntemperature. That is, as shown in the Eq. (1), the HFMR of an FM film becomes larger at high \ntemperature than at low temperature when the microwave frequency is the same. Thus, in the \nexperiments with Ni 80Fe20 films, the film was heated and the HFMR shifted to larger values compared \nwith the beginning of the experiment. Because the ESR-system parameters were kept the same in the \nexperiments, the Ni 80Fe20 film might have gone out of the FMR excitation range. Therefore, EMF was \nhardly generated and the charging of the capacitor almost stopped. Meanwhile, in the experiments \nwith Co 50Fe50 films, the film was heated under the FMR and the HFMR also became larger compared \nwith the beginning of the experiment. Similarly, the ESR-system settings were maintained in the \nexperiments. However, the Co 50Fe50 film might not be fully out of the FMR excitation range for \nCo50Fe50 films, due to the wider FMR spectrum than Ni 80Fe20. Therefore, the EMF generation in \nCo50Fe50 films under FMR was kept in a detuned state and the charging to the capacitor was maintained. \nOf course, thermally equivalent conditions for similar experiments must depend on the microwave \ncavity used and other thermal factors. That is, by appropriately controlling the thermal conditions \naround the FM film, the charging was maintained. The above result indicates that energy harvesting  \n13 \n experiments using FMR were successfully demonstrated.  \n \n \nFIG. 5. The FMR duration time dependence of discharge properties for a Co 50Fe50 film. The solid lines \nare fitted curves obtained using Eq. (2). (b) The FMR duration time dependence of V0 generated in the \nCo50Fe50 film from analysis of the data in FIG. 5(a). The dashed line is a guide for the eyes.  \n \nAt present, no diodes were connected to the circuit to rectify the electrical currents. To efficiently \ncharge the capacitor and to increase the amount of accumulated charge, the use of diodes may be \neffective because the currents generated by an FM film under the FMR are very small, and controlling \nthe flow of such micro-currents is usually difficult. The microwave power was kept the same (200 \nmW) in all experiments except for the evaluation of P-dependence. A smaller microwave power may \nbe preferable to reduce heating of the films and to precisely control the thermal conditions of the FM \nfilm under FMR. For FMR excitations, in general, a large electric power is required to apply the \nelectric current to provide a static magnetic field and a high frequency field. Thus, methods should be \n \n14 \n developed to reduce the electric power required for the excitation of FMR, using, for example, \npermanent magnets to create the static magnetic field and environmental electromagnetic waves for a \nhigh frequency magnetic field. While those might be hard to be establish, such technology is eagerly \nawaited because a lot of GHz-band microwaves exist in modern environments such as those used in \nwireless internet services. The above issues must be solved for practical use. \nIn summary, we successfully demonstrated electrical charging using the EMF generated in a FM \nfilm under FMR. In the case of Ni 80Fe20, electrical charge due to the EMF generated under the FMR \nwas stored in a capacitor ; however, the amount of charge was saturated well below the charging limit \nof the capacitor despite the increase in FMR duration time. Meanwhile in the case of Co 50Fe50, \nelectrical charge generated under the FMR was stored in a capacitor and the amount of charge \nincreased linearly with the FMR duration time. The FMR spectrum of the Co 50Fe50 films was wider \nthan that of Ni 80Fe20. In equivalent thermal states during FMR experiments, Co 50Fe50 films maintained \nFMR in a detuned condition, while Ni 80Fe20 films were outside of the FMR excitation range. The \nabove result indicated that the EMF generation phenomenon in FM films under FMR might be usable \nas an energy harvesting technology, by appropriately controlling the thermal conditions of the FM \nfilms. \n \nThis research was partly supported by a research grant from The Murata Science Foundation  \n15 \n and a research grant from The Mazda Foundation. \n \n   \n16 \n References \n1H. Akinaga, H. Fujita, M. Mizuguchi, and T. Mori, Sci. Technol. Adv. Mater. 19, 543 (2018). \n2C. Kittel, Introduction to Solid State Physics (8th ed.), Wiley (2004). \n3A. Tsukahara, Y. Ando, Y. Kitamura, H. Emoto, E. Shikoh, M.P. Delmo, T. Shinjo, and M. \nShiraishi, Phys. Rev. B 89, 235317 (2014).  \n4K. Kanagawa, Y. Teki, and E. Shikoh, AIP Advances 8, 055910 (2018). \n5E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). \n6S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002). \n7Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. Lett. 88, 117601 (2001). \n8K. Ando and E. Saitoh, J. Appl. Phys. 108, 113925 (2010). \n9E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and M. Shiraishi, Phys. Rev. Lett. 110, \n127201 (2013). \n10Y. Tani, Y. Teki, and E. Shikoh, Appl. Phys. Lett. 107, 242406 (2015). \n11Y. Tani, K. Kondo, Y. Teki, and E. Shikoh, Appl. Phys. Lett. 110, 032403 (2017). \n12Y. Tanaka, T. Kono, Y. Teki, and E. Shikoh, IEEE Trans. Magn. 55, 1400304 (2019). \n13K. Nishida, Y. Teki, and E. Shikoh, Solid State Commun. 312, 113898 (2020). \n14K. Tamura, T. Kanki, S. Shirai, H. Tanaka, Y. Teki, and E. Shikoh, AIP Advances 11, 035120 \n(2021).   \n17 \n 15C. Kittel, Phys. Rev. 73, 155 (1948). \n16F. Schreiber, J. Pflaum, Z. Frait, Th. Miihge, and J. Pelzl, Solid State Commun., 93, 965 (1995). \n17S. Baek, Y. Teki, and E. Shikoh, in preparation . \n "    },    {        "title": "1710.01534v1.Possible_evidence_for_spin_transfer_torque_induced_by_spin_triplet_supercurrent.pdf",        "content": "Possible evidence for spin-transfer  torque  induced by spin-\ntriplet supercurrent  \n \nLailai Li1,3, Yuelei Z hao2*, Xixiang Zhang2, and Young Sun1,3*  \n1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese \nAcademy of Sciences, Beijing 100190, People’s Republic of China  \n2King Abdullah University of Science and Technology (KAUST), Thuwal 23955- 6900, Saudi \nArabia  \n3School of Physical Science, University of Chinese Academy of Sciences, Beijing 100190, \nPeopl e’s Republic of China  \n*email: youngsun@iphy.ac.cn; yuelei.zhao@kaust.edu.sa   \n \nCooper pairs i n superconductor s are normally spin singlet  (spin antiparallel) . Nevertheless, recent studies \nsuggest that spin -triplet Cooper pairs  (spin parallel ) can be created at carefully engineered superconductor -\nferromagnet  interfaces1-4. If Cooper pairs are spin -polarized they would transport  not only charge  but also a \nnet spin component, but without  dissipation , and therefore  minimize  the heating e ffects associated with \nspintronic devices. Although it i s now established that triplet supercurrents  exist,  their most interesting \nproperty  – spin – is only inferr ed indirectly  from transport  measurements5-10. In conventional spintronics, it \nis well known that spin currents generate spin-transfer torques that alter magnetization dynamics  and \nswitch magnetic moments . The observation of similar e ffects due to spin -triplet supercurrents would  not \nonly confirm the net spin of triplet pairs but also  pave the way for applications  of superconducting \nspintronics . Here, we present a possi ble evidence for spin -transfer torques induced by triplet supercurrents  \nin superconductor/ferromagnet/superconductor (S/F/S) Josephson junctions . Below  the superconducting \ntransition temperature  TC, the ferromagnetic resonance (FMR) field at X-band (~ 9.0  GHz) shifts rapidly to \na lower field with decreasing temperature due to the  spin-transfer  torque s induced by triplet supercurrent s. \nIn contrast, this phenomenon is absent  in ferromagnet/superconductor  (F/S)  bilayers and \nsuperconductor/ insulator /ferromagnet /superconductor (S/I/F/S) multilayers  where no supercurrent s pass \nthrough the ferromagnetic layer . These experimental observation s are discussed  with theoretical \npredictions for  ferromagnetic  Josephson junctions with precessing magnetization .  \n \nSpin-triplet supercurrent s combining \nsuperconducting and magnetic orders provide a \ngreat opportunity  to enhance the functionality and \nperformance of spintronic devices  by offering the \npossibility  of long- range spin- polarized \nsupercurrents. As spin -one triplet Cooper pairs, \nunlike singlet pairs, can carry a net spin  component , \na spin -polarized current is naturally associated with \nthe triplet supercurrent s. Meanwhile,  spin-one \ntriplet Cooper pairs are immune to pair breaking by \nthe exchange fi eld in ferromagnets so that  triplet \nCooper pairs sustain  long-range correlations  in \nspintronic devices . To use such triplet supercurrents \nin spintronics it is  necessary to effectively  generate \nand manipulate triplet pairs in  devices.  \nIn the past decade, a number of theoretical \nmodels have been propos ed to explain how spin-\npolarized  supercurrents can be created and \ncontrolled  in S/F heterostructure s11-20, with key \ningredients  ranging from non -uniform \nsuperconductor, inhomogeneous and non-collinear \nmagnetization to strong spin-orbital coupling, etc. \nThe first experimental evidence for long- range triplet supercurrent s was reported by Keizer et al.5 \nfrom the observation of  supercurrent passing  \nthrough the half -metal lic ferromagnet  CrO 2. Then a \nseries of experiments demonstrated systematic \nevidence for triplet supercurrent s in S /F/S \nJosephson junctions6-10. Although these existing \nexperiments provide compelling evidence s for \ntriplet pairing in S /F hetero structures, they are not \ndirectly probing or using the spin carried by triplet \nsupercurrents.  \nA well-known and useful phenomenon in \nspintronic s is the spin -transfer torque  induced by  \nspin-polarized  current s, which has been widely \nused to switch magnetization  and control  \nmagnetization dynamics21. Similarly , the triplet \nsupercurrents are anticipated to induce spin -transfer \ntorques when passing through a ferromagnet . The \ndemonstra tion of spin -transfer torques due to triplet \nsupercurrents would not only confirm the net spin \nof triplet pairs but also pave the way for the \nemergence of superconducting spintronics.  \nRecently , there are quite a number  of theoretical  \nworks addre ssing on the spin- transfer torques and \n1 \n magnetization dynamics related  to triplet \nsupercurrents22-31. However , the experiment al \nstudies lie well behind  theoretical progresses.  So \nfar no clear experimental evidence for spin-transfer \ntorques induced by triplet supercurrents has been \nreported .   \nIn this work , we have investigate d \nmagnetization dynamics in  a series of S/F/S \nJosephson junctions  via FMR  technique . The \nresults demonstrate a significant influence of \nsuperconductivity on magnetization dynamics: t he \nresonance field  (Hr) shifts rapidly to a lower field \nbelow  superconduct ing transition temperature  Tc. In \ncontrast, such an effect is  absent in the control \nexperiments performed on S/F bilayers and S/I/F/S \nmultilayers where no supercurrents pass through \nthe device. Therefore, our experiments  provide an \nevidence for the spin-transfer torques induced by \ntriplet supercurrents . \n34567891011120.00.20.4b\n R (mΩ)\nT (K) warming\n cooling\nTc ~ 7.7 Ka\n \nFigure 1 Geometry of the ferromagnetic Josephson \njunction s and the configuration of FMR measurements.  \na, The Josephson junctions  in this study consist  of a \nferromagnetic layer ( Ni80Fe20, 5-30 nm  in thickness ) and \ntwo superconductor layers ( Nb, 100 nm  in thickness ). The \nFMR is measured at a fixed microwave frequency (~ 9 GHz) \nwhile scanning the DC magnetic field applied in the film \nplane. b, The resistance of a  Nb film as a function o f \ntemperature. The superconducting transition temperature TC \nis ~ 7. 7 K. \n \nThe geometry of the ferromagnetic  Josephson  \njunctions and the  schematic of FMR are shown in \nFig. 1a. When a DC magnetic field HDC is applied \nnot along the direction of magnetization , the \nmagnetization  will rotate to the direction of HDC \nalong spiral path by the driven torque  and damping \ntorque . If a microwave field with magnetic \ncomponent  hmw perpendicular to H DC is applied, \nthe magnetization  can absorb microwave energy and precess  continuously  in balance with the \ndamping torque . This is the basic principle  of FMR. \nIn our study, the ferromagnetic Josephson junctions are made of two superconducting layers of  Nb (100 \nnm in thickness ) and a  FM layer  of Ni\n80Fe20 (5 - 30 \nnm in thickness ). As shown in Fig. 1b, the transport \nmeasurement suggests a superconducting transition \ntemperature T c ~ 7.7 K  of the Nb layer .  \nThe FM R spectr a of a Nb(100 nm)/Ni 80Fe20(20 \nnm)/Nb(100 nm)  Josephson junction  measured at \nX-band (~ 9 GHz) are shown in Fig. 2a. All the \nresonance lines  exhibit a single Lorenz line -shape.  \nThe resonance field  Hr changes little  with \ntemperature above Tc. However , Hr shifts rapidly to \na lower field with decreasing temperature below T c. \nThe t emperature dependence of H r is plotted  in Fig . \n2b. As temperature decreas es from T C (7.7 K) to \n4.2 K, Hr shifts by ~ 70 mT , indicating a strong \ninfluence  on magnetization dynamics by \nsuperconductivity . We note that the shift of  Hr \nbelow TC is clearly seen in a series of S/F/S  \nJosephson junctions with different thickness of  \nNi80Fe20 layer ranging from 5 to 30 nm ( see Fig. \nS1-S4 in Supplemental Information).  \n0 30 60 90 120 150\n  dP/dH (a.u.)\nµ0H (mT) 4.2 K\n 4.8 K\n 5.2 K\n 5.8 K\n 6.3 K\n 6.6 K\n 7.1 K\n 7.6 K\n 7.7 K\n 15.3 K\n 30.8 Kdecreasing T\n0 5 10 15 20 25 3020406080100\nbµ0Hr (mT)\nT (K)a\n \nFigure 2 FMR spectra  of the Nb(100 nm)/Ni 80Fe20(20 \nnm)/Nb(100 nm) Josephson junction . a, The FMR spectra \nas a function of temperature. Below the superconducting \ntransition temperature  TC ~ 7.7 K , the resonance field H r \nshifts rapidly to a lower field with decreasing temperature.  b, \nThe resonance field  Hr as a function of temperature. The \ninset plots the structure of the sample. The significant shif t \nof Hr below TC evidences a strong influence  on \nmagnetization dynamics induced by superconductivity.  \n2 \n 20 40 60 80 100\nb\ndP/dH (a.u.)\nµ0H (mT) 4.2 K\n 8.1 K\n 10.3 Ka\n80 100 120 4.2 K\n 8.2 K\n 11.0 KdP/dH (a.u.)\nµ0H (mT)\n \nFigure 3  Control e xperiments on S/F bilayer and S/I/F/S multilayer. a , FMR spectra of a Nb(100 nm)/Ni 80Fe20(20 nm) \nbilayer. The resonance field H r shifts little  below the superconducting transition temperature T C. b, FMR spectra of a Nb (100 \nnm)/MgO (10 nm) /Ni 80Fe20(20 nm) /Nb(100 nm)  multilayer . The resonance field H r does not shift below T C. These control \nexperiments suggest that the shift of H r is caused by supercurrents passing through the ferromagnetic layer  rather than a local \neffect at one S/F interface.  \n \nFor comparison, we also measured  the FMR  of \na Nb(100 nm) /Ni80Fe20(20 nm) bilayer. As show n \nin Fig . 3a, for the S/F bilayer , Hr does not shift \nobviously below T C. From 10 K to 4.2 K Hr only \nshifts about 1 mT. This observation is similar to a \nprevious FMR study of S/F bilayers22. This control \nexperiment clarifie s that the shift of H r is closely \nrelated  to the geometry of ferromagnetic Josephson  \njunction s rather than one S/F interface. According \nto previous studies , the saturation magnetization  Ms \nof ferromagnetic  layer changes little  (< 1%) below \nTC by the interaction with superconductivity in the  \nS/F/S trilayers and multilayers32,33. Thus, i t can  not \naccount for the significant shift of H r (~ 70 mT) \nbelow T C.  \nThe shift of  Hr to a lower field indicates that an \neffective inner magnetic field parallel to the \nexternal magnetic field is produced in the \nsuperconducting state. In other words, there should \nbe an ext ra torque  induced by superconductivity  to \nassist  the external fiel d torque to keep the \nmagnetization precession. As this extra torque \nbelow TC is observed in S/F/S junctions but not in \nthe S/F bilayer, i t implies that supercurrents passing \nthrough Josephson junctions , rather than a local \neffect at one S/F interface, could play a critical role. \nTo verify this viewpoint,  we have performed \nanother control experiment in a S/I/F/S multilayer where  the supercurrent s are  blocked by the \ninsulating MgO layer .  \nFor typical S/MgO/S Josephson junctions, the \nthickness of MgO laye r is usually below 2 nm. Above 2 nm the wave function can  not overlap  and \ntunneling supercurrent s will be blocked. We then \nmade a S/ I/F/S multilayer with the thickness of \ninsulating  MgO  layer of  10 nm. The FMR spectra \nof this insulating multilayer is presented  in Fig . 3b. \nNo obvious shift of H\nr is observed below T C. From \n11 K to 4.2 K, Hr only shifts about 1 mT. This \nsecond control experiment further confirms that the \nextra torque  below TC is due to supercurrents  \npassing through the  ferromagnetic layer.  Since \nsinglet supercurrents do not carry a net spin and \nshould not cause a spin -transfer torque, it is \nconcluded that the extra torque below TC is induced \nby triplet supercurrents.   \nIn the following, we discuss the mechanism of \nspin-transfer torque s induced by triplet \nsupercurrents in the S/F/S Josephson junctions . The \nFMR experiments  in our study is a situation of \nferromagnetic Josephson junctions  with precessing \nmagnetization . Several theoretical models28-31 have \ndiscussed on this situation and predicted that the \nlong range triplet  supercurrents  can be stimulated \nby varying in time  (rather than in space ) the \norientation of the magnetization  in the ferromagnet.  \nThe microwave causes a precession of the \nmagnetization,  which corresponds to the pumping \nof a uniform  (q = 0) mode of magnons  to the  \nsystem . The presence of pumped magnons leads to \nthe long- range triplet proximity effect. The pumped  \nmagnon spin current is compensated by the spin \ncurrent  carried by the triplet Cooper pairs due to the \ntotal spin  angular momentum conservation. At the \n3 \n frequency of FMR, t he critical supercurrent is \ngreatl y amplified due to the precessing \nmagnetization  and the generated triplet \nsupercurrents through the junctions  in turn exert a \ntorque on the precessing magnetization.  \n \nFigure 4 Schematic illustration of triplet supercurrents  \n(TSC) induced spin-transfer torque in S/F/S Josephson \njunctions with precessing magnetization.  Away from the \nS/F interfaces, only s inglet Coop er pairs can exist. Triplet \ncooper pairs are generated  at the interfaces  due to the \nprecessing magnetization . The triplet pairs with up spins \n(parallel to the external DC magnetic field ) can transport  \nthrough the F layer  whereas the pairs with down spins are \nreflected back . The triplet pair s passing through the F layer  \nexert a spin -transfer torque on the magnetization, causing a \nshift of resonance field at a fixed microwave frequency.  \n \nFigure 4 presents a schematic illustration of the \ndynamic process in the ferromagnetic junctions. \nAway from the S/F interfaces, only spin -singlet \nCooper pairs can exi st below TC. A conversion \nfrom spin-singlet pairs to spin -triplet pairs due to \nthe spin-mixing or spin- flip scattering occurs at the \ninterfaces . The dynamic ally precessing \nmagnetization play s an important role not only for \nthe conversion process but also for the formation of \nCooper pair s, by  which the coherent charge and \nspin transport takes place  through the junction due \nto the conservation of total spin angular momentum \ncarried by triplet pairs and magnons28-30. For the \ntriplet pairs with up spins (parallel to the external \nDC magnetic  field) , they  can pass through the F \nlayer. However , for the triplet pairs with down \nspins , they will be reflected back to the interface.  \nThen triplet Cooper pairs passing through the ferromagnetic metal can exert a torque on the \nmagnetization. This torque has the same direction  \nas the torque generated by the DC magnetic  field. \nAs a consequence,  the resonance field  H\nr shifts to a \nlower field.  \nIn summary , our FMR experiments in a series \nof S/F/S Josephson junctions demonstrate a \nsignificant modification on magnetization \ndynamics  induced by superconductivity. In contrast, \nsuch a phenomenon is absent in S/F bilayers and \nS/I/F/S multilayers. Therefore, these results provide a strong  evidence for the existence of spin -transfer \ntorques induced by long- range triplet supercurrents. \nThe observation of spin- transfer torque associated \nwith triplet supercurrents as well as its strong \ninfluence on magnetization dynamics  pave s a way \nto the application s of superconduct ing spintronic \ndevices.  \n \nReferences  \n1. Linder, J. & Robinson, J. W. A. Superconducting \nspintronics. Nature Phys. 11, 307- 315 (2015).  \n2. Matthias, E. Spin- polarized supercurrents for \nspintronics: a review of current progress. Rep. Pro. \nPhys.  78, 104501 (2015).  \n3. Bergeret, F. S., Volkov, A. F . & Efetov, K. B. \nOdd triplet superconductivity and related phenomena in superconductor -ferromagnet \nstructures. Rev. Mod. Phys. 77, 1321- 1373 (2005).  \n4. Eschrig, M. & Löfwander, T. Triplet supercurrents in clean and disordered half -metallic \nferromagnets. N ature Phys.  4, 138- 143 (2008).  \n5. Keizer, R. S., Goennenwein, S. 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Supercurrent -\ninduced magnetization dynamics in a Josephson \njunction with two misaligned ferromagnetic layers. \nPhys. Rev. B  83, 012501 (2011). \n27. Hoffman, S., Blanter, Y. M. & Tserkovnyak, Y. \nNonlinear dynamics in a magnetic Josephson \njunction. Phys. Rev. B  86, 054427 (2012).  \n28. Takahashi, S. , Hikino, S. , Mori, M. , Martinek, J. \n& Maekawa S. Supercurrent pumping in Josephson \njunctions with a half-metallic ferromagnet . Phys. \nRev. Lett.  99, 057003 (2007) . 29. Houzet, M. Ferromagnetic Josephson junction \nwith precessing  magnetization. Phys. Rev. Lett.  101, \n057009 (2008).  \n30. Holmqvist, C., Teber, S. & Fogelström, M. \nNonequilibrium effects in a Josephson junction \ncoupled to a precessing spin. Phys. Rev. B  83, \n104521 (2011).  \n31. Holmqvist, C., Belzig, W. & Fogelström, M. \nSpin-precession -assisted supercurrent in a \nsuperconducting quantum point contact coupled to \na single -molecule magnet. Phys. Rev. B  86, 054519 \n(2012).  \n32. Wu, H., Ni, J., Cai, J., Cheng, Z. & Sun, Y.  \nExperimental evidence of magnetization \nmodification by superconductivity in a Nb/Ni\n81Fe19 \nmultilayer. Phys. Rev. B  76, 024416 (2007).  \n33. Zou, T., Wu, H., Cheng, Z. & Sun, Y. \nMagnetization modification by superconductivity \nin Nb/Ni 80Fe20/Nb trilayers. J. Magn. Magn. Mater.  \n322, 169 -172 (2010).  \n \nMethod s  \nThe superconductor -ferromagnet heterostructures  \nincluding Nb/Ni80Fe20/Nb Josephson junctions and \nNb/Ni80Fe20/Al(cap)  bilayer  are fabricated using dc \nmagnetron sputtering on glass substrates . The base \npressure  of the sputtering  system  is about  10-6 Pa. \nThe film s are deposited at an Ar pressure  of 0.5 Pa. \nThe MgO layer in the Nb/ MgO/ Ni80Fe20/Nb \nmultilayer is deposited by radio frequency \nsputter ing with a n Ar pressure about 0.8 Pa.  \nThe FMR  measurement s are performed in a JEOL \nJA-200 spectrometer with a n X-band ( f ≈ 9.0 GHz) \ncavity  resonator . The system is  equipped with a \nvariable temperature unit  down to liquid helium \ntemperature.   \n \nAcknowledgments  \nThe authors are grateful to Prof. Jian- Wang Cai, Dr. \nQin-Li Lu, and Mr. Yan Wen  for help in sample \npreparation. This work was supported by the \nNational Natural Science Foundation of China \n(Grant Nos. 11534015 and 51371192) and the \nNational Key Research and  Development Program \nof China (Grant No. 2016YFA0300700). Y.S. also \nacknowledges the support from Chinese Academy \nof Sciences (Grants No. XDB07030200 and KJZD -\nEW-M05).  \n \nAuthor contributions  \nY.S. and Y.Z. initialized this study. Y.Z. prepared \nseveral sampl es and carried out the transport \nmeasurements. L.L. performed FMR measurements. \nX.Z. contributed to materials and data analysis. Y.S. \nand Y.Z. wrote the paper, and all authors reviewed \nthe paper.  \n  \n5 \n Supplemental Information  \n \n60 70 80 90 100 110 120 130dP/dH (arb. u.)\nµ0H (mT) 4.2 K\n 4.6 K\n 5.2 K\n 5.7 K\n 6.2 K\n 6.6 K\n 7.0 K\n 7.4 K\n 7.9 K\n 8.1 K\n 9.9 K\n 20.2 KNb/Py(5 nm)/Nb\nFigure S1 FMR spectra of a Nb(100 nm)/ Ni80Fe20(5 nm)/Nb(100 nm) Josephson junction.  \n \n40 60 80 100 120 140dP/dH (arb. u.)\nµ0H (mT) 4.2 K\n 4.6 K\n 5.0 K\n 5.5 K\n 6.0 K\n 6.6 K\n 7.1 K\n 7.5 K\n 8.0 K\n 10.0 KNb/Py(10 nm)/Nb\n \n \nFigure S2. FMR spectra of a Nb(100 nm)/ Ni80Fe20(10 nm)/Nb(100 nm) Josephson junction. \n \n \n  \n6 \n 0 20 40 60 80 100 120 140 160dP/dH (arb. u.)\nµ0H (mT) 4.2 K\n 4.6 K\n 5.2 K\n 5.8 K\n 6.2 K\n 6.7 K\n 7.2 K\n 7.7 K\n 8.0 K\n 9.9 KNb/Py(30 nm)/Nb\n \n \nFigure S3. FMR spectra of  a Nb(100 nm)/ Ni80Fe20(30 nm)/Nb(100 nm) Josephson junction. \n \n \n0 4 8 12 16 200.00.20.40.60.81.0\n 5  nm\n 10 nm\n 20 nm\n 30 nmHr(T)/Hr(20 K)\nT (K)Py thickness\nFigure S4. The relative shift of resonance field, Hr(T)/Hr(20 K), for a series of Nb/ Ni80Fe20/Nb Josephson \njunction s with different thickness of Ni 80Fe20 layer.  \n \n7 \n "    },    {        "title": "0710.1974v3.Microwave_photovoltage_and_photoresistance_effects_in_ferromagnetic_microstrips.pdf",        "content": "arXiv:0710.1974v3  [cond-mat.str-el]  10 Apr 2008Microwave photovoltage and photoresistance effects in ferr omagnetic microstrips\nN. Mecking,1,2,∗Y.S. Gui,1and C.-M. Hu1,†\n1Department of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2\n2Institut f¨ ur angewandte Physik und Zentrum f¨ ur Mikrostruk turforschung,\nUniversit¨ at Hamburg, Jungiusstraße 11, 20355 Hamburg, Ge rmany\nWe investigate the dc electric response induced by ferromag netic resonance in ferromagnetic\nPermalloy (Ni 80Fe20) microstrips. The resulting magnetization precession alt ers the angle of the\nmagnetization with respect to both dc and rf current. Conseq uently the time averaged anisotropic\nmagnetoresistance (AMR) changes (photoresistance). At th e same time the time-dependent AMR\noscillation rectifies a part of the rf current and induces a dc voltage (photovoltage). A phenomeno-\nlogical approach to magnetoresistance is used to describe t he distinct characteristics of the pho-\ntoresistance and photovoltage with a consistent formalism , which is found in excellent agreement\nwith experiments performed on in-plane magnetized ferroma gnetic microstrips. Application of the\nmicrowave photovoltage effect for rf magnetic field sensing i s discussed.\n1. INTRODUCTION\nThe fact that macroscopic mutual actions exist\nbetween electricity and magnetism has been known\nfor centuries as described in many text books of\nelectromagnetism1. Now, this subject is transform-\ning onto the microscopic level, as revealed in vari-\nous spin-charge coupling effects studied in the new\ndiscipline of spintronics. Among them, striking\nphenomena are the dc charge transport effects in-\nduced by spin precession in ferromagnetic metals,\nwhich feature both academic interest and technical\nsignificance2,3. Experiments have been performed inde-\npendently by a number of groups on devices with differ-\nent configurations4,5,6,7,8,9,10,11,12,13,14,15,16. Most works\nwere motivated by the study of spin torque17,18, which\ndescribes the impact of a spin-polarized charge current\non the magnetic moment. In this context, Tulapurkar et\nal.made the first spin-torque diode4, and Sankey et al.\ndetected the spin-torque-driven ferromagnetic resonance\n(FMR) electrically5. Both measured the vertical trans-\nport across nano-structured magnetic multilayers. Along\na parallel path, a number of works19,20,21have been de-\nvoted to study the effect of spin pumping. One of the\ninteresting predictions is that injection of a spin current\nfrom a moving magnetization into a normal metal in-\nduces a dc voltage across the interface. To detect such\na dc effect induced by spin pumping20, experiments have\nbeen performed by measuring lateral transport in hybrid\ndevices under rf excitation6,7,8.\nFrom a quite different perspective, Gui et al.set out\nto explore the general impacts of the high frequency\nresponse on the dc transport in ferromagnetic metals9,\nbased on the consideration that similar links in semicon-\nductors have been extensively applied for electrical de-\ntection of both spin and charge excitations22. Guiet al.\ndetected, subsequently, photoresistance induced by bolo-\n∗Electronic address: nmecking@physnet.uni-hamburg.de\n†Electronic address: hu@physics.umanitoba.cametric effect9, as well as photocurrent10, photovoltage11,\nand photoresistance12caused by the spin-rectification ef-\nfect. A spin dynamo10was thereby realized for gener-\nating dc current via the spin precession, and the device\nwas applied for a comprehensive electrical study of the\ncharacteristics of quantized spin excitations in micro-\nstructured ferromagnets11. The spin-rectification effect\nwasindependentlyinvestigatedbyboth, Costache et al.13\nand Yamaguchi et al.14, and seems to be also responsi-\nble for the dc effects detected earlier by Oh et al.15. A\nmethodfordistinguishingthephotoresistanceinducedby\neither spin precession or bolometric effect was recently\nestablished12, which is based on the nice work performed\nby Goennenwein et al.16, who determined the response\ntime of the bolometric effect in ferromagnetic metals.\nWhile most of these studies, understandably, tend to\nemphasize different nature of dc effects investigated in\ndifferent devices, it is perhaps more intriguing to ask the\nquestions whether the seemingly diverse but obviously\nrelated phenomena could be described by a unified phe-\nnomenological formalism, and whether they might arise\nfrom a similar microscopic origin. From a historical per-\nspective, these two questions reflect exactly the spirit of\ntwo classic papers23,24published by Juretschekeand Sils-\nbeeet al., respectively, which have been often ignored\nbut have shed light on the dc effects of spin dynamics in\nferromagnets. In the approachdeveloped by Juretscheke,\nphotovoltageinduced by FMR in ferromagneticfilms was\ndescribed based on a phenomenologicaldepiction ofmag-\nnetoresistive effects23. While in the microscopic model\ndeveloped by Silsbee et al.based on the combination of\nBloch and diffusion equations, a coherent picture was\nestablished for the spin transport across the interface\nbetween ferromagnets and normal conductors under rf\nexcitation24.\nThe goal of this paper is to provide a consistent view\nfor describing photocurrent, photovoltage, and photore-\nsistance of ferromagnets based on a phenomenological\napproach to magnetoresistance. We compare the theo-\nretical results with experiments performed on ferromag-\nnetic microstrips in detail. The paper is organized in\nthe following way: in section 2, a theoretical descrip-2\ntion of the photocurrent, photovoltage, and photoresis-\ntance in thin ferromagnetic films under FMR excitation\nis presented. Sections 2.1-2.4 establish the formalism for\nthe microwave photovoltage (PV) and photoresistance\n(PR) based on the phenomenological approach to mag-\nnetoresistance. These arise from the non-linear coupling\nof microwave spin excitations (resulting in magnetiza-\ntionMprecession) with charge currents by means of the\nanisotropic magnetoresistance (AMR). Section 2.5 com-\npares our model with the phenomenological approachde-\nveloped by Juretscheke. Section 2.6 provides a discus-\nsionconcerningthemicrowavephotovoltageandphotore-\nsistance based on other magnetoresistance effects (like\nanomalous Hall effect (AHE), giant magnetoresistance\n(GMR) and tunnelling magnetoresistance (TMR)).\nExperimental results on microwave photovoltage\nand photoresistance measured in ferromagnetic mi-\ncrostrips are presented in section 3 and 4, respectively.\nWe focus in particular on their characteristic differ-\nent line shapes, which can be well explained by our\nmodel. In section 5 conclusionsand an outlook are given.\n2. MICROWAVE PHOTOVOLTAGE AND\nPHOTORESISTANCE BASED ON\nPHENOMENOLOGICAL AMR\n2.1. AMR COUPLING OF SPIN AND CHARGE\nTheAMR-couplingofspinandchargeinferromagnetic\nfilms results in microwave photovoltage and photoresis-\ntance. The photovoltage can be understood regarding\nOhms law (current I(t) and voltage U(t))\nU(t) =R(t)·I(t) (1)\nWe consider a time-dependent resistance R(t) =R0+\nR1cos(ωt−ψ) which oscillates at the microwave fre-\nquencyω= 2πfdue to the AMR oscillation arising from\nmagnetizationprecession. ψistheoscillationsphaseshift\nwith respect to the phase of the rf current I(t). For\nthe sake of generality ψwill be kept as a parameter in\nthis work and will be discussed in detail in section 3.3.\nI(t) takes the form I(t) =I1cos(ωt) and is induced by\nthe microwaves. It follows that U(t) consists of time-\ndependent terms with the frequency ω, 2ωand a con-\nstant term (time independent) which corresponds to the\ntime average voltage and is equal to the photovoltage:\nUMW=/angbracketleftbig\nR1I1cos(ωt−ψ)cos(ωt)/angbracketrightbig\n= (R1I1cosψ)/2\n(∝an}bracketle{t ∝an}bracketri}htdenotes time-averaging). A demonstrative picture\nof the microwavephotovoltage mechanism can be seen in\nfigure 1.\nThe second effect we investigate which is also based on\nAMR spin-charge coupling is the microwave photoresis-\ntance ∆RMW. This has been reported recently13with\nthe equilibrium magnetization M0of a ferromagnetic\nstripe aligned to a dc current I0. Microwave induced\nFIG. 1: (Color online). Mechanism of the AMR-induced mi-\ncrowave photovoltage: Mprecesses (period P) in phase with\nthe rf current I. (a)Mlying almost perpendicular to Iresults\nin low AMR. (b) Mlying almost parallel to Iresults in high\nAMR. The time average voltage Ubecomes non-zero.\nFIG. 2: (Color online). Mechanism of the AMR-induced pho-\ntoresistance: (a) Without microwaves (MW) Mlies perpen-\ndicular to the dc current Iand the AMR is minimal (b) With\nmicrowaves Mprecesses and is not perpendicular to Iany-\nmore. Consequently the AMR increases (higher voltage drop\nU).\nprecessionthen misalignesthe dynamicmagnetization M\nwith respect to I0and thus makes the AMR drop mea-\nsurably. In this work we present results which also show\nthat ifM0lies perpendicular to I0the opposite effect\ntakes place: Microwave induced precession causes Mto\nleave its perpendicular position what increases the AMR\n(see figure 2).\nAfter thisqualitativeintroductionwewanttogoahead\nwith a quantitative description of the AMR induced mi-\ncrowave photovoltage and photoresistance. Therefore we\ndefine an orthogonalcoordinate system (x,y,z) (see figure\n3). The y-axis lies normal to the film plane and the z-\naxis is aligned with the magnetic field Hand hence with\nthe magnetization Mwhich is always aligned with Hin\nour measurements because of the sample being always\nmagnetized to saturation.\nGeometrically our samples are thin films patterned to\nstripe shape, so that d≪w≪l, whered,wandl3\nFIG. 3: (Color online). (x,y,z) and (x′,y,z′) coordinate\nsystems in front of a layout of our Permalloy film stripe\n(200×2400µm2) with 2 contacts and 6 side junctions.\nFIG. 4: (Color online). Sketch of the magnetization preces-\nsion. The magnetic field Hencloses the angle α0with the\ncurrentI. The magnetization oscillation towards Ihas the\namplitude α1and that perpendicular to I:β1.\nare the thickness, width and length of the sample. We\napplyHalways in the ferromagnetic film plane. For\ncalculations based on the stripes geometry the coordi-\nnatesx′andz′are defined. These lie in the film plane.\nx′is perpendicular and z′parallel to the stripe. The\nfollowing coordinate transformation applies: ( x,y,z) =\n(x′cos(α0)−z′sin(α0),y,z′cos(α0) +x′sin(α0)) where\nα0is the angle between Hand the stripe.\nFor the microwave photovoltage and photoresistance\nthe longitudinal resistance R(t) =R0+RAcos2θ(t)\nof the film stripe matters. It consists of the minimal\nlongitudinal resistance R0and the additional resistance\nRAcos2θ(t) from AMR. θ(t) is the angle between the\nz′-axis (parallel to the stripe) and M.Mmoves on a\nsphere with the radius M0, which is the saturation mag-\nnetization of our sample. θ(t) can be decomposed into\nthe angleα(t) in the ferromagnetic film plane and the\nout-of-plane angle β(t) (see figure 4). Consequently:\ncosθ(t) = cosα(t)cosβ(t) (2)\nPrecession of the magnetization then yields oscillation\nofα(t),β(t) andθ(t). In our geometry the equilibriummagnetization M0encloses the in-plane angle α0with\nthe stripe. Hence in time average < β(t)>= 0 and\n<α(t)>=α0. In general the magnetization precession\nis elliptical. Its principle axis lie along the x- and y-axis\nand correspond to the amplitudes α1andβ1of the in-\nandout-of-planeangles αt\n1andβt\n1oftherfmagnetization:\nα(t) =α0+αt\n1(t) =α0+α1cos(ωt−ψ) andβ(t) =\nβt\n1(t) =−β1sin(ωt−ψ) (see figure 4). Using equation\n(2) we approximate cos2θ(t) to second order in αt\n1and\nβt\n1:\ncos2θ(t)≈cos2θ|αt\n1=βt\n1=0+αt\n1·dcos2θ\ndαt\n1|αt\n1=βt\n1=0+0\n+αt2\n1\n2·d2cos2θ\ndαt2\n1|αt\n1=βt\n1=0+βt2\n1\n2·d2cos2θ\ndβt2\n1|αt\n1=βt\n1=0(3)\nThe first orderin βt\n1vanishes becauseit is proportional\nto (sinβ)|β1=0= 0. It follows:\ncos2θ(t)≈cos2α0−α1·sin2α0cos(ωt−ψ)\n−α2\n1·cos2α0cos2(ωt−ψ)−β2\n1·cos2α0sin2(ωt−ψ)(4)\nThis equation is now used to calculate the longitudinal\nstripe voltage. To consider the general case an externally\napplied dc current I0and a microwaveinduced rf current\nI1, are included in I(t) =I0+I1cos(ωt). It follows from\nequation (1):\nU(t) = (R0+RAcos2θ(t))·(I0+I1cos(ωt)) (5)\nConsequently U(t) can be written as U(t) =U0+\nU1cos(ωt−ψ1)+U2cos(2ωt−ψ2)+U3cos(3ωt−ψ3). For\nthe photovoltage and photoresistance only the constant\ntermU0, which is equivalent to the time average volt-\nage∝an}bracketle{tU(t)∝an}bracketri}ht, matters. Combining equation (4) and (5), we\nfind:\nU0=I0(R0+RAcos2α0)−I1RAα1sin2α0cos(ψ)/2\n−I0(α2\n1cos2α0+β2\n1cos2α0)RA/2(6)\nNote that:/angbracketleftbig\nsin2(ωt−ψ)/angbracketrightbig\n=/angbracketleftbig\ncos2(ωt−ψ)/angbracketrightbig\n= 1/2,\nand∝an}bracketle{tcosωtcos(ωt−ψ)∝an}bracketri}ht= cos(ψ)/2. The first term in\nequation(6) is independent ofthe rfquantities I1,α1and\nβ1and represents the static voltage drop of I0. The sec-\nond term is the microwave photovoltage UMW. It shows\nno impact from the dc current I0. The third term repre-\nsents the microwave photoresistance ∆ RMW. It is pro-\nportional to I0and depends on the microwave quantities\nα1andβ1. By the way: It can be seen now that the\nrf resistance amplitude R1used in the beginning of this\nparagraph corresponds to: R1=RAα1sin2α0.\nTo analyze the magnetization’s angle oscillation am-\nplitudesα1andβ1it is necessary to express them by\nmeans of the corresponding rf magnetization ℜ(me−iωt).\nmis the complex rf magnetization amplitude. Its phase4\nis defined with respect to I1, so that ℜ(mxe−iωt) is in\nphase with I1cosωtat the FMR. Because M=M0+m,\nm= (mx,my,0) can (in first order approximation) only\nlie perpendicular to M0because MandM0have the\nsame length ( M0). Hence |mx|/M0= sinα1≈α1and\n|my|/M0= sinβ1≈β1forα1,β1≪90◦.\nThe microwave photovoltage and photoresistance ap-\npear whenever magnetization precession is excited. This\nmeans if the microwaves are in resonance with the FMR,\nwith standing exchange spin waves perpendicular to the\nfilm10,11,25orwith magnetostaticmodes11. In this article\nwe will analyze the FMR induced microwave photoresis-\ntance and photovoltage.\n2.2. MAGNETIZATION DYNAMICS\nTo understand the impact of the applied rf magnetic\nfieldℜ(he−iωt) on the microwave photovoltage and pho-\ntoresistance the effective susceptibilities χxx,χxyand\nχyy, which link me−iωtinside the sample with the com-\nplex external rf magnetic field he−iωt= (hx,hy,hz)e−iωt\noutside the sample, have to be calculated. Here ψis en-\ncoded in the complex phase of m.\nThe susceptibility inside the sample (magnetic field\nhine−iωt= (hin\nx,hin\ny,hin\nz)e−iωt) is determined by the\nPolder tensor26ˆχ(received from solving the Landau-\nLiftshitz-Gilbert equation28):\nm= ˆχhin=\nχLiχT0\n−iχTχL0\n0 0 0\nhin(7)\nwith\nχL=ωMωr\nω2r−ω2, χT=ωωM\nω2r−ω2\nwhereωM=γM0with the gyromagnetic ratio γ≈\nµ0·e/m= 2πµ0·28 GHz/T (electron charge eand\nmassme) andωr=γHwithout damping. Approxi-\nmation of our sample as a 2 dimensional film results in\nthe boundary conditions that hxandbyare continuous\nat the film surface meaning hx=hin\nxandby=µ0hy=\nµ0((1+χL)hin\ny−iχThin\nx). Hence:\nm=\nχxxiχxy0\n−iχxyχyy0\n0 0 0\nh (8)\nwith\nχxx=ωrωM+ωM2\nωr(ωr+ωM)−ω2\nχxy=ωωM\nωr(ωr+ωM)−ω2\nχyy=ωrωM\nωr(ωr+ωM)−ω2χxxis identical to the susceptibility describing the\npropagation of microwaves in an unlimited ferromag-\nnetic medium in Voigt geometry29(propagation per-\npendicular to M0).χxx,χxyandχyyhave the same\ndenominator, which becomes resonant (maximal) when\nω=/radicalbig\nω2r+ωrωM. This is in accordance with the FMR\nfrequency of the Kittel formula for in-plane magnetized\ninfinite ferromagnetic films30.\nThis relatively simple behavior is due to the assump-\ntion, that hinis constant within the film stripe. This\nassumption is only valid if the skin depth1δof the mi-\ncrowaves in the sample is much larger than the sam-\nple thickness. During our measurements we fix the mi-\ncrowave frequency fand sweep the magnetic field H.\nConsequently we find the FMR magnetic field H0with\nω2=γ2(H2\n0+H0M0) (9)\nand\nH0=/radicalBig\nM2\n0/4+ω2/γ2−M0/2 (10)\nNow we introduce Gilbert damping27αGby setting\nωr:=ω0−iαGωwith nowω0=γHinstead ofωr=γH.\nWe separate the real and imaginary part of χxx,χxyand\nχyy:\nχxx= (ωrωM+ωM2)·F\nχxy=ωωM·F (11)\nχyy=ωrωM·F\nwith\nF=ω0(ω0+ωM)−α2\nGω2−ω2+iαGω(2ω0+ωM)\n(ω0(ω0+ωM)−α2\nGω2−ω2)2+α2\nGω2(2ω0+ωM)2\n≈(H+H0+M0)(H−H0)+i(2H+M0)αGω/γ\n((H+H0+M0)2(H−H0)2+(2H+M0)2α2\nGω2/γ2\nThe approximation was done by neglecting the α2\nGω2\ncorrectiontotheresonancefrequency ω2=ω0(ω0+ωM)−\nα2\nGω2≈ω0(ω0+ωM) what is possible if αG≪1. Hence:\nχxx,xy,yy≈Axx,xy,yy·∆H(H−H0)+i∆H2\n(H−H0)2+∆H2(12)\nwith ∆H= ((2H+M0)/(H+H0+M0))·αGω/γ. This\ncan be approximated as ∆ H≈αGω/γif|H−H0| ≪H0.\nAxx,AxyandAyydetermine the scalaramplitude of χxx,\nχxyandχyy.\nTo analyze the FMR line shape in the following, we\nwill call the Lorentz line shape which is proportional to\n∆H/((H−H0)2−∆H2) symmetric Lorentz line shape\nandthelineshapeproportionalto( H−H0)/((H−H0)2−\n∆H2) antisymmetric Lorentz line shape. A linear com-\nbination of both will be called asymmetric Lorentz line\nshape.|H−H0| ≪H0allows us to approximate:5\nAxx≈γ(H0M0+M2\n0)\nαGω(2H0+M0)\nAxy≈M0\nαG(2H0+M0)(13)\nAyy≈γH0M0\nαGω(2H0+M0)\nThese are scalars which are independent of the DC\nmagnetic field Hand hence characteristic for the sam-\nple at fixed frequency. Indeed the assumption of Gilbert\ndamping is not essential for the derivation of equation\n(13). In the event ofa different kind ofdamping, ∆ Hcan\nalso be directly input into equation (13) replacing αGω.\nHowever because of the commonness of Gilbert damp-\ning, its usage here can provide a better feeling for the\nusual frequency dependence of Axx,xy,yy. Going ahead,\nequation (8) becomes:\nm≈∆H(H−H0)+i∆H2\n(H−H0)2+∆H2\nAxxiAxy0\n−iAxyAyy0\n0 0 0\nh(14)\nTheH-field dependencies has Lorentz line shape with\nantisymmetric (dispersive) real and symmetric (absorp-\ntive)imaginarypart,theamplitudes Axx,±iAxyandAyy\nrespectively and the width ∆ H. Note that AxxAyy≈\nA2\nxyfor|H−H0| ≪H0. Consequently the susceptibility\namplitude tensor can be simplified to:\n\nAxxiAxy0\n−iAxyAyy0\n0 0 0\nh≈\n√Axx\n−i/radicalbig\nAyy\n0\n\n\n√Axx\ni/radicalbig\nAyy\n0\n·h\n\nand equation (14) becomes:\nm=γM0\nαGω(2H0+M0)∆H(H−H0)+i∆H2\n(H−H0)2+∆H2\n·\n/radicalbig\n1+M0/H0\n−i\n0\n\n\n/radicalbig\n1+M0/H0\ni\n0\n·h\n(15)\nIt is visiblethat the ellipcity of mis independent ofthe\nexciting magnetic field h. Only the amplitude and phase\nofmare defined by h. The reason is the weak Gilbert\ndampingαGfor which much energy needs to be stored\nin the magnetization precession to have a compensating\ndissipation. Hence little energy input and impact from h\nappears.\nFrom equation (15) follows that mxandmyhave car-\ndinally the ratio:\nmx/my=i/radicalbig\n1+M0/H0 (16)\nThereforemyvanishes for ω→0 andmx=imyfor\nω→ ∞. This means that the precession of Mis ellipti-\ncal and becoming more circular for high frequencies andmore linear (along the x-axis) for low frequencies. This\ndescription applies for the case of an in-plane magnetized\nferromagnetic film. However in the case that the sample\nhas circular symmetry with respect to the magnetization\ndirection (e.g. in a perpendicular magnetized disc or in-\nfinite film10,11):α1=β1. This is the same as in the case\nthatω→ ∞. Only in these cases the magnetization pre-\ncession can be described in terms of one precession cone\nangle13. Otherwise distinct attention has to be paid to\nα1andβ1(see 3.2). Additionally it can be seen in equa-\ntion (15) that my/mxis also the ratio of the coupling\nstrength of mtohyandhxrespectively.\n2.3. MICROWAVE PHOTORESISTANCE\nThe microwave photoresistance ∆ RMWcan be de-\nduced from equation (6). First the microwave photovolt-\nage is excluded by setting the rf current I1= 0. Then we\nonly regard the microwavepower dependent terms which\ndepend onα1andβ1:\n∆RMW= (U0|I1=0−U0|I1=0,α1=0,β1=0)/I0\n=RA(−α2\n1cos2α0−β2\n1cos2α0)/2 (17)\nIf the magnetization lies parallel or antiparallel to\nthe dc current vector I0along the stripe ( α0= 0◦or\nα0= 180◦) the AMR is maximal. In this case magne-\ntization oscillation ( α1andβ1) reduces (-cos2 α0=−1)\nthe AMR by ∆ RMW=−(α2\n1+β2\n1)RA/2 (negative pho-\ntoresistance). In contrast if the magnetization lies per-\npendicular to I0(α0= 90◦, see figure 2) the resistance\nis minimal. In this case magnetization oscillation corre-\nsponding to α1will increase ( −cos2α0= +1) the AMR\n(positive photoresistance) by ∆ RMW= +α2\n1·RA/2 (os-\ncillations corresponding to β1leaveθ(t) constant in this\ncase and do not change the AMR).\nThe next step is to calculate α1andβ1. The dc mag-\nnetic field dependence of α1=|mx|/M0=|χxxhx+\niχxyhy|/M0andβ1=|my|/M0=|−iχxyhx+χyyhy|/M0\nis proportional to that of |χxx|,|χxy|and|χyy|given in\nequation (12) (imaginary symmetric and real antisym-\nmetric Lorentz line shape). Squaring this results in sym-\nmetric Lorentz line shape:\nα2\n1∝β2\n1∝/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆H(H−H0)+i∆H2\n(H−H0)2+∆H2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=∆H2\n(H−H0)2+∆H2\nHence:\nα2\n1=|Axxhx+iAxyhy|2\nM2\n0·∆H2\n(H−H0)2+∆H2\nβ2\n1=|Ayyhy−iAxyhx|2\nM2\n0·∆H2\n(H−H0)2+∆H2(18)\nUsingequation(15) and(18), equation(17) transforms\nto:6\n∆RMW=RA\n(αGω/γ)2(2H0+M0)2\n·(−(H0+M0)cos2α0−H0cos2α0) (19)\n·∆H2\n(H−H0)2+∆H2·|hx/radicalbig\nH0+M0+ihy/radicalbig\nH0|2\nThe strength of the microwave photoresistance is pro-\nportionalto1 /α2\nG. Weakdamping(small αG)istherefore\ncritical for a signal strength sufficient for detection. The\nmagnetic field dependence shows symmetric Lorentz line\nshape.\nThe dependence of ∆ RMWonα0in equation (19) re-\nveals a sign change and hence vanishing of the photore-\nsistance at\ncos2α0=1\n2/parenleftbigg\n1−H0\n3H0+2M0/parenrightbigg\n(20)\nThis means that the angle at which the photoresistance\nvanishes shifts from α0=±45◦andα0=±135◦(for\nω→0) toα0=±54.7◦andα0=±125.3◦respectively\n(forω→ ∞) when increasing ω. The reason for this\nfrequency dependence is the frequency dependence of the\nellipcity of mdescribed at the end of 2.2.\n2.4. MICROWAVE PHOTOVOLTAGE\nThe most obvious difference in appearence between\nthe microwave photoresistance discussed in paragraph\n2.3 and the microwave photovoltage discussed in this\nparagraph is that the photoresistance is proportional to\nthe square of the rf magnetization (see equation (17),\nα2\n1≈ |mx|2/M2\n0andβ2\n1≈ |my|2/M2\n0) while the photo-\nvoltageUMWis proportional to the product of the rf\nmagnetization and the rf current. Consequently the pho-\ntovoltage has a very different line shape: While the rf\nmagnetizationdepends with Lorentzlineshapeon H(see\nequation (12)), I1is independent of H. The line shape\nis hence determined by the phase difference ψbetween\nthe rf magnetization component ℜ(mxe−iωt) and the rf\ncurrentI1cosωt. This effect does not play a role in the\ncaseofphotoresistancebecausethereonlyonephasemat-\nters namely that of the rf magnetization. In contrast in\nphotovoltage measurements a linear combination of sym-\nmetric and antisymmetric Lorentz line shapes is found.\nThis will be discussed in detail in the following.\nTo isolate the microwave photovoltage in equation (6)\nthe dc current I0is set to 0:\nUMW=U0|I0=0=−I1α1RAsin2α0cosψ\n2(21)\nFrom equation (8) we follow:\nα1cosψ=ℜ(mx) =ℜ(χxxhx+iχxyhy) (22)We splithx=hr\nx+ihi\nxandhy=hr\ny+ihi\nyinto real\n(hr\nx,hr\ny) and imaginary ( hi\nx,hi\ny) part. This enables us to\nisolate the real part in equation (21) using equation (14):\nUMW=I1RAsin2α0\n2M0·{/parenleftbig\nAxyhr\ny+Axxhi\nx/parenrightbig\n∆H2\n(H−H0)2+∆H2\n+/parenleftbig\nAxyhi\ny−Axxhr\nx/parenrightbig\n∆H(H−H0)\n(H−H0)2+∆H2}(23)\nConclusively in contrast to the microwave photoresis-\ntance (∆RMW∝1/α2\nG, seeequation (19)) the photovolt-\nage is only proportional to 1 /αG∝Axx,xy,yy. Thus good\ndamping is less important for its detection31.\nTo understand the measurement results it will be nec-\nessary to transform the coordinate system of equation\n(23) to (x’,y,z’). In this coordinate system the rf mag-\nnetic field his constant during rotation as described in\nequation (33).\nTo better understand the photovoltage line shape we\nhave a closer look on ψ: When sweeping Hthe rf mag-\nnetization phase is shifted by ψmwith respect to the\nresonance case ( H=H0). The rf current has a constant\nphaseψIwhich is defined with respect to the magnetiza-\ntion’s phase at resonance. The impact ofthe dc magnetic\nfieldHon the rf current ( I1,ψI) via the FMR is believed\nto be negligible:\ncosψ= cos(ψm−ψI) = cosψmcosψI+sinψmsinψI\n(24)\nψisdeterminedbythe(complex)phaseof χxx,χxyand\nχyywith respect to the resonance case ( ℜ(χxy,yy) = 0\natH=H0) during magnetic field sweep (asymmetric\nLorentz line shape, see equation (12)):\ntanψm=ℑ/parenleftBig\n∆H(H−H0)+i∆H2\n(H−H0)2+∆H2/i/parenrightBig\nℜ/parenleftBig\n∆H(H−H0)+i∆H2\n(H−H0)2+∆H2/i/parenrightBig=H0−H\n∆H(25)\nIt should be noted that according to the Landau-\nLiftshitz equation28happlies a torque on the magne-\ntization and hence excites mttransversal. That is why\nat resonance mxshows a phase shift of 90◦with respect\ntohx. Consequently in equation (25) division by iis\nnecessary (χxxandχxybecome imaginary at resonance).\nEquation (25) means that in case that the applied mi-\ncrowave frequency is higher than the FMR frequency\n(H0> H)ψm>0 (note that mt=me−iωt),mtis de-\nlayed with respect to the resonant case. The other way\naround (H0<H) the FMR frequency is higher than that\nof the applied microwave field and mtis running ahead\ncompared to the resonance case. Using equation (25) we\nfind:\ncosψm=∆H/radicalbig\n(H−H0)2+∆H2\n(26)7\nInserting equation (18), (24), (25) and (26) into (21)\ngives:\nUMW=−RAI1sin2α0\n2·|Axxhx+iAxyhy|\nM0\n·/parenleftbigg∆H2cosψI\n(H−H0)2+∆H2−(H−H0)∆HsinψI\n(H−H0)2+∆H2/parenrightbigg\n(27)\nThe dependence on Htakes the form of a linear com-\nbination of symmetric and antisymmetric Lorentz line\nshape with the ratio 1 : tan ψI. The symmetric line\nshape contribution ( ∝∆H) arises from the rf current\ncontribution that is in phase with the rf magnetization\nat FMR and the antisymmetric from that out-of-phase.\nThis gives a nice impression of the phase ψIof the rf\ncurrent determining the line shape of the FMR.\n2.5. VECTORIAL DESCRIPTION OF THE\nPHOTOVOLTAGE\nTo complete the discussion of the microwave photo-\nvoltage we want to return to the approach used by\nJuretschke23to demonstrate that it is consistent with\nthe description above. In 2.1 we started with Ohm’s law\n(scalar equation (1)). There we integrate an angle- and\ntime-dependent resistance. Here we want to start with\nthe vectorial notation of Ohm’s law used in Juretschke’s\npublication (equation (1)23). This integrates AMR and\nanomalous Hall effect AHE. ρis the resistivity of the\nsample and ∆ ρthat additionally arising from AMR. RH\nis the anomalous Hall effect constant:\nE=ρJ+(∆ρM2)(J·M)M−RHJ×M(28)\nWe split M=M0+mtand the current density\nJ=J0+jtinto their dc ( M0andJ0) and rf contri-\nbutions ( mt=ℜ(me−iωt) andjt=jcosωt). Constance\nof|M|allowsmt= (mt\nx,mt\ny,0) in first order approxima-\ntion only to lie perpendicular to M0= (0,0,M0). To\nselect the photovoltage we set J0= 0 and approximate\nequation (28) to second order in jtandmt. The terms\nof zeroth order in both jtandmtrepresent the sample\nresistance without microwave exposure and are not dis-\ncussed here. The terms of first order in either jtormt\n(but not both) have zero time average and do not con-\ntribute to the microwave induced dc electric field EMW.\nOnly the terms that are simultaneously of first order in jt\nandmtcontribute to EMW(compare equation (4) from\nJuretschke23):\nEMW=∆ρ\nM02/angbracketleftbig/parenleftbig\njtmt/parenrightbig\nM0+/parenleftbig\njtM0/parenrightbig\nmt/angbracketrightbig\n−RH/angbracketleftbig\njt×mt/angbracketrightbig\n(29)\nThe ∆ρdependent term represents the photovoltage\ncontribution arising from AMR and the RHdependentterm that arising from AHE. Note that a second order\nofmtappears when applying a dc current J0∝ne}ationslash= 0. It\nrepresents the photoresistance discussed in 2.3. However\nit we will not be discussed here.\nIn the following we will calculate the photovoltage in\nour Permalloy film stripe considering its geometry which\nfixes the current direction. jt=jt\nz′z′along the stripe\n(z′is the unit vector along the Permalloy stripe). The\nsmall dimensions perpendicular to the stripe ( ≪L) will\nprevent the formation of a perpendicular rf current. A\nsimilar approximation of a metal grating forming a lin-\near polarizer has been considered previously9. The pho-\ntovoltageUMWis also measured along the stripe (length\nvectorL=z′·2.4mm). Whenfluctuationsof EMWalong\nthe stripe are neglected considering the large microwave\nwavelength, λ≈20 mm≫2.4 mm =L, we findUMW\nby multiplying EMWwithL:\nUMW=/integraldisplayL\n0EMWdz′≈EMW·L=\n∆ρL\nM02/angbracketleftbig\njt\nz′/parenleftbig\nz′mt/parenrightbig\n(M0z′)+jt\nz′(z′M0)/parenleftbig\nmtz′/parenrightbig/angbracketrightbig\n−0 (30)\n=∆ρL\nM0/angbracketleftbig\njt\nz′mt\nx/angbracketrightbig\nsin(2α0)\nThis is equivalent to equation (21) which can be ver-\nified by replacing ∆ ρjt\nz′L=RAI1cos(ωt) andmt\nx=\nα1M0cos(ωt−ψ). Time averaging results in the addi-\ntional factor cos( ψ)/2.\nAs discussed in 2.6 the contribution belonging to the\nanomalous Hall effect has no impact in this geometry be-\ncause it can only generate a photovoltage perpendicular\nto the rf current i.e. perpendicular to the stripe.\nComparing our results to those of Juretschke and\nEgan23,31, we note that an equation similar to (30) has\nbeen derivedin theformulafor ey0inequation(31)in Ju-\nretschke’s publication23. There the photovoltage is mea-\nsured parallel to the rf current as done in our stripe.\nHowever it has to be noted that the coordinate system\nis defined differently. The major difference compared to\nour system is that we use a stripe shaped film to litho-\ngraphically define the direction of the rf current I1, while\nthe direction of his left arbitrary. In contrast to that\nJuretschke and Egan23,31define the direction of the rf\nmagneticfield andrfcurrentbymeansoftheirmicrowave\nsetup. In equation (31) ( ey0) from Juretschke23this re-\nsults in the additional factor cos θ(which is equivalent\nto cosα0in our work) compared to equation (30). This\narises from the definition of hfixed parallel to the rf cur-\nrent (compare equation (33)).\n2.6. OTHER MAGNETORESISTIVE EFFECTS\nTHAT COUPLE SPIN AND CHARGE CURRENT\nIn this section we present other magnetoresistive ef-\nfects which can generate photovoltage and photoresis-\ntance like the AMR. This selection gives a broader view8\non the range of effects for which the photovoltage and\nphotoresistance can be discussed in terms of the analysis\npresented in this work. In principle every magnetoresis-\ntive effect can modulate the sample resistance and thus\nrectify some of the rf current to photovoltage.\nOne magnetoresistive effect is the anomalous Hall ef-\nfectAHEinferromagneticmetalsthatwas(togetherwith\nthe AMR) the basis for the discussion of Juretschke23.\nThere a current with perpendicular magnetization gener-\nates a voltage perpendicular to both. Under microwave\nexposure this alternates with the microwave frequency\nbut in an asymmetric way due to the modulated AHE\narising from magnetization precession. The asymmetric\nvoltage has a dc contribution (photovoltage)31which can\nbe measured using a 2 dimensional ferromagnetic film\nwith the magnetization neither parallel nor perpendic-\nular to it. The photovoltage induced by AHE appears\nin the film plane perpendicular to the rf current and is\nsmall25for Permalloy (Ni 80Fe20). Also a photoresistive\neffect which alters the AHE can be expected if the mag-\nnetization lies out-of-plane.\nOther examples for magnetoresistive effects are GMR\nand TMR structures which exhibit a photovoltage mech-\nanismsimilartothatinAMRfilms. Thedifferenceisthat\nthere not the direction of the ferromagnetic layer magne-\ntization with respect to the current matters. Effectively\ninstead the direction of the magnetization Mof one fer-\nromagnetic layer with respect to that of another layer is\ndecisive (see figure 5). Exciting the FMR in one layer\nyields again oscillation of the sample resistance R(t) and\nthus gives the corresponding rf voltage U(t) a non-zero\ntime average (photovoltage)4,32. This is usually stronger\nthan that from AMR films due to the generally higher\nrelative strength of GMR and TMR compared to AMR.\nIt should be noted that in current studies of the\nmicrowave photovoltages effect in multilayer struc-\ntures, the focus is on interfacial spin transfer\neffects4,5,6,7,8,19,20,21,32. It remains an intriguing ques-\ntion whether interfacial spin transfer effects and the ef-\nfect revealed in our approach based on phenomenological\nmagnetoresistance might be unified by a consistent mi-\ncroscopic model, as Silsbee et al.have demonstrated for\ndescribing both bulk and interfacial spin transport under\nrf excitation24.\nMultilayer structures also provide a nice example that\nphotovoltage generation can also be reversed when the\noscillating magnetoresistance, transforms a dc current\ninto an rf voltage33, instead of transforming an rf cur-\nrent into a dc voltage (photovoltage). This gives a new\nkind of microwave source and seems - although weaker -\nalso possible in AMR and AHE samples.\nIt can be reasoned that like microwave photovoltage\nthemicrowavephotoresistancecanalsobebasedonGMR\nor TMR instead of AMR: When aligning the 2 magneti-\nzations of both ferromagnetic layers in a GMR or TMR\nstructure microwave induced precession of one magneti-\nzation is expected to increase the GMR/TMR because\nof the arising misalignment with the other magnetiza-\nFIG. 5: (Color online). Microwave photovoltage\nin a GMR/TMR-heterostructure (ferromagnetic( M)/non-\nferromagnetic/ferromagnetic( Mf)): The dynamic magnetiza-\ntionMprecesses (period P) in phase with the current I. (a)\nMlies almost perpendicular to Mf: high GMR/TMR. (b) M\nlies almost parallel to Mf: low GMR/TMR ⇒Non-zero time\naverage of the voltage U.\ntion. With the magnetizations initially anti parallel the\nopposite effect, a microwave induced resistance decrease,\nis expected. Further work demonstrating these effects\nwould be interesting.\n3. PHOTOVOLTAGE MEASUREMENTS\n3.1. MEASUREMENT SETUP\nThe sample we use to investigate the microwave pho-\ntovoltage consists of a thin (d = 49 nm) Permalloy (Ni\n80%, Fe 20%) film stripe (200 µm wide and 2400 µm\nlong) with 300 ×300µm2bond pads at both ends (see\nfigure 3). These are connected via gold bonding wires\nand coaxial cables to a lock-in amplifier. For auxiliary\nmeasurements (e.g. Hall effect) 6 additional junctions\nare attached along the stripe (see figure 3).\nThe resistance of the film stripe is R0+RA= 85.0 Ω\nfor parallel and R0= 83.6 Ω for perpendicular mag-\nnetization. Hence the conductance is σ= 1/ρ= 2.9·\n106Ω−1m−1and the relative AMR is ∆ ρ/ρ= 1.7 %.\nThe absolute AMR is RA= 1.4 Ω. This is in good agree-\nment with previous publications9,10,11.\nThe film is deposited on a 0.5 mm thick GaAs single\ncrystal substrate, and patterned using photolithography\nand lift off techniques. The substrate is mounted on a 1\nmm polyethylene print circuit board which is glued to a\nbrass plate holding it in between the poles of an electro-\nmagnet. This provides the dc magnetic field B=µ0H\n(maximal ≈1 T). The sample is fixed 1 mm behind the\nend of a WR62 (15 .8×7.9 mm) hollow brass waveguide\nwhich is mounted normal to the Permalloy film plane.\nThe stripe is fixed along the narrow waveguide dimen-\nsion. In the K uband (12.4 - 18 GHz), that we use in9\nFIG. 6: (Color online). Sketch of the measurement geometry.\nA 1 mm thick polyethylene plate is glued on a brass holder.\nOn top of the polyethylene a GaAs substrate is glued. On\nthe substrate the Permalloy (Py) stripe is defined. This is\nelectrically wired to a voltage amplifier for photovoltage m ea-\nsurements. For photoresistance measurements an additiona l\ncurrent source is connected parallel to the voltage amplifie r,\nwhich is not shown explicitly here.\nour measurements, the WR62-waveguide only transmits\nthe TE 01mode1. The stripe was fixed with respect to\nthe waveguide but was left rotatable with respect to H.\nThis allows the stripe to be parallel or perpendicular to\nH, but keeps the magnetic field always in the film plane.\nA high precision angle readout was installed to indicate\nα0.\nThe waveguide is connected to an HP83624B mi-\ncrowave generator by a coaxial cable supplying frequen-\ncies of up to 20 GHz and a power of 200 mW. The power\nis however later significantly reduced by losses occur-\nring within the coaxial cable, during the transfer to the\nhollow waveguide and by reflections at the end of the\nwaveguide. Microwave photovoltage measurements are\nperformed sweeping the magnetic field while fixing the\nmicrowave frequency. The sample is kept at room tem-\nperature.\nTo avoid external disturbances the photovoltage was\ndetected using a lock-in technique: A low frequency\n(27.8 Hz) square wave signal is modulated on the mi-\ncrowave CW-output. The lock-in amplifier, connected\nto the Permalloy stripe, is triggered to the modulation\nfrequency to measure the resulting square wave photo-\nvoltage across the sample. Instead of the photovoltage\nalso the photocurrent can be measured10. Its strength\nI0can be found when setting U0= 0 in equation (6)\n(instead of I0= 0).\n3.2. FERROMAGNETIC RESONANCE\nThe measured photovoltage almost vanishes during\nmost of the magnetic field sweep but shows one pro-\nnounced resonance of several µV. The strength and line\nshape of this resonance are strongly depending on α0\nand will be discussed in 3.3. A line shape dependence\nFIG. 7: (Color online). Gray scale plot of the measured fre-\nquency and magnetic field dependence of the microwave pho-\ntovoltage at α0= 47◦. The dashed line shows the calculated\nFMR frequency (see equation (9)). The photovoltage inten-\nsity is strongly frequency dependent because of the frequen cy\ndependent waveguide transmission.\nof the photovoltage on the microwave frequency is also\nfound. The photovoltage with respect to the strength\nof the external magnetic field Hand the microwave fre-\nquencyf=ω/2πcan be seen in a gray scale plot in\nfigure 7, in which the resonance can be identified with\nthe FMR by the corresponding fits (dashed line) because\nthe Kittel-equation30(9) for ferromagnetic planes (our\nPermalloy film) applies. The magnetic parameters found\nareµ0M0≈1.02 T andγ≈2πµ0·28.8 GHz/T. They\nare in good agreement with previous publications9,10.\nThe exact position of the FMR is obscured by its\nstrongly varying line shape. We overcome this problem\nby the productive line shape analysis in paragraph 3.3.\nIt is found that H0is slightly dependent on α0. This can\nbe attributed to a small demagnetization field perpendic-\nular to the stripes but within the film plane arising from\nthe finite stripe dimensions in this direction. So, when\nM0lies perpendicular to the stripe, H0slightly increases\ncompared to the value fulfilling the Kittel equation for a\nplane (see equation (9)). In the parallel and perpendic-\nular case we use the approximation of our film stripe as\nan ellipsoid, where we can use the corresponding Kittel\nequation30(demagnetization factors Nx,NyandNzwith\nrespect to the dc magnetic field):\nω=γ/radicalBig\n(H0+(Nx−Nz)M0)(H0+(Ny−Nz)M0)\n(31)\nThe difference of the resonance field between the case\nthatM0lies in the film plane parallel to the stripe and\nperpendicular is 1.6 mT (0.7%) at f = 15 GHz. From this\nwe can calculate the small demagnetization factor Nx′=\n0.085% perpendicular to the Permalloy stripe within the10\nFIG. 8: (Color online). Fitting (black line) of the microwav e\nphotovoltage signal (dots) for different angles α0atf= 15\nGHz. The black horizontal bars indicate zero signal.\nfilm plane using equation (31). From the sum rule34fol-\nlows:Ny= 1−Nx′−Nz′= 1−0.085%−0 = 99.915%.\nNz′(parallel to the stripe) can be assumed to be negli-\ngibly small. This matches roughly with the dimension of\nthe height to width ratio(49 nm : 200 µm) ofthe sample.\nFor the stripe presented in section 4 similar but stronger\ndemagnetization effects are found.\nNow we will have a closer look on the magnetic prop-\nerties of the investigated film. Again at f= 15 GHz we\nfind using equation (10): H0= 0.219 T. Using asymmet-\nric Lorentz line shape fitting as described in 3.3 we get\nαG= 0.0072. Consequently Axx= 231.1,Axy= 97.1\nandAyy= 40.8 according to equation (13).\nBecause of αG= 0.0072 the magnetization precession\ndoes impressive n≈22 turns before being damped to\n1/eof its initial amplitude ( n= 1/2παG). Therefore\nthe ellipcity of mis almost independent of h(see para-\ngraph 2.2). It can be calculated from equation (16) that\nmx/my= 2.38iatω/2π= 15 GHz.\nTo check the validity of our approximation ( d≪δ,\nsee 2.2) we will now regard the skin depth δatf= 15\nFIG. 9: (Color online). Symmetric and Antisymmetric con-\ntributions tothe asymmetric Lorentz line shape fitfrom figur e\n8 (black). A small constant background is found and added\nto the antisymmetric contribution.\nGHz in our sample ( d= 49 nm). For µ=µ0(away\nfrom the FMR) we find: δ=/radicalbig\n2/ωµρ= 2.4µm. Hence\nδ≫d. This is in accordance with our approximation\nthathis almost constant within the Permalloy film (see\n2.2). However in the vicinity of the FMR: |µ| ≫µ0and\nfor the same frequency and conditions as above: µL=\n(1+χL)µ0= 133iµ0at the FMR. Thus we approximate\nδFMR=/radicalbig\n2/ω|µL|ρ= 210 nm. Hence δFMRis still\nsignificantly larger than dand our approximation is still\nvalid.\nFinally we can summerize that for samples with weak\ndamping(αG≪ω/ωM)likeourstheapproximation H≈\nH0gives results with impressive precision (see figure 8)\nbecause its discrepancies are limited to the unimportant\nmagnetic field ranges with |χxx|,|χxy|,|χyy| ≪1 which\nare far away from the FMR.11\n3.3. ASYMMETRIC LORENTZ LINE SHAPE\nAlthough in section 3.2 the frequency dependence of\nthe FMR field is verified with the gray scale plot in fig-\nure 7, it is still desirable to receive a more accurate pic-\nture of the corresponding line shape which is found to\nbe strongly angular dependent (see figure 8). In equa-\ntion (27) it is shown that the magnetic field dependence\nofUMWexhibits asymmetric Lorentz line shape around\nH=H0. HenceUMWtakes the form\nUMW=USYM\nMW+UANT\nMW\n=USYM\n0∆H2\n(H−H0)2+∆H2(32)\n+UANT\n0∆H(H−H0)\n(H−H0)2+∆H2\nThis is used to fit the magnetic field dependence of\nthe photovoltage in figure 8. For clearness the symmet-\nric (absorptive) and antisymmetric (dispersive) contri-\nbutions are shown separately in figure 9. A small con-\nstant background is found and added to the antisymmet-\nric contribution. The background could possibly arise\nfrom other weak non-resonant photovoltage mechanisms.\nThe fits agree in an unambiguous manner with the\nmeasured results. Hence they can be used to deter-\nmine the Gilbert damping parameterwith high accuracy:\nαG≈γ∆H/ω≈(0.72%±0.015%). However if the mag-\nnetization lies parallel or perpendicular to the stripe the\nphotovoltage vanishes (see equation (21)). Hence we can\nonly verify αGwhen the magnetization is neither close\nto being parallel nor perpendicular to our stripe.\nThe corresponding αG= 1/ωτin the Nickel sample\nof Egan and Juretschke31, can be estimated using the\nferromagnetic relaxation time τfrom their Table II. It\nlies in between αG= 0.12 and 0.18, so being more than\n16times higherthan the valuein oursample. This makes\nthe line shape approximation of section 2.4 invalid for\ntheir case. Consequently a much more elaborated line\nshape analysis23appears necessary.\nIn figure 8 the photovoltage along the stripe is pre-\nsented at 4 different angles α0. The signal to noise ratio\nis about 1000 because of the carefully designed measure-\nment system, where the noise is suppressed to less than\n5 nV. Because of this good sensitivity we can verify the\nmatching of our theory from section 2 with the measure-\nment results in great detail.\nIn the following we want to investigate the angular\ndependence in detail. Therefore we transform the coor-\ndinate system of equation (23) according to the transfor-\nmationpresentedinsection2.1. Doingsowecanseparate\nthe contributions from hx′,hyandhz′:\nFIG. 10: (Color online). The Bars show the angular depen-\ndence of the amplitude of the symmetric ( USYM\n0, thin bars)\nand antisymmetric ( UANT\n0, thick bars) contribution to the\nmicrowave photovoltage at f = 15.0 GHz. Note that both,\nthe symmetric and antisymmetric contribution, vanish for\nα0= 0◦,90◦,180◦and 270◦. The lines represent the cor-\nresponding fits by means of equation (34). The inlet shows\nthe geometry of the investigated Permalloy stripe and the co -\nordinate systems from figure 3 (note: z/bardblH).\nUMW=RAI1sin(2α0)\n2M0·\n{/parenleftbig\nAxyhr\ny+Axx(hi\nx′cosα0−hi\nz′sinα0)/parenrightbig\n·∆H2\n(H−H0)2+∆H2(33)\n+/parenleftbig\nAxyhi\ny+Axx(hr\nz′sinα0−hr\nx′cosα0)/parenrightbig\n·∆H(H−H0)\n(H−H0)2+∆H2}\nhx′,hyandhz′are fixed with respect to the hollow\nbrass waveguide and its microwave configuration and do\nnot change when α0is varied.\nWe find that the angular dependence of the line shape\nin equation (33) exhibits 2 aspects: an overall factor\nsin(2α0) and individual factors (sin α0, cosα0and 1) for\nthe terms belonging to the different spatial components\nofh. The overall factor sin(2 α0) arises from the AMR\nphotovoltage mechanism and results in vanishing of the\nphotovoltage signal at α0= 0◦,90◦,180◦and 270◦. This\nmeans if M0lies either parallel, antiparallel or perpen-\ndicular to the stripe axis. This is illustrated in figure 11\nand is clearly observed in our measurements (see figure\n10). We take this as a strong support for the photovolt-\nage being really AMR based.\nAnother support comes from the similarity with the\nplanar Hall effect35. The planar Hall effect generates a\nvoltageUPHEperpendicular to the current in ferromag-\nneticsamples(width W) when themagnetization M0lies\nin the current-voltage plane. It arises as well from AMR\nand vanishes when M0lies either parallel or perpendic-12\n(a)\n(b)\nFIG. 11: (Color online). When the magnetic field Hlies par-\nallel or perpendicular to the stripe, the time average volta ge\nvanishes. (a) Hlies perpendicular to I: precession of the\nmagnetization Mleaves (after half a period P/2) the angle θ\nbetweentheaxisof MandIunchanged. HencetheAMR(and\nso voltage U) is also unchanged. The photovoltage vanishes.\n(b)His parallel to I:θand the AMR stay constant during\nthe precession of Mand the time average of Iis zero. This\nmeans that only when His neither parallel nor perpendicular\nto the stripe a photovoltage is generated.\nular to the current axis.\nThesimilarityarisesbecauseofthe AMR onlygenerat-\ning a transversal resistance when the current is not lying\nalong the principle axis of its resistance matrix (parallel\nor perpendicular to the magnetization). This is the same\ngeometrical restriction as shown above for the microwave\nphotovoltage (see equation (21) and figure 11).\nWe want to emphasize the importance that in any of\nthese microwave photovoltage experiments, due to the\nunusually strong angle dependence, it is important to\npay attention to the exact angle adjustment of the sam-\nple with respect to the dc magnetic field Hwhen mea-\nsuring under high symmetry conditions ( Hparallel or\nperpendicular to the stripe) to avoid involuntary signal\nchangesdue tosmallmisalignments. Asfoundin 90◦out-\nof-plane configuration10already a misalignment as small\nas a tenth of a degree can yield a tremendous photovolt-\nage change in the vicinity of the FMR.\nFinally wewanttocome backtothe individual angular\ndependencies of the photovoltage contributions arising\nfrom the different external magnetic field components.\nIn addition to the sin(2 α0) proportional dependence of\nUMWonmx, also the strength with which mxis excited\nbyhdepends on α0. This is displayedin figure12and re-\nflected by the three terms in equation (33) depending on\nhx′,hyandhz′with cosα0, 1 and sin α0-factors respec-\ntively. Hence the symmetric USYM\n0and antisymmetric\nUANT\n0Lorentz line shape contribution to UMWare fitted\nin figure 10 with\nFIG. 12: (Color online). Angular dependent coupling of\nthe magnetization Mto the dynamic magnetic field h=\n(hx′,hy,hz′). Only the components of hperpendicular to M0\ncan excite precession of Mand therefore generate a dynamic\nm.hyis always exciting m. The excitation strength of hx′\nandhz′is angular dependent (compare equation (33)). Here\nthetwosymmetrycases areshown: M(a)perpendicular(only\nhz′andhycan excite M) and (b) parallel (only hx′andhy\ncan excite M) to the stripe.\nUSYM\n0= (US\nz′sin(α0)+US\nx′cos(α0)+US\ny)sin(2α0)\nUANT\n0= (UA\nz′sin(α0)+UA\nx′cos(α0)+UA\ny)sin(2α0)(34)\nFromUS\nz′,US\nx′andUA\nythe dynamicmagneticfield com-\nponentshi\nz′,hi\nx′,hi\nywhich are 90◦out-of-phase with re-\nspect to the rf current I1can be determined using equa-\ntion (33) and from UA\nz′,UA\nx′andUS\nywe findhr\nz′,hr\nx′and\nhr\nywhich are in phase with I1.\nIn principle I1can be separately deduced using the\nbolometric effect12as discussed in section 4.1. However\nfor the sample used here ourusage ofmultiple stipes does\nnot allow us to address the bolometric heating to one\nsinglestripe. Consequentlythe strength of I1is unknown\nso that we can not determine h, but only hI1.\nBesides,consideringthespecialdynamicmagneticfield\nconfiguration in our rectangular hollow waveguide no rf\nmagnetic field component hz′is expected to be generated\nalong the waveguides narrow dimension (z′-axis) by the\nTE01mode1(which is the microwaveconfigurationofour\nwaveguide). It follows that the sin( α0) terms in equation\n(34) vanishes. This results in the additional symmetry\nUMW(α0) =−UMW(−α0), which is clearly observed in\nour measurements (see figure 10). This symmetry was\nbroken when we used a round waveguide.\nThe vanishing of hz′in our waveguide will allow us\nto plot the direction of h2 dimensional (instead of 3\ndimensional) in figure 13. A small deviation from the\nsymmetryUMW(α0) =−UMW(−α0) is however found\nand arisesfrom a small hz′component (see table I) which\nis not displayed in figure 13. It might arise from the fact\nthat the rf microwave magnetic field hat the waveguide\nend already deviates from the TE 01mode.13\nUS\nx′,y,z′UA\nx′,y,z′AxxAxyI1·hr\nx′,y,z′I1·hi\nx′,y,z′\n(µV) (mA·µT/µ0)\nx′+2.60 +2.55231.1 -15.7 +16.4\ny+0.95 +0.30 97.1 +14.0 +4.4\nz′+0.12 0.00231.1 0.0 -0.7\nTABLE I: Determination of the rf magnetic field hat the\n200µm wide stripe at 1 mm distance from the waveguide end\nby means of equation (33). US\nx′,y,z′,UA\nx′,y,z′: Measured am-\nplitudes of the contributions to the symmetric and antisym-\nmetric Lorentz line shape of UMW(see equation (34)) with\nthe angular dependence belonging to x′,yandz′respectively\n(taken from the fitting in figure 10). Axx,xy: Correspond-\ning amplitudes of χxx,xy.hr\nx′,y,z′,hi\nx′,y,z′: rf magnetic field\nstrength calculated from US\nx′,y,z′,UA\nx′,y,z′(in-phase and 90◦\nout-of-phase contribution with respect to the current).\nFIG. 13: (Color online). Direction and ellipticity of the rf\nmagnetic field hdisplayed by showing the path I1·hpasses\nduring one cycle. This is shown at the location of the 3\nstripes (these lie normal to the picture on top of the gray\nGaAs-substrate; the 200 µm wide stripe to the right) for two\nsample positions. I1·hwas determined by means of equa-\ntion (33). The upper right path corresponds to the I1·h\nfrom table I. The hatched edges indicate metal surfaces re-\nflecting microwaves. Within the waveguide the rf magnetic\nfieldhcorresponding to the TE 10-mode is displayed in the\nbackground.\n3.4. DETERMINATION OF THE RF\nMAGNETIC FIELD DIRECTION\nUsing the different angular dependencies of the 3 sym-\nmetric and 3 antisymmetric terms in equation (33) hI1\ncan be determined. We make the assumption that the\nstripe itself does not influence the rf magnetic field con-\nfiguration,whatisatleastthecasewhenfurtherreducing\nits dimensions. Thus the film stripe becomes a kind of\ndetector for the rf magnetic field h.\nTo test this an array of 36 additional 50 µm wide and20µm distant Permalloy stripes of the same height and\nlength as the 200 µm wide stripe described above (see\nsection 3.1) was patterned beside this one. The 50 µm\nwide stripes were connected with each other at alternat-\ning ends to form a long meandering stripe9. Four stripes\nwere elongated on both ends to 300 ×300µm2Permalloy\ncontact pads. For the outer two stripes and the single\n200µm stripe hI1is calculated from the measured pho-\ntovoltage using equation (23). Table I shows the mea-\nsured voltage and the corresponding hI1for the 200 µm\nstripe at 1 mm distance from the waveguide. hI1for\nall 3 stripes is displayed in figure 13, while positioning\nthe sample at 2 distances (1 and 3.5 mm respectively)\nfrom the waveguide end. For comparison the rf magnetic\nfieldhconfigurationofthe TE 01-mode is displayedin the\nbackground. From other measurements we can estimate\nthatI1lies somewhere in the 1 mA-range.\nIt is worth noting that possible inhomogeneities of the\nrf magnetic field hwithin the Permalloy stripes will be\naveraged because UMWis linear in h. Determining the\nsign of the rf magnetic field components from the photo-\nvoltage contributions signs exhibits a certain complexity\nbecause a lot of attention has to be paid to the cho-\nsen time evolution ( eiωtore−iωt) and coordinate system\n(right hand or left hand). However the sign only reflects\nthe phase difference with respect to the rf current. The\nrf current is admittedly not identical for different stripe\npositions. Consequently the comparison of the rf magne-\ntization phase at different stripe locations is obscured.\nIt is a specially interesting point concerning microwave\nphotovoltagethatthephaseoftheindividualcomponents\nof the rf magnetic field with respect to the rf current,\nand therefore also with respect to each other can be de-\ntermined. The phase information is encoded in the line\nshape, whichisaparticularfeatureofthemicrowavepho-\ntovoltage described in this work.\nAt this point only determining hI1is possible because\nI1is unknown. However in paragraph 4.1 an approach\nto determine I1using the bolometric effect is presented.\nUsingthisapproachthe bolometricphotoresistanceisthe\nperfect supplement for the photovoltage. It delivers un-\nknownI1with almost no additional setup.\n4. PHOTORESISTANCE MEASUREMENTS\nThe principle difficulties when detecting the AMR\ninduced photoresistance are to increase the microwave\npower for a sufficient signal strength and to reduce the\nphotovoltage signal, which is in general much stronger\nand superimposes with the photoresistance. We over-\ncome the microwave power problem by using high initial\nmicrowave power (316 mW) and a coplanar waveguide\n(CPW)10which emits the microwaves as close as possi-\nble to the Permalloy film stripe (0 .137×20×2450µm3)\nwith which we detect the photoresistance. Its resistance\nis found to be R= 880 Ω and the AMR RA= 15 Ω. Its\nmagnetic properties ( γ,M0) are almost identical to that14\nof the sample investigated in section 3. We use again\nlock-in technique like in 3.1 with now an additional dc\ncurrent from a battery to measure resistance instead of\nvoltage. The strong microwave power results in strong rf\ncurrents within the sample which give a specially strong\nphotovoltage signal (see equation (27)). To achieve a\nsufficiently strong photoresitance signal the dc current\nI0and rf current I1have to be increased to the maximal\nvalue that does not harm the sample (a few mA, hence\nI0≈I1).\nIgnoring the trigonometric factors sin2 α0, cos2α0\nand cosψas well as the photoresistance term depend-\ning onβ1(that is always smaller than α1) the photo-\nvoltage signal ( UMW=α1sin(2α0)cosψRAI1/2, equa-\ntion (21)) and the photoresistance signal (∆ RMWI0≈\n−α2\n1cos(2α0)RAI0/2,(17))becomealmostidentical. But\nthe major difference is that the photoresistance is mul-\ntiplied byα2\n1and the photovoltage only by α1. Asα1\nis particularly small ( <1◦) in our experiments, this\nmeans that ∆ RMWI0is much smaller than UMW. How-\never suppressing UMWis possible because it vanishes\nforα0= 0◦,90◦,180◦,270◦(see equation (21)). A\nvery precise tuning of α0with an accuracy below 0 .1◦\nis necessary to suppress UMWbelow ∆RMWI0. Fortu-\nnately in contrast to |UMW|,|∆RMW|is maximal for\nα0= 0◦,90◦,180◦,270◦. In the following we will first\ndiscuss the bolometric photoresistance arising from mi-\ncrowave heating of the sample and afterwards the AMR\ninduced photoresistance that is discussed above.\n4.1. BOLOMETRIC (NON-RESONANT)\nThe AMR-induced ∆ RMWis not the only photoresis-\ntive effect present in our Permalloyfilm stripe. Also non-\nresonant heating by the microwave rf current I1results\nin a (bolometric) photoresistance. The major difference\ncompared to the AMR-based photoresistance is that the\nbolometric photoresistance is almost independent of the\napplied dc magnetic field Hand that its reaction time to\nmicrowave exposure is much longer (in the order of ms)\nthan that of the AMR-based photresistance (1 /αGω, in\nthe orderofns)12. The non-resonantbolometric photore-\nsistance is found with a typical strength of (∆ R/R)/P=\n0.2 ppm/mW (see figure 14).\nThe bolometric heating power Pbolarises from re-\nsistive dissipation of the rf current I1in the sample\n(Pbol=/angbracketleftbig\nRI2/angbracketrightbig\n=RI2\n0+RI2\n1/2). This can hence be used\nto determine I1, which is otherwise an unknown in equa-\ntion (27).I1can be determined for example by finding\nthe corresponding dc current I0with the same bolomet-\nric resistance change. However, especially in the sample\nwe use the thermal conductivity of the GaAs-crystal on\nwhich our Permalloystripes were deposited is so high (55\nW/m·K) that the different stripes are strongly thermally\ncoupled. Thus we can not address the bolometric signal\nof one stripe solely to the rf current of the same stripe.\nThis effect was verified comparing the resistance changes\nFIG. 14: (Color online). Photoresistance ∆ RMWmeasure-\nment (stripe resistance R). The curves show the difference\nbetween the signals ∆ UwithI0= +5 mA and I0=−5 mA\natP= 316 mW: ∆ R= (∆U(I0= +5 mA) −∆U(I0=\n−5mA))/10 mA. The subtraction suppresses the photovolt-\nage dependence on absolute |I0|(for example from bolomet-\nric AMR change). For both curves the dc magnetic field H\n(and soM) was applied within the film plane, but for a) par-\nallel to the stripe (and hence to the dc current I0) and for\nb) perpendicular. A non-resonant background of about 70\nppm from bolometric photoresistance is found. It is decreas es\nby about 1.2 ppm when the sample is turned from parallel\nto perpendicular configuration. This is caused by the 1.7 %\nAMR which changes Rand the bolometric signal proportion-\nally. The FMR signal has almost Lorentz line shape and its\nposition is significantly changing when the sample is turned\nfrom parallel to perpendicular position (see section 4.2).\nfrom one stripe while applying a dc current through an\nother stripe. Hence determination of |I1|by means of\nequation (27) is only possible when using a substrate ma-\nterial with low heat conductance (e.g. glass) or by not\ndepositing more than one stripe.\n4.2. AMR BASED (RESONANT)\nIn contrast to the non-resonant bolometric photore-\nsistance in 4.1, the typically 50 times weaker resonant\nAMR-based photoresistanceis very hard to detect. After\nvisualizing it by using the CPW and turning the sample\ninto a high symmetry position (parallel or perpendicular\ntoH) it is still necessary to regard the difference of the\nphotoresistancemeasuredwith thesamecurrentstrength\nbut with reversed current sign instead of measuring with\nonly one current direction. This eliminates the remain-\ningstill significantphotovoltagesignal, which dependson\nthe absolute current strength possibly due to bolometric\nAMR change.\nMeasurement results are presented in figure 14 for\nf= 3.8 GHz. There it can be seen that (as deduced in\n2.3), if the stripe lies parallel to the magnetization, the\nAMR is maximal and the resistance decreases when the\nFMR is excited (negative photoresistance). In contrast\nin the perpendicular case the AMR is minimal and we\nmeasure a resistance increase (positive photoresistance).\nThisbehaviorisschematicallyexplainedinfigure15. The15\nFIG. 15: (Color online). Demonstration of theangular depen -\ndence of the microwave photovoltage: Without microwaves\n(a,c) the AMR is minimal in perpendicular configuration (a)\nofMandIand maximal in parallel configuration (c). When\nthe microwaves are switched on the resistance increases in\nparallel configuration (b) and decreases in perpendicular c on-\nfiguration (d).\ncurves in figure 14 show the photoresistance at the FMR\nwith symmetric Lorentz line shape as predicted in 2.3.\nUsing equation (9) we calculate µ0H0= 16.6 mT.\nHowever a deviation of H0is found in both, parallel\n(µ0H0= 11.1mT)andperpendicular( µ0H0= 25.3mT),\nconfiguration. This is due to demagnetization which\ngives rise to an FMR shift with respect to the result from\nthe infinite film approximation (compare equation (31)).\nNx= 0.7% can be assumed because of this shift.\nUsing equation (16) we find that for our conditions\nmx/my= 7.9i. Consequently we can neglect the con-\ntribution from β1=|my|/M0in equation (17) and find\n|mx|= 13 mT using ∆ RMW= (∆R/R)·R= 1.23 mΩ\n(from figure 14) and thus α1=/radicalbig\n2·∆RMW/RA= 0.73◦\nandβ1=α1/|mx/my|= 0.09◦. The smallness of β1is\nthe reason for the resonant photoresistance strength be-\ning almost identical for M∝bardblIandM⊥I(although the\nsign is reversed). We must expect |mx|,α1andβ1to be\neven a little bit larger due to our lock-in measurement\ntechnique only detecting the sinusoidal contribution to\nthe square wave signal from the microwaves.\nThe photoresistive decrease is in accordance with that\nfound by Costache et al.13. There the magnetization is\naligned with the current ( α0= 0). Thus applying an rf\nmagneticfield decreasesthe AMR from RAtoRAcos2θc.\nThis is used to determine the precession cone angle θcby\nassumingθc=α1=β1.\nThe height to width ratio of the strip is 35 nm to\n300 nm. Because of the magnetization lying along the\nstripe,13the magnetization precession strongly deviates\nfrom being circular. Using the corresponding parameters\nµ0M0= 1.06 T,γ= 2πµ0·28 GHz/T and ω/2π= 10.5GHz), we find from equation (16) that the ratio of the\namplitudes is mx/my= 3.15i. This indicates strongly\nelliptical precession and suggests that distinguishing α1\nandβ1would provide a refined description compared to\nthat using the cone angle θc, as discussed in paragraph\n2.3.\n5. CONCLUSIONS\nWe have presented a comprehensive study of dc elec-\ntric effects induced by ferromagnetic resonance in Py mi-\ncrostrips. A theoretical model based on a phenomeno-\nlogical approach to magnetoresistance is developed and\ncompared with experiments. These provide a consistent\ndescription of both photovoltage and photoresistance ef-\nfects.\nWe demonstrate that the microwave photoresistance\nis proportional to the square of magnetization precession\namplitude. In the special case of circular magnetization\nprecession, the photoresistance measures its cone angle.\nIn the general case of arbitrary sample geometry and el-\nliptical precession, we refine the cone angle concept by\ndefining 2 different angles, which provide a precise de-\nscription of the microwave photoresistance (and photo-\nvoltage) induced by elliptical magnetization precession.\nWe show that the microwave photoresistance can be ei-\nther positive or negative, depending on the direction of\nthe dc magnetic field.\nIn contrast to the microwave photoresistance, we find\nthat the microwave photovoltage is proportional to the\nproduct of the in-plane magnetization precession compo-\nnent with the rf current. Consequently it is sensitive to\nthe magnetic field dependent phase difference between\nthe rf current and the rf magnetization. This results in a\ncharacteristicasymmetric photovoltageline shape, which\ncrosses zero when the rf current and the in-plane compo-\nnent of the rf magnetization are exactly 90◦out of phase.\nTherefore, the microwave photovoltage provides a pow-\nerful insight into the phase of magnetization precession,\nwhich is usually difficult to obtain.\nWedemonstratethattheasymmetricphotovoltageline\nshape is strongly dependent on the dc magnetic field di-\nrection, which can be explained by the directional depen-\ndence of the magnetization precession excitation. By us-\ning the model developed in this work, and by combining\nsuchasensitivegeometricaldependence ofthe microwave\nphotovoltage with the bolometric photoresistance which\nindependently measures the rf current, we are now in a\nposition to detect and determine the external rf mag-\nnetic field vector, which is of long standing interest with\nsignificant potential applications.\nACKNOWLEDGEMENTS\nWethankG.Roy,X.ZhouandG.Mollardfortechnical\nassistance and D. Heitmann, U. Merkt, and the DFG for16\nthe loan ofequipment. N. M. is supported by the DAAD.\nThis workhas been funded by NSERC and URGP grantsawarded to C.-M. H.\n1B. S.GuruandH.R.Hiziroglu, Electromagnetic Field The-\nory Fundamentals, Second Edition (Cambridge University\nPress, 2004).\n2B. A. Gurney et al., J. Shi, and R. R. Katti, in Ultrathin\nMagnetic Structures IV edited by B. Heinrich and J.A.C.\nBland (Springer, Berlin, 2004), Chapter 6,7 and 8.\n3J.-G. Zhu, and Y. 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Bauer, Bart J. van Wees, Arne\nBrataas, and Yaroslav Tserkovnyak, Phys. Rev. Lett. 97,\n216602 (2006).\n22C.-M. Hu, C. Zehnder, Ch. Heyn, and D. Heitmann, Phys.\nRev. B67, 201302(R) (2003).\n23H. J. Juretschke, J. Appl. Phys. 31, 1401 (1960).\n24R.H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,\n4382 (1979).\n25W. M. Moller, and H. J. Juretschke, Phys. Rev. B 2, 2651\n(1970).\n26D. Polder, Philosophical Magazine 40, 99 (1949).\n27T. L. Gilbert, IEEE Trans. on Magn. 40, 3443 (2004).\n28L.Landau, andL.Liftshitz, PhysikZ.Sowjet. 8153(1935).\n29R. E. Camley, and D. L. Mills, J. Appl. Phys. 82, 3058\n(1997).\n30C. Kittel, Phys. Rev. 73, 155 (1948).\n31W. G. Egan, and H. J. Juretschke, J. Appl. Phys. 34, 1477\n(1963).\n32J. N. Kupferschmidt, Shaffique Adam, and P. W. Brouwer,\nPhys. Rev. B 74, 134416 (2006).\n33S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n425, 380 (2003).\n34M. S. Sodha and N. C. 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Sengupta4\n1BLTP, Joint Institute for Nuclear Research, Dubna, Moscow R egion, 141980, Russia\n2Dubna State University, Dubna, 141980, Russia\n3Umarov Physical Technical Institute, TAS, Dushanbe, 73406 3, Tajikistan\n4School of Physical Sciences, Indian Association for the Cul tivation of Science, Jadavpur, Kolkata-700032, India\n(Dated: May 7, 2019)\nThe Josephson ϕ0junctions with the current-phase relation I=Icsin(ϕ−ϕ0), where the phase\nshiftϕ0is proportional to the magnetic moment perpendicular to the gradient of the asymmet-\nric spin-orbit potential, demonstrate a number of unique fe atures important for superconducting\nspintronics and modern informational technologies. Here w e show that a current sweep along IV-\ncharacteristic of the ϕ0junction may lead to regular magnetization dynamics with a s eries of spe-\ncific phase trajectories. The origin of these trajectories i s related to a direct coupling between the\nmagnetic moment and the Josephson oscillations in these jun ctions, and ferromagnetic resonance\nwhen Josephson frequency coincides with the ferromagnetic one. We demonstrate that an exter-\nnal electromagnetic field can control the dynamics of magnet ic moment within a current interval\ncorresponding to a Shapiro step and produce topological tra nsformation of specific precession tra-\njectories. We demonstrate the appearance of the DC componen t of superconducting current and\nclarify its role in the transformation of IV-characteristi cs in the resonance region. Good agreement\nbetween numerical and analytical results has been found in t he ferromagnetic resonance region.\nThe presented results might be used for developing novel res onance methods of determination of\nthe spin-orbit coupling parameter in the non-centrosymmet ric materials. We discuss experiments\nwhich can test our results.\nI. INTRODUCTION.\nSuperconducting spintronics is one of the intensively\ndeveloping fields of condensed matter physics today. An\nimportant place in this field is occupied by the inves-\ntigations of Josephson junctions (JJs) coupled to mag-\nnetic systems1,2. The ability to manipulate the mag-\nnetic properties by Josephson current and its opposite,\ni.e.to influence the Josephson current by the magnetic\nmoment, has attracted much recent attention3–6. The\ncentral role in these phenomena is played by spin-orbit\ninteraction. In the superconductor /ferromagnet/ super-\nconductor (S/F/S) Josephson junctions, the spin-orbit\ninteraction in a ferromagnet without inversion symme-\ntry provides a mechanism for a direct (linear) coupling\nbetween the magnetic moment and the superconducting\ncurrent. Such noncentrosymmetric ferromagnetic junc-\ntions, called hereafter ϕ0junctions, break time rever-\nsal symmetry. Consequently, the current-phase relation\n(CPR) of these junctions is given by I=Icsin(ϕ−ϕ0),\nwhere the phase shift ϕ0is proportional to the magnetic\nmoment perpendicular to the gradient of the asymmet-\nric spin-orbit potential7. This feature of the CPR allows\none to manipulate the internal magnetic moment using\nthe Josephson current7,8. The theory of the anomalous\nZeemaneffectandspin-galvaniceffectin ϕ0junctionswas\ndiscussed in Refs.9,10. Experimental realization of the ϕ0\njunction has been recently reported by Szombati et al.11.\nIn Ref.12, the authorsarguedthatthe ϕ0Josephsonjunc-\ntion is ideally suited for studying of quantum tunneling\nof the magnetic moment. They proposed that magnetic\ntunneling would show up in the ac voltage across the\njunction and it could be controlled by the bias current\napplied to the junction.Though the static properties of the S/F/Sstructures\nare well studied both theoretically and experimentally,\nmuchless isknown aboutthe magneticdynamicsofthese\nsystems13–15. Recently, the presence of an anomalous\nphase shift of ϕ0was experimentally observed directly\nthrough CPR measurement in a hybrid SNS JJ fabri-\ncated using Bi 2Se3(which is a topological insulator with\nstrong spin-orbit coupling) in the presence of an in-plane\nmagnetic field16. This constitutes a direct experimen-\ntal measurement of the spin-orbit coupling strength and\nopens up new possibilities for phase-controlled Joseph-\nson devices made from materials with strong spin-orbit\ncoupling.\nIt wasdemonstratedthatthe DC superconductingcur-\nrent applied to a S/F/S ϕ 0junction might produce a\nstrong orientation effect on the ferromagnetic layered\nmagnetic moment17. The application of DC voltage to\ntheϕ0junction would produce current oscillations and\nconsequently magnetic precession. As shown in Ref.8,\nthis precession may be monitored by the appearance of\nhigherharmonicsintheCPRaswellasbythepresenceof\na DC component of the superconducting current that is\nincreases substantially near the ferromagnetic resonance\n(FMR). The authors stressed that the magnetic dynam-\nics of the S/F/S ϕ 0junction may be quite complicated\nand stronglyanharmonic. In contrastto these results, we\ndemonstrate here that precession of the magnetic mo-\nment in some current intervals along IV-characteristics\nmay be very simple and harmonic. It is expected that\nexternal radiation would lead to a series of novel phe-\nnomena. Out of this, the possibility of appearance of\nhalf-integerShapirosteps(inadditiontotheconventional\ninteger steps) and the generation of an additional mag-\nnetic precession with frequency of external radiation was2\nalready discussed in Ref.8. However, to the best of our\nknowledge, an important problem related to the recip-\nrocal influence of Josephson current and magnetization\nat different bias current along the current-voltage (IV)-\ncharacteristics has not been investigated till now. Fur-\nthermore, the variation of the magnetic precessions in\ntheϕ0junction along the IV-characteristics has not also\nbeen addressed.\nIn this paper, we present the results on the magnetic\nprecession in the ϕ0junctions with a current sweep along\nIV-characteristic. This allows us to find specific current\nintervals with very simple magnetization dynamics. We\nshow that the origin of these trajectories is related to a\ndirect coupling between the magnetic moment and the\nJosephson oscillations, realized in these junctions, and\nmanifestation of ferromagnetic resonance features when\ntheJosephsonfrequencyisclosetotheferromagneticone.\nWe also demonstrate that the interaction of the Joseph-\nson current and the magnetic moment manifests several\ninteresting features under external electromagnetic radi-\nation. In particular, the external radiation can tune the\nnature of magnetic moment precession in a current inter-\nval corresponded to the Shapiro step. We show that such\nexternal radiation can produce a topological transforma-\ntionofmagnetizationprecessiontrajectories. Wenumeri-\ncallydemonstratethe appearanceofthe dc-componentof\nsuperconducting current and clarify its role in the trans-\nformation of IV-characteristics in the resonance region.\nThe effects of Gilbert damping and spin-orbit coupling\non IV-characteristics, magnetization precession and fer-\nromagnetic resonance features are clarified.\nII. MODEL AND METHOD.\nIn Josephson junctions with a thin ferromagnetic layer\nthe superconducting phase difference and magnetization\nof theFlayer are two coupled dynamical variables. The\nsystemofequationsdescribingthedynamicsofthesevari-\nables is obtained from the Landau-Lifshitz-Gilbert equa-\ntion and Josephson relations for current and phase dif-\nference. Particularly, the magnetization dynamics of our\nsystem is described by the Landau-Lifshitz-Gilbert equa-\ntion where the effective field depends on the phase differ-\nence\ndM\ndt=−γM×Heff+α\nM0/parenleftbigg\nM×dM\ndt/parenrightbigg\n,\nHeff=K\nM0/bracketleftbigg\nGrsin/parenleftbigg\nϕ−rMy\nM0/parenrightbigg\n/hatwidey+Mz\nM0/hatwidez/bracketrightbigg\n,(1)\nwhereγis the gyromagnetic ratio, αis a phenomenolog-\nical damping constant, M0=/bardblM/bardbl,G=EJ/(KV),Kis\nan anisotropicconstant, Vis the volume of ferromagnetic\nFlayer,l= 4hL//planckover2pi1υF,Lis the length of the Flayer, and\nhdenotes the exchange field in the ferromagnetic layer.\nBased on the equations for JJ and magnetic system,\nwe can rewrite total system of equations (to be used inour numerical studies) in normalized units\n˙mx=ωF\n1+α2{−mymz+Grmzsin(ϕ−rmy)\n−α[mxm2\nz+Grmxmysin(ϕ−rmy)]},\n˙my=ωF\n1+α2{mxmz\n−α[mym2\nz−Gr(m2\nz+m2\nx)sin(ϕ−rmy)]},\n˙mz=ωF\n1+α2{−Grmxsin(ϕ−rmy)\n−α[Grmymzsin(ϕ−rmy)−mz(m2\nx+m2\ny)]},\ndV\ndt=1\nβc[I−V−sin(ϕ−rmy)],dϕ\ndt=V,(2)\nwhereβc= 2eIcCR2//planckover2pi1is the McCumber parameter,\nmi=Mi/M0fori=x,y,z, andωF= ΩF/ωcwith\ntheferromagneticresonancefrequencyΩ F=γK/M 0and\ncharacteristic frequency ωc= 2eRIc//planckover2pi1. Here we normal-\nize time in units of ω−1\nc, external current Iin units of\nIc, and the voltage Vin units of Vc=IcR. This sys-\ntem of equations, solved numerically using the fourth or-\nder Runge–Kutta method, yields mi(t),V(t) andϕ(t)\nas a function of the external bias current I. After av-\neraging procedure18,19we can find IV-characteristics at\nfixed system’s parameters and investigate the dynamics\nof magnetization along the IV-curve20.\nThe system of equations (2) is significantly simplified\nin the absence of dissipation ( α= 0). It allows us to\nclearly understand the magnetization dynamics corre-\nsponding to the different points of the IV-curve. For\nthis purpose we calculate the temporal dependence of V\nandmiat small dissipation ( α≪1) at each value of bias\ncurrent. In parallel, we also verify the dynamics of the\nmagnetic system by solving the LLG equations at some\naveraged values of voltage. The results obtained in this\ncase are in qualitative agreement with the solutions of\nthe system (2).\nIII. MANIFESTATION OF FMR AT SMALL G\nANDr.\nWhen the currentsmallerthan the criticalone ( I < Ic)\nisapplied tothe ϕ0junction, the rotationofthe magnetic\nmoment Myis determined by sin θ= (Is/Ic)Gr(My=\nMsinθ,θis the angle between the z-axis and the M\ndirection), which signifies that superconducting current\nprovokes the rotation of Min theyz-plane8.\nFirst, we demonstrate the interaction between super-\nconducting current and magnetic moment and show the\nmanifestation of the ferromagnetic resonance in the ϕ0\njunction. Forthispurposewecalculatethe I–dependence\nof the maximal my-component at each step of bias cur-\nrent. To see effect clearly, we make simulations at very\nsmall values of the parameters Gandr.\nIn Fig. 1(a), we present the calculated one-loop IV-\ncurve (obtained by increasing and decreasing I) which\ndisplays an expected hysteresis for βc= 25. The IV-\ncharacteristic at the chosen parameters of the system3\nIV\n0.5100.51(a)\nβc=25r=0.05,G=0.05\nα=0.01,ωF=0.5\nVmymax\n0.40.800.050.10.15(b)\nmxmz\n-0.200.20.9920.99220.9924(d)\nFIG. 1. (a) IV–characteristic of the ϕ0–junction; (b) Mani-\nfestation of the ferromagnetic resonance in the voltage dep en-\ndence of mmax\ny; (c) Magnetization trajectory in the mx−my\nplane at current I= 0.505 corresponded to the maximum of\nthe resonance curve in (b); (d) The same in the mx−mz\nplane.\ndoes not react practically on the changes in the mag-\nnetization dynamics. Later, we will discuss this ques-\ntion and show the resonance manifestation in the IV-\ncurve at larger randG. Here we will concentrate on\nthe features near the ferromagnetic resonance which cor-\nresponds to the return branch around ωF=V= 0.5.\nFigure 1(b) presents the voltage dependence of the max-\nimal amplitude of magnetic moment oscillations mmax\ny\ntaken in the time domain at each value of bias current\ncalculated at small values of the Josephson to magnetic\nenergy relation G= 0.05 and the parameter of spin-orbit\ncoupling r= 0.05. We see the resonance peak around\nV=ωJ=ωF= 0.5 and its two harmonics at V=ωJ/2\nandV=ωJ/3. The trajectory of magnetization in the\nmy−mxplane at bias current I= 0.505 corresponded to\nthe resonance peak is shown in Fig. 1(c). The dynamics\nof magnetization is very simple here and corresponds to\nthe rotation of the magnetic moment around the zaxis.\nAsmalldeviationtothe yaxischangeperiodicallyduring\none rotation circle, so in the mz−mxplane the magnetic\nmoment describes the form shown in Fig.1(d).\nIV. EFFECT OF SPIN-ORBIT COUPLING.\nThe main ingredient of the considered model is the\nspin-orbit interaction in the ferromagnetic layer, so it\nplays a crucial role in the effects discovered in this paper.\nFigure 2 presentsa transformationofvoltagedependence\nofmmax\nybychangingtheparameterofspin-orbitcoupling\nr. To clarify the effect, we have taken a larger value of\nJosephson to magnetic energy of relation Gin compari-Vmymax\n0.2 0.4 0.6 0.8 100.51(a)r=0.05,G=1\nα=0.01,ωF=0.5\nβc=0.5\nVmymax\n0.2 0.4 0.6 0.8 1-1-0.500.51(c)\nr=0.7\nVmymax\n0.2 0.4 0.6 0.8 1-101(d)\nR2r=1\nR4R1R3\nFIG. 2. Transformation of the ferromagnetic resonance regi on\nbychanging the spin-orbit coupling (r). Other parameters a re\nthe same and indicated in figure (a).\nson with the results presented in Fig. 1. With increasing\ninr, the peaks are shifted and widened, reflecting the\ndamped resonance, and their intensity increases. We ob-\nserve a complex resonance region with different type of\nmagnetization trajectories along the IV-curve. Particu-\nlarly, in the insets of Fig. 2(b) we demonstrate the trans-\nformation of trajectories (in the xyplane) with decrease\nin bias current at I= 0.46 andI= 0.40. In Fig. 2(d) we\ncan clearly distinguish the regular regions ( R1,R2,R3\nandR4) indicated by arrows. One of the main questions\nstudied in this paper concerns the reaction of the mag-\nnetic system to superconducting current reflected in the\nappearance of such regular regions. So in what follows,\nwe shall concentrate on the current intervals with regular\ndynamics.\nTo clarify the dynamics in the selected regular regions,\nwe have investigated the time dependence of the magne-4\ntization component mypresented in Fig. 3 together with\nthe corresponding parts of IV-characteristic. Two differ-\nent parts are clearly pronounced in these both regions\nwhere the amplitude of mygrows with decreasing bias\ncurrent and has a jump between them. These parts are\ndenoted as R1andR2in the first region and R3and\nR4in the second one. Demonstrated in the insets the en-\nlargedtime dependence of mytakenat arbitrarycurrents\nshows a different character of oscillations. This fact ex-\nplains the observed jumps between R1andR2in the first\nregion and R3andR4in the second one. As we demon-\nstrate below, the origin of different time dependence in\nthese regions is related to the change of the character of\nmagnetic precessions.\nFIG. 3. (a) Time dependence of myin the regular region\nshown by arrow in Fig. 2(d). Insets demonstrate the charac-\nter of the my(t) oscillations in the R1andR2current inter-\nvals; (b) The same for the second region. The lines show the\ncorresponding parts of IV-characteristics.V. SPECIFIC PHASE TRAJECTORIES.\nHere we demonstrate that current intervals Riindi-\ncated in Fig. 2 and Fig. 3 are characterized by different\ntrajectories of the magnetic moment. The characteristic\ntrajectoriesintheplanes my–mx,mz–mx, andmz–my\nrealizedin theseregionsareshownin Fig.4forfourvalues\nof bias current I= 0.60, 0.54, 0.46 and 0 .42. We see dif-\nFIG. 4. Magnetization trajectories in the planes my–mx,\nmz–mx, andmz–myfor regular regions Ri\nferent specific forms of trajectories and some of them for\ndistinctnesswecall“apple”(b), “sickle”(d) “mushroom”\n(e), “fish” (g) and “moon” (h)). The first current inter-\nvalR1is characterizedby “apple”type dynamics demon-\nstratedin Fig.4(b) at I= 0.60. With decreasingbiascur-\nrent we observe a transformation of trajectories of “ap-\nple” to “mushroom”type in the mz−mxplane, while the\nthird interval R3is characterized by “fish” and “moon”\ntype trajectories. In the fourth interval R4we observe\nthe characteristic trajectory of “double fish” type. Thus,\nthe presented results demonstrate a unique possibility of\ncontrolling the magnetization dynamics via external bias\ncurrent.\nTotest whetherthe observedtemporaldependencies of\nmagnetization are related to the Josephson oscillations,\nwe have made a detailed FFT analysis at different values\nof bias current. In particular, in Fig. 5 we present the re-\nsults of FFT analysis of the time dependence of the mag-\nnetization components and voltage for JJ with and with-\nout the magnetic system at I= 0.54. Comparing the re-\nsultspresentedinthisfigure,wefindthatthedynamicsof5\nfAmp\n0.10.200.2\nf=0.0425my(t),I=0.54\nfJ+f(a)\nfJ=0.085fJ=0.085\nfAmp\n0.10.200.20.40.60.8\nf=0.0425mz(t),I=0.54fJ=0.085(b)\nfAmp\n0.10.200.050.1\nf=0.0425V(t)\nI=0.54fJ=0.085(c)\nfAmp\n0.10.200.050.1V(t)\nI=0.54\nSISJJfJ=0.085(d)\nFIG. 5. FFT analysis of time dependencies of: (a) my(t); (b)\nmz(t); (c)V(t); (d)V(t) for JJ without the magnetic system\natI= 0.54.\nmagnetizationinthiscaseisreallydeterminedbyJoseph-\nson frequency fJ=ωJ/(2π) = 0.085. The existence of\nhalf harmonics in this parameter regime indicates that\nthe excitation of magnetic dynamics happens parametri-\ncally. We also note the effect of the magnetic oscillations\non Josephsoncurrent which is manifested as a small peak\nin FFT of V(t). This peak is absent for the Josephson\njunction superconductor-insulator-superconductor with-\nout the ferromagneticlayer(SIS JJ) asit isdemonstrated\nin Figure 5(d). The Results of detailed FFT analysis of\ndynamics magnetization in the ϕ0junction at different\nvalues of bias current will be presented somewhere else20.\nVI. EFFECTS OF EXTERNAL RADIATION\nAnother important feature of the studied magnetiza-\ntion dynamics concerns the possibility of its control via\nexternal electromagnetic radiation. The presence of such\na radiation amounts to I→I(t) =I+Asin(ωt) in Eq.\n(2), where ωis the frequency and Ais the amplitude\nof the external radiation19. We find that such an exter-\nnal radiation can control the qualitative nature of the\nmagnetic precession in current interval corresponding to\na Shapiro step. To demonstrate this feature, we show\nthe IV-characteristic of ϕ0junction under external radi-\nation with frequency ω= 0.366 and amplitude A= 1\n(which demonstrates the corresponding Shapiro step at\nV= 0.366) in Fig. 6(a). The resulting magnetization\nprecession in the mz−mxplane at I= 0.475,I= 0.45\nandI= 0.385 are presented in Fig. 6(b), Fig. 6(c) and\nFig. 6(d), respectively. In sharp contrast to magnetiza-\ntion dynamics without radiation (see Fig. 4) demonstrat-\ning different specific precession dynamics with changing\nin bias current, the dynamics of magnetic precessionsalong the Shapiro step are very similar for all three cur-\nrent values.\nIV\n0.20.40.60.250.5\n2\nI=0.475(a)\nI=0.45I=0.385G=1,r=1\nω=0.366,A=1\n1\nFIG. 6. (a) IV-characteristics of JJ without (1) and with\n(2) radiation. Arrows indicate the bias current values wher e\ndynamics of magnetic precessions was investigated; (b) Dy-\nnamics of magnetic precession in the mz–mxplane in the\npresence of external radiation at I= 0.475; (c) The same at\nI= 0.45; (d) The same at I= 0.385.\nAnother central result of our work is to demonstrate\nthat the radiation may change the topology of magnetic\nprecession. In particular, we show the left-right transfor-\nmationof“mushroom”-typeprecession. AsshowninFig.\n7(a,b), such a change may be accomplished by changing\nan amplitude of radiation at a fixed DC drive current\nvalueI= 0.45. This transformation is related to a mag-\nnetizationreversalfrom −myto+myascanbe seenfrom\nchange in temporal dependence of my(t) in the presence\nof the external radiation as demonstrated in Fig. 7(c,d).\nVII. DC CONTRIBUTION TO THE\nJOSEPHSON CURRENT.\nAs it was stressed in Ref.8,20, the Gilbert damping\nplays an important role in the dynamics of S/F/S JJ.\nIt results in a DC contribution to the Josephson current\nI0(α) =αGr2ωJ\n4/parenleftbigg1\nΩ−+1\nΩ+/parenrightbigg\n, (3)\nwith Ω ±= (ωJ±1)2+α2ω2\nJ. As we see, this contribution\ndepends on the spin-orbit interaction rand relation of\nJosephson energy to magnetic energy G, and it is absent\natα= 0.\nTheresultofsuperconductingcurrentsimulationalong\nIV-characteristics with sweeping bias current down is\nshown in Fig. 8(a). It presents the voltage dependence of\nIs(V) togetherwith the analyticalcurve for I0, according6\nmxmz\n-1 0 1-1-0.500.51(a)G=1,r=1,I=0.45\nFIG. 7. Left-right transformation by changing the amplitud e\nof external radiation. (a) Dynamics of magnetic precession in\nmz–mxplanewithoutexternalradiation at I= 0.45; (b)The\nsame under radiation with frequency ω= 0.44 and amplitude\nA= 0.05; (c) Time dependence of mywithout radiation at\nI= 0.45; (d) The same under radiation.\nto (3). We see that in the resonance region the voltage\ndependence of I0is in good agreement with the result\nfor superconducting current Is(V). The Gilbert damping\nleadsto the damped ferromagneticresonanceat ωJ=ωF\nwith corresponding analytical dependence8,20formy\nmy(t) =ω+−ω−\nrsinωJt−α++α−\nrcosωJt,(4)\nwhereω±=Gr2\n2ωJ±1\nΩ±andα±=Gr2\n2αωJ\nΩ±with Ω ±=\n(ωJ±1)2+α2ω2\nJ. In Fig.8(b), we plot this analytical\ndependence together with the maximal amplitude mmax\ny\ncalculated by the system of equations (2) as a function\nof voltage. We see good agrement of both results. We\nstress that numerical calculations do not use any approx-\nimations in comparison with analytical ones (where a\nweak coupling regime was used and considered the case\nmx,my<<1 ), so the simulated dependence reflects\nadditionally harmonic of the ferromagnetic resonance at\nωJ=ωF/2.\nBased on the presented results, we may conclude that\na variation of the Josephson junction and ferromagnetic\nlayerparametersin the systemwith dampingmaylead to\nenoughstrongcoupling between superconductingcurrent\nand magnetization. Manifestation of such interaction in\nIV-characteristic of the ϕ0junction near the ferromag-\nnetic resonance is presented in Fig.8(c) where we show\nthe parts of IV-characteristics of the ϕ0junction at three\nvalues of the spin-orbit interaction at ωF= 1. The DC\ncontribution to the Josephson current manifests itself as\na deviation of IV-curve from the linear dependence in\nthe resonance region. The corresponding voltage depen-\ndencies of mmax\nyare shown in Fig.8(d). The effect ofthe spin-orbit interaction on the resonance character of\nthe presented dependence might form a theoretical foun-\ndation for developing the experimental methods for de-\ntermination of spin-orbit coupling intensity in the non-\ncentrosymmetric materials.\nVIs\n0.511.500.050.10.15I0(analytical)Is(numerical)(a)\nG=0.1,r=0.5\nα=0.1,ωF=1\nβc=25\nVmymax\n0.511.500.10.20.3\nωJ=ωF/2analytical\nωJ=ωFnumerical (b)\nIV\n0.811.20.811.2\n3.r=0.51\n2\n3(c)\n1.r=0.1\n4.r=0.7\nVmymax\n00.511.5200.20.4\n123(d)\nFIG. 8. Demonstration of the ferromagnetic resonance with\nsweeping bias current along IV-characteristics. (a) The vo lt-\nage dependence of Isand analytical I0(ωJ); (b) The voltage\ndependence of mmax\nyand analytical my(ωJ); (c) The parts\nof IV–characteristics of the ϕ0junction for G= 0.1,r= 0.5,\nα= 0.1,ωF= 1 at different values of the spin-orbit interac-\ntion; (d) The voltage dependence of mmax\nyat different r.\nVIII. CONCLUSIONS.\nTo summarize, we point out an intriguing opportunity\nto observe a different type of magnetization trajectories\nby sweeping current along the IV-characteristics of the\nϕ0junction due to a direct coupling between the mag-\nnetic moment and the Josephsoncurrent. The prediction\nin Ref.8(which is verified in our numerical simulations)\nthat the DC superconducting current in the presence of\na constant voltage Vapplied to the junction implies a\ndissipative regime can be easily detected experimentally.\nGood agreement between numerical and analytical re-\nsults found at the ferromagnetic resonance opening wide\nopportunities for further manipulation of system param-\neters and experimental verification of the magnetization\ndynamics of the materials with strong spin-orbit cou-\npling. This can be easily achieved by applying external\nradiation to the setup used in Ref.16.\nThe appropriate candidate for the experimental ver-\nification of the obtained results might be a permalloy\ndoped with Pt21. In this material the parameter that\ncharacterizes the relative strength of the spin-orbit inter-\naction is υso/υF∼1. ThePtat small doping ( up to\n10%) did not influence significantly magnetic properties7\nof permalloy21and then we may expect υso/υFto reach\n0.1 in this case also. If the length of the Flayer is of\nthe order of the magnetic decaying length /planckover2pi1/υF=h, i.e.,\nl∼1, we have r∼0.117. Another suitable candidate\nmay be a Pt/Cobilayer, ferromagnet without inversion\nsymmetry like MnSiorFeGe. In this material the spin-\norbit interaction can generate a ϕ0Josephson junction7\nwith a finite ground phase difference. The measurement\nof this phase difference may serve as an independent way\nfor the parameter revaluation11. The parameter Ghas\nbeen evaluated in Ref.8for weak magnetic anisotropy of\npermalloy K∼4×10−5K˚A−1(see Ref.22) and S/F/S\njunction with l∼1 andTc∼10KasG∼100. For\nstronger anisotropy we may expect G∼1. The typical\nferromagneticresonancefrequencyis ωF= 10GHzwhich\nis accessible in the experiments. So, we may conclude,\nthat our results can be tested experimentally.\nA very rich physics is expected if the ϕ0Josephson\njunction is exposed to microwave radiation. Addition-\nally to the features predicted in Ref.8, particularly, an\nincrease of the Shapiro steps amplitude due to spin-orbit\ncoupling near the ferromagnetic resonance, the appear-\nance of the half-integer Shapiro steps, and precession of\nthe magnetization vector with radiation frequency, we\nexpect that an external electromagnetic field can control\nqualitative features of the magnetic moment dynamicsin a current interval which corresponds to the Shapiro\nstep. Moreover, as we demonstrated in this paper, such\nradiation can also produce a topological transformation\nof precession trajectories23. We predict that the change\nin topology of the magnetization dynamics would be ob-\nserved in such systems as a function of the amplitude\nof the applied electromagnetic radiation. We consider\nthat the presented results might be used for developing\nnovel experimental resonance methods for determination\nof the spin-orbit interaction in the non-centrosymmetric\nmaterials.\nIX. ACKNOWLEDGEMENTS.\nThe authors thank A. Mazanik, I. Bobkova, A. Bobkov\nand A. Buzdin for helpful discussion. The reported study\nwas partially funded by the RFBR research projects 18-\n02-00318, 18-52-45011-IND. KS thanks DST for support\nthrough Indo-Russian grant INT/RUS/RFBR/314. Nu-\nmerical calculations have been made in the framework of\nthe RSF project 18-71-10095.\nX. REFERENCES.\n1Jacob Linder and W. A. Jason Robinson, Nature Physics\n11, 307 (2015).\n2S. Mai, E. Kandelaki, A. F. Volkov, and K. B. Efetov,\nPhys. Rev. B84, 144519 (2011).\n3A. I. Buzdin, Rev. Mod. Phys. 77, 935 2005.\n4F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys.77, 1321 2005.\n5A. A. Golubov, M. Y. Kupriyanov, and E. Ilichev, Rev.\nMod. Phys. 76, 411 2004.\n6Roopayan Ghosh, Moitri Maiti, Yury M. Shukrinov, and\nK. Sengupta, Phys. Rev. B 96, 174517 (2017).\n7A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008).\n8F. Konschelle, A. Buzdin, Phys. Rev. Lett . 102, 017001\n(2009)\n9F. Dolcini, M. Houzet, and J. S. Meyer, Phys. Rev. B92,\n035428 (2015).\n10F. Konschelle, I. V. Tokatly, and F. S. Bergeret, Phys. Rev.\nB 92, 125443 (2015).\n11D. B. Szombati, S. Nadj-Perge, D. Car, S. R. Plissard, E.\nP. A. M. Bakkers and L. P. Kouwenhoven, Nature Physics,\n12, 568–572 (2016).\n12Eugene M. Chudnovsky, Phys. Rev. B93, 144422 (2016).13X. Waintal and P. W. Brouwer, Phys. Rev. B 65, 054407\n(2002).\n14V. Braude and Ya. M. Blanter, Phys. Rev. Lett. 100,\n207001 (2008).\n15J. Linder and T. Yokoyama, Phys. Rev. B 83, 012501\n(2011).\n16Alexandre Assouline et. al., arXiv:1806.01406 (unpub-\nlished).\n17Yu. M. Shukrinov, I. R. Rahmonov, K. Sengupta, and A.\nBuzdin, Appl. Phys. Lett. 110, 182407 (2017).\n18Yu. M. Shukrinov, F. Mahfouzi, and N. F. Pedersen, Phys.\nRev.B 75, 104508 (2007)\n19W. Buckel and R. Kleiner, Superconductivity: Fundamen-\ntals and Applications , Wiley-VCH (2004).\n20Yu. M. Shukrinov, I. R. Rahmonov, A. A. Mazanik, A.\nBuzdin, and K. Sengupta, In preparation for PRB.\n21A. Hrabec, F. J. T. Goncalves, C. S. Spencer, E. Arenholz,\nA. T. N’Diaye, R. L. Stamps, and C. H. Marrows, Phys.\nRev.B 93, 014432 (2016).\n22A. Yu. Rusanov, M. Hesselberth, J. Aarts, and A. I.\nBuzdin, Phys. Rev. Lett. 93, 057002 (2004).\n23Yu. M. Shukrinov, I. R. Rahmonov, K. Sengupta, Cond-\nmat arXiv:1811.05282, (2018)"    },    {        "title": "1111.3927v1.Adiabatic_quantum_pumping_through_surface_states_in_3D_topological_insulators.pdf",        "content": "arXiv:1111.3927v1  [cond-mat.mes-hall]  16 Nov 2011Adiabatic quantum pumping through surface states in 3D topo logical insulators\nM. Alos-Palop, Rakesh P. Tiwari and M. Blaauboer\nDelft University of Technology, Kavli Institute of Nanosci ence,\nDepartment of Quantum Nanoscience, Lorentzweg 1, 2628 CJ De lft, The Netherlands.\n(Dated: November 9, 2018)\nWe investigate adiabatic quantum pumping of Dirac fermions on the surface of a strong 3D\ntopological insulator. Two different geometries are studie d in detail, a normal metal – ferromagnetic\n–normal metal (NFN)junction and aferromagnetic –normal me tal –ferromagnetic (FNF) junction.\nUsing a scattering matrix approach, we first calculate the tu nneling conductance and then the\nadiabatically pumped current using different pumping mecha nisms for both types of junctions.\nWe explain the oscillatory behavior of the conductance by st udying the condition for resonant\ntransmission in the junctions and find that each time a new res onant mode appears in the transport\nwindow, the pumped current diverges. We also predict an expe rimentally distinguishable difference\nbetween the pumped current and the rectified current.\nPACS numbers: 73.20.-r, 73.40.-c, 73.23.-b, 73.63.-b\nI. INTRODUCTION\nRecently, surface states in topological insulators have\nattracted a lot ofattention in the condensed-mattercom-\nmunity1. Both in two-dimensional (e.g., HgTe) and in\nthree-dimensional (e.g., Bi 2Se3) compounds with strong\nspin-orbit interaction the topological phase has been\ndemonstrated experimentally2–5. Although these com-\npounds are insulating in the bulk (since they have an en-\nergy gap between the conduction band and the valance\nband), their surface states support topological gapless\nexcitations. In the simplest case these low-energy excita-\ntions of a strong three-dimensional topological insulator\ncan be described by a single Dirac cone at the center of\nthe two-dimensional Brillouin zone (Γ point)2,4,6–8. The\ncorresponding Hamiltonian is given by9\nH0=/planckover2pi1vF/vector σ·/vectork−µI. (1)\nHere/vector σrepresents a vector whose three components are\nthe three Pauli spin matrices, Irepresents a 2×2 iden-\ntity matrix in spin space, vFis the Fermi velocity, and\nµis the chemical potential. The low-energy excitations\nofH0[Eq. (1)] are topologically protected against per-\nturbations10. This has prompted recent research on the\ntransport properties of surface Dirac fermions. For ex-\nample, the conductance and magnetotransport of Dirac\nfermions have been studied in normal metal – ferromag-\nnet (NF), normal metal – ferromagnetic – normal metal\n(NFN) and arrays of NF junctions on the surface of a\ntopological insulator11–13, suggesting the possibility of\nan engineered magnetic switch. An anomalous magne-\ntoresistance effect has been predicted in ferromagnetic-\nferromagnetic junctions14. Also, electron tunneling and\nmagnetoresistance have been studied in ferromagnetic –\nnormal metal – ferromagnetic (FNF) junctions15,16, for\nwhich it has been predicted that the conductance can\nbe larger in the anti-parallel configuration of the mag-\nnetizations of the two ferromagnetic regions than in the\nparallel configuration. In addition, a large research effort\nhas been devoted to studying models which predict theexistenceofMajoranafermionedgestatesattheinterface\nbetween superconductors and ferromagnets deposited on\na topological insulator9,17,18.\nIn this article we investigateadiabatic quantum pump-\ning of Dirac fermions through edge states on the sur-\nface of a strong three-dimensional topological insulator.\nQuantum pumping refers to a transport mechanism in\nmeso- and nanoscale devices by which a finite dc cur-\nrent is generated in the absence of an applied bias by\nperiodic modulations of at least two system parameters\n(typically gate voltages or magnetic fields)19–21. In or-\nder for electrical transport to be adiabatic, the period\nof the oscillatory driving signals has to be much longer\nthan the dwell time τdwellof the electrons in the system,\nT= 2πω−1≫τdwell. In the last decade, many different\naspects of quantum pumping have been theoretically in-\nvestigated in a diverse range of nanodevices, for example\ncharge and spin pumping in quantum dots22–25, the role\nof electron-electron interactions26–28, quantum pumping\nin graphene mono- and bilayers29–35as well as charge\nand spin pumping through edge states in quantum Hall\nsystems36and recently a two-dimensional topological in-\nsulator37. On the experimental side, Giazotto et al.38\nhave recently reported an experimental demonstration of\ncharge pumping in an InAs nanowire embedded in a su-\nperconducting quantum interference device (SQUID).\nOur main focus is to study quantum pumping induced\nby periodic modulations of gate voltages or exchange\nfields, which are induced by a ferromagnetic strip in two\ntopological insulator devices: a NFN and a FNF junc-\ntion, see Figs. 1 and 2. Using a scattering matrix ap-\nproach, we obtain analytical expressions for the angle-\ndependent pumped current in both types of junctions.\nWe find that the adiabatically pumped current in a NFN\ntopological insulator junction induced by periodic mod-\nulations of gate voltages reaches maximum values at spe-\ncific energy values. In order to explain the position of\nthese values, we study in detail the conductance of the\njunctions. In particular, we provide an explanation for\nresonances in the conductance that were predicted but2\nTopological insulatorPSfrag replacements\nNl NrF/vector x/vector yd\nFIG. 1. (Color online) Sketch of the N lFNrjunction on the\nsurface of a topological insulator. Pumping is induced by\napplying gate voltages (not shown) to the normal leads. In\nthe middle region a thin ferromagnetic film induces ferromag -\nnetism on the surface of the topological insulator by means\nof the exchange coupling14. The arrow in the middle region\nindicates the direction of the magnetization Min this region.\nnot explained in detail in the previous works11,12. We\nshow that each time a new resonant mode appears in the\njunction the conductance increases and the pumped cur-\nrent reaches a maximum value. For the FNF pump we\npredict a non-zero current by periodic modulation of the\nexchange magnetic coupling in the absence of external\nvoltages. We observe and analyze basic similarities and\ndifferences between the two pumps studied in this paper\nand highlight an experimentally distinguishable feature\nbetween the pumped current and the conductance.\nThe remainder of the paper is organized as follows. In\nSec. II, we describe the NFN and FNF junctions and use\na scattering matrix model to calculate the reflection and\ntransmissioncoefficientsofboth junctions. In Sec.III, we\nreview the conductance of the NFN junction and present\na detailed analysis of the plateau-like steps that appear\nin the conductance. We also analyze and compare the\nconductance of the FNF junction with parallel and anti-\nparallel configuration of the magnetization. In Sec. IV,\nwe calculate the adiabatically pumped current for the\ntwo different pumps and derive analytical expressions as\na function of the angle of incidence of the carriers. We\nalso investigate the dependence of the pumped current\nand the conductance on the width dof the middle region.\nFinally, in Sec. V we summarize our main results and\npropose possibilities for experimental observation of our\npredictions.\nII. NFN AND FNF JUNCTIONS\nWe first describe the NFN junction, see Fig. 1. The\njunction is divided into three regions: region Nl(for\nx <0), regionNr(forx > d) and the ferromagnetic\nregionFin the middle. The left and right-hand side of\nthejunctionrepresentthebaretopologicalinsulator. The\ncharge carriers (surface Dirac fermions) in these regions\nare described by the Hamiltonian H0[Eq. (1)] whoseeigenstates are given by\nψ±\nN=1√\n2/parenleftbigg\n1\n±e±iα/parenrightbigg\ne±iknxeiqy, (2)\nwhere +(−) labels the wavefunctions traveling from the\nleft (right) to the right (left) of the junction. The angle\nof incidence αand the momentum knin thex-direction\nare given by:\nsin(α) =/planckover2pi1vFq\n|ǫ+µ|, (3)\nkn=/radicalBigg/parenleftbiggǫ+µ\n/planckover2pi1vF/parenrightbigg2\n−q2. (4)\nHereǫrepresentstheenergymeasuredfromtheFermien-\nergyǫFandqdenotes the momentum in the y-direction.\nIn the normal regions NlandNra dc electrical volt-\nage can be applied via metallic top gates to tune the\nchemical potential µand thereby control the number of\ncharge carriers incident on the junction. We assume gate\nvoltages to be small compared to the bandgap for bulk\nstates (eVi≪Eg∼1 eV,i=l,r), so that transport is\nwell described by surface Dirac states8. In this case, the\neigenstates are given by\nψ±\nNl=1√\n2/parenleftbigg\n1\n±e±iαl/parenrightbigg\ne±iknlxeiqy,(5)\nψ±\nNr=1√\n2/parenleftbigg\n1\n±e±iαr/parenrightbigg\ne±iknr(x−d)eiqy,(6)\nsin(αi) =/planckover2pi1vFq\n|ǫ+µ−eVi|, (7)\nkni=/radicalBigg/parenleftbiggǫ+µ−eVi\n/planckover2pi1vF/parenrightbigg2\n−q2, (8)\nwhere the index i=l,rlabels the normal sides of the\njunction.\nIn the middle regionM ofthe junction (0 <x<d), the\npresence of the ferromagnetic strip modifies the Hamil-\ntonian by providing an exchange field. The Hamil-\ntonian that describes the surface states is now H=\nH0+Hinduced, where the induced exchange Hamiltonian\nis given by12,14\nHinduced=/planckover2pi1vFMσy, (9)\nwith the magnetization /vectorM=Mˆy. The magnitude M\ndepends on the strength of the exchange coupling of the\nferromagnetic film and can be tuned for soft ferromag-\nnetic films by applying an external magnetic field14. The\neigenstates of the full Hamiltonian Hare then given by:\nψ±\nF=1√\n2/parenleftbigg\n1\n±e±iαm/parenrightbigg\ne±ikmxeiqy,(10)3\nTopological insulatorPSfrag replacements\nFl Fr\nN/vector x/vector yd\nFIG. 2. (Color online) Sketch of the F lNFrjunction. Ferro-\nmagnetic films are placed on top of the topological insulator\non the left and right providing exchange fields in these re-\ngions. The arrows indicate the direction of the correspondi ng\nmagnetizations MlandMr, see the text for further explana-\ntion.\nwith\nsin(αm) =/planckover2pi1vF(q+M)\n|ǫ+µ|, (11)\nand\nkm=/radicalBigg/parenleftbiggǫ+µ\n/planckover2pi1vF/parenrightbigg2\n−(q+M)2. (12)\nFrom Eq. (12) we see that for a given energy there exists\na critical magnetization\nMc=±2|ǫ+µ|/(/planckover2pi1vF), (13)\nbeyond which for all transverse ( q) modes the wave-\nfunction changes from propagating to spatially decaying\n(evanescent) along the x-direction11,12.Now we describe the FNF junction, see Fig. 2. Region\nFl(x <0) and region Fr(x > d) are modeled as ferro-\nmagnetic regions, respectively, with different magnetiza-\ntionsMl,Mralong they-axis and corresponding wave-\nfunctionψF[Eq. (10)]. The Dirac fermions in the middle\nregion N (0 <x<d) are described by the wavefunctions\nψN[Eq.(2)]. When calculating transport properties of\nthe FNF junction, we focus on two different alignments\nof the magnetizations of the ferromagnetic regions: the\nparallel configuration (M l/bardblMr), where the magnetiza-\ntions in the ferromagnetic regions point in the same di-\nrection, and the anti-parallel configuration (M l/bardbl- Mr),\nin which the magnetizations are in opposite directions.\nUsingEqns.(2)-(12)wecancalculatethereflectionand\ntransmissioncoefficientsforaDiracfermionwithenergy ǫ\nand transversemomentum qincident from the left on the\njunction, for both the NFN and the FNF junctions. To\nthis end, we consider a general F lFmFrjunction, where\nthe wavefunctions in each of the three regions left ( l),\nmiddle (m) and right ( r) are given by:\nψl=ψ+\nl+rllψ−\nl,\nψm=pψ+\nm+qψ−\nm, (14)\nψr=trlψ+\nr.\nHereψ±\nj(j=l,m,r) are the wavefunctions (5), (6) or\n(10) (depending on the junction considered) and rlland\ntrldenote the corresponding reflection and transmission\ncoefficients. By requiring continuity of the wavefunction\natthe interfaces x= 0andx=d, weobtainthereflection\nand transmission coefficients:\nrll=eiαle2ikmd(1+ei(αm+αl))(eiαm−eiαr)+(eiαl−eiαm)(1+ei(αm+αr))\ne2ikmd(eiαm−eiαl)(eiαm−eiαr)+(1+ei(αm+αl))(1+ei(αm+αr)), (15)\ntrl=eikmd(1+e2iαm)(1+e2iαl)\ne2ikmd(eiαm−eiαl)(eiαm−eiαr)+(1+ei(αm+αl))(1+ei(αm+αr)). (16)\nHereαjdenotes the polar angle of the wavevector in re-\ngionj=l,m,r[Eqns. (7) and (11)]. When considering\nan electronincident from the rightlead, one cansimilarly\nobtainrrrandtlr. These expressions for the reflection\nand transmission coefficients form the basis of our cal-\nculations of the conductance and the pumped current in\nSecs. III and IV respectively.\nIII. CONDUCTANCE\nThe conductance GNFNof a topological insulator NFN\njunction has been studied in earlier work by Mondal et\nal.11,12, who predicted oscillatory behavior of GNFNas\na function of the applied bias voltage (see also Fig. 3).In this section we first briefly review their results and\nthen add a quantitative explanation for the oscillations\noftheconductance. Thisexplanationiscrucialforunder-\nstanding the behavior of the pumped current in the next\nsection. We also calculate and analyze the conductance\nin a FNF junction.\nThe general expression for the conductance Gacross\nthe junction in terms of the transmission probability\nT(α)≡|trl(α)|2is given by\nG= (G0/2)/integraldisplayπ/2\n−π/2T(α)cosαdα. (17)\nHereG0=2e2\nhρ(eV)/planckover2pi1vFW,ρ(eV) =|µ+eV|/(2π(/planckover2pi1vF)2)\ndenotes the density of states, Wis the sample width,4\n00.511.522.5300.10.20.30.40.5\n  \nPSfrag replacements\nGNFN/G0\nǫ/µ(1)\n(2)\n(3)\n(4)(1)→˜M= 3\n(2)→˜M= 3.5\n(3)→˜M= 4\n(4)→˜M= 4.5\nFIG. 3. (Color online) The conductance GNFNof the NFN\njunction [Eq.(17)] as a function of ǫ/µforVl=Vr= 0\nand for different values of the effective magnetization ˜M≡\n/planckover2pi1vFM/µ= 3 (solid blue line), 3.5 (dashed green line), 4 (dot-\ndashed red line) and 4.5 (dotted light-blue line). The effect ive\njunction width ˜d≡µd/(/planckover2pi1vF) = 5.\nand the integration is over all the angles of incidence α.\nForVl=Vr= 0 (αl=αr=α) the angle-dependent\ntransmission probability TNFN(α) is given by11,12\nTNFN(α) = cos2(α)cos2(αm)//bracketleftbig\ncos2(kmd)cos2(α)cos2(αm)\n+sin2(kmd)(1−sin(α)sin(αm))2/bracketrightbig\n,(18)\nwhereαmis the polar angle of the wave vector in the\nmiddle region as defined in Eq. (11). This angle can be\nexpressedintermsof αusingthefactthatthemomentum\nis conserved along y-axis as:\nsin(αm) = sin(α)+M/planckover2pi1vF\n|ǫ+µ|. (19)\nFigure 3 shows the conductance of the NFN junction\n[obtained from Eqns. (17) and (18)] as a function of the\nenergy of the incoming carriers ǫ/µfor different values of\nthe effective magnetization ˜M≡/planckover2pi1vFM/µ. For a given\nmagnetization M, the conductance is zero for ǫ<ǫc, i.e.,\nGNFN(ǫ) = 0, with the critical energy ǫc≡/planckover2pi1vFM/2−µ.\nBelow this energy there are no traveling modes inside the\nbarrier. Ourresults agreewith the previousresults in the\nliterature11,12.\nFrom Fig. 3 it can be observed that the conductance\nchanges from plateau-like to oscillatory as ǫ/µincreases.\nIn order to provide an explanation for this behavior we\nfirst analyze the plateau like regime in detail. After set-\ntingαl=αr≡αinEq.(15), webeginbyfindingthecon-\nditions when the reflection coefficient is zero, i.e., rll= 0.\nThe first, trivial, condition α=αm+ 2πncorresponds\nto the situation of an entirely normal junction (i.e., no\nferromagnetic region). The second and more interesting\ncondition is sin( kmd) = 0. This is the case when trans-\nmission occurs via a resonant mode of the junction andPSfrag replacements\n−π/2−π/40 π/4π/2TNFN(α)\nα1\n0.8\n0.6\n0.4\n0.2\n0(a)n=1→\nPSfrag replacements\n−π/2−π/40 π/4π/2TNFN(α)\nα1\n0.8\n0.6\n0.4\n0.2\n0(b)n=2→\nPSfrag replacements\n−π/2−π/40 π/4π/2TNFN(α)\nα1\n0.8\n0.6\n0.4\n0.2\n0(c)n=3→\nPSfrag replacements\n−π/2−π/40 π/4π/2TNFN(α)\nα1\n0.8\n0.6\n0.4\n0.2\n0(d)←n=4\nPSfrag replacements\n−π/2−π/40 π/4π/2TNFN(α)\nα1\n0.8\n0.6\n0.4\n0.2\n0(e)n=5→\nPSfrag replacements\n−π/2−π/40 π/4π/2TNFN(α)\nα1\n0.8\n0.6\n0.4\n0.2\n0(f)n=6→\nFIG. 4. (Color online) The transmission probability TNFN(α)\n[Eq. (18)] as a function of the angle of incidence αfor differ-\nent values of energy ǫ/µ, (a)ǫ/µ= 0.7, (b)ǫ/µ= 0.9, (c)\nǫ/µ= 1.2, (d)ǫ/µ= 1.6, (e)ǫ/µ= 2.4, and (f) ǫ/µ= 2.9.\nParameters used are ˜d= 5 and ˜M= 3.\ncan be written as (using Eqns. (3) and (12))\nkmd=|ǫ+µ|\n/planckover2pi1vFd/radicalBigg\n1−/parenleftbigg\nsinα+/planckover2pi1vFM\n|ǫ+µ|/parenrightbigg2\n=nπ.(20)\nEq. (20) indicates that for a given Mandǫthere are\ncertainprivilegedangles αcforwhichthe barrierbecomes\ntransparent:\nsin(αc) =±/radicaltp/radicalvertex/radicalvertex/radicalbt1−/parenleftBigg\nnπ\n˜d(1+ǫ\nµ)/parenrightBigg2\n−˜M\n|1+ǫ\nµ|,(21)\nwith˜d≡dµ/(/planckover2pi1vF) being the dimensionless barrier\nlength. These modes are referred to as resonant modes\nin this article.\nFigure 4 shows the transmission probability [Eq. (18)]\nas a function of the angle of incidence αfor different\nvalues of energy ǫ/µ. The dashed (red) vertical lines\ncorrespond to the angles satisfying Eq. (21) for differ-\nentn. It can be seen that as the energy increases more\nresonant modes become available for transmission. It is\nalso worth noting that for energies at which only one\nmode is present ( n= 1, see Fig. 4(a)) the transmission5\n00.511.522.53PSfrag replacements\n−π/2−π/40π/4αc\nǫ/µn= 1→n= 2→\n←n= 3←n= 4←n= 5n= 6→\nFIG. 5. (Color online) The real part of the angle of incidence\nαc[Eq. (21)] versus energy ǫ/µfor the modes n= 1,2,3,4,5.\nParameters used are the same as in Fig. 4.\nis strongly localized at one particular angle. This prop-\nerty could be exploited to fabricate single-mode filters.\nFor low energy excitations only negative angles α(i.e.,\nq-momenta anti-parallel to M) contribute to the conduc-\ntance, see Figs. 4(a)-(d). As the energy increases, the\nresonant modes move from the left to the right and also\npositive angles α(i.e.,q-momenta parallel to M) begin\nto contribute, see Figs. 4(e) and (f).\nNow we address the question why a mode becomes\nresonant in the barrier. Figure 5 shows the angle of inci-\ndenceαc[Eq. (21)] for different values of nas a function\nof energyǫ/µ. For a given n, the resonant mode does not\ncontribute to the conductance if the energy ǫsatisfies the\nconditionǫc<ǫ<ǫαnc, becausethe imaginarypartofthe\nmomentum kmis nonzero and thus the mode is decaying\nalong thex-direction. This critical energy ǫαncfor each\nmode is given by:\nǫαnc\nµ=n2π2\n2˜d2˜M+˜M\n2−1. (22)\nWhenǫ > ǫαnc,kmbecomes real and the mode be-\ncomes resonant. As we increase the energy, all the modes\nasymptotically reach their saturation angle α=π/2.\nFinally, we analyze the effect of the appearance of sub-\nsequent resonant modes on the conductance. For small\nenergies, the conductance increases in plateau-like steps\n(see Fig. 3). The first plateau corresponds to the sit-\nuation in which the first transmission mode appears at\nα=−π/2. As the energy increases, a new resonant\nmode appears and the conductance increases in a step-\nlike manner. The plateaus are not sharp due to the fact\nthat each new mode appearing is not sharply peaked,\nbut rather has a certain distribution around a particular\nangle of incidence, see Fig. 4. Once the energy is large\nenough for there to be contributions from both positiveand negative angles of incidence, the conductance be-\ncomes oscillatory. For very large energies ( ǫ≫ǫc), the\n00.511.522.5300.20.40.60.8\n  \nPSfrag replacements\nGNFN/G0\nǫ/µ(1)\n(2)\n(3)(4)(1)→eV/µ= 0\n(2)→eV/µ= 1\n(3)→eV/µ= 2\n(4)→eV/µ= 3\nFIG. 6. (Color online) The conductance GNFNof the NFN\njunction as a function of ǫ/µfor different values of gate volt-\nages,eV/µ= 0 (solid blue line), 1 (dashed green line), 2\n(dashed-dot red line) and 3 (double-dotted light-blue line ).\nAs before, ˜M= 3 and ˜d= 5.\neffect of the magnetic barrier disappears and the conduc-\ntance becomes unity ( GNFN=G0).\nFigure6 showsthe conductance asa function ofenergy\nfor several values of applied bias voltages Vl=Vr≡V.\nAs expected, the features of the conductance remain the\nsame for finite V. As we increase eV, the critical energy\nǫc=/planckover2pi1vFM/2+eV/2−µfor the onset of the conductance\nincreases and the spacing between two consecutive reso-\nnant modes decreases. As a result the plateaus become\nnarrower.\nIn the remaining part of this section we study the\nconductance in a topological insulator FNF junction, as\nshown in Fig. 2. We consider both the junction with\nparallel and with anti-parallel magnetization in the fer-\nromagnetic regions. In the parallel configuration, us-\ningαl=αr≡α= sin−1(/planckover2pi1vF(q+M)/|ǫ+µ|) and\nαm= sin−1(/planckover2pi1vFq/|ǫ+µ|) in Eq. (16), we find that the\nconductanceis similarto the conductanceofaNFN junc-\ntion, asdisplayedinFig.3. However,intheFNFjunction\nthe first resonant mode becomes resonant for positive α\n(i.e., transverse q-momentum parallel to M) and as the\nenergy increases, the resonances move towards negative\nvalues of the angle α. This, however, does not affect the\ntotal conductance, as we sum over all possible angles of\nincidence, and the same analysis as for the NFN junction\npresented above can be applied to understand the FNF\njunction with parallel magnetization.\nIn the case of anti-parallel alignment of the magne-\ntization in the two ferromagnetic regions, we substitute\nsin(αl) =/planckover2pi1vF(q+M)/|ǫ+µ|,sin(αr) =/planckover2pi1vF(q−M)/|ǫ+µ|\nand sin(αm) =/planckover2pi1vFq/|ǫ+µ|in Eq. (16). The conduc-\ntance of this junction was studied previously in Refs.15,16\nand the transmission probability TFNF,AP(αl,αr)≡\n|trl(αl,αr)|2is given by:6\nTFNF,AP(αl,αr) =cos2(αl)cos2(αm)\ncos2(kmd)cos2(αl+αr\n2)cos2(αm)+sin2(kmd)/bracketleftbig\ncos/parenleftbigαl−αr\n2/parenrightbig\n−sin(αl+αr\n2)sin(αm)/bracketrightbig2.(23)\n22.533.544.5500.10.20.30.40.5\n  \nPSfrag replacements\nGFNF,AP/G0\nǫ/µ(1)\n(2)\n(3)(1)→˜M= 3\n(2)→˜M= 3.5\n(3)→˜M= 4\nFIG. 7. (Color online) The conductance GFNF,APof the FNF\njunction in the anti-parallel configuration as a function of ǫ/µ,\nfor˜M= 3 (blue solid line), 3 .5 (green dashed line) and 4 (red\ndotted line). The parameter ˜d= 5.\nThe total conductance is obtained by multiplying\nTFNF,AP(αl,αr) with cos(αr)/cos(αl) and then integrat-\ningovertheallowedanglesofincidence39, i.e., fromαc1=\nsin−1(2/planckover2pi1vFM/(|ǫ+µ|)−1) toαc2= sin−1(2/planckover2pi1vFM/(|ǫ+\nµ|)+1). Thus we can write\nGFNF,AP=G0/2/integraldisplayπ/2\nαc1GFNF,AP(αl,αr)cos(αl)dαl,\n(24)\nwhere\nGFNF,AP(αl,αr) =cos(αr)\ncos(αl)TFNF,AP(αl,αr).(25)\nFig. 7 shows the conductance of the FNF junction in\nthe anti-parallel configuration. From the horizontal axis\nwe see that the critical energy ǫcfor the onset of the\nconductance is larger than in the corresponding parallel\nconfiguration. Moreover, as the energy ǫ/µincreases the\nconductance exhibits no plateau behavior: it increases in\nan oscillatory fashion. This oscillatory behavior can be\nunderstoodfromFig.8, whichshows GFNF,AP(α) forfour\ndifferent values of ǫ/µ. Note that all the angles ( αl,αm\nandαr) can be expressed in terms of one angle, which we\nchoose to be αl≡α. As the energy increases, the area\nunder the curve oscillates resulting in oscillations in the\nconductance.\nSummarizing, we have obtained a quantitative expla-\nnation for the behavior of the conductance in topological\ninsulator NFN and FNF-junctions in terms of the num-\nber of resonant modes in the junction. This explanation\nforms the basis for understanding the behavior of the\npumped current in the next section.00.20.40.60.81\n00.20.40.60.81\n00.20.40.60.81\n00.20.40.60.81\n00.20.40.60.81\nPSfrag replacements\nGFNF,AP(αl)/G0 GFNF,AP(αl)/G0 GFNF,AP(αl)/G0 GFNF,AP(αl)/G0 GFNF,AP(αl)/G0\nαlαlαlαlαl(a) (a) (a) (a) (a)\n0 0 0 0 0 π/4π/4π/4π/4π/4 π/2π/2π/2π/2π/200.20.40.60.81\n00.20.40.60.81\nPSfrag replacements\nGFNF,AP(αl)/G0 GFNF,AP(αl)/G0\nαlαl(b) (b)\n0 0 π/4π/4 π/2π/2\n00.20.40.60.81\n00.20.40.60.81\n00.20.40.60.81\nPSfrag replacements\nGFNF,AP(αl)/G0 GFNF,AP(αl)/G0 GFNF,AP(αl)/G0\nαlαlαl(c) (c) (c)\n0 0 0 π/4π/4π/4 π/2π/2π/200.20.40.60.81\n00.20.40.60.81\n00.20.40.60.81\n00.20.40.60.81\nPSfrag replacements\nGFNF,AP(αl)/G0 GFNF,AP(αl)/G0 GFNF,AP(αl)/G0 GFNF,AP(αl)/G0\nαlαlαlαl(d) (d) (d) (d)\n0 0 0 0 π/4π/4π/4π/4 π/2π/2π/2π/2\nFIG. 8. The angle-dependent total transmission\nTFNF,AP(αl,αr) for the anti-parallel configuration of\nthe FNF junction as a function of the angle of incidence αl\nfor (a)ǫ/µ= 2.47, (b)ǫ/µ= 2.57, (c)ǫ/µ= 3.10 and (d)\nǫ/µ= 3.40. Parameters used are ˜M= 3 and ˜d= 5.\nIV. ADIABATICALLY PUMPED CURRENT\nIn this section we investigate adiabatically pumped\ncurrents through NFN and FNF junctions in a topo-\nlogical insulator40. In general, a pumped current is\ngenerated by slow variation of two system parameters\nX1andX2in the absence of a bias voltage19,20. For\nperiodic modulations X1(t) =X1,0+δX1cos(ωt) and\nX2(t) =X2,0+δX2cos(ωt+φ), the pumped current Ip\nintotheleftleadofthejunctioncanbeexpressedinterms\nof the area Aenclosed by the contour that is traced out\nin (X1,X2)-parameter space during one pumping cycle\nas20:\nIp=ωe\n2π2/integraldisplay\nAdX1dX2/summationdisplay\nmΠ(X1,X2) (26a)\n≈ωe\n2πδX1δX2sinφ/summationdisplay\nmΠ(X1,X2),(26b)\nwith\nΠ(X1,X2)≡Im/parenleftbigg∂r∗\nll\n∂X1∂rll\n∂X2+∂t∗\nlr\n∂X1∂tlr\n∂X2/parenrightbigg\n.(27)\nHererllandtlrrepresent the reflection and transmission\ncoefficients into the left lead and the index msums over\nall modes (a similar expression can be obtained for the\npumped current into the right lead). Eq. (26b) is valid7\nin the bilinear response regime where δX1≪X1,0and\nδX2≪X2,0and the integral in Eq. (26a) becomes inde-\npendent of the pumping contour.\nFirst we analyze the NFN pump, where the pumped\ncurrent is generated by adiabatic variation of gate volt-\nagesVlandVrwhich change the chemical potential in\nthe normal leads on the left and right of the junction, re-spectively (see Fig. 1). Calculating the derivatives of the\nreflection and transmission coefficients rll[Eq. (15)] and\ntrlwith respect to αlandαr, substituting into Eq. (27)\nand using∂αj/(e∂Vj) = tan(αj)/|ǫ+µ−eVj|(j=l,r),\nthe pumped current for V1=V2≡Vand for a specific\nangle of incidence αis given by:\nINFN\np(α) =−INFN\n0cos3(αm)sin2(α)cos(α)sin(2kmd)\n(1+ǫ/µ−eV/µ)2(cos2(α)cos2(αm)cos2(kmd)+sin2(kmd)(1−sin(α)sin(αm))2)2.(28)\n00.511.522.53−0.08−0.0400.040.08\nPSfrag replacements\nIFNFp/I0\nǫ/µ (a)\nFIG. 9. The pumped current INFN\np[Eq. (30)] in the NFN\njunction as a function of ǫ/µforV= 0,˜d= 5 and ˜M= 3.\nHereINFN\n0≡ωe/(8π)sin(φ)(eδV1/µ)(eδV2/µ) and\nsin(αm) is given by Eq. (19). In the limit M→0 (i.e.,\nαm→α) in an entirely normal junction, we obtain from\nEq. (28) the angle-dependent pumped current as:\nINFN\np|˜M=0=INFN\n0/integraldisplayπ/2\n−π/2sin(2kmd)sin2α\n(1+ǫ/µ−eV/µ)cos3αdα.\n(29)\nOn the other hand, the transmission TNFN(α) [Eq. (18)]\nin this limit is given by TNFN|˜M→0→1, independent\nof the angle of incidence α. We notice that even if the\nprobability for transmission is one, it is possible to pump\na current in the adiabatic driving regime. The total\npumped current INFN\npis then obtained by integrating\noverα:\nINFN\np=/integraldisplayπ/2\n−π/2INFN\np(α)cosαdα. (30)\nIn general, this integral cannot be evaluated analyticallyand we have obtained our results numerically. Figure 9\nshows the total pumped current INFN\np(in units of INFN\n0)\nat zero bias V=V1=V2= 0 for ˜M= 3. Comparing\nFigs. 3 and 9 we see that there is a correlation between\nthe pumped current and the conductance for the NFN\njunction: for low energies, the pumped current INFN\npis\nzero as no traveling modes are allowed in the junction.\nAs we increase the energy, each time a resonant mode\nappears [see Eq. (20)], the pumped current diverges and\nchanges sign. For energies where both positive and neg-\native angles of incidence contribute to the conductance,\nthe pumped current remains finite but keeps changing\nits sign. From Figs. 3 and 9 it can also be seen that\nthe pumped current vanishes for energies at which sub-\nsequent resonant modes become fully transmitting. In\nordertogainfurtherinsightweplottheanalogueofFig.4\nfor the pumped current. Figure 10 shows the pumped\ncurrent [Eq. 30] as a function of the angle of incidence α\nfor different values of ǫ/µ. The chosen values of ǫ/µare\nsameasin Fig.4. Weseethat the featuresin Fig.10have\na direct correlation with the features in Fig. 4: whenever\nthere is a sharp peak in the transmission the pumped\ncurrent diverges and changes sign. The key feature that\ndistinguishes between the pumped current and the con-\nductance is that the pumped current changessign at par-\nticular values of the energy, while the conductance does\nnot.\nNow we analyze the pumped current in the FNF junc-\ntion with parallel orientation of the magnetizations. In\nthis system, the driving parameters are the magneti-\nzationsMlandMrin the left and right contacts, re-\nspectively, see Fig. 2. After calculating the derivatives\nof the reflection and transmission coefficients, obtaining\nthe imaginary part of Eq. (27), and using ∂αj/∂Mj=\n/planckover2pi1vF/(|ǫ+µ|cos(αj)) (j=l,r), the pumped current\nIFNF\np(α) forM1=M2=Mis:\nIFNF\np(α) =−IFNF\n0cos3(αm)cos(α)sin(2kmd)\n(1+ǫ/µ)2(cos2(α)cos2(αm)cos2(kmd)+sin2(kmd)(1−sin(α)sin(αm))2)2.(31)8\n−0.3−0.2−0.100.10.20.3\n−0.3−0.2−0.100.10.20.3\nPSfrag replacements\nINFNp(α)/I0 INFNp(α)/I0\nα α(a) (a)\n−π/2−π/2−π/4−π/40 0π/4π/4π/2π/2\n−0.2−0.100.10.2\n−0.2−0.100.10.2\n−0.2−0.100.10.2\n−0.2−0.100.10.2\nPSfrag replacements\nINFNp(α)/I0 INFNp(α)/I0 INFNp(α)/I0 INFNp(α)/I0\nα α α α(b)\n−π/2−π/2−π/2−π/2−π/4−π/4−π/4−π/40 0 0 0π/4π/4π/4π/4π/2π/2π/2π/2\n−0.2−0.100.10.2\n−0.2−0.100.10.2\nPSfrag replacements\nINFNp(α)/I0 INFNp(α)/I0\nα α(c) (c)\n−π/2−π/2−π/4−π/40 0π/4π/4π/2π/2\n−0.4−0.3−0.2−0.100.1\n−0.4−0.3−0.2−0.100.1\nPSfrag replacements\nINFNp(α)/I0 INFNp(α)/I0\nα α(d) (d)\n−π/2−π/2−π/4−π/40 0π/4π/4π/2π/2\n−12−8−404x 10−3\n−12−8−404x 10−3\nPSfrag replacements\nINFNp(α)/I0 INFNp(α)/I0\nα α(e) (e)\n−π/2−π/2−π/4−π/40 0π/4π/4π/2π/2\n−0.8−0.6−0.4−0.200.2\n−0.8−0.6−0.4−0.200.2\nPSfrag replacements\nINFNp(α)/I0 INFNp(α)/I0\nα α(f) (f)\n−π/2−π/2−π/4−π/40 0π/4π/4π/2π/2\nFIG. 10. The pumped current INFN\np(α) [Eq. (28)] as a func-\ntion of the angle of incidence αfor different values of energy\nǫ/µ, (a)ǫ/µ= 0.7, (b)ǫ/µ= 0.9, (c)ǫ/µ= 1.2, (d)ǫ/µ= 1.6,\n(e)ǫ/µ= 2.4, and (f) ǫ/µ= 2.9. Parameters used are ˜d= 5\nand˜M= 3.\n00.511.522.530.40.60.81\nPSfrag replacements\nGNFN/G0\n˜d(a)\n00.511.522.53−0.5−0.2500.250.5\nPSfrag replacements\nINFNp/I0\n˜d(b)\nFIG. 11. (a) The conductance of the NFN junction as a\nfunction of ˜d. (b) The pumped current for the NFN junction\nas a function of ˜d. Parameters used are ǫ/µ= 2.5,˜M= 3\nandV= 0.\nHereIFNF\n0=ωe/(8π)sin(φ)(/planckover2pi1vFδMl/µ)(/planckover2pi1vFδMr/µ)\nand sin(αm) = sin(α)−/planckover2pi1vFM/(|ǫ+µ|).\nThe behavior of the current IFNF\npis similar to that\nof the pumped current INFN\npin a NFN-junction (shown\nin Fig. 9). This can also be seen by comparing the de-\nnominators in Eqns. (28) and (31). Again we observe\nthat the pumped current diverges at exactly the samelocations where the conductance changes sharply. But\nthere is an important difference between both pumped\ncurrents. The pumped current in an NFN-junction at\nnormal incidence vanishes, INFN\np(α= 0) = 0, while\nIFNF\np(α= 0)/negationslash= 0. This difference arises because the\ntwo pumps are driven by two different parameters (volt-\nages in the NFN pump and magnetizations in the FNF\npump).\nFinally, we briefly analyze the behavior of the pumped\ncurrent as a function of the width dof the middle region.\nFor energies below ǫc, the pumped current of the NFN\njunction decays to zero as the width dincreases (there\nareno resonantmodes in the system). Forenergieslarger\nthanǫ > ǫc, the pumped current INFN\nposcillates as a\nfunction of width ˜d. Fig. 11 shows the conductance and\nthe pumped current as a function of ˜dforǫ/µ= 2.5. The\npeaks in the conductance correspond to the resonance\ncondition Eq. (20). The pumped current INFN\npchanges\nits sign at exactly the same values of ˜dwhere the con-\nductance has a maximum. This analysis holds as well for\nthe FNF junction.\nV. SUMMARY AND DISCUSSION\nTo summarize, we have analyzed quantum transport\nby Dirac fermion surface states in NFN and the FNF\njunctions in a 3D topological insulator. We have shown\nthat for low energies the appearance of a new resonant\nmode results in a plateau-like increment of the conduc-\ntance and a diverging pumped current in these junctions\nwhich also changes sign. This is our key result, and rep-\nresents an experimentally distinguishable signature be-\ntween conductance and the pumped current. We high-\nlighted an interesting difference between the two differ-\nent pumping mechanisms for the NFN and FNF junc-\ntions, observing different behaviors for normal incidence\n(α= 0). Experimentally, the NFN pump could be re-\nalized using current technology. The FNF pump will be\nmore difficult to realize since it requires oscillating mag-\nnetizations. A possible way to realize a FNF pump could\nbe by moving the two ferromagnetic layers coherently\nusing a nanomechanical oscillator41. Experimental veri-\nficationofourpredictionswillprovidefurtherinsightinto\nquantum transport through these junctions.\nACKNOWLEDGMENTS\nThis research was supported by the Dutch Science\nFoundation NWO/FOM.\n1M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).2M. K¨ onig, S. Wiedmann, C. Br¨ une, A. Roth, H. Buhmann,\nL. W. Molenkamp, X.-L. Qi, andS.-C. Zhang, Science 318,9\n766 (2007).\n3D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava,\nand M. Z. Hasan, Nature 452, 970 (2008) .\n4Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A.\nBansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan,\nNature Phys. 5, 398 (2009).\n5Y. Xia, D. Qian, D. Hsieh, R. Shankar, H. Lin, A. Bansil,\nA. V. Fedorov, D. Grauer, Y. S. Hor, R. J. 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Giazotto, P. Spathis, S. Roddaro, S. Biswas, F. Taddei,\nM. Governale and L. Sorba, Nature Physics (2011).\n39This restriction of the angles of incidence comes from the\nfact that the minimum and the maximum value of the an-\ngleαrfor the transmitted wavefunction is −π/2 andπ/2\nrespectively.\n40A fundamental difference between these pumps and the\nones in graphene29–34is the nature of the spinor in the\nHamiltonian (1) which in our case represents a real spin\ndue to the spin-orbit interaction, while in graphene the\nspinor represents a pseudo-spin (or the sub-lattice vari-\nable).\n41A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.\nRev. Lett. 94, 167201 (2005)."    },    {        "title": "1108.2108v1.Bias_induced_destruction_of_ferromagnetism_and_disorder_effects_in_GaMnAs_heterostructures.pdf",        "content": "arXiv:1108.2108v1  [cond-mat.mes-hall]  10 Aug 2011Bias-induced destruction of ferromagnetism and disorder e ffects in GaMnAs\nheterostructures\nChristian Ertler∗1and Walter P¨ otz1\n1Institute of Theoretical Physics, Karl-Franzens Universi ty Graz, Universit¨ atsplatz 5, 8010 Graz, Austria\nThe magneto–electric properties of resonant tunneling dou ble barrier structures using GaMnAs\nfor the quantum well is investigated within a self-consiste nt Green’s function approach and a tight–\nbinding electronic structure model. The magnetic state of t he well is determined self–consistently by\nthetunnelingcurrentwhichcontrolstheholespindensitya nd, hence, thedegreeofexchangesplitting\nof the subbands inside the well. Prompted by recent experime nts we compare model systems of\nincreasing defect concentration (substitutional disorde r) regarding their I-V curve, magnetic state,\nand spin polarization. We predict that, near resonance, the ferromagnetic order which may be\npresent at zero bias in the GaMnAs well tends to be destroyed. Resonance peaks are found to be\nmore sensitive to disorder than ferromagnetic ordering and spin polarization of the steady–state\ncurrent.\nPACS numbers: 85.75.Mm, 73.23.Ad, 73.63.-b, 72.25.Dc\nI. INTRODUCTION\nThe realization of electric control of ferromagnetism\nin nanostructures is of great interest both for spintronic\ndevice application and for achieving a better understand-\ning of the physical mechanisms and dynamics underlying\nthe formation of ferromagnetic order in Mn doped semi-\nconductors. Dilute magnetic semiconductors (DMS) are\nmademagneticbydopingofZnS–structuredsemiconduc-\ntors with transition metal elements, which provide local\nmagnetic moments arising from open electronic dorf\nshells [1, 2]. A prototype is bulk Ga 1−xMnxAs in which\nMn residing on the Ga site (Mn Ga) donates both a hole\nand a local magnetic moment. Mn Gais an at least mod-\nerately deep acceptor and associated levels lie about 100\nmeV above the valence band edge [3]. In a recent scan-\nning tunneling microscopy experiment the radius of the\nMn acceptor wave function has been determined to be\nabout 2 nm [4]. Due to an antiferromagnetic exchange\ncoupling between the itinerant holes and the local Mn\nd-electrons, an effective ferromagnetic ordering among\nthe Mn-ions, known ascarrier–mediatedferromagnetism,\ncan be established [5–7]. Due to the hole–concentration–\ndependent effective exchange field the spin degeneracy\nof the holes is lifted resulting in a self–consistently spin\npolarized hole gas. The critical temperature Tcfor the\noccurrence of ferromagnetism in bulk GaMnAs is typi-\ncally below ∼150 K and depends on Mn concentration\nand sample preparation [1, 2, 6].\nProbably due to the degree of (unwanted) defects in\nGaMnAs samples depending on growth conditions, ex-\nperimental evidence has led to somewhat conflicting con-\nclusions about the electronic structure in the vicinity of\nthe Fermi level. While all experimental studies on ferro-\nmagnetic bulk GaMnAs confirm it to be p–type, there is\n∗email:christian.ertler@uni-graz.atsome debate as to the precise position of the Fermi en-\nergy [2]. Some experiments can be interpreted by placing\nit into the top of a GaAs–like valence band edge which,\nat most, broadened by disorder [1]. Others suggest the\nexistence of an isolated impurity band (of high “effective\nmass”) which forms at Mn concentrations above ∼1.5%\nleading to a metal–insulator transition in high–quality\nGaMnAs [4, 8, 9].\nIrrespective of the detailed electronic structure it is\nclear that, due to disorder, the localization length of va-\nlence band eigenstates will tend to decrease as one moves\nfrom the top of the valence band towards the energy\ngap [10]. Near the band edge a coexistence of local-\nized and delocalized (Bloch–like) eigenstates may be ex-\npected, similar to amorphous Si [11, 12]. If Mn Gais the\nmain defect to provide modification of the valence band\nedgefrom what it is forGaAs, quantizationeffects can be\nexpected for a layer thickness ≤3 nm. Recent tunneling\nspectroscopyofGaMnAs quantum well structures hasin-\ndicated such effects [9]. However, the signatures in the\ncurrent–voltage characteristics appear to be rather weak\nand no regions of negative differential conductivity due\nto resonances associated with GaMnAs well layers have\nbeen observed yet, with the exception of an asymmet-\nric magnetoresistance resonant tunneling structure [13].\nThis suggests that a significant concentration of defects\nmay be present, as it is the case, e.g., in thin layers of\namorphous Si, in which similarly weak signatures have\nbeen found [11, 12]. Disorder on the other hand, as we\nknow from bulk Ga 1−xMnxAs, goes hand in hand with\nferromagnetic order and, indeed, ferromagnetic behavior\nhas been verified experimentally for thin layers of GaM-\nnAs [9, 14].\nDue to spin–selective hole tunneling in and out of a\nferromagnetic Mn-doped GaAs quantum well, regardless\nwhether occurring sequentially or resonantly, one can ex-\npect hole population and spin polarization to become de-\npendent upon the bias applied to a double barrier struc-\nturecontainingsuchawell. Thismechanismshouldallow\nan electric control of ferromagnetic order in the quan-2\ntum well and is evaluated in this paper with particu-\nlar attention to the qualitative effect of disorder in the\nMn doped well region. An important question addressed\nis whether disorder and unwanted defects can suppress\nspin–selective tunneling . Disorder has a strong influence\nonthe electronicpropertiesandthey, inturn, stronglyin-\nfluence(resonant)tunneling. High–qualityferromagnetic\nlayersintegratedin semiconductorheterostructures, have\nbeenenvisionedtoallowspin-dependentcarriertransmis-\nsion [15]. Magnetic resonant tunneling structures of high\nstructuralqualitymayallowtherealizationofspinvalves,\nspin filtering, and spin switching devices, as proposed in\nseveral studies [13, 16–22].\nIn this article we investigate spin–selective hole trans-\nport in GaAs/AlGaAs/GaMnAs/AlGaAs/GaAs hetero-\nstructures within the limit of moderately thin samples so\nthat an effective independent particle model provides a\ngood first approximation. We apply a non-equilibrium\nGreen’s function formalism based on a tight–binding\nHamiltonian for the electronic structure, including self-\nconsistency regarding the charge density and the ex-\nchange splitting of the effective potential, as well as\ncharge transfer to the contacts. The carriers’ Coulomb\ninteraction and the exchange coupling with the magnetic\nions are described within a mean-field picture. Details of\nour model are exposed in Sect. II. Since disorder seems\nto play a major role in actual samples we study the ef-\nfect of substitutional disorder on the I-V characteristics,\nthe ferromagnetic state of the heterostructure, and spin–\npolarizationofthe current density. Results and relevance\nto experiment are discussed in Sect. III. Summary and\nconclusions are given in Sect. IV.\nII. SELFCONSISTENT TRANSPORT MODEL\nThebasicfeaturesofthe semiconductordouble–barrier\nstructure near the top of the valence band edge are\nmapped onto a two–band tight-binding Hamiltonian for\nthe heavy holes ( J3=±3/2)\nHs=/summationdisplay\ni,σεi,σ|i,σ/angbracketright/angbracketlefti,σ|\n+/summationdisplay\ni,σσ′ti,σσ′|i,σ/angbracketright/angbracketlefti+1,σ′|+h.c.,(1)\nwhereεi,σis the spin-dependent ( σ=↑,↓≡ ±1) onsite\nenergy at lattice site i,ti,σσ′denotes the hopping-matrix\nbetween neighboring lattice sites, and h .c.abbreviates\nthe Hermitean conjugate term. Spin conserving hopping\ngives a diagonal matrix ti,σ,σ′=tδσσ′with the hopping\nparameter t=−/planckover2pi12/(2m∗a2) depending on the effective\nmassm∗andthelatticespacing abetweentoneighboring\nlattice sites. The onsite energy\nεi,σ=Ui−eφ−σ\n2∆i (2)\nincludes the intrinsicholeband profile Uidue to theband\noffset between different materials, the electrostatic po-tentialφwithedenoting the elementary charge, and the\nlocal exchange splitting ∆ i. Near the band-edges this\nmodel is equivalent to an effective–mass model, however,\nithastheadvantagethatstructuralimperfections, aswell\nasspin–flip processes, can be readilybe modeled byvary-\ning the onsite and hopping energies. A more realistic de-\nscription can be achieved by introducing a larger set of\norbital basis functions at each lattice site [15, 23–25].\nWithin a mean-field approach the exchange coupling\nbetween holes and magnetic impurities can be described\nby two interrelated effective magnetic fields, respectively,\noriginating from a nonvanishing mean spin polarization\nof the ions’ d–electrons /angbracketleftSz/angbracketrightand from the hole spin den-\nsity/angbracketleftsz/angbracketright= (n↑−n↓)/2 [26–28]. The exchange splitting\nof the hole bands is given by\n∆(z) =−Jpdnimp(z)/angbracketleftSz/angbracketright(z), (3)\nwithzdenoting the longitudinal (growth) direction of\nthe structure, Jpd>0 is the exchange coupling between\nthe impurity spin and the carrier spin density (in case\nof GaMnAs p-like holes couple to the d-like impurity\nelectrons), and nimp(z) is the impurity density profile\nof magnetically active ions. The magnetic order between\nthe impurities is mediated by the holes and the effective\nimpurityspin polarizationdepends onthe meanholespin\npolarization via\n/angbracketleftSz/angbracketright=−SBS/parenleftbiggSJpd/angbracketleftsz/angbracketright\nkBT/parenrightbigg\n, (4)\nwhere, respectively, kB,T,BSis the Boltzmann con-\nstant, the lattice temperature, and the Brillouin function\nof order S, here with S= 5/2 for the Mn impurity spin.\nCombation of Eq. (3) and Eq. (4) gives a self-consistent\neffectiveHamiltonianfortheholes Heff=−σ∆(z)/2with\n∆(z) =Jpdnimp(z)SBS/braceleftbiggSJpd[n↑(z)−n↓(z)]\n2kBT/bracerightbigg\n.(5)\nThis shows that a manipulation of the hole spin density\n/angbracketleftsz/angbracketrightby the applied bias is the key to the control of fer-\nromagnetic order in the heterostructure.\nWithin a Hartree mean-field picture space-charge ef-\nfects are taken into account self-consistently by calculat-\ning the electric potential from the Poisson equation,\nd\ndzǫd\ndzφ=e[Na(z)−n(z)], (6)\nwhereǫandNa, respectively, denote the dielectric con-\nstant and the Mn Gadensity. The local hole density at\nsite|i/angbracketrightis computed as\nn(i) =−i\nAa/summationdisplay\nk||,σ/integraldisplaydE\n2πG<(E;iσ,iσ),(7)\nwithAandk||, respectively, being the in-plane cross sec-\ntional area of the structure and the in-plane momentum.3\nThe non-equilibrium “lesser” Green’s function G<is cal-\nculated from the equation of motion\nG<=GRΣ<GA(8)\nwhereGRandGA= [GR]+denotes the retarded and\nadvanced Green’s function, respectively. The scattering\nfunction Σ<= Σ<\nl+Σ<\nrdescribes particle inflow of from\nthe left ( l) and right ( r) reservoir and [29]\nΣ<\nl,r=f0(E−µl,r)(ΣA\nl,r−ΣR\nl,r), (9)\nwheref0(x) = [1 + exp( x/kBT)]−1is the Fermi dis-\ntribution function and µlandµr, respectively, denote\nthe quasi–Fermi energies in the contacts. The retarded\nand advanced self-energy terms ΣR= ΣR\nl+ ΣR\nrand\nΣA= [ΣR]+, respectively, couple of the simulated sys-\ntem region to the left and right contacts, modeled as\nsemi–infinite chains, for which analytic expressions are\navailable [29, 30]. The retarded Green’s function, finally,\nis given by\nGR=/bracketleftbig\nE+iη−Hs−ΣR/bracketrightbig−1. (10)\nA self–consistent calculation of the spin–dependent ef-\nfective (one–particle) potential, based on the band split-\nting given by Eq. (5), the Poissonequation Eq. (6,7), and\nthekineticequationsEqs.(8)and(10), alsoentailsanad-\njustment of the quasi–Fermi energies relative to the band\nedges in the contacts to ensure asymptotic charge neu-\ntrality [31]. These loops must be solved self-consistently\nuntil convergence to a steady–state solution is reached.\nA small external magnetic field is applied initially to aid\nspontaneous symmetry breaking. For the next bias iter-\nation, the self–consistent solution from the previous bias\nvalue is used for an initial guess. In most cases a sta-\nble self–consistent solution is found. However, due to\nthe nonrelativistic nature of the model with orbital mo-\ntion decoupled from the spin degree of freedom, ”spin\nup” and ”spin down” are degenerate. Moreover, there\nare bias regions where no single stable solution exists re-\ngarding charge distribution (intrinsic charge bistability)\nand/or magnitude of the exchange interaction (magnetic\nmulti-stability) [31, 32].\nHaving obtained the self-consistent potential pro-\nfile the spin-dependent transmission probability Tσ′σ(E)\nfrom the left to the right reservoir is calculated from spe-\ncial matrix elements of the retarded Green’s function [24]\nTσ′σ=Tσ′←σ(E) =vr,σ′|GR(E;rσ′,lσ)|2\nvl,σ|G0(E;lσ,lσ)|2(11)\nwithG0denoting the free Green’s function of the asymp-\ntotic region, and vl,σandvr,σ, respectively, are the spin-\ndependent group velocities in the leads. GR(E;rσ′,lσ)\nis computed most conveniently by adding one layer af-\nter the other which requires solely 2x2 matrix inversions\nwithin the present two–band model [30].The steady–state current is calculated within station-\nary scattering theory (a generalized Tsu-Esaki formula),\njσ′σ=em∗kBT\n(2π)2/planckover2pi13/integraldisplay∞\n0dE Tσ′σg(E)\ng(E) = ln/braceleftbigg1+exp[( µl−E)/kBT]\n1+exp[( µr−E)/kBT]/bracerightbigg\n.(12)\nThe applied bias V= (µl−µr)/eis defined as the dif-\nference in quasi-Fermi levels of the contacts. In such\na nonequlibrium situation the hole concentration in the\nGaMnAs layer is not determined not simply by its ferro-\nmagnetic state, but also depends on other physical quan-\ntities, such as doping level in the contact regions, struc-\nturalparametersofthe heterostructure,andappliedbias.\nIII. RESULTS AND DISCUSSION\nIn this section we employ the model outlined above\nand study the influence of disorder on the resonant tun-\nneling characteristics and the magnetic state of a double-\nbarrierstructure featuring a GaMnAs quantum well. For\nthis simulation we use parameters for GaAs and GaM-\nnAs which are wewell established in the literature: m∗=\n0.4m0,ǫr= 12.9,Vbar= 300 meV, µl=µr= 70 meV,\nd= 20˚A,w= 30˚A,nimp= 1×1020cm−3,Jpd= 0.15\neV nm3[33],T= 4.2 K, where m0denotes the free\nelectron mass, ǫris the relative permittivity, Vbaris the\nbare barrier height of AlGaAs relative to GaAs, dand\nw, respectively, are the barrier and quantum well width.\nThe background charge Naarises from an assumed 10%\nof the Mn doping nimp, since GaMnAs is known to be a\nheavilycompensatedsemiconductor[6,34]. Dopinginthe\ncontacts, experimentallyBemaybe used, issuchthat the\nthermal equilibrium position of the Fermi energy µl=µr\nlies near the first heavy–hole resonance, thus, providing\nferromagnetic order at zero bias.\nSince GaMnAs represents a ternary alloy, disorder\nfrom Mn Gasubstitution must be accounted for to cap-\nture its electronic structure. In other ternary alloys, such\nas AlGaAs or CdZnSe, disorder effects lead to features,\nsuch as direct to indirect gap transitions and band bow-\ning [35, 36].\nIn addition, unwanted defects, such as antisites, inter-\nstitials, Mn complexes, vacancies, etc, as well as addi-\ntional defects arising from the heterointerfaces must be\nexpected or have been observed [2, 4]. Clearly, the com-\nplexity of the material is such that a realistic predictive\nelectronic structure calculation from first principles is\npractically impossible. Hence we must take a completely\ndifferent approach and model disorder phenomenologi-\ncally, taking doping level and basic properties of Mn Ga\nfor guidance. Here disorder effects in the GaMnAs layer\naremodeled byperformingaconfigurationalaverageover\nstructures with randomly selected onsite and hopping\nmatrix elements of the tight-binding Hamiltonian in the\nMn doped region. We choose a 5% Mn Gaconcentration4\n−0.4−0.200.20.40.60.810−1010−5100\nE (eV)DOS (arb. units)\nFIG. 1: Averaged total density of states near the center of\nthe GaMnAs well region for zero bias and maximum hopping\ndisorder ( σt= 20%). For valence bandstates ( E >0) integra-\ntionoverin-planemomentumis takenintoaccountandthere-\nsult is normalized to the 2D-density of states D0=m∗/(π/planckover2pi12).\nin the well and model random (uncorrelated) substitu-\ntional disorder. If a Mn ion is present at a given lattice\nsite in the well the onsite energy is shifted according to a\nGaussian distribution around a mean onsite energy–shift\nof 80 meV and a standard deviation of 20 meV, which\nare reasonable values according to data available for the\nisolated Mn Gaacceptor, as well as recent experimental\nresults for Ga 1−xMnxAs for x 2-6% [4, 9]. The hopping\nmatrix element is sampled according to a Gaussian with\n5, 10, and 20% standard deviation ( σt) of its bulk value\nt. This increase is in hopping matrix variation is used to\nsimulate an increasing level of defect concentration, for\nfixed Mn Gaconcentration. For each such randomly se-\nlected Hamiltonian the transport problem is solved self–\nconsistently and the I-V curve is computed. Final re-\nsults for I-V curve, magnitude of magnetization and spin\ncurrent polarization, etc., are obtained by averagingover\nindividual resultsobtained forthese configurations. Typ-\nically300configurationsareusedtoperformthisaverage.\nIn this simulation we keep constant the number of active\nMn spins so that the formation and degree of ferromag-\nnetic order at given bias is determined self–consistently\nfrom the hole spin polarization in the well region.\nFigure 1 shows the averaged density of states near\nthe band edge in the Ga 1−xMnxAs well region for zero\nbias and maximum degree of hopping disorder modeled\nhere (σt= 20%). It is seen that the main effect of our\nmodel for substitutional disorder leads to to a broad-\nenedisolatedimpuritystateabovethe valencebandedge,\nwhereby the latter clearly displays the steps characteris-\ntic for 2d quantization effects in spite of disorder (note\nthe logarithmicscale!). Thedefect levelsdonot providea\ngenuinebandsincethey areconfinedtothe Ga 1−xMnxAs\nwell region. These states may trap holes and contribute00.1 0.2 0.3 0.410−1510−1010−5100\nE (eV)Transmission (arb. units)\n  \nT↑↑\nT↑↑ disorder\nT↓↓\nT↓↓ disorder\nFIG. 2: (Color online) Spin-dependent transmission proba-\nbility of the double barrier structure at zero bias with and\nwithout disorder ( σt= 5%).\nto ionized impurity scattering but are not actively in-\nvolved in tunneling here since they lie below both quasi\nFermi levels at up to moderate bias. The transmission\nprobability versus energy of the incident holes at zero\nbias is displayed in Fig. 2. In absence of disorder, the\nGaMnAs top valence band structure is modeled as that\nof GaAs plus a self–consistent exchange and the reso-\nnances, indicated as dashed lines in Fig. 2, are spin-split\nby about 30 meV. Taking into account disorder, see solid\ncurves for σt= 5% in Fig. 2) leads to spectral broad-\nening and a shift of the resonances deeper into the va-\nlence band (anti–bonding effect) of the resonances. An\nincrease in overlap of the transmission peaks for spin–\nup and spin–down holes under disorder is particularly\npronounced for the first heavy–hole resonance since it is\nmost sensitive to potential fluctuations. For low contact\ntemperatures, however, this effective spin splitting en-\nsures spin–dependent tunneling rates even at a level of\ndisorderwherethenon–monotonicincreaseofthecurrent\nwith applied bias is practically lost, as shown below.\nContributions from the light–hole band lead to ad-\nditional resonances. Their inclusion would call for a\nhigher–dimensional tight–binding model which properly\ncaptures nonparabolicity parallel to the heterointerfaces\nand goes beyond the scope of this paper. We just point\nout that, for the present structure a light–hole–band res-\nonance would be expected somewhere between the first\ntwo heavy–hole–associated resonances, contributing to a\nfurther masking of negative differential conductance [9].\nDiscontinuous second derivatives of the IV-curve ob-\ntained from tunneling spectroscopy have been attributed\ntoquantizationeffectsintheGaMnAsquantumwell. Re-\ngions of negative differential conductivity, however, have\nnot been observed directly in the IV-curve [9]. This sug-\ngests that disorder may play a considerable role, similar\nto amorphous silicon quantum wells for which weak sig-5\n0 0.1 0.2 0.300.511.522.53x 105\nV (V)j (A/cm2)\n  \n00.05 0.1024x 104\nV (V)j (A/cm2)σt = 0%\nσt = 5%\nσt = 10%\nσt = 20%\nFIG. 3: (Color online) IV-characteristics of a magnetic dou-\nble barrier structure due to heavy–hole associated bands fo r\ndifferent degrees of disorder, i.e., standard deviations σtof\nthe hopping matrix elements, as explained in the text. The\ninset shows the IV-curve in the voltage range in which the\ntransport takes place via the first heavy-hole subband.\n0 0.1 0.2 0.30102030\nV (V)|∆| (meV)\n  \nσt = 0%\nσt = 5%\nσt = 10%\nσt = 20%\nFIG. 4: (Color online) The configuration averaged spin split -\nting|∆|in the quantum well as a function of the applied bias\nfor different degrees of disorder.\nnatures of resonances have been predicted and observed\n[11, 12]. The current-voltage I–V characteristics for an\nincreasing degree of disorder is plotted in Fig. 3. Our\nmodel reveals that the first region of negative differen-\ntial resistance, corresponding to the first heavy hole res-\nonance, does not disappear until considerable hopping\ndisorder of about 10%, modeled as variance in t, is as-\nsumed. This relatively high value needed to flattened the\nIV-curve in our simulations suggests that in real samples\ndefectsotherthanMn Ga, suchasinterstitials, antisitede-\nfects or voids, may play a considerable role for blurring\nthe resonances.atomic sitesE (eV)\n  \n110203040506000.10.20.3\n−25−20−15−10−50\nµl\nµr\nFIG. 5: Logarithmic local density of states (LDOS) as a func-\ntion of energy at the bias V= 0.04 V (before the first current\nmaximum). The self-consistent band profile is indicated by\nthe solid line.\natomic sitesE (eV)\n  \n110203040506000.10.20.3\n−25−20−15−10−50\nµl\nµr\nFIG. 6: Logarithmic local density of states (LDOS) as a func-\ntion of energy at the bias V= 0.052 V (at the first current\npeak). The self-consistent band profile is indicated by the\nsolid line.\nIn Fig. 4 the average exchange band spin splitting |∆|\nin the quantum well is plotted versus applied bias, show-\ning that the degree of ferromagnetic order in this struc-\nture can be controlled by the applied bias. At the po-\nsitions of the current maxima (compare to Fig. 3) the\nquantum well is populated by the contacts with unpolar-\nized holes, resulting in a breakdown of the ferromagnetic\norder. The degree of bias control is moderated for in-\ncreasing defect concentrations. However, our averaging\nprocedure may exaggerate this reduction since it corre-\nsponds to a physical situation where, in plane, there is\na coexistence of (uncorrelated) ferromagnetic and non–\nferromagnetic domains. If on the other hand, in–plane\nferromagneticorderisestablishedordestroyedcoherently\n(in correlated fashion), the degree of bias control is un-\nderestimated by our sampling procedure. This makes us\nsuggestto performspin–sensitivetunneling spectroscopy,6\natomic sitesE (eV)\n  \n110203040506000.10.20.3\n−25−20−15−10−50\nµl\nµr\nFIG. 7: Logarithmic local density of states (LDOS) as a func-\ntion of energy at the bias V= 0.06 V (at off-resonance con-\nditions). The self-consistent band profile is indicated by t he\nsolid line.\n0 0.1 0.2 0.300.20.40.60.81\nV (V)|Pj|\n  \nσt = 0%\nσt = 5%\nσt = 10%\nσt = 20%\nFIG. 8: (Color online) Averaged currentspin polarization |Pj|\nversus applied bias Vfor different degrees of disorder.\nideallyin presenceofan externalmagneticfield, since the\ncurrent spin polarization is predicted to be a more sen-\nsitive signature to determine the degree of ferromagnetic\norder in the sample than resonances in the I–V curve. In\nfact the latter are predicted to occur at a bias when the\nsample is in the nonmagnetic state only.\nTo illustrate the changes in the magnetic state of the\nquantum well near the first current peak at V= 0.052 V,\nwe plot the local density of states (LDOS) closely be-\nlow resonance ( V= 0.04 V, Fig. 5), at resonance ( V=\n0.052 V, Fig. 6), and above resonance ( V= 0.06 V,\nFig. 7). At zero bias (not shown in Fig. 5) ferromagnetic\norder is present, placing the Fermi energy between the\nspin split subband edges. The spin up level is found at a\nlower energy than the spin down level. (Note that we use\nan inverted energy scheme for the valence band.) This\nsituationisachievedbymatchingthecontactdopinglevelto the lowest (heavy–hole) resonance in the heterostruc-\nture. As bias is increased both spin–up and spin–down\nsubband become accessible from the emitter side, how-\never, only the lower spin–up subband is accessible from\nthe collector side. This maintains hole spin polarization,\nhowever, it decreases with increasing bias, leading to a\ncontinuousdecreasein∆showninFig.4. Whileholespin\npolarization decreases, hole density increases in the well\nregion partially screening the applied bias. As the ap-\nplied bias is increased resonance is reached. Hole charge\nand exchange splitting in the well are no longer suffi-\ncient to keep the spin–down subband above the collector\nquasi–Fermi level and ferromagnetic order collapses, as\nshown in Fig. 6. Both subbands are equally flooded from\nboth contacts leading to an unpolarized hole gas in the\nwell and, hence, to a destruction of ferromagnetic order.\nNote the change in the effective potential profile in the\nwellfromconvexupinFig. 5toalmostlinearinFig.6. If\nthe bias is increased further the levels are pushed closely\nbelow the emitter valence band edge and resonance to\nthe emitter is suppressed. However, any small pertur-\nbation triggers the system into a ferromagnetic state, in\nwhich the spin up level preferrentially is filled from the\ncollectorside (see Fig. 7). Thus, inspection of the LDOS-\nplots shows that the ferromagnetic state in the quantum\nwell is determined by the relative position of the well\nsubband edges and the contact quasi–Fermi levels. An\nappropriate tailoring of these levels allows to change the\nhole gas polarization under bias and thereby provides\nelectrical control of the ferromagnetic state in the well.\nInourcaseweconsideronlyatwo–terminalconfiguration\nwith source and drain contacts but the use of additional\ngates in transverse direction (multi–terminal configura-\ntions) provides an additional control knob to move the\nsubbands. Although these structures are very difficult\nto realize in practice, they have been studied in a recent\nexperiment [22].\nThe change of the magnetic state in the well is directly\nreflected in the (collector) current spin polarization, as\nshown in Fig. 8. If the entire quantum well becomes\nnonmagnetic, the spin density in the well vanishes and\nthe collector current becomes unpolarized. A polarized\ncurrent indicates exchange–split subbands in the well,\ndemonstrating ferromagnetism in the well. The exper-\nimental probing of the current spin polarization at the\ncollector side therefore would give important additional\ninformation to confirm the interpretation of recent ex-\nperiments regarding size quantization effects in GaMnAs\nquantum wells [9].\nIV. CONCLUSIONS AND OUTLOOK\nIn summary, we have used a steady–state transport\nmodel to investigate the role of structural disorder on\nthe interplay of ferromagneticorder and resonant tunnel-\ning in double barrier structures with a GaMnAs quan-\ntum well. Ferromagnetic exchange, as well as the hole7\nCoulomb interaction are treated within a self–consistent\nmean–field approximation. Disorder effects are modeled\nby random variation of onsite and hopping matrix ele-\nments of the tight-binding Hamiltonian according to ba-\nsic experimental findings on the Mn Gaacceptor. In this\nwork we have modeled an electronic structure with an\nisolated impurity band, as supported by most of the re-\ncent experiments.\nFor samples which, at zero bias, exhibit ferromagnetic\norder in the GaAsMn well we predict that ferromagnetic\norder is destroyed under bias near (the first heavy-hole)\nresonance. While at resonance the well region is flooded\nby holes these are overall unpolarized thus prohibiting\nthe communication of ferromagnetic ordering amongst\nMnGasites in the well. Within our model we thus are\nable to provide a possible explanation for the absence\nof exchange splitting near resonances, as observed in re-\ncent tunneling spectroscopy measurements on thin GaM-\nnAs layers [9]. Furthermore we find that an experimen-\ntal investigation of the spin polarization of the collec-\ntor current can give information about the presence of\nmagnetism in the quantum well. Such a measurement\nis more revealing regarding the ferromagnetic state thanthe search for resonances in the I–V curve, since the for-\nmer is more robust against disorder than the latter. We\nfind that disorder tends to suppress negative differential\nresistance region in the IV-curve. Although our model\nis merely qualitative it indicates that substitution disor-\nder from Mn doping alone is not sufficient to explain the\nabsence negative differential conductivity in experiment.\nThere are a number of open question which should be\naddressed in the future. One pertains to the correlation\nlength of ferromagnetic order parallel to the heteroint-\nerface. It has direct influence on the spin–valve action\nand its bias–control of ferromagnetic heterostructures.\nFurthermore, one may ask to what extent magneto–\ntransport experiments can distinguish between the two\nmain electronic structure models (with and without iso-\nlated impurity band) discussed in the literature.\nV. ACKNOWLEDGMENT\nThis work has been supported by the FWF project\nP21289-N16.\n[1] T. Jungwirth, J. Sinova, J. Maˇ sek, J. Kuˇ cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809 (2006).\n[2] K. S. Burch, D. D. Awschalom, and D. N. Basov, J.\nMagn. Magn. Mat. 320, 3207 (2008).\n[3] J. Schneider, U. Kaufmann, W. Wilkening, M. Baeumler,\nand F. K¨ ohl, Phys. Rev. Lett. 59, 240 (1987).\n[4] A. Richardella, P. Roushan, S. Mack, B. Zhou, D. A.\nHuse, D. D. Awschalom, and A. Yazdani, Science 327,\n665 (2010).\n[5] H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. End,\nS. Katsumoto, and Y. Iye, Appl. Phys. Lett. 69, 363\n(1996).\n[6] A. Van Esch, L. Van Bockstal, J. De Boeck, G. Ver-\nbanck, A. S. van Steenbergen, P. J. Wellmann, B. Gri-\netens, R. Bogaerts, F. Herlach, and G. Borghs, Phys.\nRev. B56, 13103 (1997).\n[7] T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Fer-\nrand, Science 287, 1019 (2000).\n[8] K. S. Burch, D. B. Shrekenhamer, E. J. Singley,\nJ. Stephens, B. L. Sheu, R. K. 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Datta, Electronic Transport in Mesoscopic Sys-\ntems(Cambridge University Press, Cambridge, England,\n1995).\n[30] E. N. Economou, Green’s Functions in Quantum Physics\n(Springer, Berlin, 1983).8\n[31] W. P¨ otz, J. Appl. Phys. 66, 2458 (1989).\n[32] C. Ertler, W. P¨ otz, and J. Fabian, Appl. Phys. Lett. 97,\n042104 (2010).\n[33] B. Lee, T. Jungwirth, and A. H. MacDonald, Phys. Rev.\nB61, 15606 (2000).[34] S. Das Sarma, E. H. Hwang, and A. Kaminski, Phys.\nRev. B67, 155201 (2003).\n[35] Z. Q. Li and W. P¨ otz, Phys. Rev. B 43, 12670 (1991).\n[36] Z. Q. Li and W. P¨ otz, Phys. Rev. B 46, 2109 (1992)."    },    {        "title": "1201.0285v3.Ferromagnetic_Ordering_in_Carbon_Nanotubes__Incorporated_in_Diamond_Single_Crystals.pdf",        "content": "arXiv:1201.0285v3  [cond-mat.str-el]  2 Jun 2013Ferromagnetic Ordering in Carbon Nanotubes, Incorporated in Diamond Single\nCrystals\nDmitri Yerchuck (a), Vyacheslav Stelmakh (b), Alla Dovlatova (c), Yauhen Yerchak (b), Andrey Alexandrov (c)\n(a) - Heat-Mass Transfer Institute of National Academy of Scien ces of RB, Brovka Str., 15, Minsk, 220072, RB,\ndpy@tut.by\n(b) Belarusian State University, Nezavisimosti Avenue 4, Minsk, 22 0030,\nRB (c) - M.V.Lomonosov Moscow State University, Moscow, 119899\n(Dated: November 17, 2018)\nThe physical origin of the mechanism of the formation of ferr omagnetic ordering in carbon nan-\notubes (NTs), produced by high energy ion beam modification o f diamond single crystals in /angbracketleft110/angbracketright\nand/angbracketleft111/angbracketrightdirections has been found. It is concluded from analysis of e xperimental results on fer-\nromagnetic spin wave resonance observed, that the only π-electronic subsystem of given NTs is\nresponsible for the appearance of ferromagnetism. It is det ermined by asymmetry in spin density\ndistribution in Su-Schrieffer-Heeger (SSH) topological so liton lattice. The formation of SSH topo-\nlogical soliton lattice is considered in the frames of gener alized SSH-model of organic conductors, in\nwhichπ-electronic subsystem is represented being to be 1D quantum Fermi liquid.\nPACS numbers: 71.10.-w, 73.63.Fg, 78.30.-j, 76.30.-v, 76. 50.+g, 78.67.-n\nKeywords: Ferromagnetism, Carbon, Nanotubes, Spin Wave Re sonance, Quantum Fermi Liquid\nI. INTRODUCTION AND BACKGROUND\nIt is well known that all substances on the whole are\nmagnetics. At the same time, it is also well known, that\nclassical magnetic ordering is existing in the substances,\nwhich arebuilt from the atomswith unfilled inner atomic\nd- orf-shells or include given atoms in their elemen-\ntary units. In other words, classical magnetics are the\nsubstances, elementary units of which include transition\nchemical elements with unfilled atomic 3d-, 4d-, 5d-, 6d-\nshells, or rare earth elements with unfilled atomic 4f, 5f-\nshells. Carbon does not refer to given group. Neverthe-\nless, there are at present a number of communications on\nmagnetic ordering in carbon and carbon based materials.\nThe first report on the experimental revealing of mag-\nnetic orderingin carbonstructurallyorderedsystemswas\npresented on the IBMM-Conference in Knoxville, TN,\nUSA [1] in 1990. Given result was confirmed in report\non E-MRS Conference in Strasbourg, France [2] also in\n1990. Let us remark, that the first report on magnetic\nordering in structurally non-ordered carbon materials is\nappeared almost in the same time. It is the work [3],\nwhere ferromagnetic ordering in pyrolytic carbon, pro-\nduced by chemical vapour deposition (CVD) method us-\ning adamantane to be source material, was found. Let\nus also remark, that simultaneously, the reports [1],\n[2] were the first reports on the formation by high en-\nergy ion beam modification (HEIBM) of diamond single\ncrystals structurally and magnetically ordered quasi-one-\ndimensional (quasi-1D) system along ion tracks, that is\nnew carbon allotropic form, which was identified with\nnanotubes NTs, incorporated in diamond matrix in di-\nrection, precisely coinciding with ion beam direction.\nGiven system possesses by a number of interesting phys-\nical properties, reported in [4], [5], [6], [7]. When con-\ncerne the magnetic ordering, it was established from thestudy of temperature dependence of electron spin res-\nonance (ESR) absorption intensity, that, for instance,\nincorporated nanotubes, produced by neon HEIBM of\ndiamond single crystal along /an}bracketle{t100/an}bracketri}htcrystallographic di-\nrection, possess by weak antiferromagnetic ordering [4],\n[6], [7]. At the same time, copper HEIBM with implan-\ntation direction along /an}bracketle{t111/an}bracketri}htcrystal axis, nickel HEIBM\nwith implantation direction along /an}bracketle{t110/an}bracketri}htaxis, [4], [6], [7],\nand boron HEIBM of polycrystalline diamond films with\nimplantation direction transversely to film surface, [5],\nlead to formation of NTs, incorporated in diamond ma-\ntrix, which are possessed by ferromagnetic ordering. It\nwas established directly by observation of ferromagnetic\nspin wave resonance (FMSWR) [5], [6], [7]. The first ob-\nservationsofFMSWR in carbonmaterialsandin the ma-\nterials, which are non-traditional magnetic materials, on\nthe whole, were reported in above cited works. It was es-\ntablished, that magneticorderingis inherentpropertyfor\ncarbonelectronic system, that is, it is not connected with\npresence of magnetic impurities, since starting samples\nwere selected in that way, that the absolute spin number\nof paramagnetic impurities and paramagnetic structural\nimperfections at all did not exceed the value 1012spins.\nVery recently, [8], antiferroelectric ordering has been\nfound in the same pure carbon allotropic form - quasi-\n1Dcarbonzigzag-shapednanotubes(CZSNTs), obtained\nby boron- and copper-HEIBM of diamond single crystals\nin/an}bracketle{t111/an}bracketri}ht-direction. It was established by means of the de-\ntectionofnewopticalphenomenon-antiferroelectricspin\nwave resonance (AFESWR), which was theoretically de-\nscribed and experinmentally confirmed for the first time\nby infrared (IR) spectroscopy study of carbynes in [9]. It\nseems to be very new property of pure carbon allotropic\nforms - quasi-1D CZSNTs and carbynes. Moreover, on\nthe observation of antiferroelectric ordering has been re-\nported in [9] for the first time for all carbon and carbon2\nbased systems on the whole. Given results mean, that\npure carbon in the form of quasi-1DCZSNTs or carbynes\nis multiferroic system. Especially significant was the ob-\nservation of AFESWR with linear k-dipersion law, where\nkis magnitude of wave vector /vectork. It was the first work,\nin which the general spin wave theory was experimen-\ntally confirmed. Let us remark, that given theory was\nstarting from the work [10] in 1936 and was develoved in\nmany subsequent works, for instance, in [11], [12], [13],\n[14], [15], [16], however for antiferromagnetic spin wave\ncase. At the same time, it has been shown in [9], that\nthe conclusions of the antiferroelectric spin wave theory\nand antiferromagnetic spin wave theory are qualitatively\nidentical, in particular, dipersion law is the same.\nLet us also remark, that experimental observation\nof multiferroicity in quasi-1D CZSNTs and carbynes\nmeans the breakdown of space inversion symmetry along\nCZSNT hypercomplex symmetry axis and respectively\nalong carbyne chain symmetry axis. In the case of CZS-\nNTs, it agrees well with the model of quasi-1D CZSNTs\n[17], [8], based on bond dimerization in all chain compo-\nnents of quasi-1D CZSNT along its hypercomplex sym-\nmetry axis z, which actually leads to inversion symmetry\nbreakdown along given axis. Therefore, the experimental\nobservation of antiferroelecricity of quasi-1D CZSNTs,\nnecessary condition for which is the evident prediction of\nthe model (appearance of nonzero polarisation by atomic\ndisplacements), proposed in [17], can be considered to be\nadditional argument in favour of given theoretical model.\nThe idea of the formation of ferromagnetic ordering (or\nferroelectric ordering) in quasi-1D carbon systems was\nproposed in [9]. It has been shown, that by asymmetric\ndeviation of spin density relatively the chain direction\nthe constant component is appeared. It can lead in the\ncase of density wave formation along 1D axis to ferro-\nmagnetic ordering, if density wave distribution is mag-\nnetic spin distribution and to ferroelectric ordering by\nelectric own moments or electric dipole distribution. At\nthe same time, which subsystem (or subsystems), that is\nπsubsystem or σsubsystem (or even both subsystems) is\n(are) responsible for ferromagnetic ordering in quasi-1D\nCZSNT was not established.\nThe aim of given work is to study in more details the\npropertiesofcylindricalnanotubes, producedin diamond\nsinglecrystalsbyhighenergyionimplantation, whichare\npossessing by C∞symmetry axis, and to experimentally\nestablish the mechanisms of formation of ferromagnetic\nand ferroelectric ordering in given NTs.\nII. EXPERIMENTAL TECHNIQUE\nSamples of type IIa natural diamond, implanted by\nhigh energy ions of nickel (the energy of ions in ion beam\nwas 335 MeV, ion beam dose was 5 ×1014cm−2) and\nimplanted by high energy ions of copper and boron (the\nenergy of ions in ion beam was 63 MeV and 13.6 MeV\nforcopperandboronionscorrespondingly,ionbeamdosewas 5×1014cm−2) have been studied. Nickel implanted\nsample was representing rectangle with the sides ≈5 and\n4 mm in based (110) plane. Copper and boron implanted\nsampleswererepresentingin theirgeometryprismeswith\nquilateral triangle in their bases, coinciding with (111)\ncrystallographic plane, side of base triangle was equal\nto≈5mm. Thickness of all samples was equal to ≈\n1mm. Ion implantation of copper and boron was per-\nformed along [111] crystal direction, that is transversely\nto triangle prism base uniformly along all the surface.\nNickel implantation was performed along [110] crystal\ndirection, that is transversely to based (110) rectangle\nplane also uniformly along all the plane surface. The\nsamples studied were magnetically pure samples, since\nthey have been selected so that the absolute spin num-\nber did not exceed the value ≈1012spins in each of the\nsample used before implantation. The temperature of\nthe samples during the implantation was controlled and\nit did not exceed 400 K.\nESR spectra were registered on X-band ESR-\nspectrometer ”Radiopan” at room temperature by using\nofTE102mode rectangular cavity. The ruby standard\nsample was permanently placed in the cavity on its side-\nwall. One of the lines of ESR absorption by Cr3+point\nparamagnetic centers (PC) was used for the correct rel-\native intensity measurements of ESR absorption, for the\ncalibration of the amplitude value of magnetic compo-\nnent of the microwave field and for precise phase tuning\nof microwave field. It was possible owing to unsaturat-\ning behavior of ESR absorption in ruby in the range of\nthe microwave power applied, which was ≈100 mW in\nthe absence of attenuation. Unsaturable character of the\nabsorption in a ruby standard was confirmed by means\nof the measurements of the absorption intensities in two\nidentical ruby samples in dependence on the microwave\npower level. The first sample was standard sample, per-\nmanently placed in the cavity, the second sample was\nplaced in the cavity so that the resonance line intensity\nwas about 0.1 of the intensity of corresponding line of\nthe first sample. They were registered simultaneously\nbut their absorption lines were not overlapped owing to\nslightly different sample orientations. The foregoing in-\ntensity ratio was preserved for all microwave power val-\nues in the range used, which indicates, that really ruby\nsamples are good standard samples in ESR spectroscopy\nstudies.\nIII. RESULTS\nThere has been established, that copper and boron\nHEIBM with implantation direction along /an}bracketle{t111/an}bracketri}htcrystal\naxis, and boron HEIBM of polycrystalline diamond films\nwith implantation direction transversely to film surface\nlead to formation of NTs, incorporated in diamond ma-\ntrix, which possess by relatively weak ferromagnetic or-\ndering. The experimental results were in details repre-\nsented in [4], [5], [6], [7], and they will be not reproduced3\nin presented paper. The studies of FMSWR in the sam-\nple implanted along /an}bracketle{t110/an}bracketri}htcrystal direction were also de-\nscribed in [7], however we will summarise in the paper\npresented the experimental results with more details and\nsomenew resultswill be given, whichseem tobe essential\nto establish the detailed processes, leading to ferromag-\nnetic ordering formation, since the concrete subsystem,\nwhich is responsible for the mechanism of the ferromag-\nnetically ordered state was not proposed in the papers\nabove cited.\nThe strong anisotropy of FMSWR excitation has been\nobserved in implanted unannealed samples. FMSWR\nwas registered the only in the case, when the implan-\ntation plane was perpendicular to the vector /vectorH1of mag-\nnetic component of the microwave field. The intensity\nof FMSWR absorption was increasing with temperature\nof isochronal 20-minute’s annealing up to ≈700 Celsius\ndegrees. It is interesting, that after the annealing at 350\nCelsius degrees and by higher temperatures FMSWR ab-\nsorption was detected also by the vector /vectorH1of magnetic\ncomponent of the microwave field position, being to be\nparallelto sampleimplantationplane. The parametersof\nFMSWR absorption, that is splitting parameter, which\ncan be characterised by, for instance, effective g-values of\nits modes (since central FMSWR-line, which corresponds\nto ferromagnetic resonance has the only weak anisotropy\nand it can be considered approximately isotropic if to\ncompare its anisotropy with rather strong anisotropy of\nFMSWR modes). We will use the g-values of the third\nmode, Figure 1 for given characteristics. It is seen, that\namplitudes, linewidths and intensities of FMSWR modes\nare also strongly anisotropic, see Figures 2 to 4, where\nangular dependencies of the linewidth, FMSWR absorp-\ntion amplitude and intensity of the third FMSWR-mode\nare presented.\nThe splitting parameter anisotropy seems to be non-\ntrivial, it characterises by two maxima - at 54.7 degrees,\nthat is, in [111] direction of diamond lattice, and in di-\nrection about 17 degrees from [100] crystal direction. It\nis interesting, that the angular dependence of linewidth\nhas the resemblance with the angular dependence of g-\nvalue, it characterises also by two maxima at the same\nangles, see Figure 2. Minimal effective g-value and mini-\nmal value of linewidth for the third mode are achieved in\nthecaseof /an}bracketle{t110/an}bracketri}htionbeammodificationat[110]direction.\nThey are equal to 2.1526and equal to 100G respectively,\nmaximaleffective g-valueand maximalvalue oflinewidth\ncorrespond to [111] direction of diamond lattice and are\nequal respectively to 2.3845 and 360 G. Coincidence of\nthedirectionsofanisotropymaximawith diamondlattice\ncharacteristic directions is indication of the role of host\ndiamond lattice on the characteristics of magnetic or-\ndering, determining the magnetic symmetry, that seems\nto be possible by relative softness of the corresponding\nelectronic subsystems. Therefore we obtain the strong\nevidence, that ferromagnetic ordering is determined by\nelectron-electron correlations in valence π-subsystem of\nelectronic system (but not in σ-subsystem), since the/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s44/s49/s53/s50/s44/s50/s48/s50/s44/s50/s53/s50/s44/s51/s48/s50/s44/s51/s53/s50/s44/s52/s48/s69/s70/s70/s69/s67/s84/s73/s86/s69/s32/s103/s45/s86/s65/s76/s85/s69\n/s91/s49/s48/s48/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s49/s93/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s48/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s65/s78/s71/s76/s69/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFigure1: Angulardependenceofeffectiveg-valuefor thethi rd\nFMSWR-mode, observed in NTs, incorporated in diamond\nsingle crystal by nickel ion beam direction transversely (1 10)\nsample plane\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48\n/s91/s49/s48/s48/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s49/s93/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s48/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s65/s78/s71/s76/s69/s32/s40/s68/s101/s103/s114/s101/s101/s41/s76/s73/s78/s69/s87/s73/s68/s84/s72/s32/s40/s71/s97/s117/s115/s115/s41\nFigure 2: Angular dependence of the linewidth of the third\nFMSWR-mode, observed in NTs, incorporated in diamond\nsingle crystal by nickel ion beam direction transversely (1 10)\nsample plane\nonlyπ-subsystem can be subjected to so strong response\n(which is mapped by the strong anisotropy of FMSWR-\nmodesaboveanalysedwith the axescoincidingwith sym-\nmetry axes of the host lattice) to the surrounding lattice\npresencebyNT-formation. The σ-subsystem ofNTs pro-\nduced is expected to be not so sensitive to the presenceof\nhost diamond lattice. Given prediction is confirmed by\nformation of own magnetic symmetry with axes, which\nare not coinciding with symmetry axes of host diamond\nlattice in NTs, produced by /an}bracketle{t100/an}bracketri}htion beam modification.4\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s65/s77/s80/s76/s73/s84/s85/s68/s69/s32/s40/s114/s46/s117/s46/s41\n/s91/s49/s48/s48/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s49/s93/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s48/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s65/s78/s71/s76/s69/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFigure 3: Angular dependence of FMSWR absorption ampli-\ntude of the the third FMSWR-mode, observed in NTs, incor-\nporated in diamondsingle crystal bynickel ion beam directi on\ntransversely (110) sample plane\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s73/s78/s84/s69/s78/s83/s73/s84/s89/s32/s40/s114/s46/s117/s46/s41\n/s91/s49/s48/s48/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s49/s93/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s48/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s65/s78/s71/s76/s69/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFigure 4: Angular dependence of the FMSWR absorption\nintensity of the third FMSWR-mode, observed in NTs, incor-\nporated in diamondsingle crystal bynickel ion beam directi on\ntransversely (110) sample plane\nIt is explained by direct participation of σ-subsystem in\nmagnetic ordering formation, [18].\nIt is interesting, that, although NT-formation is start-\ning at much more lower ion fluences, the given strong\nresponse to diamond lattice presence by initial NT-\nformation process is preserved by entire HEIBM of near\nsurface region in diamond single crystal. Further, ampli-\ntude dependence, Figure 3, is characterisedby the clearly\npronounced maximum near 35.3 degrees from [100] crys-\ntal direction of diamond lattice, that is, it coincides also\nwith symmetry direction of diamond lattice. It is seen,that amplitude dependence is also strongly anisotropic.\nReally, the ratio of maximal amplitude value to its mini-\nmal value is equal to 4.8. It is additional indication, that\nmechanism of ferromagnetic ordering is connected with\nthe formation of ordered structures just in π-subsystem\nof incorporated NTs. Simultaneously, it is the display of\nthe fact, that anisotropy of FMSWR absorption by mag-\nneticallyorderedNT-structureandtheanisotropyofNT-\nstructure itself, which is determined by relatively weak\ng-value anisotropy of main resonance mode, are quanti-\ntatively different.\nTwo not very pronounced maxima at 54.7 degrees and\nin direction near 17 degrees from [100] crystal are ap-\npeared additionally to the maximum near 35.3 degrees\ndirection in the angular dependence of the intensity, Fig-\nure 4. It is clear, that additional maxima are determined\nby contribution of linewidth anisotropy, compare Figure\n4 and Figure 2.\nIV. DISCUSSION\nLet us indicate once again, that although it was es-\ntablished, that copper and boron HEIBM with implan-\ntation direction along /an}bracketle{t111/an}bracketri}htcrystal axis, nickel HEIBM\nwith implantation direction along /an}bracketle{t110/an}bracketri}htaxis, and boron\nHEIBM of polycrystalline diamond films with implan-\ntation direction transversely to film surface lead to for-\nmation of NTs, incorporated in diamond matrix, which\npossess by ferromagnetic ordering, [4], [5], [6], [7], the\nnature of given state and mechanism, leading to its for-\nmation was not strictly theoretically described. To solve\ngiven task, it seems to be necessary to know the nature\nof charge and spin carriers and the mechanisms of car-\nrier transport and interactions of charge and spin carri-\ners both between themselves and with phonons and pho-\ntons. There seems to be paramount significant the same\ntask for nanoelectronics, spintronics and for the other\nbranches of nanotechnology. There is existing in the the-\noryof 1Delectronic systems in particular in the theory of\nconducting NTs the following concept, which was start-\ning with the work of Tomonaga in 1950, [19], and with\nthe work by Luttinger in 1963, [20], when it has become\nclear that the electron-electron interaction destroys the\nsharpFermi surfaceand leads toa breakdownofthe Lan-\ndau Fermi liquid (LFL) theory. The resulting non-LFL\nstate is commonly called Luttinger liquid (LL), or some-\ntimes Tomonaga-Luttinger liquid to describe the univer-\nsal low-energy properties of 1D conductors. LL behavior\nis characterized by pronounced power-law suppression of\nthe transport current and the density of states, and by\neffect of spin-charge separation. The nature of the spin\nand charge carriers was also proposed to be the follow-\ning. The chargeless spin 1/2 quasiparticles - spinons and\nspinless quasiparticles with the charge ±e - holons were\nintroduced. It was concluded from the universality of LL\ndescription that the physical properties do not depend\non details of the model, the interaction potential, and5\nso on, but instead they are only characterized by a few\nparameters - critical exponents. Further, the LL concept\nwas believed to hold for arbitrary statistical properties\nof the particles, that is, both for fermions and bosons. It\nprovided a paradigm for non-Fermi liquid physics. When\nconcern the carbon NTs, we have to remark, that there\nis the following viewpoint. The single-wall carbon nan-\notubes (SWCNTs) are considered in many works to be\n1D objects (it is not always correct, especially for stan-\ndardNTswith diameterinseveralnanometersandmore)\ncan be described the only in the frame of LL concept.\nMoreover, SWCNTs are considered even to be the best\nmodel system of the LL state demonstration. However,\ngiven viewpoint is based the only on experimental ob-\nservation of power-law behavior by measuring the tun-\nneling conductance of SWNTs in dependence on tem-\nperature and voltage andv ballistic nature of transport,\nwhich was found by electron force microscopic measure-\nments At the same time spin-charge separation by mech-\nanism of spinons and/or holons’ formation has not been\nobserved so far. Therefore, the existing viewpoint seems\nto be insufficiently grounded, since power-lawbehaviorof\nthetunnelingconductanceindependenceontemperature\nand voltage and ballistic nature of transport phenomena\ncan be explained in the frames ofthe quite otherconceps.\nIt has to be also remarked, that both the models LL\nandLFLarethemodelsofideal(andevenoversimplified)\nquantum liquids, since they do not take into account the\nnonlinearityofthefermionspectrumontheonehandand\nelectron-phonon interactions on the other hand. In fact\nboththemodelsdescribenotstronglyadequatelythereal\nprocesses, since the changes in charge state of arbitrary\natom in 1D chain to be the result of electron-electron\ninteraction are always accompanied by the changes in\nphonon subsystem (and vice versa). It is consequence of\ngeneric coupling between operators of creation and anni-\nhilation in electron subsystem and in phonon field. Let\nus also remember, that key argument for insertion of the\nnotion ”Luttinger liquid” is in fact the simplification, de-\ntermined by linearization of the generic spectrum of par-\nticles in neighborhood of Fermi points in k-space. Just\ngiven simplification has led to divergencies arising in the\nperturbation theory in 1D-case and it does not means\nthat 1D Fermi liquid description is incorrect in general\ncase. Consequently, the description of NTs the only in\nthe frames of LL concept seems to be also oversimplifi-\ncation. Moreover, it is showed in [21], that the concept\nof description of 1D correlated electronic systems in the\nframe of 1D Fermi liquid (FL) can be renewed. The most\ninteresting result is that, that the Fermi liquid concept\ncan be applied just to quasi-1D carbon NTs. It was con-\nsidered in [21] the concept of 1D FL on the example of\nwell known 1D system - trans-polyacetylene (t-PL), that\nis, it is in fact the generalization of well known model de-\nveloped by Su, Schrieffer, Heeger (SSH-model), [22], [23],\n[24]. SSH-model, indistinctionfromLFLandLLmodels,\ntakes into account the electron-phonon interaction. The\nsubsequent generalization, for instance, for applicationof given model immediately to quasi-1D carbon zigzag\nshaped nanotubes can be easily obtained by using of hy-\npercomplex number theory like to its application in the\nworks, [8], [17], where hypercomplex number theory was\nused for the interpretation of quantum optics effects in\nquasi-1D NTs.\nLet us summarise briefly for the convenience of the\nreaders the results, presented in [21].\nThe Born-Oppenheimerapproximationwasconsidered\nand starting Hamiltonian was the following\nˆH(u) =ˆH0(u)+ˆHπ,t(u)+ˆHπ,u(u).(1)\nThe first term in (1) is\nˆH0(u) =/summationdisplay\nm/summationdisplay\ns(ˆP2\nm\n2M∗ˆa+\nm,sˆam,s+Ku2\nmˆa+\nm,sˆam,s).(2)\nIt representsthe sum of operatorof kinetic energy ofCH-\ngroup motion (the first term in (2)) and the operator of\ntheσ-bondingenergy(the secondterm). Coefficient Kin\n(2) is effective σ-bonds spring constant, M∗is total mass\nof CH-group, umis configuration coordinate for m-th\nCH-group, which corresponds to translation of m-th CH-\ngroup along the symmetry axis zof the chain, m=1,N,\nNis number of CH-groups in the chain, ˆPmis operator\nof impulse, conjugated to configuration coordinate um,\nm=1,N, ˆa+\nm,s, ˆam,sare creation and annihilation op-\nerators of creation or annihilation of quasiparticle with\nspin projection son them-th chain site in σ-subsystem\nof t-PA.\nThe second term in (1) can be represented in the form\nof two components and it is\nˆHπ,t(u) =ˆHπ,t0(u)+ˆHπ,t,α1(u) =\n/summationdisplay\nm/summationdisplay\ns[(t0(ˆc+\nm+1,sˆcm,s+ˆc+\nm,sˆcm+1,s)]+\n(−1)m2α1u)(ˆc+\nm+1,sˆcm,s+ˆc+\nm,sˆcm+1,s)],(3)\nwhere ˆc+\nm,s, ˆcm,sare creation and annihilation operators\nof creation or annihilation of quasiparticle with spin pro-\njectionson them-th chain site in π-subsystem of t-PA.\nIt is the resonance interaction (hopping interaction in\ntight-binding model approximation) of quasiparticles in\nπ-subsystem of t-PA electronic system, which is consid-\nered to be Fermi liquid, and in which the only constant\nand linear terms in Taylor series expansion of resonance\nintegral about the dimerized state are taking into ac-\ncount.\nThe expression for the operator ˆHπ,u(u), which de-\nscribes the part of electron-phonon interaction, deter-\nmined by interaction between quasiparticles in Fermi liq-\nuid state of π-subsystem in terms of {ˆc(c)\nk,s}and{ˆc(v)\nk,s}is\nthe following\nˆHπ,u,α 2(u) =/summationdisplay\nk/summationdisplay\nk′/summationdisplay\nsα2(k,k′,s)ˆc+(c)\nk′,sˆc+(v)\nk′,sˆc(v)\nk,sˆc(c)\nk,s.\n(4)6\nIt has to be indicated, that the operators {ˆc+\nm,s},{ˆcm,s},\nm=1,N, were represented in the form\n{ˆcm,s}={ˆc(c)\nm,s}+{ˆc(v)\nm,s},\n{ˆc+\nm,s}={ˆc+(c)\nm,s}+{ˆc+(v)\nm,s},(5)\nrelated to π−c- andπ−v-band correspondingly, and\nthen/vectork-space operators were defined\n{ˆc(c)\nk,s}={i√\nN/summationdisplay\nm/summationdisplay\ns(−1)m+1exp(−ikma)ˆc(c)\nm,s},\n{ˆc(v)\nk,s}={1√\nN/summationdisplay\nm/summationdisplay\nsexp(−ikma)ˆc(v)\nm,s},(6)\nm=1,N. Theconsiderationwasrestrictedbythetaking\ninto account the contribution of the term, correspond-\ning to interaction between the quasiparticles in differ-\nent bands, which seems to be the most essential. Phys-\nically the identification of linear on displacement upart\nof resonance interaction (hopping) and the pairwise in-\nteraction of quasiparticles in π-subsystem between them-\nselves with electron-phonon interaction, which was done\nin cited work, is understandable, if to take into account,\nthat by atomic CHgroup displacements the phonons are\ngenerated, which in its turn can by release of the place\non, for instance, ( CH)mgroup, to deliver the energy and\nimpulse, which are necessary for transfer of the quasipar-\nticle (electron) from adjacent ( m−1)- or (m+1)-position\nin chain in the case of resonance interaction (hopping).\nFor the case the pairwise interaction of quasiparticles, it\nmeans, that its linear on displacement upart is realized\nby means of phonon field, which transfers the energy and\nimpulse from one quasiparticle to another (which can be\nnot inevitable adjacent). Mathematically it was proved\nin the following way. The processes of interaction in c\n(v) band can be considered to be independent on each\nother. It means, that transition probability from the\n/an}bracketle{tkl,s|-state to /an}bracketle{tkj,s|-state in c-band and from /an}bracketle{tk′\nl,s|-state\nto/an}bracketle{tk′\nj,s|-state in v-band, which is proportional to coeffi-\ncientα2(k,k′,s), can be expressed in the form of product\nof real parts of corresponding matrix elements, that is in\nthe form\nα2(k,k′,s)∼Re/an}bracketle{tkl,s|ˆV(c)|kj,s/an}bracketri}htRe/an}bracketle{tk′\nl,s|ˆV(v)|k′\nj,s/an}bracketri}ht=\n/summationdisplay\nkphRe/an}bracketle{tkl,s|ˆV(c)|kph/an}bracketri}ht/an}bracketle{tkph|kj,s/an}bracketri}ht×\n/summationdisplay\nkphRe/an}bracketle{tk′\nr,s|ˆV(v)|kph/an}bracketri}ht/an}bracketle{tkph|k′\nn,s/an}bracketri}ht,(7)\nwhereˆV(v)=V0(v)ˆe(ˆeisunitoperator)isthefirsttermin\nTaylorexpansionofpairwiseinteractionofquasiparticles,\nfor instance, with wave vectors k′\nr,k′\nnand spin projec-\ntionsinv-band, that is, in ground state, ˆV(c)=V1(c)uˆe\nis the second term in Taylor expansion of pairwise inter-\naction in excited state (in c-band), that is, it is product\nof configuration coordinate uand coordinate derivativeatu= 0 of operator of pairwise interaction of quasipar-\nticles with wave vectors kl,kjand spin projection sin\nc-band,kphis phonon wavevector, and the summation is\nrealized over all the phonon spectrum. At that, since the\nlinear density ofpairwise interactionis independent on k,\nwhich is the consequence of translation invariance of the\nchain,V0(v),V1(c)are constants. Therefore, the pairwise\ninteraction is considered to be accompanying by process\nof phonon generation, when electronic quasiparticles are\nalready in excited state, that is, in c-band (retardation\neffect of phonon subsystem is taken into account). Then\nit will be ˆV(c)=V0(c)uˆe,ˆV(v)=V0(v)ˆe. A number of\nvariants are possible along with process of phonon gener-\nation, corresponding to states of electronic quasiparticles\ninc-band above described. The result will mathemati-\ncally be quite similar, if to change the energetic place of\nexcitation, that is, if to interchange the role of candv\nbands for given process. There seem to be possible the\nrealization of both the stages (that is phonon generation\nand absorption) for electronic quasiparticles in single c\norvband states and simultaneous realisation both the\nstages in both the bands. Mathematical description will\nbe for all possible variants similar and for distinctness,\nit was considered only the first variant. For the sim-\nplicity, the processes were considered, in which the spin\nprojection is keeping to be the same. Since in z-direction\nthe impulse distribution is quasi-continuous (the chain\nhas the macroscopic length L=Na) the standard way/summationtext\nkph→L\n2π/integraltext\nkphhas been used. Further, phonon states\nwere described by wave functions /an}bracketle{tkph|=v0exp(ikphz),\nwherez∈[0,L],kph∈[−π\n2a,π\n2a],v0is constant. Then,\nit was obtained from (7) the expression\nα2(k,k′,s) =b|v0v|2|v0c|2V0(c)uV0(v)|φ0cs|2|φ0vs|2×\nN\n2π(ql−qj)(qr−qn)Re{exp[i(klml−kjmj)a]expika}×\nRe{exp[i(k′\nrmr−k′\nnmn)a]expik′a},\n(8)\nwhere|φ0cs|2,|φ0vs|2are squares of the modules of the\nwave functions |kj,s/an}bracketri}htand|k′\nj,s/an}bracketri}htrespectively, k=kph(ql−\nqj),k′=k′\nph(qr−qn)ql,qj,qr,qn∈Nwith additional\nconditions ( ql−qj)a≤L, (qr−qn)a≤L,b- is aspect\nratio, which in principle can be determined by compari-\nson with experiment. Here the values ( ql−qj), (qr−qn)\ndetermine the steps in pairwise interaction with phonon\nparticipation and they are considered to be fixed. The\nprocesses, for which k=k′, were considered. Conse-\nquently, ( qr−qn) = (ql−qj). Then the Hamiltonian\nˆHπ,u,α 2(u) was represented in the form\nˆHπ,u,α 2(u) =\n/summationdisplay\nk/summationdisplay\nk′/summationdisplay\ns4α2usinkasink′aˆc+(c)\nk′,sˆc+(v)\nk′,sˆc(v)\nk,sˆc(c)\nk,s,(9)7\nwhere 4α2(s) is\nb|v0v|2|v0c|2V0(c)V0(v)|φ0cs|2|φ0vs|2×\nN\n2π[(ql−qj)]2= 4α2(s)(10)\nLet us remark, that the Hamiltonian ˆHπ,u,α 2(u) de-\nscribes the attraction between the electrons, it can lead\nto formation of Cooper pairs in a π-subsystem and to\nsuperconductivity.\nTwo values for the energy of quasiparticles, indicating\non the possibility of formation of the quasiparticles of\ntwokinds both in c-bandand v-band havebeen obtained.\nThey are the following\nE(c)\nk(u) =Q2∆2\nk−ǫ2\nk/radicalbig\nǫ2\nk+Q2∆2\nk,\nE(v)\nk(u) =ǫ2\nk−Q2∆2\nk/radicalbig\nǫ2\nk+Q2∆2\nk(11)\nand\nE(c)\nk(u) =/radicalBig\nǫ2\nk+Q2∆2\nk,\nE(v)\nk(u) =−/radicalBig\nǫ2\nk+Q2∆2\nk(12)\nThe factor Qis determined by relation\n[1+α2\n2α1/summationdisplay\nk/summationdisplay\nsQ∆ksinka/radicalbig\nǫ2\nk+Q2∆2\nk(n(c)\nk,s−n(v)\nk,s)] =Q,(13)\nwheren(c)\nk,sis eigenvalue of density operator of quasipar-\nticles’ number in c-band,n(v)\nk,sis eigenvalue of density\noperator of quasiparticles’ number in v-band. The quasi-\nparticles with the energy, determined by (12) at Q= 1\nare the same quasiparticles, that were obtained in known\nSSH-model.\nSubsequent analysis has showed, that SSH-like solu-\ntion (12) is inapplicable for the description of standard\nprocesses, passing near equilibrium state by any parame-\nters. The quasiparticles, described by SSH-like solution,\ncan be created the only in strongly nonequilibrium state\nwith inverse population of the levels in c- andv-bands.\nAt the same time the solution, the energy of quasipar-\nticles for which is determined by (11) can be realised\nboth in near equilibrium and in strongly non-equilibrium\nstates of the π-subsystem of t-PA, which is considered to\nbe quantum Fermi liquid.\nThe continuum limit for the ground state energy of\nthet-PA chain with SSH-like quasiparticles will coincide\nwith known solution, [23], [24], if to replace ∆ kQ →∆k.\nThe calculation of the ground state energy E[u]\n0(u) of the\nt-PA chain with quasiparticles’ branch, which is stable\nnear equilibrium by taking into account, that in ground\nstatenc\nk,s= 0,nv\nk,s= 1, in the continuum limit gives\nE[u]\n0(u) =−2Na\nππ\n2a/integraldisplay\n0(Q∆k)2−ǫ2\nk/radicalbig\nQ∆k)2+ǫ2\nkdk+2NKu2.(14)Then, calculation of the integral results in the expression\nE[u]\n0(u) =4Nt0\nπ{F(π\n2,1−z2)+\n1+z2\n1−z2[E(π\n2,1−z2)−F(π\n2,1−z2)]}+2NKu2,(15)\nwherez2=2Qα1u\nt0,F(π\n2,1−z2) is the complete elliptic\nintegral of the first kind and E(π\n2,1−z2) is the complete\nelliptic integral of the second kind. Approximation of\nground state energy at z≪1 for the solution, which is\nstable near equilibrium position, gives\nE[u]\n0(u) =N{4t0\nπ−6\nπln2t0\nQα1u4(Qα1)2u2\nt0+\n28(Qα1)2u2\nπt0+...}+2NKu2.(16)\nIt is seen from (16), that the energy of quasiparticles,\ndescribed by given solution, has the form of Coleman-\nWeinberg potential with two minima at the values of\ndimerization coordinate u0and−u0like to the en-\nergy of quasiparticles, described by SSH-solution. It is\nunderstandable, that subsequent considerations, includ-\ning electrically neutral S = 1/2 soliton and electrically\ncharged spinless soliton formation, that is the appear-\nance of the phenomenon of spin-charge separation, by\nFL description of 1D systems will be coinciding in its\nmathematical form with those ones in SSH-model.\nItisthe mainresultofthework[21], whichmeans, that\nthe physical properties of 1D systems can be described in\ntheframesof1DquantumFLincludingthemechanismof\nappearance of the most prominent feature of 1D systems\n- the phenomenon of spin-charge separation.\nSince the model proposed takes into consideration the\nelectron-electron correlations in explicit form, it seems\nto be ground for its application to electronic systems,\nin which electron-electron correlations are rather strong.\nIn particular, it can be used for analysis of the physical\nproperties including ESR and FMSWR spectra analysis\nin quasi-1D carbon NTs. It can be done by above indi-\ncated manner, that is by using of hypercomplex number\ntheoryanalogouslytotheoreticalanalysisofquantumop-\ntics effects in [17] and analysis in [8] of Raman spectra\nin given objects.\nIt is substantial, that the mechanism of the phe-\nnomenon of spin-charge separation in 1D FL is topo-\nlogical soliton mechanism, which is quite different from\nAnderson spinon-holon mechanism.\nThe results obtained show, that the shape of π-solitons\n(orσ-solitons) is given by the expressionhaving the same\nmathematical form both in SSH-model and in its FL gen-\neralisation. It is\n|φ(n)|2=1\nξπ(σ)sech2[(n−n0)a\nξπ(σ)−vπ(σ)t]cos2nπ\n2,(17)\nwheren,n0are variableand fixed numbers of CH-unit in\nCH-chain,aisC−Cinteratomic spacing projection on8\nchain direction, vπ(σ)isπ(σ)-soliton velocity, tis time,\nξπ(σ)isπ(σ) coherence length. It has been shown in [21],\nthatπ-solitons ( σ-solitons) differ in fact by numerical\nvalue of coherence length in SSH-model and in its FL\ndescription.\nThe observation of FMSWR phenomenon in the NTs,\nformed by /an}bracketle{t110/an}bracketri}hthigh energy nickel ion implantation in\nnatural diamond single crystal (at ion beam dose in\n5×1014cm−2) unambiguously means, that the SSH π-\nsoliton lattice formation takes place, that is periodical\nnon-harmonic spin-density wave. For the simplicity we\ncan approximate the function of the shape of given den-\nsity wave(let us designateit with abbreviationASLSDW\nand ASLCDW (Approximation of Soliton Lattice Spin\n(Charge) Density Wave) by trapeziform function, Fig-ure 5. Real function of the envelope shape of soliton\nlattice spin density wave will be between given trapezi-\nform function and harmonic sinusoidal function for den-\nsity wave (HSDW - Harmonic Spin Density Wave). We\nwill show, that given approximation is rather good for\nqualitative and in the principle for quantitative estima-\ntion of characteristics of ferromagnetic order formation\nin carbon NTs. The calculation in [9] will be adapted for\ngiven case. Actually, from mathematical viewpoint the\ndifference between ASLSDW and HSDW states is min-\nimal, since ASLSDW can be represented in the form of\nsuperposition of HSDWs, where amplitude of main har-\nmonic is strongly exceeding the others. It follows from\nthe expansionofASLSDWprofilein Fourierseries, which\nis\nf(x′) =A(2L0+L1)∞/summationdisplay\nm=0\n\n/bracketleftBig\n2(2L0+L1)/parenleftBig\ncosπmL0\n2L0+L1−cosπm(L0+L1)\n2L0+L1/parenrightBig\n−πL1msinmπ/bracketrightBig\n2L1π2m2·cos2πmx′\nL\n\n,(18)\nf(x) =A(2L0+L1)\n1\n4+(2L0+L1)∞/summationdisplay\nm=1/parenleftBig\ncosπmL0\n2L0+L1−cosπm(L0+L1)\n2L0+L1/parenrightBig\n2L1π2m2·cos2πmx\nL\n. (19)\nFigure 5: Approximation of soliton lattice spin (charge) de n-\nsity wave profile along the chain direction\nThe sense of the parameters L 0, L1, L is seen from\nFigure 5. The functions f(x),f(x′) denote spin den-\nsity distributions S(x),S(x′) in the case of ASLSDW\nand charge density distributions ρ(x),ρ(x′) in the case\nof ASLCDW. Expression (18) corresponds to position\nof chain atoms along X′-direction, Figure 5. Expres-\nsion (19) corresponds to position of chain atoms along\nX-direction, Figure 5. We see, that by asymmetric de-\nviation of spin (or charge) density relatively the chain\ndirection, the constant component is appeared. It will\nlead in the case of soliton lattice formation to ferromag-\nnetic ordering if density wave distribution is magnetic\nspin density wave distribution and to ferroelectric order-\ning by similar electric dipole density distribution. It is\nclear from (18) and (19), that the amplitude of compo-\nnents with m >1 is inversely proportional to m2, thatis, it diminishes ratherquicklywhen number mincreases.\nIn the first order approximation, the relations (18) and\n(19) can be represented by\nf(x′) =A(2L0+L1)αcos2πx′\nL+o/parenleftbigg1\nm2/parenrightbigg\n,(20)\nf(x) =A(2L0+L1)/parenleftbigg1\n4+1\n2αcos2πx\nL/parenrightbigg\n+o/parenleftbigg1\nm2/parenrightbigg\n,\n(21)\nwhereαis\nα=/bracketleftBig\n(2L0+L1)/parenleftBig\ncosπL0\n2L0+L1−cosπ(L0+L1)\n2L0+L1/parenrightBig/bracketrightBig\nL1π2.(22)\nIt is seen from (17), that envelope function of single soli-\nton spin density distribution corresponds precisely to the\ncase with xaxis position, that is, it is strongly asym-\nmetric relatively the chain axis. By the formation of the\nsoliton lattice the periodic envelope function of soliton\nspin density distribution in soliton lattice will be deter-\nmined by overlapping of individual envelope functions\nand the asymmetry extent can be even more pronounced\ndepending on overlapping extent.\nLet us also remark, that given conclusion is preserved,\nwhen simultaneously with Peierls metal-insulator tran-\nsition the Mott-Hubbard metal-insulator transition also\ntakes place It leads to modification of oscillating factor9\nin (17). Instead of factor cos2nπ\n2the factor ( b1cos2nπ\n2−\nb2sin2nπ\n2) is appeared, at that in real case of t-PA the\nvalue of b1is substantially exceeds b2, although b2is\nnonzero, [25].\nThe viewpoint above developed agrees well with re-\nsults of the work [26], where the real lineshapes of SSH-\nsolitonline in t-PA, registeredbyboth ESR andelectron-\nnuclear double resonance (ENDOR), were approximated\neven with rectangular spin-density distribution (that is,\nby particular case of trapeciform distribution at L1= 0).\nIt seems to be essential, that real ESR lineshape was\nfitted by given distribution very well by ξπ= 24aand\nb2/b1= 0.3 and qualitatively the lineshape of ENDOR\nspectrum, which is substantially more sensitive to choose\nof shape function, was also reproduced at the same pa-\nrameters ξπ= 24aandb2/b1= 0.3.\nFurther, it was found also, that there is numerical\ncoincidence of the characteristics of SSH topological π-\nsolitons in t-PA and in quasi-1D CZSNTS, formed both\nby/an}bracketle{t111/an}bracketri}htand/an}bracketle{t110/an}bracketri}htion implantation. So, the values of\ng-tensor components in Cu-implanted sample are g1=\n2.00255 (it is minimal g-value and it is g||principal di-\nrection), g2=g3=g⊥= 2.00273,the accuracyofrelative\ng-value measurements is ±0.00002, [4]. It is substantial,\nthatg-value of paramagnetic π-solitons in t-PA, equaled\nto 2.00263, [27], gets in the middle of given rather nar-\nrow interval of g-value variation of PC in ion produced\nNTs. Although anisotropy of paramagnetic π-solitons\nint-PA, mapping the distribution of π-electron density\nalong the whole individual t-PA chain, was not resolved\nby ESR measurements in [27] directly, there are indirect\nevidences on axial symmetry of ESR absorption spectra\nint-PA too, [26], [28]. Consequently, the value 2.00263\nis average value and it coincides with accuracy 0.00002\nwith average value of aforecited principal g-tensor val-\nues of PC in quasi-1D CZSNTs. Taking into account\ngiven high precision coincidence of g-values in t-PA and\nquasi-1DCZSNTswecanconcludethatSSH–solitonden-\nsity distribution in quasi-1D CZSNTs is also asymmetric\nalong hypercomplex x-axis.\nTherefore, the appearance of ferromagnetic ordering\nin quasi-1D CZSNTs, formed by /an}bracketle{t111/an}bracketri}htand/an}bracketle{t110/an}bracketri}htion im-plantation, for which the only π-subsystem is responsible\nis experimentally confirmed.\nThus we have found the physical origin of the mecha-\nnism of the formation of ferromagnetic ordering in car-\nbon NTs. It is determined the only by π-subsystem of\nfull electronic system, which agrees well with experimen-\ntal results on FMSWR above summarised and explains\nqualitatively the sensitivity of the symmetry of FMSWR\nparameters, Figures 1 to 4, to symmetry directions of\nsurroundingdiamond crystal lattice in NTs, produced by\n/an}bracketle{t110/an}bracketri}htimplantation by relative softness of π-subsystem in\ncomparison with σ-subsystem. Given conclusion agrees\nwell with the results of the work [18], where the partici-\npation of σ-subsystem in magnetic ordering formation in\nNTs, produced by /an}bracketle{t100/an}bracketri}htimplantation, leads to own mag-\nnetic symmetry, the main symmetry axes for which are\nnot coinciding with symmetry axes of the host diamond\nlattice.\nV. CONCLUSIONS\nThe physical origin of the mechanism of the formation\nof ferromagnetic ordering in carbon nanotubes produced\nby high energy ion beam modification of diamond sin-\ngle crystals in /an}bracketle{t110/an}bracketri}htand/an}bracketle{t111/an}bracketri}htdirections has been es-\ntablished. It is determined by asymmetry of spin den-\nsity distribution of Su-Schrieffer-Heeger topological soli-\nton lattice, which is formed in 1D Fermi quantum liquid\nstate of the only π-electronic subsystem of given NTs.\nThe conclusions are based on detailed analysis of an-\ngular dependencies of the parameters of ferromagnetic\nspin wave resonance, numerical values and symmetry of\ng-values of main FMSWR moxdes, taking into account\nthe theory of 1D Fermi liquid and adaptation of the cal-\nculation, which gives the qualitative evaluation of total\nmagnetization and indicates on the appearance of ferro-\nmagnetic ordering by the formation of π-electronic SSH-\nsoliton lattice (produced by spin 1/2 SSH solitons in 1D\nFermiπ-electronic liquid) with asymmetric spin distribu-\ntion relatively 1D hypercomplex symmetry axis of NTs\nstudied.\n[1] Erchak D.P, Penina N.M, Stelmakh V.F, Tolstykh VP,\nZaitsev AM, The 7th Int.Conf.IBMM 90, Abstracts,\nKnoxville, USA, 1990, p.313\n[2] EfimovV.G,ErchakD.P,Gelfand R.B,PeninaN.M,Stel-\nmakhV.F,VSVarichenko, UlyashinA.G,ZaitsevAM,E-\nMRS 1990 Fall Meeting, Abstracts, Strasbourg, France,\n1990, p.C-V/P 12\n[3] Kawataba K., Mizutani M., Fukuda M., Mizogami S.,\nSynthetic Metals, 33(1989) 399–402\n[4] Erchak D.P, Efimov V.G, Zaitsev AM, Stelmakh V.F,\nPenina N.M, Varichenko VS, Tolstykh VP, Nuclear In-\nstrum.Meth.in Phys.Res.,B, 69(1992) 443-451\n[5] Erchak D.P, Guseva M.B, Alexandrov A.F, AlexanderH, Pilar v.Pilchau A, Pis‘ma v Zhurnal Experimentalnoi\ni Teoreticheskoi Fiziki, 58, N 4 (1993) 268-271, JETP\nLetters, 58, N 4 (1993) 275-278\n[6] Ertchak D.P, Efimov V.G, Stelmakh V.F, Review, Zhur-\nnal Prikladnoi Spectroskopii, 64, N 4 (1997) 421-449,\nJ.Applied Spectroscopy, 64, N 4, (1997) 433-460\n[7] Ertchak D.P, Efimov V.G, Stelmakh V.F, Martinovich\nV.A, Alexandrov A.F, Guseva M.B, Penina N.M, Kar-\npovich I.A, Varichenko V.S, Zaitsev A M, Fahrner W.R,\nFink D, Physica Status Solidi, b, 203, N2 (1997) 529-548\n[8] Yerchuck D, Dovlatova A, J.Phys.Chem.,C, DOI:\n10.1021/jp205549b, 116, N 1 (2012) 63-80\n[9] Yearchuck D, Yerchak Y, Alexandrov A, Phys.Lett.A,10\n373, N 4 (2009) 489-495\n[10] Hulthen L, Proc.Roy.Acad.Sci.(Amsterdam), 39(1936)\n190\n[11] Anderson P W, Phys.Rev., 86(1952) 694\n[12] Keffer F, Thesis, Berkeley, January 1952\n[13] Keffer F, Kaplan H, Yafet Y, American Journal of\nPhysics, 21, N 4 (1953) 250-257\n[14] Ziman J M, Proc.Phys.Soc.(London), A65(1952) 540\n[15] Nakamura T., Progr.Theor.Phys., 7(1952) 539\n[16] Tani K., Progr.Theor.Phys., 31, N 3 (1964) 335-356\n[17] DovlatovaA, Yearchuck D, Chem.Phys.Lett., 511(2011)\n151-155\n[18] Yerchuck D, Stelmakh V, Dovlatova A, Yerchak Y,\nAlexandrov A, in press\n[19] Tomonaga S., Prog.Theor.Phys., 5(1950) 544\n[20] Luttinger J M, J.Math.Phys., 4(1963) 1154[21] A.Dovlatova, D.Yerchuck, F.Borovik, in press\n[22] Su W.P., Schrieffer J.R, and Heeger A.J, Phys.Rev.Lett. ,\n42(1979) 1698\n[23] SuW.P., Schrieffer J.R, andHeeger A.J, Phys.Rev.B, 22,\n(1980) 2099\n[24] Heeger A.J, Kivelson S, Schrieffer J.R, Su W-P,\nRev.Mod.Phys., 60(1988) 781-850\n[25] Thomann H, Dalton L R, Grabowsky M, Clarke T C,\nPhys.Rev.B, 31(1985) 3141\n[26] Kahol P K, Mehring M, J.Phys.C, 19(1986) 1045-1054\n[27] Goldberg, I B; Crowe, H R; Newman, P R; Heeger, A.J;\nMacDiarmid, A G, J.Chem.Phys., 70(1979) 1132-1135\n[28] Kuroda S., Tokumoto M., Kinoshita N., Shirakawa H.,\nJ.Phys.Soc.Jpn., 51(1982) 693-694arXiv:1201.0285v3  [cond-mat.str-el]  2 Jun 2013Room Temperature Superconductivity and Uncompensated Ant iferromagnetic\nOrdering in Carbon Nanotubes\nDmitri Yerchuck (a), Vyacheslav Stelmakh (b), Alla Dovlatova (c), Yauhen Yerchak (b), Andrey Alexandrov (c)\n(a) - Heat-Mass Transfer Institute of National Academy of Scien ces of RB, Brovka Str.15, Minsk, 220072, dpy@tut.by\n(b) - Belarusian State University, Nezavisimosti Avenue 4, Minsk, 2 20030, RB\n(c) - M.V.Lomonosov Moscow State University, Moscow, 119899\n(Dated: November 17, 2018)\nThe phenomenon of formation of uncompensated antiferromag netic ordering coexisting with su-\nperconductivity at room temperature in carbon nanotubes, p roduced by high energy ion beam\nmodification of diamond single crystals in /angbracketleft100/angbracketrightdirection is argued.\nPACS numbers: 71.10.-w, 73.63.Fg, 78.30.-j, 76.30.-v, 76. 50.+g, 78.67.-n\nKeywords:\nI. INTRODUCTION AND BACKGROUND\nDiscovery of new types of superconducting materials\nhas accelerated in 21th century. The commencement of\n21th century was commemorated by the discovery of su-\nperconductivity, which was observed at relatively high\ntemperature Tc= 40 K in the simple (structurally and\nelectronically) compound MgB2[1]. The origin of its\nis understood to be arising from charge carriers, which\nturn out to be placed into very strongly bonding states.\nThey in its turn respond very sensitively to the bond-\nstretching vibrational modes, see, for instance [2], [3],\n[4]. The boron-boron bonds in the graphite-like layers\nofMgB2are rather strong, and it is argument to the\nappearance of superconductive state. At the same time,\nthe graphite itself and diamond are materials that have\neven stronger bonds (in graphene plane in the case of\ngraphite).\nConsequently, it allows to consider the carbon and\ncarbon-based materials to be perspective materials for\nthe realization of superconducting states.\nReally, the second step in the field was the discovery\nof superconductivity at 4 K in very heavily boron-doped\ndiamond, reported in 2004 by Ekimov et al [5]. Confir-\nmation has been provided by Takano et al, who reported\nthe value of transition temperature to superconducting\nstateTcequaled to 7 K in B-doped diamond films [6].\nThe origin of of superconductivity in diamond was dis-\ncussed in a number of theoretical works, see for exam-\nple [7], [8]. In [7], an ab initio study of the supercon-\nductivity of boron doped diamond within the framework\nof a phonon-mediated pairing mechanism was presented.\nIt has been shown in [7], that the role of the dopant,\nin substitutional position, is unconventional in that half\nof the coupling parameter αoriginates in strongly lo-\ncalized defect-related vibrational modes, yielding a very\npeaked Eliashberg function (spectral decomposition of\nα). The electron-phonon coupling potential was found\nto be extremely large, however Tcremained to be low\nbecause of the low value of the density of states at the\nFermi level (hat is connected with 3D nature of the net-\nwork). The authors of [7] have invited to study the caseof doped diamond surfaces, where both the contraction\nof the reconstructed bonds and the 2D nature of the sur-\nface states may lead to much larger Tc. We will show,\nthat given idea, concerning of 2D nature of the carbon\nstates by preservation of bond strength (that allows to\ngenerate high frequency phonons) is actually true. The\nsame idea (however in implicit form) is presented in [8],\nwhere the superconductivity of boron-doped diamond is\nstudied in comparison with its analogy with MgB2. So,\nit was found, that the deformation potential of the hole\nstates arising from the C-C bond stretch mode in dia-\nmond is 60 percents larger than the corresponding quan-\ntity inMgB2that drives its high Tc. It leads to very\nlarge electron-phonon matrix elements. The evaluated\ncoupling strength coefficient αby using in [8] of rather\nsimplified approach leads nevetheless to Tcvalues in the\nonly 5-10 K range, in agreement with experiment (al-\nthough in [8] the rather simplified approach has been\nused). Hence, it makes phonon coupling to be the likely\nmechanism. Really, let us to represent the key points for\ngiven conclusion.\n(1) The carrier states are the very strongly covalent\nbonding states, that makes diamond so hard.\n(2) The carrier states should be sensitively coupled to\nthe bond-stretching mode, which lies at the very high\nfrequency of 1332 cm−1(0.16 eV) in diamond.\nBoth ingredients are the same ones prevailing in\nMgB2. There are differences, both of a positive and neg-\native nature. In MgB2, the only two of the nine phonon\nbranches are bond-stretching, whereas in diamond three\nof the six phonon branches are bond-stretching. On the\nother hand, MgB2is strongly two dimensional in its sig-\nnificantσ-bands, which means a near-step-function in-\ncrease in the density of participating states by doping,\nthe states in diamond are three-dimensional and their\nFermi level density of states N(0) increases with doping\nlevel essentially more slowly.\nAuthors of [8] conclude, that higher doping should in-\ncreaseTcsomewhat, but effects of three dimensionality\nprimarily on the density of states will keep doped dia-\nmond from having a Tccloser to that of MgB2.\nTherefore, authors of above cited works come indepen-2\ndently to the same conclusions concerning the nature of\nsuperconductivity in heavily boron-doped diamond. It\nhas to be remarked, that discovery of superconductivity\nin diamond followed the discovery of superconductivity\nin doped silicon clathrates [9] ( Tc= 8 K), a cage-like sil-\nicon material which crystallizes in the same sp3environ-\nment. Let us also remark, that even though the reported\ntemperatures are rather low by sp3environment, the su-\nperconducting transition of column IV semiconductors is\nof much interest, since it concerns very common materi-\nals, in which column IVa elements in Mendeleev Periodic\nTable are based elements.\nThe aforesaid idea to use 2D-modification of column\nIVa elements was successfully realized relatively recently\n(in 2008) in the work [10] and the essential progress in Tc\nenhancement up to 145 K was achieved. In [10] the tran-\nsiton tothe superconductingstatein the siliconsandwich\nS-Si-QW-S nanostructures prepared by short time diffu-\nsion of boron after preliminary oxidation of the n-type Si\n(100)-surface has been found. The sandwich S-Si-QW-S\nstructures represent themselves the p-type high mobil-\nity silicon quantum wells (QW) confined by the nanos-\ntructured δ-barriers heavily doped with boron on the n-\ntype Si (100)-surface. The studies of the cyclotron reso-\nnance angular dependences, the scanning tunneling mi-\ncroscopy images and the electron spin resonance (ESR)\nhave shown, that the nanostructured δ-barriers consist\nof a series of alternating undoped and doped quantum\ndots, with the doped dots containing the single trigonal\n(C3v-symmetry) dipole centers, B++B−, which are pro-\nduced by the negative-U reconstruction of the shallow\nboron acceptors, 2 B0→B++B−. The temperature and\nmagneticfielddependenciesoftheresistance,thermo-emf\n(Seebeck coefficient), specific heat and magnetic suscep-\ntibility were studied and gave clear evidence of the high\ntemperature superconductivity, Tc= 145 K. It, accord-\ning to [10], seems to be resulting from the transfer of the\nsmall hole bipolarons through the B++B−dipole cen-\nters at the Si-QW- δ-barrier interfaces. The value of the\nsuperconductor energy gap has been found to be equal\n0.044 eV. The extremely low value of the hole effective\nmassinthesandwichS-Si-QW-Sstructuresthathasbeen\nderived from the measurements of the Shubnikov - de\nHaas oscillations is considered by authors to be the prin-\ncipal argument for the bipolaronic mechanism of high\ntemperature superconductor properties that is based on\nthe coherent tunneling of bipolarons.\nThe next success of the first decade of 21th century\nin the field of superconductivity studies was the discov-\nery of superconductivity coexisting with antiferromag-\nnetic ordering in the iron-based layered pnictide com-\npound LaFeAsO (that is, also in material with prevailed\n2D-dimensional strucure). It was repoted approximately\nin the same time with the discovery of Bagraev et al\n(in 2008) in [11]. Next, the superconductivity has been\ndiscovered in both oxygen containing RFeAsO (R = La,\nNd, Sm) compounds and in oxygen free AFe2As2(A =\nBa, Sr, Ca) compounds. It is interesting, that the su-perconductivity occurs upon doping into the FeAs lay-\ners of either electrons or holes. Let us remark, that\nowing to the highly two-dimensional structure the pnic-\ntides are like to the cuprates. It gave rise to the view-\npoint that the physics of the pnictides is similar to the\ncuprates, and involves insulating behavior. However,\nthere is a growing consensus among researchers that\nMott-transition physics does not play a significant role\nfor the iron pnictides, and there are strong indications,\nthat magnetic order is of spin-density wave (SDW) type\nrather than Heisenberg antiferromagnetism of localized\nspins. In particular, it is evidenced by a relatively small\nvalue of the observed magnetic moment per Fe atom,\nwhich is around 12–16 percents of 2 µB. In another\ndistinction to the cuprates, electronic structure, which\nwas proposed by band-structure calculations and was\nsupported by angle-resolved photoemission spectroscopy,\nconsists of two small hole pockets centered around Γ\npoint,/vectork= (0,0) and of two small electron pockets cen-\ntered around M point /vectork=/vectorQ= (π,π) in the folded Bril-\nlouin zone (BZ) (two Fe atoms in the unit cell).\nMany theoretical studies are devoted at present to the\nstudy of superconductivity state (SSt) formation in pnic-\ntides. For instance, the authors of the paper [12] have\npresented Fermi-liquid analysis of SDW magnetism and\nsuperconductivity in given compounds. They considered\na two-band model with small hole and electron pockets\nlocated near Γ and M points in the folded BZ and ar-\ngued, that for the geometry indicated, particle-hole and\nparticle-particle channels are nearly identical, and the\ninteractions logarithmically increase at low energies. It\nhas been found, that the interactionsin the SDW and ex-\ntended s-wave ( s+- wave) channels /vectork= (0,0),/vectork=/vectork+/vectorQ\nbecome comparable in strength being to be the result\nof the increase in the intraband pair hopping term and\nthe reduction in the Hubbard-type intraband repulsive\ninteraction. The authors also argued, that at zero dop-\ning, SDW instability comes first, but at a finite doping,\ns+(s±in designation by other authors) superconducting\ninstability occurs at a higher temperature.\nThes+pairing bears similarity to magnetically medi-\nateddx2−y2pairing in systems with large Fermi surface\n(FS) with an idea that in both cases the pairing comes\nfrom repulsive interaction, peaked at /vectorQ, and requires the\ngap to change its sign under /vectork→/vectork+/vectorQ. The difference\nis that for small pockets, the gap changes sign away from\nthe Fermi surface and remains constant along the FS.\nSpin response of a clean and doped s+superconductor is\nanalysed in [12] and it has been found that\n(i) it possesses a resonance mode which disperses like\nto Anderson-Bogolyubov mode, that is, with the same\nvelocity,\n(ii) intraband scattering by nonmagnetic impurities\ndoes not affects the system, but interband scattering af-\nfects the system in the same way like to magnetic impu-\nrities in an s-wave superconducting state.\nLet us touch now on the nature of magnetic ordering3\nin carbon and carbon based materials too. It is well\nknown, that all substances on the whole are magnetics\nand that classical magnetic ordering is existing in the\nsubstances, which are built from the atoms with unfilled\ninner atomic d- orf-shells or include given atoms in their\nelementary units. In other words, magnetically ordered\nsolid substances are the groups ofsubstances, elementary\nunits of which include transition chemical elements with\nunfilled atomic 3d-, 4d-, 5d-, 6d-shells, or 4f, 5f-shells\nof rare earth elements. Carbon does not refer to given\ngroups. At the same time, there are at present a number\nof reports on magnetic ordering in carbon and carbon\nbased materials.\nOn the experimental revealing of magnetic ordering\nin carbon structurally ordered systems was reported for\nthe first time during the IBMM-Conference in Knoxville,\nTN, USA [13] and it was confirmed in report on E-MRS\nConference in Strasbourg, France [14]. Let us remark,\nthat the first report almost in the same time on magnetic\nordering in structurally non-ordered carbon materials is\nthe work [15], where ferromagnetic ordering in pyrolytic\ncarbon, produced by chemical vapour deposition method\nwas found. Let us also remark, that simultaneously, the\nreports [13], [14] were the first reports on the formation\nby high energy ion beam modification (HEIBM) of di-\namond single crystals structurally and magnetically or-\ndered quasi-one-dimensional(quasi-1D) system along ion\ntracks, that is, on the formation of new carbon allotropic\nform, which was identified with nanotubes (NTs), incor-\nporated in diamond matrix. It was shown, that axes of\nincorporated NTs are very precisely coinciding with ion\nbeam direction [16]. Given NTs were found to be pro-\nduced also in polycrystalline diamond films with implan-\ntation direction transversely to film surface [17]. They\npossessbyanumberofinterestingphysicalproperties,re-\nportedin[16], [17], [19], [18]. Whenconcernthemagnetic\nordering, it was established from the study of temper-\nature dependence of electron spin resonance absorption\nintensity,that, forinstance,incorporatednanotubes, pro-\nduced by neon HEIBM of diamond single crystal along\n/an}bracketle{t100/an}bracketri}htcrystallographic direction, possess by weak antifer-\nromagnetic ordering [16], [19], [18]. At the same time,\ncopper HEIBM with implantation direction along /an}bracketle{t111/an}bracketri}ht\ncrystal axis, nickel HEIBM with implantation direction\nalong/an}bracketle{t110/an}bracketri}htaxis [16], [19], [18] and boron HEIBM of\npolycrystalline diamond films with implantation direc-\ntion transversely to film surface [17] lead to formation\nof NTs, incorporated in diamond matrix, which possesss\nby ferromagnetic ordering. It was established directly by\nobservation of ferromagnetic spin wave resonance (FM-\nSWR) [17], [19], [18]. It was found, that magnetic or-\ndering is inherent property for given carbon electronic\nsystem and it is not connected with magnetic impurities,\nsince starting samples were selected in that way, that\nthe absolutespin number ofparamagneticimpurities and\nthe other paramagnetic structural imperfections in the\nsamples studied did not exceed the value ∼1012spins.\nVery recently [20], antiferroelectric ordering has beenfound in the same pure carbon allotropic form - quasi-\n1Dcarbonzigzag-shapednanotubes(CZSNTs), obtained\nby boron- and copper-HEIBM of diamond single crys-\ntals in/an}bracketle{t111/an}bracketri}ht-direction. It was established by means of\nthe detection of new optical phenomenon - antiferroelec-\ntric spin wave resonance (AFESWR), which was theoret-\nically described and experinmentally confirmed for the\nfirst time by infrared (IR) spectroscopy studies of car-\nbynes and polyvinilidenhalogenides in [21]. Let us in-\ndicate on some significant conclusions, which were done\non base of foregoing results. Given results mean, that\npure carbon in the form of quasi-1D CZSNTs and car-\nbyne chains are multiferroic systems. In its turn, the\nexperimental observation of multiferroicity in quasi-1D\nCZSNTs and carbynes means the breakdown of space\ninversion symmetry along CZSNT hypercomplex (that\nis, along n-dimensional symmetry axis z, see [22], [20],\nwherenis the number on the chain in CZSNT) andalong\ncarbyne chain symmetry axis. In the case of CZSNTs, it\nagreeswellwith themodelofquasi-1DCZSNTs[22], [20],\nbased on bond dimerization in all chain components of\nquasi-1D CZSNT along its hypercomplex symmetry axis\nz, which actually leadsto inversionsymmetry breakdown\nalong given axis. It is evident, that inversion symmetry\nbreakdown gives necessary condition for the appearance\nof nonzero polarisation by atomic displacements, that is\nfor antiferroelecricity (however it seems to be not suffi-\ncient condition in general case).\nQualitatively, the appearance of magnetic ordering in\ncarbon systems can be understandable, if to take into\naccount, that free carbon atoms have spin value S = 1\nand orbital moment value L = 1 with opposite direction,\nresulting in compensation of each other. It is clear, that\nin condensed carbon compounds given situation can be\nchanged by both the change of orbital moment direction\nand/or its value. It means, that on carbon base (and on\nthe base of the other IV-group elements - Si, Ge, free\natoms of which have also spin value S = 1 and orbital\nmoment value L = 1 with opposite direction) can be pro-\nduced the materials with magnetic properties to be com-\nparable with those ones, which possess the substances,\nelementary units of which include transition chemical el-\nements with unfilled atomic 3d-, 4d-, 5d-, 6d-shells, or\n4f-, 5f-shells of rare earth elements. The mechanisms\nto achieve given goal can be very different. One of the\nmechanisms was discussed in [23].\nThe aim of given work is to study in more details the\nproperties of non-cylindrical nanotubes, produced in di-\namond single crystals by high energy ion implantation,\nwhich are possessing instead of C∞symmetry axis the\nonly by C4symmetry axis and to establish the mech-\nanisms of formation of magnetic and electric ordering\nin given NTs. They seems to be the appropriate can-\ndidates for high temperature superconducting systems,\nsince both the mechanisms of superconductivity like to\nthose ones established in MgB2and in pnictides, briefly\nreviewed above, can be realized (see Section IV). More-\nover, it will be theoretically shown, that usual s-wave4\nmechanism, proposed by Bardeen, Cooper, Schrieffer\n(BCS) [24] can also be realized. In other words, mul-\nticannel superconductivity is predicted in given NTs.\nLet us remark, that non-cylindrical nanotubes, incor-\nporated in diamond single crystals, are representing the\nquite new class of carbon structures, since they cannot\nbe considered to be limit case of fullerene series (whereas\nit takes place for cylindrical nanotubes). It is the conse-\nquence of the alternating-sign curvature of the four-petal\nNT-surface in the direction, being to be transversal to\nNT-axis (the curvature of cylindrical nanotubes like to\nfullerenes is not alternating-sign).\nII. EXPERIMENTAL TECHNIQUE\nSamples of type IIa natural diamond, implanted by\nhigh energy ions of nickel (the energy of ions in ion beam\nwas335MeV)havebeenstudied. Paramagneticallypure\nsampleshavebeenselectedsothattheabsolutespinnum-\nber did not exceed the value ≈1012spins in each of the\nsamples used before implantation. Ion implantation was\nperformed along /an}bracketle{t100/an}bracketri}htcrystal direction (ion beam dose\nwas 5×1013cm−2) transversely to sample (100)-plane\nuniformly along all the plane surface. The temperature\nof the samples during the implantation was controlled\nand it did not exceed 400 K. ESR spectra were regis-\ntered on X-band ESR-spectrometer ”Radiopan” at room\ntemperature by using of TE102mode rectangular cavity.\nTherubystandardsamplewaspermanentlyplacedin the\ncavity on its sidewall. One of the lines of ESR absorp-\ntion byCr3+point paramagnetic centers (PC) in ruby\nwas used for the correct relative intensity measurements\nof ESR absorption, for the calibration of the amplitude\nvalue of magnetic component of the microwave field and\nfor precise phase tuning of modulation field. The cor-\nrect relative intensity measurements become to be possi-\nble owing to unsaturating behavior of ESR absorption in\nruby in the range of the microwave power applied, which\nwas≈100 mW in the absence of attenuation. Unsat-\nurable character of the absorption in a ruby standard\nwas confirmed by means of the measurements of the ab-\nsorption intensities in two identical ruby samples in de-\npendence on the microwavepower level. The first sample\nwas standard sample, permanently placed in the cavity,\nthe second samplewasplacedin the cavityawayfrom the\nloop of magnetic component of microwave field so, that\nits resonance line intensity was about 0.1 of the inten-\nsity of corresponding line of the first sample. Both the\nsamples were registered simultaneously but their absorp-\ntion lines were not overlapped owing to slightly different\nsample orientations. The foregoing intensity ratio was\nprecisely preserved for all microwave power values in the\nrange used, which indicates, that really ruby samples are\ngood standard samples in ESR spectroscopy studies.Figure 1: Spectral distribution of ESR absorption intensit y\nin diamond single crystal, implanted by high energy nickel\nions by beam direction transversely (100) sample plane, the\nsample was rotated in (0 11) plane, /vectorH0||[100] crystal axis,\nleftmost line belongs to ruby standard\n/s49/s54/s48/s48 /s50/s52/s48/s48 /s51/s50/s48/s48 /s52/s48/s48/s48/s53/s48/s54/s48/s55/s48/s56/s48/s65/s66/s83/s79/s82/s80/s84/s73/s79/s78/s32/s40/s114/s46/s117/s46/s41\n/s77/s65/s71/s78/s69/s84/s73/s67/s32/s70/s73/s69/s76/s68/s32/s40/s71/s97/s117/s115/s115/s41\nFigure 2: Spectral distribution of ESR absorption intensit y\nin diamond single crystal, implanted by high energy nickel\nions by beam direction transversely (100) sample plane, the\nsample was rotated in (0 11) plane, /vectorH0||[111] crystal axis\nIII. RESULTS\nThe ESR spectra observed in carbon nanotubes, pro-\nduced by nickel high energy /an}bracketle{t100/an}bracketri}htion beam modification\nof natural diamond single crystals, are presented in Fig-\nures1to 3in crystaldirections[100], [111]and 60degrees\nfrom [100] correspondingly. The line in the range (1865 -\n1981)G(givenfieldrangeisindicatedbyarrowsinFigure\n1) is the absorption line by ruby standard, it is shifted to\nbotton in Figure 1 and it is removed in the same range in5\n/s49/s53/s48/s48 /s50/s48/s48/s48 /s50/s53/s48/s48 /s51/s48/s48/s48 /s51/s53/s48/s48 /s52/s48/s48/s48/s52/s53/s54/s48/s55/s53/s65/s66/s83/s79/s82/s80/s84/s73/s79/s78/s32/s40/s114/s46/s117/s46/s41\n/s77/s65/s71/s78/s69/s84/s73/s67/s32/s70/s73/s69/s76/s68/s32/s40/s71/s97/s117/s115/s115/s41\nFigure 3: Spectral distribution of ESR absorption intensit y\nin diamond single crystal, implanted by high energy nickel\nions by beam direction transversely (100) sample plane, the\nsample was rotated in (0 11) plane, the angle between /vectorH0and\n[100] crystal axis is 60 degrees\n/s48/s44/s48/s48 /s48/s44/s50/s53 /s48/s44/s53/s48 /s48/s44/s55/s53 /s49/s44/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s65/s66/s83/s79/s82/s80/s84/s73/s79/s78/s32/s65/s77/s80/s76/s73/s84/s85/s68/s69/s32/s40/s114/s46/s117/s46/s41\n/s78/s79/s82/s77/s65/s76/s73/s90/s69/s68/s32/s77/s65/s71/s78/s69/s84/s73/s67/s32/s67/s79/s77/s80/s79/s78/s69/s78/s84/s32\n/s79/s70/s32/s77/s73/s67/s82/s79/s87 /s65/s86/s69/s32/s70/s73/s69/s76/s68/s32/s91/s72\n/s49/s47/s72\n/s49/s40/s48/s41\n/s93\nFigure 4: Dependence of absorption amplitude of the left lin e\nL in ESR spectrum of NTs incorporated in diamond single\ncrystal on magnetic componentof microwave field at /vectorH0||[100]\ncrystal axis\nFigure 2. The most intensive two lines, belonging to the\nsample studied, were excited spontaneously the only by\nvery precise orientation of external static magnetic field\n/vectorH0in the plane coinciding with the plane, transversal\nto implantation plane and containing the implantation\ndirection. Therefore, resulting spectrum was consisting\nof three lines, at that two new lines have rather large\nanisotropic linewidths. Let us designate given lines by\nRbfor the right broad line and by L for the left line./s48/s44/s48/s48 /s48/s44/s50/s53 /s48/s44/s53/s48 /s48/s44/s55/s53 /s49/s44/s48/s48/s48/s50/s53/s53/s48/s55/s53/s49/s48/s48/s65/s66/s83/s79/s82/s80/s84/s73/s79/s78/s32/s65/s77/s80/s76/s73/s84/s85/s68/s69/s32/s40/s114/s46/s117/s46/s41/s89/s32/s61/s32/s65/s32/s43/s32/s66/s49/s42/s88 /s32/s43/s32/s66/s50/s42/s88 /s94/s50/s32/s43/s32/s66/s51/s42/s88 /s94/s51/s32/s43/s32/s66/s52/s42/s88 /s94/s52\n/s80/s97/s114/s97/s109/s101/s116/s101/s114/s32/s32/s32/s32/s32/s86/s97/s108/s117/s101\n/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45\n/s65 /s32/s32/s32/s32/s32/s49/s44/s49/s52/s54/s49/s56\n/s66/s49 /s32/s32/s32/s49/s53/s53/s44/s50/s56/s56/s48/s54\n/s66/s50 /s32/s32/s32/s45/s54/s51/s44/s49/s53/s52/s52/s56\n/s66/s51 /s32/s32/s45/s50/s52/s48/s44/s49/s56/s56/s53/s52\n/s66/s52 /s32/s32/s32/s50/s50/s48/s44/s53/s53/s48/s56/s55\n/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45\n/s78/s79/s82/s77/s65/s76/s73/s90/s69/s68/s32/s77/s65/s71/s78/s69/s84/s73/s67/s32/s67/s79/s77/s80/s79/s78/s69/s78/s84/s32\n/s79/s70/s32/s77/s73/s67/s82/s79/s87 /s65/s86/s69/s32/s70/s73/s69/s76/s68/s32/s91/s72\n/s49/s47/s72\n/s49/s40/s48/s41\n/s93\nFigure 5: Dependence of absorption amplitude of the right\nbroad line Rbin ESR spectrum of NTs incorporated in di-\namond single crystal on magnetic component of microwave\nfield at /vectorH0||[100] crystal axis\nThe right broad line was overlapped with relatively nar-\nrow almost isotropic line, designated by Rn(given line\nwas observed by usual sample orientation). Addition-\nally, very broad strongly intensive anisotropic absorption\nwas observed. It consists of two backgrounds with two\ndip positions (in integrated spectrum) at ∼2410 G and\n∼2892 G by spectrum registration in the direction cor-\nresponding to [111] diamond lattice direction, Figure 2.\nDip positions for given background absorption were co-\ninciding by static magnetic field direction in 60 degrees\nfrom [100] diamond crystal direction, Figure 3. It seems\nto be the display of the fact, that the symmetry of the in-\nteraction, leading to the appearance of very strong back-\ngroundabsorptionisdeterminedbyinherentsymmetryof\nNTs, produced by [100] HEIBM, which is not connected\nwith potential effect of diamond lattice presence.\nDependencies of absorption amplitudes of L-line and\nRbline on magnetic component of microwave field at\nfixed orientation of polarising magnetic field /vectorH0||[100]\ncrystal axis have been studied, Figures 4 and 5. It is\nseen from comparison of the Figures 4 and 5, that given\ndependencies are quite different. The dependence, pre-\nsented for L-line in Figure 4, is superlinear. It is similar\nto the dependencies, which were earlier observed in the\nsamples, modified by HEIBM with copper, neon, nickel\nions (however with dose 5 ×1014) [16], [19], [18], that\nis, in the case of entire modification of diamond layer,\nwhich is localised near implantation surface. It means,\nthat the layerwas consisting then the only of NTs, which\nseem to be interacting each other. In the studied sample\n(integral dose is 5 ×1013), individual NTs are isolated by\ndiamond structure, nevertheless the superlinear depen-\ndence is taking place, which seems to be unexpected. Let6\n6\nFigure 6: Angular dependence of g-factor of the left line L in\nESR spectrum of NTs incorporated in diamond single crystal,\nthe sample was rotated in (0 11) plane\n/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s65/s66/s83/s79/s82/s80/s84/s73/s79/s78/s32/s73/s78/s84/s69/s78/s83/s73/s84/s89/s32/s40/s114/s46/s117/s41\n/s91/s49/s48/s48/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s49/s93/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s48/s49/s49/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s65/s78 /s71/s76/s69/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFigure 7: Angular dependence of ESR absorption intensity\nof the left line L in ESR spectrum of NTs incorporated in\ndiamond single crystal, the sample was rotated in (0 11) plane\nus remark, that the initial part of the curve, presented in\nFigure4, caninthe principlebe approximatedbyalinear\ndependence (dashed line), although it is clear from com-\nparison with the approximation of the whole curve, solid\nline in Figure 4, that even initial part, strongly speaking,\nis not linear. Solid line in Figure 4 is the polinomial fit\nwith the function f(x) =b0+b1x+b2x2+b3x3+b4x4,\nwhereb0=−0.17117,b1= 208.92305,b2=−139.06624,\nb3= 159.14424,b4=−33.90983.\nDependence ofabsorptionamplitude ofthe right broad\nlineRbin ESR spectrum of NTs on magnetic component\nof microwave field is strongly nonlinear. It is charac-\nterised for the values of relative magnetic component of/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s53/s48/s55/s53/s49/s48/s48/s49/s50/s53/s49/s53/s48/s76/s73/s78/s69/s87/s73/s68/s84/s72/s32/s40/s71/s97/s117/s115/s115/s41\n/s91/s49/s48/s48/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s91/s49/s49/s49/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s48/s49/s49/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s65/s78/s71/s76/s69/s32/s40 /s68/s101/s103/s114/s101/s101/s41\n1\nFigure 8: Angular dependence of linewidth of the left line L i n\nESR spectrum of NTs incorporated in diamond single crystal,\nthe sample was rotated in (0 11) plane\n/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s48/s44/s50/s53/s48/s44/s53/s48/s48/s44/s55/s53/s49/s44/s48/s48/s49/s44/s50/s53/s49/s44/s53/s48/s65/s83/s89/s77/s77/s69/s84/s82/s89/s32/s69/s88/s84/s69/s78/s84/s32/s40/s65/s47/s66/s41\n/s91/s49/s48/s48/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49 /s49/s49/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s48/s49/s49/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s65/s78/s71/s76/s69/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFigure 9: Angular dependence of asymmetry extent A/B of\nthe left line L in ESR spectrum of NTs incorporated in dia-\nmond single crystal, the sample was rotated in (0 11) plane\nmicrowave field H1/H(0)\n1in the range (0-0.75) by usual\nsaturating law, but in the range (0.75-1) it acquires\nprominent superlinear nonsaturating character. The de-\npendence for ESR absorption kinetics in the form, pre-\nsented in Figure 5, is observed in ESR-spectroscopy for\nthe first time. It can be approximated by the solid line,\nwhich represents itself the polynomial fit in accordance\nwith relation Y(x) =A+B1x+B2x2+B3x3+B4x4,\nwhereA= 1.14618,B1= 0.77956,B2=−0.00159,\nB3=−3.03868e−5,B4= 1.40072e−7. Angular depen-\ndence of g-factor of the left line L in ESR spectrum of7\n/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s49/s54/s52/s49/s54/s56/s49/s55/s50/s49/s55/s54/s49/s56/s48/s49/s56/s52\n/s89/s32/s61/s32/s65/s32/s43/s32/s66/s49/s42/s88 /s32/s43/s32/s66/s50/s42/s88 /s94/s50/s32/s43/s32/s66/s51/s42/s88 /s94/s51/s32/s43/s32/s66/s52/s42/s88 /s94/s52\n/s80/s97/s114/s97/s109/s101/s116/s101/s114/s32/s32/s32/s86/s97/s108/s117/s101\n/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45\n/s65/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s49/s54/s51/s44/s50/s50/s49/s56/s49\n/s66/s49 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s44/s53/s48/s50/s50/s54\n/s66/s50 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s49/s44/s49/s56/s50/s54/s54/s69/s45/s52\n/s66/s51 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s45/s49/s44/s52/s52/s49/s52/s51/s69/s45/s52\n/s66/s52 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s49/s44/s50/s48/s50/s51/s57/s69/s45/s54\n/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s45/s81/s45/s70/s65/s67/s84/s79/s82/s32\n/s91/s48/s49/s49/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s49/s49/s93/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s91/s49/s48/s48/s93\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s65/s78/s71/s76/s69/s32/s40/s68/s101/s103/s114 /s101/s101/s41\nFigure 10: Angular dependence of the cavity Q-factor with\nthe sample implanted by high energy nickel ions by beam\ndirection transversely (100) sample plane, the sample was r o-\ntated in (0 11) plane\nNTs in the sample studied is presented in Figure 6. It\nconsist of two branches. One branch is in the angle range\n0-50 degrees from [100] crystal lattice direction (which is\ncoinciding with NT axis direction), the second branch is\nin the angle range 50-90 degrees. Let us remark, that\nthe connection point of two branches, equaled to 50 de-\ngrees for the g-values of L-line is not coinciding with the\npoint of the junction of two dips in the very broad (and\nconsequentlyveryintensive) absorption, testifying on the\nexistence of two different resonance processes, which are\nresponsible for the appearance of L-line and very broad\nlines. The deviation of g-values from free electron value\ng = 2.0023 is very large, at that the minimal value is\nachieved in the range 16-20 degrees from the [011] direc-\ntion in diamond lattice and it is equal to ≈2.0719, max-\nimal g-value corresponds to NT axis direction, that is to\n[100] crystal lattice direction and it is equal to ≈2.3120.\nGiven values are characteristic for the systems with the\nstrong magnetic ordering. Consequently, we have ob-\ntained the direct proof of the spontaneous transition of\nNT system, incorporated in diamond lattice, in the state\nwith the strong magnetic ordering. Angular dependence\nof ESR absorption intensity of the left line L has quali-\ntatively opposite character to g-factor dependence. The\nmaximal absorption value corresponds to the direction,\nbeing to be transversal to NT axis, which is coincides\nwith [011] direction in diamond lattice, Figure 7. Ad-\nditional maximum is observed at 60 degrees from given\ndirection. Let us remark, that both the maxima in an-\ngular dependence of ESR absorption intensity of the line\nL are observed also in angular dependence of linewidth,\nFigure 8, indicating, that the main features in angular\ndependence of ESR absorption intensity are governed by\nangular dependence of linewidth. It is confirmed also bythat, the very pronounced maximum in the angular de-\npendence ofabsorptionamplitude ofline Lis in the range\n10-20degreesfrom [100]crystal lattice direction (the cor-\nresponding Figure is not presented). At the same time it\nis not pronounced in angular dependence of absorption\nintensity, see Figure 7. It is seen from angular depen-\ndence of linewidth of the line L, Figure 8, that its value\nis strongly dependent on direction of the static magnetic\nfield applied. There are four maxima, main maximum is\nachieved at [011] direction with linewidth value, which is\nequal to 140.9 G, very pronounced maximum is observed\nat the angle in 40 degrees from [100] lattice direction,\nlinewidth value is equal to 139.1 G, that is, it is com-\nparable with main maximum value. Two, not very pro-\nnounced maxima are observed correspondingly at at the\nangle between 10 and 20 degrees and at the angle in 60\ndegrees from [100] lattice direction, linewidth values are\nequal correspondingly to 68.7 G and 100 G. Main min-\nimum is achieved near NT axes’ direction, more stricly\nat≈5 degrees from given direction with linewidth value,\nwhich is equal to 49.1 G. The foregoing linewidth values\nare characteristic for the states with strong magnetic or-\ndering, that is, it is additional argument in favour of the\nconclusion on the formation of strong magnetic ordering\nin the sample studied. It is also seen from Figure 8, that\ntwobranchesoflinewidth growth, starting at ≈5 degrees\nand in the range 70-75 degrees from [100] direction, have\nthe resemblance. That seems to be the indication on the\nsame originoflinewidth broadeningprocess, determining\nforegoing growth branches.\nEspecially interesting, that the line L is asymmetric,\nFigure 9. However, the angular dependence of the ratio\nA/Bof the asymmetry extent is disagreeding with the\nangular dependence, which has to be observed by usual\nDyson effect in metals or semiconductors [25]. The value\nB/Aisequal2.55forstatic(immobile)paramagneticcen-\nters(PC)inconductivemediainthecaseofthicksamples\nand it is determined by the space dispersion contribution\n[26], which is appeared in conductive media. It corre-\nsponds to the ratio of space dispersion contribution and\nabsorption contribution to resulting ESR response equal\nto (1 : 1) [26]. The value B/Afor absorption derivative\nis increasing from 2.55 to more than 19 for mobile PC\n(or by presence of spin diffusion) in dependence on the\nvelocity of mobile PC (or on the rate of the spin diffu-\nsion) [27]. In the case of thin samples, the ratio B/Ahas\nintermediate values, between 1 and above indicated, de-\npending on the thickness of the samples. It is seen from\nFigure 9, that the ratio A/B is anisotropic. The maximal\nA/B value (correspondingly, minimal B/A) is near [111]\ncrystal lattice direction and it is equal to 0.83, Figure 9.\nThe minimal A/B (maximal B/A) value is near 60 de-\ngrees from [011] crystal lattice direction and it is equal to\n0.49, Figure 9. Let us remark, that by usual Dyson effect\nin conducting thin samples (in particular in the samples\nwith metallic NTs, producing the network) the maximal\ndeviation from the ratio A/B = 1 has to be observed by\nmicrowave field propagation direction along the sample8\nside with maximal size of implanted region in rotation\nplane, that is by H0along [100] crystal direction, in the\ncase, when the network is opaque for microwave field in\ndirection, transverseto NT axisdirection, orby H0along\n[011] in the case, when the network is opaque the only\nfor microwave field propagation in direction, coinciding\nwith NT axes. The observed maximal deviation of ratio\nA/B from A/B = 1 at ≈60 degrees from [011] confirms\nthe conclusion on nontrivial nature of Dyson-like effect\nin the sample studied.\nThe presence of the ruby standard allowed to control\nthecavity Q-factor,Figure10. It seemstobesubstantial,\nthatQ-factor is increasing in the ranges, where deviation\nof ratio A/B from A/B = 1 is also increasing, that is,\nincrease is starting near 60 degrees from [011] crystal\nlattice direction and increase takes place in the range\nnear 10-30 degrees from the same [011] crystal direction,\ncompare Figures 10 and 9. For usual Dyson effect it\nhas to be conversely, Q-factor has to be minimal in the\ndirection of maximal deviation of ratio A/B from A/B\n= 1, that is near 30 degrees from [100] crystal lattice\ndirection, Figure 9. It is seen from Figure 10, that Q-\nfactor has in given direction the maximal value, which\nindicatessimultaneously, that theapproximationbysolid\nlinehastobemoreaccurate(moreofexperimentalpoints\nis required).\nIV. DISCUSSION\nIt will be further argued, that the results above de-\nscribed are agreed with spontaneous transition of the\nsystem to the state which characterises by coexistence\nsimultaneously of antiferromagnetic (AFM) uncompen-\nsated ordering and superconductivity, which is realized\nin electron spin resonance conditions and it is absent\nwithout resonance. Given specific conditions seem to be\nindicating, that the nature of given state and mecha-\nnisms, leading to its formation cannot be entirely coin-\nciding with known ones, including above reviewed. To\nsolve given task, it seems to be necessary to know the\nnature of charge and spin carriers and the mechanisms\nof carrier transport and interactions of charge and spin\ncarriers both between themselves and with phonons and\nphotons in NTs studied. There seems to be paramount\nsignificant the same task for nanoelectronics, spintronics\nand for the other branches of nanotechnology. Let us\nremark, that in the theory of 1D electronic systems, in\nparticular,inthetheoryofconductingNTsisexistingthe\nfollowing concept. It is based on the work of Tomonaga\n[28]andontheworkofLuttinger[29], fromwhichfollows,\nthat the electron-electron interaction destroys the sharp\nFermi surface and leads to a breakdown of the Landau\nFermi liquid (LFL) theory for 1D systems. The result-\ning non-LFL state is commonly called Luttinger liquid\n(LL), or sometimes Tomonaga-Luttinger liquid (TLL).\nGiven approach was used upto now for description of\nthe universal low-energy properties of all 1D conductors.The theory of LL (TLL) predicts the pronounced power-\nlaw suppression of the transport current and the density\nof states, and the effect of spin-charge separation. The\nnature of the spin and charge carriers accorging to LL\n(TLL) theory is the following. They are chargeless spin\n1/2 quasiparticles - spinons and spinless quasiparticles\nwith the charge ±e - holons. The universality of LL\n(TLL) description means that the physical properties of\n1D systems have to be not depending on details of the\nmodel, of the interaction potential, and so on. They are\nonly characterized by a few parameters - critical expo-\nnents. Quite remarkably, that the LL (TLL) concept is\nbelieved to be valid for arbitrary statistical properties of\nthe particles, that is both for fermions and bosons. It\nis interesting that along with a paradigm for non-Fermi\nliquid physics for description of any 1D systems the con-\ncept of LL (TLL) was extended for description of 2D and\n3D correlated electrons in systems with linear dispersion\nlaw.\nConcerning NTs, let us remark, that the single-wall\ncarbon nanotubes (SWCNTs) are considered in many\nworks to be 1D objects (it is not always correct, espe-\ncially for standard NTs with diameter in severalnanome-\nters and more), which can be described the only in the\nframe of LL (TLL) concept. Moreover, SWCNTs are\nconsidered to be the best model systems for the LL state\ndemonstration. The arguments used to confirm given\nconclusion are the following. It is the experimental ob-\nservation of power-law behavior by measuring the tun-\nneling conductance of SWNTs in dependence on temper-\nature and voltage. It has to be remarked however, that\npower laws are widely spread in the physics. They can\napproximate some other dependences or can follow from\nthe other theories too. Electron force microscopic mea-\nsurements showedalsothe ballistic nature oftransport in\nconducting SWNTs, predicted by the LL (TLL) model.\nBallistic behavior of transpot phenomena can also be de-\ntermined by the other causes, not connected with LL\n(TLL) model. Insufficiency of the substantiation of the\napplicability of LL (TLL) model to SWCNTs becomes\nto be evident, if to take into account, that main feature,\nof given model - spin-charge separation by spinon-holon\nmechanism has not been observed.\nIthastobealsotakenintoaccount,thatboththemod-\nels LL (TLL) and LFL are the models of ideal quantum\nliquids. Moreover, both the models are oversimplified,\nsince they do not take into account the nonlinearity of\nthe fermion spectrum on the one hand and the presence\nof electron-phonon interactions on the other hand. In\nfact both the models describe not strongly adequately\nthe real processes, since the changes in the charge state\nof arbitrary atom in 1D chain, being to be the result of\nelectron-electron interaction, are always accompanied by\nthe changes in phonon subsystem (and vice versa). It\nis consequence of generic coupling between operators of\ncreation and annihilation in electron subsystem and in\nphonon field (see futher for more details). Let us also\nremember, that key argument for insertion of the no-9\ntion ”Luttinger liquid” itself is in fact also based on the\nsimplification, determined by linearization of the generic\nspectrum of particles in neighborhood of Fermi points in\nk-space. At the same time the divergencies arising in the\nperturbation theory in 1D-case are the consequence just\nof given simplification. From here it is not follows, that\n1D Fermi liquid description is incorrect in general case,\nwhich takes into account the electron-phonon interaction\nand/or nonlinearity of the generic spectrum of particles\nin neighborhood of Fermi points in k-space.\nSo we come to conclusion, that the description of NTs\nthe only in the frames of LL concept seems to be also\noversimplification. Moreover,it is showedin [30] that the\nconceptofdescriptionof1Dcorrelatedelectronicsystems\nwihtin the framework of 1D Fermi liquid (FL) can be\nrestored.\nIt was considered in [30] the concept of 1D FL on the\nexample of well known 1D system - trans-polyacetylene\n(t-PL). It is in fact the generalization of well known\nmodel of organic 1D conductors proposed by Su, Schri-\neffer, Heeger (SSH-model) [31], [32], which is formally\nFermi gas model. It will be futher shown that SSH-\nmodel takes the intermediate position between Fermi gas\nand Fermi liquid models, since it takes into considera-\ntion the electron-electron correlations in implicit form.\nThe further generalization, for instance, for application\nof 1D FL model immediately to quasi-1D carbon zigzag\nshaped nanotubes can be obtained by usingof hypercom-\nplex number theory like to the works [22], [20], where hy-\npercomplex number theory was applied for the interpre-\ntation of quantum optics effects in carbon zigzag shaped\nNTs.\nLet us represent for the convenience of the readers the\nmain moments of calculation and the results of the work\n[30].\nThe Born-Oppenheimerapproximationwasconsidered\nand starting Hamiltonian was the following\nˆH(u) =ˆH0(u)+ˆHπ,t(u)+ˆHπ,u(u).(1)\nThe first term in (1) is\nˆH0(u) =/summationdisplay\nm/summationdisplay\ns(ˆP2\nm\n2M∗ˆa+\nm,sˆam,s+Ku2\nmˆa+\nm,sˆam,s).(2)\nIt represents itself the sum of operator of kinetic energy\nof CH-group motion (the first term in (2)) and the op-\nerator of the σ-bonding energy (the second term). Co-\nefficient Kin (2) is effective σ-bonds spring constant,\nM∗is total mass of CH-group, umis configuration coor-\ndinate for m-th CH-group, which corresponds to trans-\nlation of m-th CH-group along the symmetry axis zof\nthe chain, m=1,N,Nis number of CH-groups in the\nchain,ˆPmis operator of impulse, conjugated to configu-\nration coordinate um,m=1,N, ˆa+\nm,s, ˆam,sare creation\nand annihilation operators of creation or annihilation of\nquasiparticle with spin projection son them-th chain\nsite inσ-subsystem of t-PA. The second term in (1) can\nbe represented in the form of two components and it isˆHπ,t(u) =ˆHπ,t0(u)+ˆHπ,t,α1(u) =\n/summationdisplay\nm/summationdisplay\ns[(t0(ˆc+\nm+1,sˆcm,s+ˆc+\nm,sˆcm+1,s)]+\n(−1)m2α1u)(ˆc+\nm+1,sˆcm,s+ˆc+\nm,sˆcm+1,s)],(3)\nwhere ˆc+\nm,s, ˆcm,sare creation and annihilation operators\nof creation or annihilation of quasiparticle with spin pro-\njectionson them-th chain site in π-subsystem of t-PA.\nIt is the resonance interaction (hopping interaction in\ntight-binding model approximation) of quasiparticles in\nπ-subsystem of t-PA electronic system, which is consid-\nered to be Fermi liquid, and in which the only constant\nand linear terms in Taylor series expansion of resonance\nintegral about the dimerized state are taking into ac-\ncount.\nOperator ˆH(u) is invariant under spatial translations\nwith period 2 a, whereais projection of spacing between\ntwo adjacent CH-groups in undimerized lattice on chain\naxis direction, and which is equal to 1.22 ˚A. It means,\nthat all various wave vectors /vectorkin/vectork-space will be in re-\nduced zone with module of /vectorkin the range −π\n2a≤k≤π\n2a\n[32]. Reduced zone is considered like to usual semicon-\nductors to be consisting of two subzones - conduction ( c)\nband and valence ( v) band. Then the operators {ˆc+\nm,s},\n{ˆcm,s},m=1,N, were represented in [30] in the form\n{ˆcm,s}={ˆc(c)\nm,s}+{ˆc(v)\nm,s},\n{ˆc+\nm,s}={ˆc+(c)\nm,s}+{ˆc+(v)\nm,s},(4)\nrelated to π−c- andπ−v-band correspondingly, and\n/vectork-space operators were defined\n{ˆc(c)\nk,s}={i√\nN/summationdisplay\nm/summationdisplay\ns(−1)m+1exp(−ikma)ˆc(c)\nm,s},\n{ˆc(v)\nk,s}={1√\nN/summationdisplay\nm/summationdisplay\nsexp(−ikma)ˆc(v)\nm,s},(5)\nm=1,N.\nTheσ-operators {ˆa+\nm,s}and{ˆam,s},m=1,Nwere\nalso represented in the form like to (4) for π-operators\nand analogous to (5), transforms was defined. It leads to\nthe following expression for the operator ˆH0(u)\nˆH0(u) =/summationdisplay\nk/summationdisplay\ns(ˆP2\n2M∗+Ku2)(ˆnσ,c\nk,s+ ˆnσ,v\nk,s),(6)\nwhere ˆnσ,c\nk,sand ˆnσ,v\nk,sare operators of number of σ-\nquasiparticlesin σ-c-band and σ-v-band correspondingly.\nThe independence of |um|onm,m=1,N, was taken\ninto consideration.\nThe expression for ˆHπ,t0(u) in terms of {ˆc(c)\nk,s}and\n{ˆc(v)\nk,s}is coinciding with known corresponding expression\nin [31], [32] and it is\nˆHπ,t0(u) =/summationdisplay\nk/summationdisplay\ns2t0coska(ˆc+(c)\nk,sˆc(c)\nk,s−ˆc+(v)\nk,sˆc(v)\nk,s)(7)10\nThe expression for the second part of operator ˆHπ,t(u)\nin terms of {ˆc(c)\nk,s}and{ˆc(v)\nk,s}is also coinciding in its form\nwith known corresponding expression in [31], [32] and it\nis given by\nˆHπ,t,α1(u) =/summationdisplay\nk/summationdisplay\ns4α1usinka(ˆc+(v)\nk,sˆc(c)\nk,s+ˆc+(c)\nk,sˆc(v)\nk,s),\n(8)\nwhere subscript α1in Hamiltonian designation indicates\non the taking into account the part of electron-phonon\ninteraction, connected with resonance interaction (hop-\nping)processes. Theexpressionfortheoperator ˆHπ,u(u),\nwhich describes the part of electron-phonon interac-\ntion, determined by interaction between quasiparticles\nin Fermi liquid state of π-subsystem in terms of {ˆc(c)\nk,s}\nand{ˆc(v)\nk,s}is the following\nˆHπ,u,α 2(u) =/summationdisplay\nk/summationdisplay\nk′/summationdisplay\nsα2(k,k′,s)ˆc+(c)\nk′,sˆc+(v)\nk′,sˆc(v)\nk,sˆc(c)\nk,s.\n(9)\nThe contribution of the term, corresponding the only to\ninteraction between the quasiparticles in different bands,\nwhich seems to be the most essential, was considered.\nThe expression for α2(k,k′,s) was obtained in the form\nα2(k,k′,s) =b|v0v|2|v0c|2V0(c)uV0(v)|φ0cs|2|φ0vs|2×\nN\n2π(ql−qj)(qr−qn)Re{exp[i(klml−kjmj)a]expika}×\nRe{exp[i(k′\nrmr−k′\nnmn)a]expik′a},\n(10)\nwhere|φ0cs|2,|φ0vs|2are squares of the modules of the\nwave functions |kj,s/an}bracketri}htand|k′\nj,s/an}bracketri}htrespectively, k=kph(ql−\nqj),k′=k′\nph(qr−qn)ql,qj,qr,qn∈Nwith additional\nconditions ( ql−qj)a≤L, (qr−qn)a≤L,b- is aspect ra-\ntio, which in principle can be determined by comparison\nwithexperiment. Herethevalues( ql−qj), (qr−qn)deter-\nmine the steps in pairwise interaction with phonon par-\nticipation and they are considered to be fixed. The pro-\ncesses, for which k=k′, are considered. Consequently,\n(qr−qn) = (ql−qj) and the operator ˆHπ,u,α 2(u) is\nˆHπ,u,α 2(u) =\n/summationdisplay\nk/summationdisplay\nk′/summationdisplay\ns4α2(s)usinkasink′aˆc+(c)\nk′,sˆc+(v)\nk′,sˆc(v)\nk,sˆc(c)\nk,s,(11)\nwhereα2(s) is\nα2(s) =b\n4|v0v|2|v0c|2V0(c)V0(v)|φ0cs|2|φ0vs|2×\nN\n2π[(ql−qj)]2(12)\nThe system of operators ˆ c+(c)\nk′,s, ˆc+(v)\nk′,s, ˆc(v)\nk,s, ˆc(c)\nk,scorre-\nsponds to noninteracting quasiparticles, and it is under-\nstandable, that in the case of interacting quasiparticlestheir linear combination has to be used\n/bracketleftBigg\nˆa(v)\nk,s\nˆa(c)\nk,s/bracketrightBigg\n=/bracketleftbigg\nαk,s−βk,s\nβk,sαk,s/bracketrightbigg/bracketleftBigg\nˆc(v)\nk,s\nˆc(c)\nk,s/bracketrightBigg\n,(13)\nThen it has been shown, that the diagonalpart of Hamil-\ntonianˆHπ,t,α1(u), whichcorrespondstoSSHone-electron\ntreatment of electron-phonon coupling, can be repre-\nsented in the form\nˆHd\nπ,t,α1(u) =/summationdisplay\nk/summationdisplay\ns2∆kαk,sβk,s(ˆn(c)\nk,s−ˆn(v)\nk,s),(14)\nwhere ∆ k= 4α1usinka, ˆn(c)\nk,sis density of operator of\nquasiparticles’ number in c-band, ˆn(v)\nk,sis density of oper-\nator of quasiparticles’ number in v-band.\nThe diagonal part ˆHd\nπ,u,α 2(u) of operator ˆHπ,u,α 2(u)\nof pairwise interaction, which is linear in displacement\ncoordinate uand leads to participation of the phonons,\nis given by the expression\nˆHd\nπ,u,α 2(u) = 4α2u/summationdisplay\nk/summationdisplay\nk′/summationdisplay\nsαk′βk′(ˆn(v)\nk′,s−ˆn(c)\nk′,s)\n×αk,sβk,s(ˆn(v)\nk,s−ˆn(c)\nk,s)sink′asinka(15)\nLet us remark, that the Hamiltonian ˆHπ,u,α 2(u) de-\nscribes the attraction between the electrons, it can lead\nto formation of Cooper pairs in a π-subsystem and to su-\nperconductivity of both of usual s-wave type, described\nin [24], that is, with Cooper pairs in singlet S = 0 state\nand with Cooper pairs in triplet S = 1 state. It is like\nto well-known possibility of the formation of singlet and\ntriplet excitons and it seems to be substantial conclusion\nbeing to be the key moment for coexistence of super-\nconductivity and magnetic ordering. In fact, the new\nmechanism for superconductivity was proposed.\nThe diagonal part ˆHd\nπ,t0(u) of operator ˆHπ,t0(u) in\nterms of operator variables ˆ a(c)\nk,sˆa(v)\nk,sis given by the rela-\ntion\nˆHd\nπ,t0(u) =/summationdisplay\nk/summationdisplay\nsǫk(α2\nk,s−β2\nk,s)(ˆn(c)\nk,s−ˆn(v)\nk,s),(16)\nwhereǫk= 2t0coska.\nThe operator transformation for the σ-subsystem,\nanalogous to (13) shows, that the Hamiltonian ˆH0(u) is\ninvariant under given transformation, that is, it can be\nrepresented in the form, given by (6).\nTofindthevaluesofelementsofmatrixinrelation(13),\nthe Hamiltonian ˆH(u) has been tested for conditional\nextremum in dependence on the variables αk,βkwith\ncondition α2\nk,s+β2\nk,s= 1.\nTwo values for the energy of quasiparticles, indicating\non the possibility of formation of the quasiparticles of\ntwokindsboth in c-band and v-band havebeen obtained.11\nThey are the following\nE(c)\nk(u) =Q2∆2\nk−ǫ2\nk/radicalbig\nǫ2\nk+Q2∆2\nk,\nE(v)\nk(u) =ǫ2\nk−Q2∆2\nk/radicalbig\nǫ2\nk+Q2∆2\nk(17)\nand\nE(c)\nk(u) =/radicalBig\nǫ2\nk+Q2∆2\nk,\nE(v)\nk(u) =−/radicalBig\nǫ2\nk+Q2∆2\nk(18)\nThe factor Qis determined by relation\n[1+α2\n2α1/summationdisplay\nk/summationdisplay\nsQ∆ksinka/radicalbig\nǫ2\nk+Q2∆2\nk(n(c)\nk,s−n(v)\nk,s)] =Q,(19)\nwheren(c)\nk,sis eigenvalue of density operator of quasipar-\nticles’ number in c-band,n(v)\nk,sis eigenvalue of density\noperator of quasiparticles’ number in v-band. The quasi-\nparticles with the energy, determined by (18) at Q= 1\nare the same quasiparticles, that were obtained in known\nSSH-model.\nSufficient conditions for the minimum of the functions\nE(αk,sβk,s) were obtained by standard way, which was\nused also in [20]. It consist in that, that the second dif-\nferential of the energy being to be the function of three\nvariables αk,s,βk,sandλk,shas to be positively defined\nquadratic form. From the condition of positiveness of\nthree principal minors of quadratic form coefficients the\nthree sufficient conditions for the energy minimum have\nbeen obtained. Their analysis has showed, that SSH-like\nsolution is inapplicable for the description of standard\nprocesses, passing near equilibrium state by any param-\neters [30]. The quasiparticles, described by SSH-like so-\nlution, can be created the only in strongly nonequilib-\nrium state with inverse population of the levels in c- and\nv-bands. At the same time the solution, the energy of\nquasiparticles for which is determined by (17) can be\nrealised both in near equilibrium and in strongly non-\nequilibrium states of the π-subsystem of t-PA, which is\nconsidered to be quantum Fermi liquid [30].\nThe continuum limit for the ground state energy of\nthet-PA chain with SSH-like quasiparticles will coincide\nwith known solution [32], if to replace ∆ kQ →∆k. The\ncalculation of the ground state energy E[u]\n0(u) of the t-\nPAchainwithquasiparticles’branch,whichisstablenear\nequilibrium by taking into account, that in ground state\nnc\nk,s= 0,nv\nk,s= 1, in the continuum limit gives\nE[u]\n0(u) =−2Na\nππ\n2a/integraldisplay\n0(Q∆k)2−ǫ2\nk/radicalbig\nQ∆k)2+ǫ2\nkdk+2NKu2.(20)Then, calculation of the integral results in the expression\nE[u]\n0(u) =4Nt0\nπ{F(π\n2,1−z2)+\n1+z2\n1−z2[E(π\n2,1−z2)−F(π\n2,1−z2)]}+2NKu2,(21)\nwherez2=2Qα1u\nt0,F(π\n2,1−z2) is the complete elliptic\nintegral of the first kind and E(π\n2,1−z2) is the complete\nelliptic integral of the second kind. Approximation of\nground state energy at z≪1 for the stable near equilib-\nrium solution gives\nE[u]\n0(u) =N{4t0\nπ−6\nπln2t0\nQα1u4(Qα1)2u2\nt0+\n28(Qα1)2u2\nπt0+...}+2NKu2.(22)\nIt is seen from (22), that the energy of quasiparticles,\ndescribed by given solution, has the form of Coleman-\nWeinberg potential with two minima at the values of\ndimerization coordinate u0and−u0like to the energy\nof quasiparticles, described by SSH-solution. It is un-\nderstandable, that further considerations, including elec-\ntrically neutral S = 1/2 soliton and electrically charged\nspinless soliton formation, that is the appearance of the\nphenomenonofspin-chargeseparation,byFLdescription\nof1Dsystems will be coinciding in its mathematical form\nwith those ones in SSH-model.\nThus, in [30] was established the possibility to describe\nthe physical properties of 1D systems in the frames of\n1D quantum FL including the mechanism of appearance\nof the most prominent feature of 1D systems - the phe-\nnomenonofspin-chargeseparation. Itwasalsoshownthe\npossibility of simultaneous formation of superconducting\nstate and the state with magnetic ordering in 1D FL.\nLet us remark,that the model proposedtakesinto con-\nsideration the electron-electron correlations in explicit\nform,whichseemstobegroundforitsapplicationtoelec-\ntronic system of quasi-1D NTs, where electron-electron\ncorrelations are known to be rather strong. In particu-\nlar, it can be used for analysis of ESR spectra. It can\nbe done by above indicated manner, that is by using of\nhypercomplex number theory analogously to theoretical\nanalysis of quantum optics effects in [22] and analysis in\n[20] of Raman spectra in quasi-1D NTs. It is essential,\nthat the FL soliton spin-charge separation mechanism in\nquasi-1DcarbonNTshasexperimentalconfirmation[16],\n[19], [18], whereas the LL (FLL) spin-charge separation\nmechanism has not been found. It seems to be the di-\nrect confirmation of the applicability of the theory of FL\nabove considered to carbon quasi-1D NTs. Given results\nmean, that the explanation of the results, presented in\nSection III, has to take into consideration the FL behav-\nior of electronic system of NTs, incorporated in diamond\nsingle crystalin [100]direction. It is the main consequece\nof given part for the subsequent analysis. On the other\nhand, it justifies the brief representation of the results of\nthe work [30].12\nThe results above considered show, that the shape of\nπ-solitons (or σ-solitons) is given by the expression with\nthe same mathematical form both in SSH-model and in\nits FL generalization. It is\n|φ(n)|2=1\nξπ(σ)sech2[(n−n0)a\nξπ(σ)−vπ(σ)t]cos2nπ\n2,(23)\nwheren,n0are, correspondingly, variable and fixed num-\nbers ofCH-unit in CH-chain,aisC−Cinteratomic\nspacing projection on chain direction, vπ(σ)isπ(σ)-\nsoliton velocity, tis time,ξπ(σ)isπ(σ) coherence length.\nIt is seen, that π-solitons ( σ-solitons) differ in fact the\nonly by numerical value of coherence length in SSH-\nmodel and in its FL generalization. Really, the coherence\nlengthsξ0πandξ0σare determined by the relation [33]\nξ0π=/planckover2pi1vF\n∆0π,ξ0σ=/planckover2pi1vF\n∆0σ, (24)\nwhere ∆ 0σ, ∆0πareσ−andπ-bandgap values at T=\n0K,vFis Fermi velocity. In SSH-model vFis propor-\ntional to t0, in SSH-FL-model, here presented, vFis\nproportional to sum t0+t1. It allows to explain some\ndiscrepancy between theoretical value in SSH-model and\nexperimental values for ξπand its dispersion, depend-\ning on production technology. Theoretical value in SSH-\nmodel for ξπin t-PA is 7 a, and it is low boundary in the\nrange 7a−11a, obtained for ξπfrom experiments [34].\nIt means, that in the samples with π-soliton coherence\nlength, equaled to 11 a,t1is equal to 0 .57t0at the same\nt-PA band gap value (it is possible, since factor Qis in-\ndependent on t1and can be close to 1).\nConsequently, we come to conclusion, that the con-\nstant component in Taylor expansion of electron-electron\ninteraction potential with the term, proportional to t1, is\nsubstantial and that the value of t1can depend on the\npreparation technology.\nAbovedescribedexperimentalresultsobtainedbyESR\nstudyofNTs, formedindiamondsinglecrystalsinthere-\nsult of the /an}bracketle{t100/an}bracketri}htion beam modification indicate, that for\nthecorrectdescriptionofNTs’propertiesthe σ-electronic\nsubsystem has to be taken into consideration. It follows\nimmediately from the appearance of inherent magnetic\nsymmetry directions, which are not coinciding with host\nlattice symmetry directions. From the other hand, the\nanalysis of numerical values of g-factor and linewidth\nvalues, Figure 6, Figure 8, means, that magnetic interac-\ntions are strong and their strength values are comparable\nwith the corresponding values in usual magnetic systems\nwith unfilled inner shells. At the same time, it is shown\nin [23], that in the case, when magnetic ordering is de-\ntermined the only by π-subsystem of the NTs, the mag-\nnetic interactions are relatively weak, magnetic ordering\nsymmetry characteristics are governed by symmetry di-\nrections of surrounding diamond lattice (with accuracy\nof implantation direction relatively diamond lattice axes\nsymmetry directions). It takes place in NTs produced\nby means of /an}bracketle{t110/an}bracketri}htion beam modification and /an}bracketle{t111/an}bracketri}htionbeam modification. It seems to be evident that by /an}bracketle{t110/an}bracketri}ht\nion beam modification we have the case of strong anti-\nferromagnetic (AFM) ordering Really, the conclusion on\njust AFM ordering (but not ferromagnetic) is in agree-\nment with observation of two both very broad and two\nmoderately broad lines. The appearance of two AFM-\nresonance lines (if linearly polarised microwave field is\nused by detection, that was the case in our experiments)\nwas established by Kittel in the work [35], which was the\nfirst work on the theory of AFM-resonance. It has been\nfound in related our work [36], that magnetic moments of\ntwo sublattices being to be opposite directed are uncom-\npensated in their magnitude, that is, strongly speaking,\nwe are dealing with uncompensated AFM-resonance or,\nin other words, with ferrimagnetic resonance. This is so\nindeed, since the ratio of intensities of the absorption,\ncorresponding to L and Rb-lines is equal to ≈3.5 [36].\nLet us consider the following Hamiltonian\nˆH(u) =ˆH0(u)+ˆH1(u)+ˆH2(u)+ˆH3(u)+ˆH4(u),(25)\nwhereuis configuration coordinate along the symmetry\naxiszof the individual chain of NT. It is suggested to\nbe independent on site position and on subsystem kind.\nOperator ˆH0(u) is\nˆH0(u) =/summationdisplay\n/vectork/summationdisplay\nm/summationdisplay\nq/summationdisplay\nsεmq(u)ˆc+\n/vectorkmsˆc/vectorkqs,(26)\nin which subscripts m,q={π,σ},sis spin projection, /vectork\nis wave vector, ˆ c+\n/vectorkms, ˆc/vectorkmsare operators of creation and\nannihilation of the quasiparticle with spin projection s\nand wave vector /vectorkinmth (qth) subsystem, εmq(u) are\nthe resonance interaction integrals (hopping interaction\nin tight-binding model approximation) of quasiparticles\ninπ-subsystem of electronic system, in σ-subsystem of\nelectronic system, which are considered to be 1D quan-\ntum Fermi liquids, and between π- andσ-subsystems.\nThe operator ˆH1is\nˆH1(u) =/summationdisplay\nm/summationdisplay\njU1(j,m,u)ˆc+\njms(↑)ˆcjms(↑)ˆc+\njms(↓)ˆcjms(↓),\n(27)\nwherej=1,Nisthesiteposition, U1(j,m,u)isintrasub-\nsystem Coulomb coupling parameter, which is dependent\nin general case on j,m,u. The operator ˆH2(u) is\nˆH2(u) =/summationdisplay\nm>q/summationdisplay\njU2(j,m,q,u)/summationdisplay\nsˆc+\njmsˆcjms/summationdisplay\nsˆc+\njqsˆcjqs\n(28)\nwhereU2(j,m,q,u) is intersubsystem Coulomb cou-\npling parameter, which is dependent in general case on\nj,m,q,u. The operator ˆH3(u) is\nˆH3(u) =/summationdisplay\nm>q/summationdisplay\nj/summationdisplay\ns/summationdisplay\ns′J1(j,m,q,u)ˆc+\njmsˆc+\njqs′ˆcjms′ˆcjqs,\n(29)\nwhereJ1(j,m,q,u) is the inter-subsystem Hund’s rule\ncoupling, which is dependent in general case on j,m,q,u.13\nThe operator ˆH4(u) is\nˆH4=/summationdisplay\nm/negationslash=q/summationdisplay\njJ2(j,m,q,u)ˆc+\njms(↑)ˆc+\njms(↓)ˆcjqs(↓)ˆcjqs(↑)\n(30)\nwhereJ2(j,m,q,u) is pair hopping parameter, which is\ndependent in general case on j,m,q,u.\nLike to foregoing consideration the Hamiltonians\nˆH1(u) andˆH2(u) can be expanded in Taylor series about\nthe dimerized state. So, restrcting by two first terms in\nTaylor expansion, we have\nˆH1(u) =/summationdisplay\nm/summationdisplay\nj(U(0)\n1+\n(−1)j2αm\n1u)ˆc+\njms(↑)ˆcjms(↑)ˆc+\njms(↓)ˆcjms(↓),(31)\nwhere{αm\n1},m={π,σ}are constants of electron-\nphonon interactions, accompanying the processes of in-\ntrasubsystem Coulomb interations.\nˆH2(u) =/summationdisplay\nm>q/summationdisplay\nj(U(0)\n2+\n(−1)j2αmq\n2u)/summationdisplay\nsˆc+\njmsˆcjms/summationdisplay\nsˆc+\njqsˆcjqs,(32)\nwhere{αmq\n2},m,q={π,σ}are constants of electron-\nphonon interactions, accompanying the processes of in-\ntersubsystem Coulomb interactions. It is clear, that the\nsecond terms in (31) and in (32) describe the attraction\nbetween strongly correlated electrons, it can explain the\nnature of the pairing mechanism in high temperature su-\nperconductors.\nThe Hamiltonian ˆH(u) can be considered to be basic\nHamiltonian for its generalizationto describe the proper-\ntiesofcarbonNTs, producedby /an}bracketle{t100/an}bracketri}htionbeammodifica-\ntion of diamond single crystals, in particular for analysis\nof ESR data above described. The generalization of the\nHamiltonian ˆH(u) can be done by the way, proposed in\n[22] on the basis of hypercomplex number theory, at that\nit has to be taken into account, that, strongly speaking,\nNTs, produced by /an}bracketle{t100/an}bracketri}htimplantation can be described\nin the framework of hypercomplex number theory by its\ngeneralization too, since C4symmetry indicates on in-\nequivalence of the chains along the NT axis.\nLet us remark, that the Hamiltonian (25) is similar\nto two-orbital-Hamiltonian, proposed in [37] for spectral\nanalysis of the iron-based superconductors. It will be al-\nmost coinciding in the case when {αm\n1}= 0,m={π,σ},\n{αmq\n2}= 0,m,q={π,σ},J1(j,m,q,u),J2(j,m,q,u)\nare independent on j,m,q,u. The difference in given\ncase consists in inequivalence of σandπ-subsystems, in\ncomparison with equivalence of Fe orbitals dxzanddyz,\nconsidered in [37]. However, even in given more simple\ncase the task was solved the only by numerical methods.\nThe main result is represented in Figure 8 in [37].\nThe magnetic excitation spectrum carries information\non the nature of magnetism and the characteristics of\nsuperconductivity. It has been discussed in the liter-\nature, that an observation of a sharp quasiparticle-likeresonance peak in the spin fluctuation spectrum with the\nonset of superconductivity may strongly indicate a sign\nchange in the gap structure caused by the superconduct-\ning coherence factors. It has been established, that in\niron pnictides a strong spin resonance occurs in the s+-\nwave SSt. The comparison of the ESR spectra, Figures 1\nto 3, with theoretically calculated spectral function, pre-\nsentedin Figure8in [37]allowsto suggest,that the spon-\ntaneous transition in ESR response in the sample studied\nindicates on transition to SSt-state, which is coexisting\nwith antiferromagnetic ordering. Therefore, it is addi-\ntional confirmation, that two lines - L-line and Rb-line -\nare assigned with AFM-resonace observed in SSt-state.\nInequivalence of the main characteristics of given lines\ncan be attributed to strong inequivalence of two subsys-\ntems in NTs in comparison with theoretically calculated\nin [37]. It seems to be essential the result in given work\nindicating on the appearance of absorption with very\nbroad spectral distribution and peak-dip-hump feature\nwhen the system becomes supercononducting. We have\nobserved the derivative of spectral function, which cor-\nresponds by its integration to spectral function with two\npeak-dip-hump features. It seems to be consequence of\ndifferent coupling of the resonance mode to fermions in π\nandσ-subsystems. Spectralfunction, presentedinFigure\n8 in [37], was calculated numerically and physical nature\nof the appearance of absorption with very broad spectral\ndistribution has not been established. It has been done\nin the work [38]. The authors have been studied theoret-\nically the spin response in the normal and superconduct-\ning states of Fe-based pnictide superconductors. They\nshowed that the resulting magnetic fluctuation spectrum\ncalculated within random-phase approximation consists\nof two contributions. The first contribution is deter-\nmined by the antiferromagnetic spin fluctuations peaked\nat wave vector /vectorQAFMarising in the result of the inter-\nbandscattering. Thesecondcontributioncomesfromthe\nintraband scattering and results in a broad continuum of\nthe SDW fluctuations with a small momenta.\nFurther the authors of [37] indicate, that ”detailed\nstudy of the magnetic and the electronic spectrum shows\nthat the dispersion of the magnetic resonance mode in\nthe nearly isotropic s+superconducting state exhibits\nanisotropic propagating behavior in an upward pattern”.\nGiven conclusion is also in agreement with experiment\n[36].\nFurther, the observation of superlinear dependence in\nabsorption kinetics,corresponding to L-line, Figure 4, is\nstrong evidence of the mobility of spin carriers [19]. The\nswitching from the saturating behavior to nonsaturat-\ning behavior with superlinear absorption kinetics of Rb-\nline, Figure 5, can be attributed to decrease of screening\nof microwave magnetic field (and static magnetic field\ntoo) byπsubsystem when microwave power is increased\nin the range H1/H(0)\n10.75−1. Here we take in mind\nthe reasonable suggestion that screening effect by πsub-\nsystem is substantially more strong in comparison with\nscreening effect by σ-subsystem and microwavefield pen-14\netrates more effectively at low microwave power through\nσ-subsystem. In fact, the spin carriers in πsubsystem\nare pinned at low power, it is the consequence of short\npenetration depth. Sharp increase of absorption in the\nrangeH1/H(0)\n10.75−1 is explained then by two factors\n- by depinning and by increasing of the number of spin\ncarrirers, interacting with microwavefield in given range.\nIn the favour of the SSt formation indicates the ob-\nservation of Dyson-like effect with unusual angular de-\npendence of asymmetry extent A/B of L-line.It is qual-\nitatively explained in [36]. It seems to be also under-\nstandable the presence of some angular dependence of\nQ-factor. It is seen from Figure 10, that relative change\nof Q-factor is small (although it is surely detectable),\nQ-factor is nonmonotonically increased by the change of\nsamplerotationanglefrom[011]to[100]theonlyin1.115\ntimes. Some decrease of Q-factor in the range (60 - 90)\ndegrees from [100] can be explained by the existence of\nnonsuperconducting part of NT-network, [which is con-\nfirmed by the detection of practically isotropic narrow\nlineRn] at simultaneous decrease of the contribution in\nthe total superconducting state of intraband transitions\n(see for details further), detected by very broad lines,\nwhich is decreasing in given range (corresponding Fig-\nures are not represented). In other words, in given range\nthe redistribution of resonance absorption contribution\nbetween superconducting and nonsuperconducting parts\nin the favour of nonsuperconducting part, which charac-\nterises by some cavity Q-losses, although they are small,\ntakes place. The fact, that the maximal Q-factor value is\nachieved in [100] direction can be explained in the follow-\ning way. In given direction the part of microwave power\ncan penetrate through free unmodified diamond space\nbetween NTs to all sample volume, which is insulating\nand it is free from any magnetic impurities. That means,\nthattherelativecontributionofnonsuperconductingpart\nof NT-network with some small Q-losses into total res-\nonance and nonresonance parts of the interaction of all\nsample electronic subsystems with microwavefield has to\nbe minimal, which is really observed.\nThe very pronounced angular dependence of linewidth\nof L-line seems to be the most clear demonstration of\nMeissner effect. Meissner effect is expected to be very\nanisotropicin the sample studied, since, on the one hand,\nthe superconductivity is suggested to be multicannel (see\nfurther), that is, it is determined by different mecha-\nnismssimultaneously. Ontheotherhand, thereareinthe\nsample unmodified regions between NTs in NT-network,\nwhich strengthen anisotropy of Meissner effect. So, near\nthe [100] direction Meissner effect seems to be minimal,\nsince static magnetic field H0can penetrate along NT\naxesboth ininnerNTspaceandinoutersurroundingun-\nmodified diamond regions. That ensures the minimal in-\nhomogeneity of magnetic field along all tube surface and\nminimal value of linewidth, which really takes place. In\nother words, effective thickness of superconducting layer\nseems to be less than penetation depth value in Meiss-\nner effect. Then, the linewidth increase, which is start-ing from 5 degrees [given value is the only approximate\nvalue, more precise measurements were not provided] by\nincrease of effective thickness of superconducting layer,\nsince individual static magnetic field line will intersect\nthebignumberofNTsevenbysmalldeviationfromstrict\nimplantation direction. Let us remark, that our previous\nmeasurements show, that inaccuracy in implantaton di-\nrecton by implantation process does not exceed 1 degree,\nat the same time we have to remark, that some inaccu-\nracy can be in given experiment in determination of [100]\ndirection in sample rotation plane (which, however does\nnot exceed 2 degrees). Then the appearance of branch\nin (5-40) degree range, where the growth of linewidth of\nline L isobserved, is explainedby inhomogeneityincrease\nalong individual static magnetic field line determined by\nMeissner effect. Given viewpoint correlates well with\ndata on angular dependence of absorption intensity. It is\nseen from Figure 7, that growth of absorption intensity\nin (5-40) degree range is not pronounced, since, although\nthe effective thickness of operating region for absorption\nprocess is increased, but inhomogeneity of amplitude of\nmagnetic component of microwave field is also increased,\nbeing to be the consequence of Meissner effect too. In\nfact, the average value of amplitude of magnetic com-\nponentH1of microwave field is decreased, resulting in\nalmostcompensationofeffectofgrowthofeffective thick-\nness of absorbing layer. Starting from 40 degrees upto ≈\n70 degrees the processes of intraband transions become\nvery effective [in AFM cannel the very broad lines are\ncorresponding to given processes, it really takes place,\nthe corresponding Figures are not represented, although\nthe readers can compare the Figures 2 and 3 with the\nFigure 1]. It leads to substantial decrease of penetration\ndepth, resulting in substantial decrease of the effective\nthickness of absorbing layer. It means, that part of NTs\nalong individual static magnetic field line drop out from\nresonance conditions at all, being to be consequence of\nscreening both static magnetic field and microwave field.\nIt is understandable, that both absorption intensity and\nlinewidth of line L have to be decreasing, at that the\nappearance of absorption intensity decreasing is evident\n(both average value of H1and effective number of ab-\nsorbing spin carriers are decreasing). Linewidth decreas-\ning is explained by decrease of range of static magnetic\nfield, where resonance conditions can be restored by H0\nscan, being to be consequence on more sharp H0-field\nstrength damping to the same near zero value. Given\nconclusions correspond to experimental data, see Figures\n7 and 8. The second branch of linewidth growth, starting\nin the range 70-75 degrees from [100] direction, Figure 8,\nand corresponding branch of intensity growth, Figure 7,\nhavethe same origin, whichhas linewidth and absorption\nintensity growth, starting at ≈5 degrees. It is direct con-\nsequence of the damping of intraband transions, taking\nplace in given range. It explains the resemblance of two\nbranchesoflinewidth growth, startingat ≈5degreesand\nin the range 70-75 degrees from [100] direction, seen in\nFigure 8. The difference of analogous branches of ab-15\nsorption intensity growth can be explained by different\nscreeningofmagneticcomponent H1ofmicrowavefieldin\ngiven ranges. Actually, when H0is near [100] direction,\nthen propagation direction for microwave field is near\n[011] direction, that is, near the direction, which is trans-\nverse to tube axes. Given direction is characterised by\nstrong reflection and backscattering of microwave power.\nAt the same time, when H0is near [011] direction, then\npropagation direction for microwave field is near [100]\ndirection, that is, near the direction, which is parallel to\ntube axes. In given case reflection and backscattering\nof microwave power leads to its propagation along NT\nsurface in intertube space.\nTherefore, the observed angular dependences of\nlinewidth of line L and absorption intensity, correspond-\ning to given line, become clear qualitative explanation by\ntaking into consideration the Meissner effect.\nLet us give some simple evaluation of penetration\ndepth, based on given experimental results. The effec-\ntive diameter of NTs can be evaluated from near surface\nlayer modification extent with ion beam dose increase.\nIt was surely established in previous studies, see, for in-\nstance, [20] that at 5 ×1014cm−2ion integral fluence\nthe entire modification takes place. Then in the approx-\nimation of uniform tube distribution and by neglecting\nof track canneling, we obtain the diameter value, equal\nto≈4.5˚A. The effective distance between NT centers\nby the same suggestions at 5 ×1013cm−2ion integral\nfluence is ≈14˚A. Then assuming that by direction in 5\ndegrees intersection length by individual static magnetic\nfield line is achieved the penetration depth value, we ob-\ntain the number of the NTs intersected, equal to ≈2500\n[The length of superconducting part of NT was taken to\nbe equal 20 µm. Although the strict value of the ratio\nof the lengths of superconducting and nonsuperconduct-\ning parts is unknown, given value seems to be suitable\nfor approximate evaluation.] Then by using of effective\nsuperconducting depth for individual NT, equal to inter-\natomic distance in graphene layer, that is 1.42 ˚Ainstead\nof intertube distance we obtain the value of penetration\ndepth in ≈34 nm. Naturally, given evaluation gives the\nonly order for penetration depth value, however given\nevalution is coinciding in its order with well known pene-\ntration depth, in particular, with Londons’ length, which\nis equal to (in its order) ∼10 nm in superconducting\nmetals.\nLet us represent some additional arguments in favour\noftheinterpretationaboveproposed. Theverystrongad-\nditional argument is the observation of very pronounced\nDyson-like effect itself, which, what is more, is observed\nby unconventional A/B angular dependence. For com-\nparison, in very similar NT-system, which was formed\ninside the channels of a non-magnetic insulating SAPO\n5 zeolite crystal Dyson effect was not observed [39]. Let\nus give a detailed description of given system prepara-\ntion. According to [39] the sample preparation method\ninvolves heat treatment of a SAPO 5 in an inert atmo-\nsphere(pyrolysis)and filling its poreswith a suitable car-bon source. It results in the presence of the NTs with the\nonly three chiralities (5,0) (4,2) and (3,3), thus minimiz-\ningchiraldistribution. Ithasalsobeeninferredthat(5,0)\nand (3,3) chiral tubes are metallic (it seems to be essen-\ntial for comparison with our results) and (4,2) tubes are\nsemiconducting. Raman radial breathing mode (RBM)\nfeatures indicate an average inner diameter of 0.4 nm for\ngiven single walled carbon nanotubes. From the optical\npolarized photoluminescence data, the arrays of SWNTs\nare found to align according to the channels of the ze-\nolite crystal. The ESR samples studied in [39] imply\nthat single walled carbon nanotubes are occluded inside\nthe channels of a non-magnetic insulating SAPO 5 zeo-\nlite crystal. For reasonsof comparison, ESR observations\nhave also been carried out on free standing SWCNTs ob-\ntained through dissolution of the zeolite matrix in aque-\nous acidic solution. At all the temperatures covered, a\nsymmetric isotropic ESR signal was observed at zero-\ncrossing g-value gc≈2.0025, indicating on the absence\nof Dyson effect.\nGiven comparison seems to be correct since track sur-\nface is in fact SWCNT, at that the diameter of NTs is\nalso comparable. Let us remark once again, that two\nkinds of SWCNTs in [39] were identified to be metallic.\nTherefore, even given comparison seems to be sufficient\nto confirm the conclusion on the reality of AFM-SSt of\nNTs in our sample, since to observe Dyson-like effect the\nconductivity has to be better than metallic.\nTo explain the symmetry character of angular depen-\ndence of strong absorption with very broad lines, let us\nconsider the band model of NTs. For qualitative conclu-\nsions, it seems to be sufficient to considerthe band model\nof graphene.\nThe first calculation of electronic states in a 2D lat-\ntice of carbon atoms with a honeycomb symmetry have\nbeen undertaken by Wallace [40] in 1947. Wallace used\ngraphenetobeastartingelementfordescriptionofbands\nin bulk graphite. Taking into account the strong hy-\nbridization of 2 s2p2orbitals in the graphene plane, Wal-\nlaceconsideredjust theremaining porbital(orientedper-\npendicular to the crystal plane) to be responsible for the\nelectronicbandstructureinthevicinityofthe Fermilevel\nand suggested a standard tight-binding approach. Con-\nsidering the only the nearest-neighbour hopping param-\neterγ0, a pair of π-bands is obtained [41]\nE∗\nπ(/vectork) =−Eπ(/vectork) =\nγ0/radicalBigg\n1+4cos24kya0\n2+4cos4kx√\n3a0\n2cos4kya0\n2,(33)\nwhich distinctly cross (touch) at two inequivalent Kand\nK′points of the Brillouin zone. The strength of the\nnearest-neighbour hopping is 3.2 eV and the lattice con-\nstanta0= 0.246 nm is by a factor of√\n3 larger than the\ndistance between the nearest carbon atoms.\nSo, in pristine graphene, the Fermi level lies just at the\ntouching (crossing) point (the Dirac or charge neutrality16\npoint) of π∗andπbands and graphene has a charac-\nter of zero-band-gap semiconductor (semimetal). Band\nstructure on some distance from Fermi level consist of\nsix symmetric Dirac cones in the approach above con-\nsidered, with vertices, which produce regular hexagon,\nthat is with angle distance from each other in π/3 rel-\natively the hexagon center. To the first approximation\ngiven structure is retained for NTs, that seems to be sub-\nstantial for the explanation of the observed experimental\ndata in/an}bracketle{t100/an}bracketri}ht-incorporated NTs, which are displaing own\nsymmetry, different from diamond lattice symmetry (see\nfurther).\nClose to a given crossing (touching) point, the elec-\ntronic bands are nearly linear and practically rotation-\nally symmetric. In other words, the carrier dispersion\nrelations take a simple form\nE∗\nπ=−Eπ≈vF/planckover2pi1|/vectork|, (34)\nwhere the momentum /vectorkis measured with respect to\nK(K′) point. The parameter vF, having dimension of\na velocity, is directly related to the coupling strength\n(hopping integral) between the nearest carbon atoms:\nvF=√\n3a0γ0/(2/planckover2pi1). It is known, that the linearity of\nbands in graphene (in the vicinity of the KandK′\npoints) implies, on the one hand, that charge carriers be-\nhavior in pristine graphene is like to relativistic particles\nwith zero rest mass and constant velocity vF, equaled to\n≈106cms−1in given case. They are often attributed to\nmassless Dirac fermions, and their behaviour is described\nby the effective Hamiltonian [41]\nˆH=vF/bracketleftbigg\n0 ˆpx−iˆpy\nˆpx+iˆpy0/bracketrightbigg\n=vFˆ/vector σˆ/vector p,(35)\nwhich is equivalent to the Hamiltonian in the Weyl equa-\ntion for real relativistic particles with zero rest mass\n(originally for neutrinos) derived from the Dirac equa-\ntion. On the other hand, the dispersion relation (34) is\nkey relation for LL-behavior of electronic system. There-\nfore, in the first approximation the electronic system\nof graphene is considered in the literature to be 2D-\nLuttinger liquid system, which seems to be incorrect,\ntaking into account foregoing discussion.\nTherefore, the relativistic-like image of electronic\nstates in graphene given by Hamiltonian (35) remains\nto our opinion an very approximate model. Even in the\ncase of electronic states in the vicinity of the Dirac point\nthe interaction with phonon subsystem has to be taken\ninto consideration. It can lead qualitatively to the same\nsimple model proposed, however numerical characteris-\ntics will be other. Naturally, the deviations from this\nrelativistic model become significant for states far away\nfrom the Dirac point, even if only the nearest neighbours\nin thetight-binding calculationareconsidered. Otherde-\nviations may arise when including the hopping integrals\nbetween next-nearest neighbours. For example, when\ntaking into account the non-zero values of next-nearest\nhopping integrals,the nonlinearityisenhanced and Diracconesbecome asymmetricwith respecttothe chargeneu-\ntrality point.\nQualitatively the characteristic features of angular de-\npendencies of the parameters of ESR-spectra observed\ncan be explained now in the following way. The quasi-\nparticle spectral function, which describes the ESR spec-\ntrum observed is [37]\nA(/vectork,ω) =−1\nπIm[/summationdisplay\naGaa(/vectork,iωn→ω+iδ)] (36)\nwith the dressed normal single-particle propagators Gaa\ndetermined by solving the coupled Dyson-Gorkov’sequa-\ntions,ωnis bosonic Matsubara frequency, ωn= 2nπT.\nWe see, that spectral function depend on /vectorkin explicit\nform. The value of AFM vector /vectorQis determined by\n/vectork-differences between inequivalent KandK′points of\nDirac cones, that is by π/3, which is really experimen-\ntally displaying in angular dependencies of absorption\nspectral distribution. In particular, it becomes to be un-\nderstandable, why the absorption with very broad two\nlines is observed in the range of the angles near π/3 with\nintensity maximum at π/3 and coincidence of peak-dip-\nhump features of both very broad lines at given angle.\nIt is taken into account, that periodical function in wave\nvector k-space has the main mode with the same period\nin frequency ω-space, which is equivalent to H0space,\nrealized by scanning of static magnetic field, by means of\nwhich the spectra were registered.\nWe see from foregoing theoretical consideratin, that in\nthe caseconsideredtheadvantagesofseveralmechanisms\nof SSt formation can be joined. On the one hand, s-wave\nmechanism, mediated by the coupling of charge carri-\ners with stretched phonon modes like to MgB2, heavily\nboron doped diamond and sandwich S-Si-QW-S struc-\ntures can be taking place. Moreover, just crimped cylin-\ndrical shape allows to increase the strength of C-C bonds\nby preservation of high density of the states on FS, re-\nsulting from low dimensionality (which seems to be in-\ntermediate between 1D and 2D). On the other hand, the\nmultiband structure of valence and conductivity bands\nallows to realise the formation of AFM-SSt by means of\nthes+-wave formation like to pnictides and additionally\np-wave formation. It seems to be new mechanism - joint\ns+-p-wave mechanism. Just given mechanism is experi-\nmentally proved. The independent on dimerization coor-\ndinate (which can be both in static and dynamic states)\nelectron-electronrepulsiontermcangivethecontribution\nto AFM-SSt formation by given mechanism. The forego-\ning theoretical consideration allow to suggest also, that\nusual s-wave BCS mechanism with S = 0 Cooper pairing\nprocess of quasiparticlescan produce additional indepen-\ndent superconducting cannel. Given mechanism cannot\nbe detected, however, by magnetic resonance technique\ndirectly. Along with given mechanism, the s-wave BCS-\nlike mechanism with S = 1 Cooper pairing process of\nquasiparticles can in principle also take place. The at-\ntractive terms, which are proportional to dimerization17\ncoordinate, seem to be contributing to given phonon-\nmediated mechanisms and to s-wave mechanisms, me-\ndiated by the coupling of charge carriers with stretched\nphonon modes like to those ones established in MgB2,\nheavily boron doped diamond and sandwich S-Si-QW-\nS structures. Further, the formation of σ-polaron lattice\nwithAFM-ordering,whichtakesplace,forinstanceinthe\nsamples, implanted in /an}bracketle{t111/an}bracketri}htdirection [20], leads also to\nnew possible mechanism of AFM-SSt formation. It will\nbe pures+-wavemechanism, like to those taking place in\nmany pnictides. Main feature, which differ given mech-\nanism from known ones is the other spatial distribution\nof delocalised spins. It is σ-polaron lattice instead SDW.\nTherefore, all the terms in Taylor expansion of\nelectron-electron interaction above considered can con-\ntribute to formation of SSt by different channels.\nThe switch to the SSt allows to explain the substantial\nbroadening of ESR-lines, both, the rather large minimal\nvalue of the linewidth ofL-line and Rb-line in comparison\nwithRn-line in nonsuperconducting state, which seems\nto be partly coexisting in the sample studied (let us re-\nmember, that it is indicated by the presence of Rn-line\nin ESR-spectra). Really, in the SSt with a momentum\ndependent SSt-gap ESR linewidth ∆ His determined by\nspin-lattice relaxation time T1, ∆H∼1/T1, which is\ngenerally given by (see, for instance, [42])\n(T1T)−1\n(T1T)−1\nT=Tc=\n2\nkBTc∞/integraldisplay\n0[N2\ns(E)+αcM2\ns(E)]f(E)[1−f(E)]dE,(37)\nwhere\nNs(E) =1\n4π2π/integraldisplay\n0π/integraldisplay\n0E/radicalbig\nE2−|∆(φ,θ)|2sinθdφdθ (38)\nMs(E) =1\n4π2π/integraldisplay\n0π/integraldisplay\n0∆(φ,θ)|/radicalbig\nE2−|∆(φ,θ)|2sinθdφdθ, (39)\nNs(E) andMs(E) are the density of states (DOS) for\nquasiparticles and the anomalous DOS originating from\nthe coherence effect of the transition probability in the\nSSt, respectively, ∆( φ,θ) is SSt-gap. In conventional s-\nwaveSSt, thepresenceof Ms(E) givesriseto∆ Hjustbe-\nlowTcsince it usually has an isotropic gap with the same\nsign on the all Fermi surfaces. By contrast, in uncon-\nventional d-wave and/or p-wave SSt-states, the Ms(E)\nterm is cancelled out by integrating over the momen-\ntum space on the SSt-gap. It can explain the difference\nin linewidth values for L- andRblines, which seems\nto be connected with s- andp-wave SSts correspond-\ningly, simultaneously realized in the sample studied. In\nthe multiband system, the Ns(E) andMs(E) terms in(37) are represented in the form ( Nh\ns(E) +Ne\ns(E)) and\n(Mh\ns(E) +Me\ns(E)), respectively, where the Nh\ns(E) and\nNe\ns(E) are the DOS of the hole and electron FSs, re-\nspectively. In the case of the Fe-pnictides the Ms(E) is\nnegligibly small. It was theoretically proposed that this\nresult is accounted for on a basis of a nodeless s+-wave\npairing scenario assuming a sign reversal gap function,\n+∆hand -∆ eon the hole and electron FSs, respectively.\nIn cases, where the ∆ hand ∆ ehave opposite signs, it\nis noteworthy that the 2 Mh\ns(E)×Me\ns(E) component in\n(Mh\ns(E) +Me\ns(E))2becomes negative. In particular,\nwhen assuming the well- nested FSs, it is anticipated\nthat the sign-nonconserving interband scattering process\n(+∆h↔ −∆e) may exceed the sign-conserving intra-\nband scattering process (+∆ h↔+∆hand ∆ e↔∆e).\nThe former process reduces the M2\ns(E) term through the\nnegative contribution of the 2 Mh\ns(E)×Me\ns(E), whereas\nthe latterprocessdoesnot. Here, todealwith convoluted\nintraband and interband contributions in the spin relax-\nation process the coefficient αcin expression (37) is in-\ntroduced phenomenologically. It takes a value αc≤1 de-\npending on the weight of the interband contribution. Re-\nally, the substantial increase of linewidth Lin AFM-SC\nstate in comparison with Rn-line means, that coefficient\nαcin (37) is nonzero. Moreover, increase of linewidth\nofRb-line in comparison with Rn-line means, that there\nis the additional mechanism of line broadening in addi-\ntion to above considered. It is determined by Meissner\neffect and always will be take place by transition to SSt\nindependently on superconductivity mechanism.\nMore detailed studies are necessary to clarify all the\nprocesses leading to room temperature superconductiv-\nity. In particular, all known models cannot explain the\nobservation of the transition to AFM-SSt just in mag-\nnetic spin resonance conditions. To explain the role of\nspin resonanceconditions forswitch to AFM-SSt wehave\nto take into account the quantum nature of EM-field in\nradiospectroscopy range. Given task has been solved in\n[43], where matrix-operator difference-differential equa-\ntions for dynamics of spectroscopic transitions in 1D\nmultiqubit exchange coupled (para)magnetic and opti-\ncal systems by strong dipole-photon and dipole-phonon\ncoupling are derived within the framework of quantum\nfield theory. It has been established, that in the model\nconsidered the relaxation processes are of pure quantum\ncharacter, which is determined by the formation of the\ncoherent system of the resonance phonons and by the\nappearence along with absorption process of EM-field\nenergy the coherent emission process, acompanying by\nphonon Rabi quantum oscillations, which can be time-\nshared. For the case of radiospectroscopy it corresponds\nto the possibility of the simultaneous observation along\nwith (para)magntic spin resonance the acoustic spin res-\nonance.\nLet us represent the brief review for given results with\nthe same aim, that is, for convenience of readers. In the\nwork [44] the system of difference-differencial equations\nfor dynamics of spectroscopic transitions for both radio-18\nandopticalspectroscopyforthemodel, representingitself\nthe 1D-chain of N two-level equivalent elements coupled\nby exchange interaction (or its optical analogue for the\noptical transitions) between themselves and interacting\nwith quantized EM-field and quantized phonon field has\nrecently been derived. The model presented in [44] dif-\nfers from Tavis-Cummingsmodel [45] the most essentialy\nby inclusion into consideration of quantized phonon sys-\ntem, describing the relaxation processes from quantum\nfield theory position. Seven equations for the seven oper-\nator variables, describing joint system {field + matter }\nwerepresented in matrix form by three matrix equations.\nThey are the following\n∂\n∂t\nˆσ−\nl\nˆσ+\nl\nˆσz\nl\n= 2/bardblg/bardbl\nˆF−\nl\nˆF+\nl\nˆFz\nl\n+||ˆR(λ)\n/vector ql||,(40)\n∂\n∂t\nˆa/vectork\nˆa+\n/vectork\n=−iω/vectork||σz\nP||\nˆa/vectork\nˆa+\n/vectork\n\n+i\n/planckover2pi1\n−N/summationtext\nl=1(ˆσ+\nl+ ˆσ−\nl)v∗\nl/vectork\nN/summationtext\nl=1(ˆσ+\nl+ ˆσ−\nl)vl/vectork\n,(41)\n∂\n∂t\nˆb/vectork\nˆb+\n/vector q\n=−iω/vector q||σz\nP||\nˆb/vector q\nˆb+\n/vector q\n+\ni\n/planckover2pi1\n−N/summationtext\nl=1ˆσz\nlλ/vector ql\nN/summationtext\nl=1ˆσz\nlλ/vector ql\n,(42)\nwhere\n\nˆσ−\nl\nˆσ+\nl\nˆσz\nl\n=ˆ/vector σl= ˆσ−\nl/vector e++ ˆσ+\nl/vector e−+ ˆσz\nl/vector ez(43)\nis vector-operator of spectroscopic transitions for lth\nchain unit, l=2,N−1 [44]. Its components, that is,\nthe operators\nˆσjm\nv≡ |jv/an}bracketri}ht/an}bracketle{tmv| (44)\nare set up in correspondence to the states |jv/an}bracketri}ht,/an}bracketle{tmv|,\nwherev=1,N,j=α,β,m=α,β. For instance, the\nrelationships for commutation rules are\n[ˆσlm\nv,ˆσpq\nv] = ˆσlq\nvδmp−ˆσpm\nvδql. (45)Further\n\nˆF−\nl\nˆF+\nl\nˆFz\nl\n=ˆ/vectorF=/bracketleftBig\nˆ/vector σl⊗ˆ/vectorGl−1,l+1/bracketrightBig\n, (46)\nwhere vector operatorsˆ/vectorGl−1,l+1,l=2,N−1, are given\nby the expressions\nˆ/vectorGl−1,l+1=ˆG−\nl−1,l+1/vector e++ˆG+\nl−1,l+1/vector e−+ˆGz\nl−1,l+1/vector ez,(47)\nin which\nˆG−\nl−1,l+1=−1\n/planckover2pi1/summationdisplay\n/vectorkˆfl/vectork−J\n/planckover2pi1(ˆσ−\nl+1+ ˆσ−\nl−1),(48a)\nˆG+\nl−1,l+1=−1\n/planckover2pi1/summationdisplay\n/vectorkˆfl/vectork−J\n/planckover2pi1(ˆσ+\nl+1+ ˆσ+\nl−1),(48b)\nˆGz\nl−1,l+1=−ωl−J\n/planckover2pi1(ˆσz\nl+1+ ˆσz\nl−1).(48c)\nHere operator ˆfl/vectorkis\nˆfl/vectork=vl/vectorkˆa/vectork+ˆa+\n/vectorkv∗\nl/vectork. (49)\nIn relations(48) Jis the exchangeinteractionconstantin\nthe case of magnetic resonance transitions or its optical\nanalogue in the case of optical transitions, the function\nvl/vectorkin (49) is\nvl/vectork=−1\n/planckover2pi1pjm\nl(/vector e/vectork·/vector e/vectorPl)E/vectorke−iω/vectorkt+i/vectork/vector r,(50)\nwherepjm\nlis matrix element of operator of magnetic\n(electric) dipole moment /vectorPlofl-thchain unit between\nthe states |jl/an}bracketri}htand|ml/an}bracketri}htwithj∈ {α,β},m∈ {α,β},\nj/ne}ationslash=m,/vector e/vectorkis unit polarization vector, /vector e/vectorPlis unit vec-\ntor along /vectorPl-direction, E/vectorkis the quantity, which has the\ndimension of magnetic (electric) field strength, /vectorkis quan-\ntized EM-field wave vector, the components of which get\nadiscretesetofvalues, ω/vectorkisthefrequency,corresponding\nto/vectorkth mode ofEM-field, ˆ a+\n/vectorkand ˆa/vectorkare EM-fieldcreation\nand annihilation operators correspondingly. In the sug-\ngestion, that the contribution of spontaneous emission\nis relatively small, pjm\nl=pmj\nl≡pl, wherej∈ {α,β},\nm∈ {α,β},j/ne}ationslash=m. Further, matrix ||ˆR(λ)\n/vector ql||is\n||ˆR(λ)\n/vector ql||=1\ni/planckover2pi1\n2ˆB(λ)\n/vector qlˆσ−\nl\n−2ˆB(λ)\n/vector qlˆσ+\nl\n0\n(51)\nHereˆB(λ)\n/vector qlis\nˆB(λ)\n/vector ql=/summationdisplay\n/vector qλ/vector ql(ˆb+\n/vector q+ˆb/vector q), (52)19\nˆb+\n/vector q(ˆb/vector q) is the creation (annihilation) operator of the\nphonon with impulse /vector qand with energy /planckover2pi1ω/vector q,λ/vector qlis\nelectron-phonon coupling constant. In equations (41)\nand (42) /bardblσz\nP/bardblis Pauli z-matrix, /bardblg/bardblin equation (40)\nis diagonal matrix, numerical values of its elements are\ndependent on the basis choice. It is at appropriate basis\n/bardblg/bardbl=\n1 0 0\n0 1 0\n0 0 1\n. (53)\nRight hand side expression in (46) is vector product of\nvector operators. It can be calculated in accordance with\nexpression\n/bracketleftBig\nˆ/vector σl⊗ˆ/vectorGl−1,l+1/bracketrightBig\n=1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vector e−×/vector ezˆσ−\nlˆG−\nl−1,l+1\n/vector ez×/vector e+ˆσ+\nlˆG+\nl−1,l+1\n/vector e+×/vector e−ˆσz\nlˆGz\nl−1,l+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle′\n,(54)\nthat is by using of known expression for usual vector\nproduct with additional coefficient1\n2the only, which is\nappeared, since the products of two components of two\nvector operators are replaced by anticommutators of cor-\nresponding components. Given detail is mapped by sym-\nbol⊗in (46) and by symbol′in determinant (54).\nIt follows from comparison with semiclassical Landau-\nLifshitz (L-L) equation for dynamics of spectroscopic\ntransitions for a chain of exchange coupled centers [46],\n[44], that the equation, which is given by (40) is its QED-\ngeneralization. In comparison with semiclassical descrip-\ntion, where for the description of dynamics of spectro-\nscopic transitions is sufficient the only one vector equa-\ntion (L-L equation or L-L based equation), in the case\nof completely quantum consideration L-L type equation\ndescribes the only one subsystem of three-part-system,\nwhich consist of EM-field, dipole moments’ (magnetic or\nelectric) matter subsystem and phonon subsystem. It\nwas concluded in [44], that the presence of additional\nequations for description of transition dynamics by QED\nmodel in comparison with semiclassical model leads to a\nnumberofneweffects, whichcanbepredictedtheonlyby\nQED consideration of resonance transition phenomena.\nThe terms like to right hand side terms in (42) were\nused in so called ”spin-boson” Hamiltonian [47] and in\nso called ”independent boson model” [48]. Given models\nwere used to study phonon effects in a single quantum\ndot within a microcavity [49], [50], [51], [52], [53]. So,\nit has been shown in [52], [53], that the presence of the\nterm in Hamiltonian [44]\nˆHCPh=N/summationdisplay\nj=1/summationdisplay\n/vector qλ/vector q(ˆb+\n/vector q+ˆb/vector q)ˆσz\nj, (55)\nwhich coincides with corresponding term in Hamiltonian\nin [52], [53] at N= 1 [contribution of given term to\nthe equations for spectroscopic transitions is ±N/summationtext\nl=1ˆσz\nlλ/vector q,see equation (42), (note that the equations for spectro-\nscopic transitions were not derived in above cited works\n[49], [50], [51], [52], [53])] leads the only to exponential\ndecrease of the magnitude of quantum Rabi oscillations\nwith increase of electron-phonon coupling strength and\neven to their supression at relatively strong electron-\nphonon coupling. However, it is shown in [43], that\nby strong electron-photon coupling and strong electron-\nphonon coupling quite other picture of quantum relax-\nation processes becomes to be possible. It is argued in\ngivenworkthefollowing. Thedefinitionofthewavefunc-\ntion of the chain system, interacting with quantized EM-\nfield and with quantized lattice vibration field, to be vec-\ntorofthestateinHilbertspaceoverquaternionring, that\nis quaternion function of quaternion argument, leads to\nLorentz invariance of the equations (40) to (42) and to\npossibility of the transfer to observables. In fact, in the\nwork cited, the main role of spin vector for the quantum\nstate description was taken into account. Since spin vec-\ntor is vectorof the state [in Hilbert spaceoverquaternion\nring with accuracy to normalization factor] of 1D quan-\ntum system, interacting with quantized electromagnetic\nfield, all the components ofthe vector ofthe state, that is\nthe components of spin vector, being to be peer compo-\nnents, have to be taken into consideration. At the same\ntime, the Hamiltonian, given by (55) describes in fact\nthe only part of interaction with phonon field, which cor-\nresponds the only to z-component of the vector of the\nstate. The interaction of dipole subsystem with phonon\nfield, corresponding to x- and y-components of the vec-\ntor of the state of dipole subsystem (that is, S+- andS−\ncomponents of the spin of matter subsystem, since they\nare proportional to two linear combinations of peer x-\nandy-components of vector of the state of the system\nconsidered) was taken into consideration in [43]. There-\nfore, the following Hamiltonian was obtained in a natural\nway\nˆH=ˆHC+ˆHF+ˆHCF+ˆHPh+ˆHCPh,(56)\nwhereˆHCis chain Hamiltonian by the absence of the in-\nteraction with EM-field, ˆHFandˆHPhare photon and\nphonon field Hamiltonians correspondingly, ˆHCFand\nˆHCPhare, accordingly, Hamiltonians, describing the in-\nteraction between quantized EM-field and electronicsub-\nsystem of atomic chain and between quantized phonon\nfield and electronic subsystem of atomic chain. Then the\nequationsofthe motionfor spectroscopictransitionoper-\natorsˆ/vector σl, for quantized EM-field operators ˆ a/vectork, ˆa+\n/vectorkand for\nphonon field operators ˆb/vector q,ˆb+\n/vector qare the following. Instead\nequation (40) the equation\n∂\n∂t\nˆσ−\nl\nˆσ+\nl\nˆσz\nl\n= 2/bardblg/bardbl\nˆF−\nl\nˆF+\nl\nˆFz\nl\n+||ˆR(λz)\n/vector ql||+||ˆR(λ±)\n/vector ql||(57)20\ntakes place, where matrix ||ˆR(λz)\n/vector ql||is\n||ˆR(λz)\n/vector ql||=1\ni/planckover2pi1\n2ˆB(λz)\n/vector qlˆσ−\nl\n−2ˆB(λz)\n/vector qlˆσ+\nl\n0\n(58)\nwithˆB(λz)\n/vector ql, which is given by\nˆB(λz)\n/vector ql=/summationdisplay\n/vector q[(λz\n/vector ql)∗ˆb+\n/vector q+λz\n/vector qlˆb/vector q]. (59)\nMatrix||ˆR(λ±)\n/vector ql||is\n||ˆR(λz)\n/vector ql||=1\ni/planckover2pi1\n−ˆB(λ±)\n/vector qlˆσz\nl\nˆB(λ±)\n/vector qlˆσz\nl\nˆB(λ±)\n/vector ql(ˆσ+\nl−ˆσ−\nl)\n,(60)\nwhereˆB(λ±)\n/vector qlis\nˆB(λ±)\n/vector ql=/summationdisplay\n/vector q[(λ±\n/vector ql)∗ˆb+\n/vector q+λ±\n/vector qlˆb/vector q]. (61)\nThe equation (41) remains without changes. The equa-\ntion (42) is\n∂\n∂t\nˆb/vectork\nˆb+\n/vector q\n=−iω/vector q||σz\nP||\nˆb/vector q\nˆb+\n/vector q\n+\ni\n/planckover2pi1\n−N/summationtext\nl=1{λz\n/vector qlˆσz\nl+λ±\n/vector ql(ˆσ+\nl+ ˆσ−\nl)}\nN/summationtext\nl=1{λz\n/vector qlˆσz\nl+λ±\n/vector ql(ˆσ+\nl+ ˆσ−\nl)}\n.(62)\nHereλz\n/vector qandλ±\n/vector qare electron-phonon coupling con-\nstants, which characterise respectively the interaction of\nelectron subsystem of jth chain unit, corresponding to z-\ncomponent of its vector of state (or Sz\nj) and the interac-\ntion of electron subsystem of jth chain unit, correspond-\ning to±- componenst of its vector of state (or S+\nj- and\nS−\njcomponents of the spin of jth chain unit). It seems to\nbe understandable, that they can be different in general\ncase. Moreover, in order to take into account the interac-\ntion with both equilibrium and nonequilibrium phonons\nboth the electron-phonon coupling constants have to be\ncomplex numbers.\nThus, QFT model for dynamics of spectroscopic tran-\nsitions in 1D multiqubit exchange coupled system was\ngeneralized by taking into account, that spin vector is\nproportional to quaternion vector of the state of anyquantum systen in Hilbert space defined over quaternion\nring and consequently all the spin components has to be\ntaken into account. New quantum phenomenon was pre-\ndicted in [43]. The prediction results from the structure\nof the equations derived and it consists in the following.\nThe coherent system of the resonance phonons, that is,\nthe phonons with the energy, equaled to resonance pho-\nton energy can be formed by resonance, that can lead to\nappearance along with Rabi oscillations determined by\nspin (electron)-photon coupling with the frequency ΩRF\nofRabioscillationsdeterminedbyspin(electron)-phonon\ncoupling with the frequency ΩRPh. In other words, QFT\nmodel predicts the oscillation character of quantum re-\nlaxation, that is quite different character in comparison\nwith phenomenological and semiclassical Bloch models.\nMoreover,if |λ±\n/vector ql|< gthe second Rabi oscillationprocess\nwill be observed by stationary state of two subsystems\n{EM-fied + magnetic (electric) dipoles }, that is, it will\nbe registered in quadrature with the first Rabi oscilla-\ntion process. It can be experimentally detected even by\nstationary spectroscopy methods.\nThe second quantum Rabi oscillation process is gov-\nerned by the formation of the coherent system of the\nresonancephonons. Thereforealongwithabsorptionpro-\ncess ofEM-field energythe coherentemission process can\ntake place. Both the quantum Rabi oscillation processes\ncan be time-shared. For the case of radiospectroscopy\nit corresponds to the possibility of the simultaneous ob-\nservation along with (para)magnetic spin resonance the\nacoustic spin resonance.\nThe predicted phenomenon of the formation of the\ncoherent system of the resonance phonons can find the\nnumber of practical applications, in particular it can be\nused by elaboration of various logic quantum systems in-\ncluding quantum computers and quantum communica-\ntion systems. The appearance of coherent system of the\nresonance hypersound phonons with high energy seems\nto be crucial for the switch of electronic system of NTs\nto AFM-SSt. Really, let us consider the most simple ex-\nample BCS s-wave mechanism of superconductivity. It is\ntaking place, when the interaction between electrons, re-\nalised through phonon subsystem, will be attractive. In\nits turn, given interaction is attractive, when the energy\ndifferencebetweentheelectronstatesinvolvedislessthan\nphonon energy /planckover2pi1ωph[24]. In other words, the most sig-\nnificant contribution to the attractive interaction energy\nis given by short-wavelength phonons.\nTherefore, the appearance of coherent system of high\nenergy hypersound phonons in resonance conditions\nseems to be having key role for switch of the NTs-\nnetwork in the sample studied to the state, characterised\nby superconductivity and uncompensated antiferromag-\nnetism. On the other hand, it is strong argument, that\nphonon-mediated mechanisms are also give contribution\nto total superconducting state.\nLet us remark, that there are additional results in\nfavour of model proposed, represented in [36]. The\nphenomenon of ferrimagnetic spin wave resonance [un-21\ncompensated antiferromagnetic spin wave resonance] has\nbeen established [for the first time in magnetic resonance\nspectroscopy]bymoredetailedanalysisofthespectraob-\nserved. The fact itself of observation of uncompensated\nantiferromagnetic spin wave resonance (SWR) is direct\nproof of the formation of antiferromagnetic ordering [un-\ncompensated]. Spin wave resonance observed has two\nmain peculiarities.\n1.Theoppositedeviationoftheasymmetryextentratio\nA/B from 1 of resonance modes in comparison with main\nAFM mode, at that given deviation increases with mode\nnumber increase. It is the result, which allows to exclude\nfrom the consideration the Dyson effect. Given peculiar-\nity of ferrimagnetic spin wave resonance was explained\nqualitatively by existence of nodes like to explanation of\nthe asymmetry extent of the resonance lines in a dx2−y2\nsuperconductors.\n2.The substantial increase of the intensity of ferrimag-\nnetic spin wave resonance modes with mode number in-\ncrease. Let us remark, that intensity conservationlaw for\nSWR modes was found for NTs incorporated in diamond\nmatrix with other implantation directions [18], carbynes\nand for some organic quasi-1D substances (polyvinyli-\ndenehalogenides-PVDF)[21]. Intheotherearlierknown\ncases, for instance, by SWR in ferromagnetic metals, the\nintensity of SWR modes is decreasing with mode number\nincreasing, see, for example, Figure 1 in [54]. The pecu-\nliarity observed in the sample studied is explained by\ntaking into account the presence of the magnetic fluctu-\nation spectrum consisting of the continuum of the AFM\nspin fluctuations peaked at AFM vector /vectorQ. For SWR\nmodes wave vector |/vector q| /ne}ationslash= 0 and |/vector q|is increasing with\nmode number increase, coming near to the value of /vectorQ.\nThen the dynamical magnetization will be determined\nby Fourier component of the magnetic fluctuation field\nwith the frequency, coinciding with the operating mi-\ncrowave frequency of the spectrometer. Given compo-\nnent is added to dynamical magnetization produced by\nmagnetic component of microwave field used and it de-\ntermines mode intensity growthwith unusual asymmetry\nextent.\nTheobservationofthe onlypeculiaritiesofSWR above\nindicated seems to be sufficient to insist on the for-\nmation in NTs’ network of the sample studied of s+-\nsuperconductivity at room temperature, coexisting with\nuncompensated antiferromagnetic ordering.\nTheresultsabovediscussedcanbeconsideredtobethe\nbasis for the method of identification of superconducting\nstates, coexisting with magnetism.\nV. CONCLUSIONS\nThe formation in carbon NTs, produced by high en-\nergy ion beam modification of diamond single crystals in\n/an}bracketle{t100/an}bracketri}htdirection and representing themselves the surface\nof ion tracks, of uncompensated antiferromagnetic order-\ning coexisting with superconductivity at room tempera-ture is argued. It is based on ESR studies. A number\nof peculiarities has been observed for the first time in\nradiospectroscopy. They are the following.\n1.It is the fact itself of the switch in resonance con-\nditions to other rather stable state. It was shown, that\nnewstate isdefined by uncompensatedantiferromagnetic\nordering coexisting with superconductivity. It is charac-\nterisedspectroscopicallybyappearanceoftwonewrather\nbroadanisotropiclines, designatedLand Rb, whichhave,\nhowever, quite different spectroscopic properties, and by\ntwo very broad intensive lines.\n2.Dependence of absorption amplitude of the right\nbroad line Rbin ESR spectrum of NTs on magnetic com-\nponent of microwave field is strongly nonlinear. It is\ncharacterised for the values of relative magnetic compo-\nnent of microwave field H1/H(0)\n1in the range (0-0.75) by\nusual saturating law, but in the range (0.75-1)it acquires\nprominent superlinear nonsaturating character.\n3.Unusual angular dependence of asymmetry extent,\nwhich cannot be described within the framework of\nDyson theory.\nMain details in very pronounced angular dependencies\nof linewidth of the left line L and intesity of absorp-\ntion, corresponding to given line are exlained by corre-\nsponding angular dependence of Meissner effect. It has\nbeen showed, that broadening mechanism, determined\nby Meissner effect, will take place for any paramagnetic,\nor magnetically ordered system, localised in supercon-\nducting region, that is, given broadening mechanism is\nuniversal. It is established for the first time in radiospec-\ntroscopy.\nPenetration depth of static magnetic field was evalu-\nated to be equal ≈34 nm.\nDifferenceinlinewidthsofthelineLand Rbisanalysed\nwithintheframesofrelaxationtheoryinsuperconducting\nstate (SSt), which takes into account the anomalous den-\nsity of states (DOS) originatingfrom the coherence effect\nof the transition probability in the SSt. DOS, originat-\ning fromthe coherenceeffect givesrise to linewidth ofthe\nline L, which is responsible for s+branch of mixed s+p-\nwavesuperconductivity. At the same time, in p-waveSSt\nthe coherence effect is cancelled out by integrating over\nthe momentum space on the SSt-gap, that is, it does not\ngive rise to linewidth of the line Rb, which is responsible\nforpbranch.\nHamiltonian for mathematical description of the phe-\nnomenon observed is built. It is based on the concept of\n1D Fermi liquid for electronic states of quasi-1D systems,\nthe concept was developed earlier, however, it is shortly\nreviewed in given paper.\nThe analysis of the concept of 1D Fermi liquid allowed\nto propose a number of the other possible mechanisms of\nSSt formation in the sample studied. On the one hand, s-\nwavemechanism, mediated bythe couplingofchargecar-\nriers with stretched phonon modes like to MgB2, heavily\nboron doped diamond and sandwich S-Si-QW-S struc-\ntures can be taking place. Moreover, just crimped cylin-\ndrical shape of NTs allows to increase the strength of22\nC-C bonds by preservation of high density of the states\non FS, resulting from low dimensionality. On the other\nhand, the multiband structure of valence and conductiv-\nity bands allows to realise the formation of AFM-SSt by\nmeans of the s+-wave formation like to pnictides and ad-\nditionally p-wave formation. It seems to be new mecha-\nnism-joint s+-p-wavemechanism. Justgivenmechanism\nis experimentally proved. The independent on dimer-\nization coordinate electron-electron repulsion terms in\nthe Hamiltonian proposed can give the contribution to\nAFM-SSt formation by given mechanism. The forego-\ning theoretical consideration allow to suggest also, that\nusual s-wave BCS mechanism with S = 0 Cooper pairing\nprocess of quasiparticlescan produce additional indepen-\ndent superconducting cannel. Given mechanism cannot\nbe detected, however, by magnetic resonance technique\ndirectly. Along with given mechanism, the s-wave BCS-\nlike mechanism with S = 1 Cooper pairing process of\nquasiparticles can in principle also take place. The at-\ntractive terms Hamiltonian, which are proportional to\ndimerization coordinate, can contribute to given phonon-\nmediated mechanisms and to s-wave mechanisms, medi-\nated by the coupling of charge carriers with stretched\nphonon modes like to those ones established in MgB2,heavily boron doped diamond and sandwich S-Si-QW-S\nstructures. Further, the formation of σ-polaron lattice\nwith AFM-ordering, which can take place in the NTs,\nleads to new possible mechanism of AFM-SSt formation.\nIt will be pure s+-wave mechanism, like to those taking\nplace in many pnictides. Main feature, which differ given\nmechanism from known ones is the other spatial distri-\nbution ofdelocalized spins. It is σ-polaron lattice instead\nspin density wave.\nEspecially interesting seems to be the role of external\nquantized EM-field, which proposed to be responsible for\nthe switch to SSt by means offormationofcoherentlong-\nlivedsystemsofresonancehypersoundphonons. Thecor-\nrespondingquantumfieldtheorywasproposedsomething\nearlier, however, brief review is given. Based on given\nresult, we can conclude, that quantized radiospectrospy-\nrange EM-field seems to be working constituent for re-\nalization of room temperature SSt. On the other hand,\nit is considered to be strong argument of the participa-\ntionintheSSt-formationofBCSorBCS-likemechanisms\n(maybe the only at the stage of a transitional process).\nThus, the room temperature SSt in /an}bracketle{t100/an}bracketri}ht-NTs, incorpo-\nrated in diamond matrix can be formed in the result of\nparticipation of several mechanisms.\n[1] Nagamitsu J., Nakagawa N., Muranaka T., and Akimitsu\nJ., Nature, 410(2001) 63\n[2] AnJM andPickett WE, Phys.Rev.Lett., 86(2001) 4366\n[3] Kortus et al., Phys.Rev.Lett., 86(2001) 4656\n[4] Kong Y. et al., Phys.Rev.B, 64(2001) 020501\n[5] Ekimov E A, Sidorov V A, Bauer E D, Mel’nik N N,\nCurro N J, Thompson J D, and Stishov S M, Nature,\n428(2004) 542\n[6] Takano Y., Nagao M., Kobayashi K., Umezawa H., Sak-\naguchi I., Tachiki M., Hatano T., and Kawarada H.,\nAppl.Phys.Lett., 85(2004) 2581\n[7] Blase X, Adessi Ch, Connetable D, Phys.Rev.Lett., 93\n(2004) 237004\n[8] Lee K.-W. and Pickett W E, Phys.Rev.Lett., 93(2004)\n237003\n[9] Kawaji H., Horie H.-O., Yamanaka S., and Ishikawa M.,\nPhys.Rev.Lett., 74(1995) 1427\n[10] Bagraev N T, Gehlhoff W, Klyachkin L E, Malyarenko\nA M, and Romanov V V, arXiv:0806.2800v1 [cond-\nmat.supr-con]\n[11] Kamihara Y., Watanabe T., Hirano M., and Hosono H.,\nJ.Am.Chem.Soc., 130(2008) 3296\n[12] ChubukovAV,Efremov DV,andEremin I,Phys.Rev.B,\n78(2008) 134512-134512-10\n[13] Erchak D.P, Penina N.M, Stelmakh V.F, Tolstykh VP,\nZaitsev AM, The 7th Int.Conf.IBMM 90, Abstracts,\nKnoxville, USA, 1990, p.313\n[14] EfimovV.G,ErchakD.P,Gelfand R.B,PeninaN.M,Stel-\nmakhV.F,VSVarichenko, UlyashinA.G,ZaitsevAM,E-\nMRS 1990 Fall Meeting, Abstracts, Strasbourg, France,\n1990, p.C-V/P 12\n[15] Kawataba K., Mizutani M., Fukuda M., Mizogami S.,\nSynthetic Metals, 33(1989) 399–402[16] Erchak D.P, Efimov V.G, Zaitsev AM, Stelmakh\nV.F, Penina N.M, Varichenko VS, Tolstykh VP,\nNucl.Instrum.Meth.in Phys.Res.,B, 69(1992) 443-451\n[17] Erchak D.P, Guseva M.B, Alexandrov A.F, Alexander H,\nPilar v.Pilchau A, Pis‘ma Zh.Experiment.Teor.Fiz., 58,\nN 4 (1993) 268-271, JETP Letters, 58, N 4 (1993) 275-\n278\n[18] Ertchak D.P, Efimov V.G, Stelmakh V.F, Re-\nview, Zh.Prikladn.Spectr., 64, N 4 (1997) 421-449,\nJ.Appl.Spectr., 64, N 4, (1997) 433-460\n[19] Ertchak D.P, Efimov V.G, Stelmakh V.F, Martinovich\nV.A, Alexandrov A.F, Guseva M B, Penina N.M, Kar-\npovich I.A, Varichenko V S, Zaitsev A M, Fahrner W R,\nFink D, Phys.Stat.Sol.,b, 203, N2 (1997) 529-548\n[20] Yerchuck D, Dovlatova A, J.Phys.Chem.,C, DOI:\n10.1021/jp205549b, 116, N 1 (2012) 63-80\n[21] Yearchuck D, Yerchak Y, Alexandrov A, Phys.Lett.A,\n373, N 4 (2009) 489-495\n[22] Dovlatova A, YearchuckD, Chem.Phys.Lett., 511(2011)\n151-155\n[23] Dmitri Yerchuck, Vyacheslav Stelmakh, Alla Dovlatova ,\nYauhen Yerchak, Andrey Alexandrov, in press\n[24] Bardeen J, Cooper L N, Schrieffer J.R, Phys.Rev., 108,\nN 5 (1957) 1175-1204\n[25] Dyson F D Phys.Rev. 98(1955) 349-359\n[26] Erchak D.P, Zaitsev A M, Stel’makh V.F, Tkachev\nV D, Phys.Tekhn.Polupr., 14, N 1 (1980) 139-143,\nSov.Phys.Semicond., USA, 14, N 1 (1980) 79-82\n[27] Poole C P, Jr, Technique of EPR-spectroscopy, Moscow,\nMir, 1970, 557 pp\n[28] Tomonaga S., Prog.Theor.Phys., 5(1950) 544\n[29] Luttinger J M, J.Math.Phys., 4(1963) 1154\n[30] Alla Dovlatova, Dmitri Yerchuck, Felix Borovik, in pre ss23\n[31] Su W.P., Schrieffer J.R, and Heeger A.J, Phys.Rev.Lett. ,\n421698 (1979)\n[32] SuW.P., Schrieffer J.R, andHeeger A.J, Phys.Rev.B, 22,\n(1980) 2099\n[33] Lifshitz E.M, Pitaevsky L.P, Statistical Physics, par t 2,\nM., Nauka, 1978, 448 pp\n[34] Heeger A.J, Kivelson S, Schrieffer J.R, Su W.-P.,\nRev.Mod.Phys., 60(1988) 781-850\n[35] Kittel C, Phys.Rev., 82(1951) 565\n[36] DmitriYerchuck,YauhenYerchak,VyacheslavStelmakh ,\nAlla Dovlatova, Andrey Alexandrov, in press\n[37] Zhang J, Sknepnek R and Schmalian J, Phys.Rev.B, 82\n(2010) 134527\n[38] Korshunov M.M, Eremin I, Phys.Rev.B, 78(2008)\n140509(R)\n[39] Rao S S, Stesmans A, Noyen J V, Jacobs P, Sels B, Eu-\nrophys.Lett., 90, N 5 2010 57003\n[40] Wallace P R, Phys.Rev., 71(1947) 622–634\n[41] Castro Neto A H, Guinea F, Peres N M R, Novoselov K\nS, and Geim A K, Rev.Mod.Phys., 81(2009) 109\n[42] Hidekazu Mukuda, Mariko Nitta, Mitsuharu Yashima,\nYoshio Kitaoka, Parasharam M. 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Kent\nDepartment of Physics, New York University, 4 Washington Place, New York, New York 10003, USA\n(Dated: August 22, 2018)\nWe investigate the effect of eddy currents on ferromagnetic resonance (FMR) in ferromagnet-\nnormal metal (FM/NM) bilayer structures. Eddy-current effects are usually neglected for NM layer\nthicknesses below the microwave (MW) skin depth ( /similarequal800 nm for Au at 10 GHz). However, we show\nthat in much thinner NM layers (10-100 nm of Au or Cu) they induce a phase shift in the FMR\nexcitation when the MW driving field has a component perpendicular to the sample plane. This\nresults in a strong asymmetry of the measured absorption lines. In contrast to typical eddy-current\neffects, the asymmetry is larger for thinner NM layers and is tunable through changing the sample\ngeometry and the NM layer thickness.\nINTRODUCTION\nEddy currents are induced currents in conductors by\nchanging magnetic fields. These currents flow in closed\nloops perpendicular to the driving fields, and produce\nadditional Oersted fields that partially compensate the\nexternal driving fields. The effects of eddy currents on\nFMR in conducting films is well known in the limit of film\nthickness approaching their electro-magnetic skin depth\n(/similarequal800 nm for bulk Au at 10 GHz). In those cases,\neddy-current effects can lead to linewidth broadening and\ngive rise to spin-wave excitations due to inhomogenous\nmicrowave fields [1, 2].\nThe microwave frequency spin dynamics in nanostruc-\ntures usually involves stacks of layers combining FM and\nNM at the nanometer scale [3, 4]. Although eddy-current\neffects are usually neglected in metals with thicknesses\nbelow their skin depth, some studies have shown that this\nmay be important also for normal metal (NM) films far\nbelow their skin depth [5–10]. In these studies it was pre-\ndominantly microwave-screening effects that were consid-\nered, and little attention was paid to how the induced\nOersted fields can affect the magnetization dynamics in\nan adjacent ferromagnetic thin film.\nFerromagnetic resonance (FMR) spectroscopy experi-\nments probe static and dynamic properties of magnetic\nmaterials. The technique relies on measuring the mi-\ncrowave absorption associated to the precession of the\nmagnetization. In FMR experiments, position and width\nof absorption lines carry valuable information about ma-\nterial parameters such as anisotropy fields and magnetic\ndamping [11].\nMany experimental setups used for such studies have\nthe main component of the MW driving field oriented in\nthe sample plane (coplanar waveguide (CPW)/striplineFMR). Non-uniformity of the MW field, and sample po-\nsition with respect to the CPW/stripline center would\nstill lead to field components perpendicular to the sam-\nple plane, which would enhance the effects of eddy cur-\nrents. On the other hand, in cavity-FMR setups, the MW\nfields can be oriented either parallel or perpendicular to\nthe sample plane depending on the cavity.\nDifferences in symmetry of FMR lines have been used\nto study the spin pumping from a magnetic material to a\nnormal metal [12–14]. We notice here that a recent study\nhas reported different values for the voltage induced by\nthe inverse spin hall effect, depending on the cavity mode\nused [15]. In such studies, lineshape symmetry is one of\nthe main parameters used to analyze the results.\nHence, to correctly interpret experimental data in-\nvolving FMR it is important to understand how eddy\ncurrents—even in very thin films—can cause modifica-\ntions in the measured FMR lineshape.\nIn this study we investigate the contribution of\neddy currents to the FMR absorption lineshapes in\nferromagnet-normal metal (FM/NM) bilayer structures.\nWe have systematically studied how the sample geome-\ntry and NM thickness affects the coupling between mi-\ncrowave (MW) fields and eddy-current-induced fields,\nand we show that this coupling is tunable through chang-\ning the sample geometry and the NM layer thickness.\nTHEORETICAL MODEL FOR THE OBSERVED\nLINESHAPES\nThe ferromagnetic resonance is usually driven directly\nby the MW field from a cavity or from a coplanar waveg-\nuide/microstrip line. However, capping a FM sample\nwith a NM layer leads to circulating eddy currents inarXiv:1412.1385v3  [cond-mat.mes-hall]  31 Mar 2015ii\nthe NM, and additional Oersted fields in the FM. These\nOersted fields have a different phase with respect to\nthe MW fields—there is a relative phase lag between\nthe MW fields and the Oersted fields from the induced\ncurrents—, and this results in a distortion of the FMR\nlineshape. A sketch of the FM/NM bilayer geometry\nand the path of the induced eddy currents is shown in\nFig. 1a. The induced currents flow in closed loops in\nthe sample plane, with highest current density along the\nsample edges [16, 17]. Figures 1b and c compares two\nrepresentative FMR lineshapes for a 10 nm Py sample\nbefore and after capping it with 10 nm Au; although res-\nonance frequency and linewidth stay constant, the line-\nshape changes considerably.\ny\nzxFMNM\nh(t)\nH0\nIEddy\ndm/dtm(t)\nxa7\nb7 c7 Py Py-10Au\nhEddy\n500 1000 1500−505\nHnfield,n[Gauss]Intensityn[A.U.]β:n−0.3\nΓ:n 45.3\nH0:n987.6\nExp.\nFit\nFIG. 1. (a) A schematic of the sample geometry showing the\npath of the induced eddy currents flowing in closed loops around\nthe sample, with highest current density along the sample edges\n[16, 17]. (b) and (c): FMR lineshapes and fitted parameters from\nEq.(14) for a sample with 10 nm Py (b), and the same sample after\nbeing capped with 10 nm Au (c).\nTo understand the origin of the distorted lineshapes\ndue to the induced eddy currents, we consider a model\ndescribing the magnetization dynamics of the FM, start-\ning from the Landau Lifshitz Gilbert equation [18]:\n∂M\n∂t=−γM×Heff+α\nMs/parenleftBigg\nM×∂M\n∂t/parenrightBigg\n, (1)\nwhereγis the gyromagnetic ratio, αis the Gilbert damp-\ning parameter, and Msis the saturation magnetization.\nThe effective magnetic field, Heff, includes the external\nfieldH0, the anisotropy field, HA(we neglect the effects\nof dipole and exchange fields) and a driving oscillatory\nfield, hac, composed of MW fields and fields from the\neddy currents. In the following, the contribution fromthe anisotropy field to the effective field is neglected, as\nthis is negligible compared to the resonance field for Py.\nWe assume the external applied field, H0, is along z\nin the film plane (see Fig. 1a) and only consider per-\nturbations of the oscillatory field, hac, in the x-y plane,\nperpendicular to the external applied field (the compo-\nnents in the direction of the applied field do not directly\nperturb the dynamics of M).\nThe phase of the microwave excitation in any FMR\nexperiment is arbitrary and can depend on many factors.\nHowever, as we are only interested in relative phase dif-\nferences, we can set the reference phase of the MW field\nto zero. The combined driving field can thus be written\nin the form:\nhac(t) =hMWeiωt+hindei(ωt−φ)\n= [hMW+hind(cosφ−isinφ)]eiωt≡heiωt[1−βi],(2)\nwhereφis the relative phase difference between the MW\nfield and the induced field, and h=hMW+hindcosφ.\nThe parameter βis thus defined as:\nβ= (hindsinφ)/(hMW+hindcosφ) = (βx,βy),(3)\nand accounts for the relative magnitude of the two fields\nand their phases in the x/ydirection respectively. The\nparameterβwill thus approach zero when the induced\nfield is small compared to the MW field, or the phase\ndifference between the MW field and the induced field is\nclose to 0 degrees. There will also be a maxima for βfor\nsome value of the phase difference in the range between\n90-180 degrees, which depends on the magnitude of the\ninduced field compared to the MW field.\nThe magnetization M(t) is then taken of the form\nM(t) =M0z+meiωt, where m⊥z. The magnetic re-\nsponse to small excitation fields, m=χh, is determined\nby the Polder susceptibility tensor χ[19].\nThe elements of χwere determined by solving Eq. (1),\nand discarding higher order terms. Setting m=χhand\nintroducing ω0=γHandωM=γM0, one obtains:\nχ=/parenleftbiggχxxiχxy\n−iχyxχyy/parenrightbigg\n, (4)\nwhere the matrix elements are given by\nχxx/yy =(1−iβx/y)ωM(ω0+iαω)\nω2\n0−ω2(1 +α2) + 2iαωω 0, (5)\nχxy/yx =(1−iβy/x)ωωM\nω2\n0−ω2(1 +α2) + 2iαωω 0. (6)iii\nThe observable quantity in our FMR experiments is\nthe MW power absorption, which is given by an integral\nover the sample volume V[26]:\nPabs=1\n2/Rfractur/integraldisplay\nViω(χh)·h∗dV (7)\nSplitting χinto its real and imaginary part and using\nthathxandhyare orthogonal, one obtains:\nPabs=1\n2/Rfractur/integraldisplay\nViω[χ/prime+iχ/prime/prime]/parenleftbigghx\nhy/parenrightbigg\n·/parenleftbig\nh∗\nx,h∗\ny/parenrightbig\ndV\n=−1\n2/integraldisplay\nVω/parenleftbiggχ/prime/prime\nxxhx+χ/prime/prime\nxyhy\nχ/prime/prime\nyxhx+χ/prime/prime\nyyhy/parenrightbigg\n·/parenleftbig\nh∗\nx,h∗\ny/parenrightbig\ndV\n∝ω(χ/prime/prime\nxxh2\nx+χ/prime/prime\nyyh2\ny),(8)\nThe MW power absorption is thus given by the imag-\ninary part of the diagonal elements χ/prime/prime\nxxandχ/prime/prime\nyy, for the\nfield components in the x/y direction respectively.\nUsing that χyy/xx is written in the form χyy/xx =\nZ1/Z2, whereZiare complex numbers, one can seper-\nate the real and imaginary part by multiplying the ex-\npression by the complex conjugate of the denominator:\nχyy/xx =Z1Z∗\n2\nZ2Z∗\n2. Assuming low damping, ( α2≈0) this\ngives:\n/Rfractur(χxx/yy ) =ω0ωM(ω2\n0−ω2)−βx/yαωωM(ω2\n0+ω2)\n(ω2\n0−ω2)2+ (2αωω 0)2,\n(9)\n/Ifractur(χxx/yy ) =−αωωM(ω2\n0+ω2)−βx/yω0ωM(ω2\n0−ω2)\n(ω2\n0−ω2)2+ (2αωω 0)2.\n(10)\nAs the FMR linewidth for permalloy films is small com-\npared to the resonance frequency, one can assume that\none does not need to deviate far from the resonance in\norder to observe the shape of the curve. That being the\ncase,ω2\n0+ω2≈2ω2\n0, and\n(ω2\n0−ω2)2= (ω0+ω)2(ω0−ω)2≈4ω2\n0(ω0−ω)2.(11)\nHence, for narrow linewidths, Eq.(10) is well approxi-\nmated by:\n/Ifractur(χxx/yy )≈/parenleftbigg−ωMΓw\n4/parenrightbigg1 +βx/y(ω0−ω)/Γw\n(ω0−ω)2+ (Γw/2)2,(12)\nwhere the parameter Γ w= 2αωhas been introduced to\ndescribe the linewidth. This expression consists of two\ncomponents: a symmetric absorption lineshape arising\nfrom the in-phase driving fields, and an antisymmetricdispersive lineshape proportional to βarising from out-\nof-phase driving fields. The βparameter is thus deter-\nmined by the ratio between the absorptive and dispersive\ncontributions to the FMR lineshape. [20, 21].\nIn our set-up the microwave frequency is fixed at 9.4\nGHz, and the magnetic field H0is then swept to lo-\ncate the ferromagnetic resonance at the resonance field,\nH0=HR, satisfying the condition for the resonance fre-\nquency,ωR=γ/radicalbig\nHR(HR+ 4πMs). To extract βfrom\nour experiments, we thus use an expression of same func-\ntional form as Eq. (12), but expressed in terms of field\nrather than frequency.\n/Ifractur(χxx/yy ) =A1 +βx/y(HR−H0)/ΓH\n(HR−H0)2+ (Γ H/2)2, (13)\nwhere A is an unimportant proportionality factor, and\nΓHhas been introduced to describe the linewidth. In this\nform, Eq.(13) describes what is known in the litterature\nas Dysonian lineshapes [20–22].\nIn our experiments, we measure the field derivative of\nthe MW absorption. The experimental data is thus fitted\nto the derivative of Eq. (13) with respect to the external\nfield, which is given by:\nd\ndH0/Ifractur(χxx/yy ) =A/bracketleftBigg\n−βx/y/ΓH\n(HR−H0)2+ (Γ H/2)2\n+2(HR−H0)[1 +βx/y(HR−H0)/ΓH]\n[(HR−H0)2+ (Γ H/2)2]2/bracketrightBigg\n.(14)\nThrough the βparameter, FMR lineshapes in an other-\nwise unperturbed system is thus a measure of the ampli-\ntudes and relative phase of the MW field and the induced\nfields from eddy currents.\nEXPERIMENTAL SETUP\nExperiments were performed with Permalloy\n(Py=Fe 20Ni80) as the ferromagnet layers, and gold\n(Au) and copper (Cu) as NM layers. The Py was grown\nby E-beam evaporation on oxidized silicon substrates,\nand the Au and Cu layers were grown by DC Magnetron\nsputter deposition. We controlled the thickness of\nthe deposited NM layers using a Veeco Dektak 150\nprofilometer, and we cut our samples using a Dynatex\nDX-III combined scriber and breaker to obtain well\ndefined sample geometries. Ferromagnetic resonance\nmeasurements were carried out in a commercial EPR\nsetup (Bruker Bio-spin ELEXSYS 500, with a cylindrical\nTE-011 microwave cavity).\nThe sample is attached to a quartz rod connected to\na goniometer, allowing to rotate the sample 360 degrees.\nThe MW field is oriented perpendicular to the sampleiv\nhmwSampleSample holder\nFIG. 2. Schematic of the cylindrical TE-011 microwave cavity,\nshowing the sample position and field geometry.\nplane and is rotationally symmetric due to the cylindrical\nshape of the cavity, as shown in Fig. 2.\nOur FMR experiments were performed with a low am-\nplitude ac modulation of the static field, which allows\nlock-in detection to be used in order to increase the sig-\nnal to noise ratio. The measured FMR signal is then\nproportional to the field derivative of the imaginary part\nof the susceptibility. The experimental data was thus fit-\nted to Eq.(14), dχ/dH 0(i.e., we obtained an absorption\nline as in Fig.1b when the driving field had only the MW\ncomponent; we obtained an absorption line as in Fig.1c\nwhen the driving field had a strong component from the\neddy-current induced fields).\nRESULTS AND DISCUSSION\nEffect of sample geometry\nWe first focus on the effect of the sample geometry. A\nfull in-plane 360 degrees rotation of a sample of dimen-\nsions 1×3 mm with a thickness of 10 nm Py capped with\n10 nm Au is shown in Fig. 3a, where θ= 0 corresponds\nto an applied field, H0, parallel to the short side of the\nsample. We note that although capping the sample with\na thin NM layer affects the lineshape asymmetry consid-\nerably, the resonance field HRand linewidth Γ stay con-\nstant. Thicker NM layers of materials with considerable\nspin orbit coupling would lead to a linewidth broaden-\ning due to loss of spin angular momentum through spin\npumping effects, but for thin Cu/Au layers this effect is\nnegligible [23, 24].\nThe microwave field in the cavity can be considered\nuniform on the length scale of the sample, and rota-\ntionally symmetric due to its cylindrical shape. If the\nmicrowave excitaton is inhomogenous when rotating a\nlong sample, it could be possible to excite magnetostatic\nmodes in the FM film [25]. However, if this was the case\na)\nb)\n0 1 2 3 4−20020406080\nLength,([mm]Asymmetry(parameter,β Length(series\nRotation,(plotted(as(L(θ)\n0 100 200 30001020304050\nRotation(angleθ,([degrees]Asymmetry(parameter,βFIG. 3. (a) Angular dependence of the βparameter describing the\nFMR lineshape for a sample of 10 nm Py capped with 10 nm Au of\ndimension 1 ×3 mm; the applied field is rotated 360 degrees in the\nfilm plane. (b) Sample length dependence of βfor samples of 10\nnm Py capped with 10 nm Cu (and 5nm Ta to prevent oxidation)\nof dimensions 1 ×Lmm, i.e each datapoint in the ”Length series”\ncorresponds to a separate sample of length L. We also plotted in\n(b) the rotational measurements shown in (a) for a single sample,\nconsidering that the effective length in the direction of the applied\nfield,H0, is approximated by L(θ) =lsin(θ) +wcos(θ),l= 3 mm\nandw= 1 mm.\nin our experiments, one should observe the same asym-\nmetry for the FM without the NM capping layer. To\nrule out conclusively this as a cause of the asymmetry,\nwe performed control experiments where we re-positioned\nthe sample with an offset from the centre of the cavity\n(offset of the same order as the sample dimension). This\ndid not affect the asymmetry of the lineshape, indicating\nthat an inhomogenous MW field in the cavity could not\nbe the cause of the observed effect.\nTo investigate the effect of sample geometry further, a\nset of samples of dimensions 1 ×L mm, where L ranged\nfrom 0.5 to 4 mm was studied. The samples were again of\n10 nm Py capped with 10 nm Cu, and 5 nm Ta to prevent\noxidation. We notice that the sample length in the direc-\ntion parallel to the applied field is the main parameter\nthat determines the asymmetry of the FMR lineshapes,\ngiven by the parameter β. Figure 3b shows the depen-v\ndence of sample length when the varying dimension is\nparallel to the applied field. We see that the asymmetry\nincreases with the sample’s length and reaches a value\nwhereβappears to diverge at a length of about 3.3 mm.\nSamples with a length below 1 mm have lineshapes al-\nmost identical to samples with no NM capping.\nWe consider now the basic physics to describe the\nabove results. The induced eddy currents flow in closed\nloops in planes perpendicular to the MW magnetic field,\nwhich is perpendicular to the film plane in our experi-\nment. Thus, to obtain circulating eddy currents as shown\nin Fig. 1a, it is required to have the MW field perpendic-\nular to the film plane. We have conducted control experi-\nments where the MW fields were applied in the film plane\nand we observed that the FMR lineshapes were always\nsymmetric, indicating there were no observable effect of\nthe eddy currents.\nIn our experimental geometry, the induced eddy cur-\nrents flow mainly in circulating paths, with highest cur-\nrent density along the sample edges [16, 17]. The induced\nOe fields have a component in the film plane and another\nperpendicular to the film plane. As indicated in Fig. 1a,\nfor the sample edges that are parallel to the applied field,\nthe Oe fields will have the main in-plane component per-\npendicular to the applied field and could thus affect the\nFMR of the Py film. On the other hand, currents along\nsample edges perpendicular to the applied field will give\nrise to an in-plane Oersted field that is parallel to the\napplied field, and should not affect the FMR response.\nThe observed strong rotational dependence (See, Fig.\n3a), suggest that the effective driving field has the dom-\ninating contribution oriented in the sample plane; the\ncontribution from the component perpendicular to the\nsample plane should not depend on the direction of the\nsample edges with respect to the applied field. As the\neffective driving field appears to be dominated by the in\nplane components, this indicates that the induced local\nfield perturbing the FM is larger than the external field.\nThis could be possible due to the close proximity to the\ninduced currents at the FM/NM interface.\nWe now compare the length series with the rotational\nmeasurements by using a simple geometric approxima-\ntion: we consider that the length of the sample parallel\nto the applied field is given by L(θ) =lsin(θ) +wcos(θ),\nwherelandware the length (3 mm) and width (1 mm)\nof the sample, and θ= 0 corresponds to the applied field\nparallel to the short side of the sample. We have plotted\nin Fig. 3b the rotational measurements following this ap-\nproach and we can see that the resulting curve is almost\nidentical to the length series.\nTo investigate the effect of sample size closer, we de-\nsigned a control experiment that consisted of taking a\nlarge sample of dimensions 1 ×3 mm and dividing it into\nelectrically isolated regions of 1 ×1 mm. This was per-\nformed using an automated scriber that scratched the\nsample without breaking it—we limited the size of thepossible current loops (as illustrated in Fig. 4).\nm(t)\ndm/dt\ny\nzxFMNM\nh(t)\nH0\nIEddy IEddy\nFIG. 4. Scratching the sample limits the size of the posible current\nloops, reducing the magnitude of the induced fields.\nWe tested this for samples with no NM capping, and\nthe FMR signal was not affected. However, the same\nprocedure on a sample capped with 10 nm Au presented\na remarkable effect: the lineshape before scratching the\nsample was strongly asymmetric, but after scratching the\nfilm it returned to being symmetric again and matched\nthe lineshapes for a sample of dimensions 1 ×1 mm.\nThe asymptotic behaviour of βas the sample length\nincreases, can be understood by considering how the sam-\nple size affects the magnitude of the induced field. As\na simplified model, we approximate the current path\nas a rectangular loop around a sample of length land\nwidthw. The induced electromotive force (EMF) is then\ngiven by the rate of change of magnetic flux through the\narea enclosed by the loop; its absolute value is given by\n|/epsilon1|=lw/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂h\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle∝lw h MW2πf, wherefandhMW are\nthe MW frequency and amplitude respectively. The re-\nsistance of such a loop is given by R= 2Rs(l+w)/ζ,\nwhereRsis the sheet resistance and ζis the width of\nthe current path. The induced current is finally given\nbyI=/epsilon1/R. We consider as an approximation that the\nmagnetic field resulting from a current Iin such a plane\nis given by hind=µ0I/2ζ, whereµ0is the vacuum per-\nmeability. The expression for the induced field is thus\nproportional to the sample size.\nAs an estimate for the strength of the induced field,\nwe calculated this for a square sample, w=l, using the\nabove expression for hind. The sheet resistance was mea-\nsured to be Rs≈50 Ω for samples with 10 nm Py capped\nby 10 nm Cu and 5 nm Ta. Considering the width of the\ncurrent path along the sample edge as ζ=w/4, one can\nestimate the sample size where hind=hMW. In the pres-\nence of a MW field of 9.4 GHz and 6 µT (values used\nduring our experiments), we obtained that at a dimen-\nsion of about 2×2 mm the induced field equals the MW\nfield. Our estimate corresponds with what we see in the\nexperiments: when the sample size approaches the mm\nscale, the effects become increasingly important.vi\nThickness of NM layer\nNext, we focus on another important parameter that\ngoverns the effect of eddy currents: the thickness of the\nNM layer. We prepared samples with a NM (both with\nAu and Cu) thickness ranging from 10 nm to 1 µm (Au)\nand 10-50 nm (Cu). (The experiments presented here\nwere also performed in two more samples, where we ob-\ntained the same results. The measured asymmetry pa-\nrameterβas a function of NM thickness is shown in Fig.\n5 for a sample of dimensions 1 ×3 mm, with the applied\nfield parallel to the long side of the sample. Replacing\nthe Au layer by Cu in the range 10-50 nm shows a sim-\nilar behavior; the thicker the NM, the more symmetric\nthe FMR lineshapes. In the thick film limit one observes\nasymmetric lineshapes again, but with an opposite sign\nof theβparameter.\n101102103−10010203040506070\nNM thickness, [nm]Asymmetry parameter,β\nAu\nCu\n1011021030102030\nNM thickness, [nm]βx\nFIG. 5. NM thickness dependence of the βparameter describing\nthe FMR lineshape for a sample of dimension 1 ×3 mm, with the\napplied field parallel to the long side of the sample. Comparing\nAu and Cu as NM layer in the region 0-50 nm. Inset: Calculated\nthickness dependence of βx, given by Eq. (3) and (15)\nTo explain the thickness dependence, we use a simpli-\nfied model where we assume that the induced eddy cur-\nrents circulate in two dimensional planes in the NM layer.\nThe Oersted fields originated by the eddy currents have\na relative phase lag, φ, compared to the external MW\nfield, which in the ideal case of no inductance is expected\nto beφ=−90 degrees ( IEddy∝∂h\n∂t). However, due to\nthe inductance and resistance of the NM film, there will\nbe an additional phase between the MW field and the\ninduced field that depends strongly on the NM thickness\n(due to the low conductivity of Py compared to Au/Cu,\nwe consider the currents to circulate mainly in the NM\nlayer). At larger thicknesses, one also needs to take into\naccount phase shifts due to the skin effect. Consideringthis, one can write the relative phase lag as a function of\nNM thickness as [26]:\nφ(d) =−/bracketleftbigg\n90 + tan−1/parenleftbiggωL(d)\nR(d)/parenrightbigg\n+d/δ/bracketrightbigg\n, (15)\nwhereωis the microwave angular frequency, LandR\nare the inductance and resistance of the film, dis the\nNM thickness and δis the MW skin depth ( /similarequal800 nm for\nAu at 10 GHz).\nTo estimate values for the inductance, we consider a\nrectangular current path along the edges of the NM layer\n[27], and sample dimensions of w=1 mm,l=3 mm with\nthicknessd(L≈10−8H for a thickness of 10 nm).\nL(d) =µ0µr\nπ/bracketleftBigg\n−2(w+l) + 2/radicalbig\nl2+w2−l·ln/parenleftBigg\nl+√\nl2+w2\nw/parenrightBigg\n−w·ln/parenleftBigg\nw+√\nl2+w2\nl/parenrightBigg\n+l·ln/parenleftbigg2l\nζd/parenrightbigg\n+w·ln/parenleftbigg2w\nζd/parenrightbigg/bracketrightBigg\n,\n(16)\nwhereµRis the relative permeability of the NM film\n(≈1), andζis the width of the current path, set to w/2\nin this calculation.\nAs mentioned in the previous section, we measured a\nsheet resistance of Rs≈50 Ω for samples with 10 nm\nPy capped by 10 nm Cu and 5 nm Ta. However, for NM\nlayers in this thickness regime, the conductivity depends\nstrongly on the thickness. This is due to increased in-\nterface scattering in thin films when the thickness is of\nthe same order as the electron mean free path. Due to\nthis we estimate the film resistance, R(d), as a function\nof thickness by introducing a correction factor, η, which\ndescribes a correction to the film conductivity compared\nto its bulk value. In [6] it was found that the increase\nin conductivity is close to linear in film thickness be-\nlow 20-30 nm, before reaching an asymptotic value for\nthicker films. In our calculations we thus considered η\nto be a linear function of the film thickness below 20nm:\nη= min/parenleftBig\n1,d\nlmfp/parenrightBig\n, wheredis the NM thickness and we\nsetlmfp= 20 nm.\nFrom the rotational measurements, we argued that the\neffective driving field is dominated by the in plane com-\nponent of the induced field. For thicker NM layers the\nsheet resistance is reduced, and the magnitude of the in-\nduced field should thus increase, as |hind|∝1/Rs. Due\nto this, the effective driving field should be dominated by\nthe in plane component, hx, also for thicker NM layers.\nFrom Eq. (8) and (14), the asymmetry of the lineshape\nis then given by the parameter βx.\nUsing these approximations we computed the phase\nshift between the MW field and the induced field, and\ncalculated the thickness dependence of βx, given by Eq.\n(3) and (15), illustrated in the inset of Fig. 5. As thevii\nNM thickness increase, the phase difference approaches\na value of 180 degrees, which then corresponds to βx= 0\n(i.e., the asymmetric lineshapes disappear quickly as the\nNM thickness increases). For thicker NM layers, one gets\nan additional contribution to the phase difference due to\nthe skin effect. At a certain thickness the phase shift will\nthus be larger than 180 degrees, which corresponds to an\nopposite sign of βx.\nThese main features of our simple model agrees well\nwith the experimental data in Fig. 5, where the asym-\nmetry drops off quickly with the thickness for thin NM\nlayers. As the thickness of the NM layer is increased fur-\nther, one also observes the expected transition to asym-\nmetric lineshapes again, but with an opposite sign of the\nβparameter. The thick film limit corresponds to the\nregime where one usually assumes eddy-current effects to\nbecome important, i.e. when the NM thickness approach\nits MW skin depth.\nExperimentally, we observed the strongest lineshape\nasymmetry in films with a NM thickness of 10 nm. We\nalso investigated thinner NM layers of 5 nm, and the\nFMR lineshapes were similar to single Py films. We be-\nlieve this is because we had non-continuous metal films\nfor these thicknesses; Au films tend to be granular and\nthe Cu films might have oxidized.\nSUMMARY\nTo summarize, we have shown that induced eddy cur-\nrents can play an important role in FM/NM bilayer struc-\ntures for certain sample geometries. In contrast to what\nis usually assumed about eddy currents, our results in-\ndicate that these effects can be important also for film\nthicknesses far below their skin depth. In FMR measure-\nments, the influence on lineshape asymmetries has to be\ntaken into account for NM layers below 50 nm and sample\ndimensions above approx. 1 mm2when the MW field has\na significant component perpendicular to the film plane.\nThe dynamics of the system is determined by the\ninterplay of the MW fields and induced fields by eddy\ncurrents, and we have shown that this coupling is tunable\nthrough changing the sample geometry and the NM layer\nthickness. The tunability of the coupling opens up possi-\nbilities to use patterned NM structures to tailor the local\nfield geometry and phase of the induced microwave fields,\nwhich could be of importance for magnonics applications.\nACKNOWLEDGEMENTS\nWe are grateful for insightful discussions with A.\nBrataas and H. Skarsv˚ ag. This work was supported by\nthe Norwegian Research Council (NFR), project num-\nber 216700. V.F acknowledge long term support fromNorFab Norway, and partial funding obtained from the\nNorwegian PhD Network on Nanotechnology for Mi-\ncrosystems, which is sponsored by the Research Coun-\ncil of Norway, Division for Science, under contract no.\n221860/F40. FM acknowledges support from Cata-\nlan Government through COFUND-FP7 and support\nfrom MAT2011-23698. A. D. K acknowledges sup-\nport through the US-National Science Foundation, NSF-\nDMR-1309202.\n∗vegard.flovik@ntnu.no\n[1] C. Kittel, Physical review. 73, 2, (1948)\n[2] P. Pincus, Physical review. 118, 3, (1960)\n[3] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, B. I.\nHalperin, Rev. Mod. Phys. 77, 1375, (2005)\n[4] V. V. Kruglak, S. O. Demokritov, D. Grundler, J. Phys.\nD: Appl. Phys 43 264001, (2010).\n[5] S. Fahy, C. Kittel, S. G. Louie, American Journal of\nPhysics 56, 989 (1988):\n[6] I. V. Antonets, L. N. Kotov, S. V. Nekipelov, and E. N.\nKarpushov, Technical Physics, November 2004, Volume\n49, Issue 11, pp 1496-1500\n[7] M. Bailleul, Appl. Phys. Lett. 103, 192405 (2013)\n[8] M. Kostylev, J. Appl. Phys. 106, 043903 (2009)\n[9] I. S. Maksymov and M. Kostylev, J. Appl Phys. 116,\n173905 (2014)\n[10] I. S. Maksymov, Z. Zhang, C. Change, and M. Kostylev,\nIEEE MAGNETICS LETTERS, Volume 5 (2014)\n[11] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M.\nHu, Phys. Rev. B 84, 054423 (2011)\n[12] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin,\nG. E. W. Bauer, S. D. Bader, and A. Hoffmann, Phys.\nRev. B 82, 214403 (2010)\n[13] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104,\n046601 (2010)\n[14] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011)\n[15] S. I. Kim, M. S. Seo, S. Y. Park J. Appl. Phys. 115,\n17C501, (2014)\n[16] M.Krakowski, Archiv f¨ ur Elektroteknik 64 (1982) 307-\n311\n[17] G.De Mey, Archiv f¨ ur Elektrotechnik, 1974, Volume 56,\nIssue 3, pp 137-140\n[18] L. Landau, E. Lifshitz. Phys. Z. Sowjetunion 8, 153\n(1935).\n[19] D. Polder Phil. Mag. 40, 99-115 (1949)\n[20] C. J. Oates, F. Y. Ogrin, S. L. Lee, P. C. Riedi, G. M.\nSmith, T. Thomson J. Appl. Phys. 91, 1417, (2002).\n[21] C. P. Poole, ”Electron Spin Resonance-A Comprehensive\nTreatise on Experimental techniques”. Wiley, New York,\n(1967).\n[22] F. J. Dyson Phys. Rev. 98, 349 (1955)\n[23] Y. Tserkovnyak, A. Brataas and G. W. Bauer, Phys. Rev.\nB. 66, 224403 (2002)\n[24] S. Mizukami, Y.Ando and T. Miyazak, Phys. Rev. B. 66,\n104413 (2002)\n[25] P. Wolf, Z. Angew, Phys. 14, 212 (1962)\n[26] J. D. Jackson, Classical Electrodynamics, (1998)viii\n[27] Frederick W. Grover, Inductance Calculations, (2004)"    },    {        "title": "2305.04744v1.Large_magnetocaloric_effect_in_the_kagome_ferromagnet_Li__9_Cr__3__P__2_O__7____3__PO__4____2_.pdf",        "content": "arXiv:2305.04744v1  [cond-mat.mtrl-sci]  8 May 2023Large magnetocaloric effect in the kagome ferromagnet Li 9Cr3(P2O7)3(PO4)2\nAkshata Magar,1Somesh K,1Vikram Singh,1J. J. Abraham,2,3Y. Senyk,2\nA. Alfonsov,2B. Büchner,2,4V. Kataev,2A. A. Tsirlin,5and R. Nath1,∗\n1School of Physics, Indian Institute of Science Education an d Research Thiruvananthapuram-695551, India\n2Leibniz IFW Dresden, D-01069 Dresden, Germany\n3Institute for Solid State and Materials Physics, TU Dresden , 01069 Dresden, Germany\n4Institute for Solid State and Materials Physics and Würzbur g-Dresden\nCluster of Excellence ct.qmat, TU Dresden, D-01062 Dresden , Germany\n5Felix Bloch Institute for Solid-State Physics, Leipzig Uni versity, 04103 Leipzig, Germany\nSingle-crystal growth, magnetic properties, and magnetoc aloric effect of the S= 3/2kagome\nferromagnet Li 9Cr3(P2O7)3(PO4)2(trigonal, space group: P¯3c1) are reported. Magnetization\ndata suggest dominant ferromagnetic intra-plane coupling with a weak anisotropy and the onset\nof ferromagnetic ordering at TC≃2.6K. Microscopic analysis reveals a very small ratio of inter-\nlayer to intralayer ferromagnetic couplings ( J⊥/J≃0.02). Electron spin resonance data suggest\nthe presence of short-range correlations above TCand confirms quasi-two-dimensional character\nof the spin system. A large magnetocaloric effect characteri zed by isothermal entropy change of\n−∆Sm≃31J kg−1K−1and adiabatic temperature change of −∆Tad≃9K upon a field sweep of 7 T\nis observed around TC. This leads to a large relative cooling power of RCP≃284J kg−1. The large\nmagnetocaloric effect, together with negligible hysteresi s render Li 9Cr3(P2O7)3(PO4)2a promising\nmaterial for magnetic refrigeration at low temperatures. T he magnetocrystalline anisotropy con-\nstantK≃ −7.42×104erg cm−3implies that the compound is an easy-plane type ferromagnet with\nthe hard axis normal to the ab-plane, consistent with the magnetization data.\nI. INTRODUCTION\nKagome lattice hosts a plethora of interesting phe-\nnomena. Its frustrated nature renders antiferromagnetic\nkagome insulators a natural playground for the experi-\nmental realization of quantum spin liquid [ 1]. Whereas\nkagome ferromagnets are not frustrated and develop\nmagnetic order, they are no less interesting because\nflat bands and Dirac fermions expected in this set-\nting have far-reaching implications for transport prop-\nerties. Recent work on ferromagnetic kagome metals\nexposed anomalous Hall and Nernst effects, as well as\nchiral edge states, in several intermetallic compounds,\nsuch as Fe 3Sn2[2], Co3Sn2S2[3], LiMn 6Sn6[4], and\nUCo0.8Ru0.2Al [5]. Concurrently, insulating kagome fer-\nromagnets were actively studied in the context of magnon\nHall effect and other exotic properties associated with\nDirac magnons [ 6–8].\nFerromagnets are further interesting as materials with\nthe large magnetocaloric effect (MCE) that can be in-\nstrumental in cooling via adiabatic demagnetization for\nreaching temperatures in the sub-Kelvin range [ 9–11].\nThis magnetic refrigeration technique is often consid-\nered as the most energy-efficient, cost-effective (as3He\nand4He are expensive), and environment-friendly re-\nplacement for the conventional refrigeration based on\ngas compression/expansion technique. For this purpose,\nmaterials with large magnetic moment, low magnetic\nanisotropy, low magnetic hysteresis, and extremely low\ntransition temperature are desirable [ 12,13]. The nature\n∗rnath@iisertvm.ac.inof the magnetic transition and the specific form of the\nmagnetic structure are also deciding factors for the per-\nformance of a MCE material. Ferromagnetic insulators\nwith second-order phase transition are proposed to be\nexcellent MCE materials, as only a small change in ap-\nplied magnetic field is sufficient to yield a large entropy\nchange and adiabatic temperature change, compared to\nany paramagnetic salt [ 10,13]. A very few ferromagnetic\ninsulators with low transition temperature are reported\nto satisfy the above prerequisites and qualify for low-\ntemperature applications [ 14–16].\nIn the following, we report the magnetic properties of\nLi9Cr3(P2O7)3(PO4)2(LCPP) which is a structural sib-\nling of the recently reported S= 5/2Heisenberg kagome\nantiferromagnet Li 9Fe3(P2O7)3(PO4)2(LFPP) with a\ntrigonal space group P¯3c1[17]. LFPP shows the onset of\nan antiferromagnetic (AFM) ordering below TN≃1.3K\nand a characteristic 1/3magnetization plateau below\nT∗≃5K. The NMR spectra along with the NMR spin-\nlattice relaxation time reveal the presence of an exotic\nsemiclassical nematic spin liquid regime between TNand\nT∗. On the contrary, LCPP is found to be a ferromag-\nnet and it undergoes a ferromagnetic (FM) ordering at\nTC≃2.5K. LCPP exhibits a large MCE around TCand\nappear to have strong potential for cryogenic applications\nsuch as low temperature sensors in space research, achiev-\ning sub-kelvin temperatures for basic research, hydrogen\nand helium gas liquefaction etc [ 9,18].\nII. METHODS\nPlatelet single crystals of LCPP with the lateral size\nof 0.5 mm to 1 mm were synthesized by a self-flux tech-26.9034 Å\nJ┴\nJ\nabc(a)\nFIG. 1. (a) Corner-sharing equilateral triangles of Cr3+form\na kagome lattice. The intraplane ( J) and interplanar ( J⊥)\ncoupling are shown. The kagome lattice layers are well sepa-\nrated from each other with interlayer distance of 6.9034 Å. ( b)\nInteractions between the magnetic Cr3+ions via PO 4tetra-\nhedra are shown.\nnique as reported in Ref. [ 19]. The mixture of starting\nmaterials, Li 3PO4, Cr2O3, and NH 4H2PO4in the molar\nratio 15:1:9 was kept in an alumina crucible and heated\ngradually to 900◦C. The cooling process involves three\nsteps. At first, the sample was cooled down to 850◦C at\na rate of 50◦C per hour and then to 600◦C at a slow rate\nof2◦C per hour. Finally, the sample was allowed to cool\nnaturally to room temperature. In order to dissolve the\nflux and separate the crystals, the sample was treated\nwith 1 M solution of acetic acid for five days followed by\nthe treatment with saturated NaCl solution and distilled\nwater. The final product after the treatment yields the\nmixture of mm-sized single crystals and polycrystalline\nsample. The large-sized crystals were hand-picked and\nthe remaining part is grinded to get the polycrystalline\nsample.\nRoom-temperature single-crystal x-ray diffraction\n(XRD) was performed on a good quality single crystal\nusing the Bruker KAPPA APEX-II CCD diffractome-\nter equipped with graphite monochromated Mo Kα1ra-\ndiation ( λ= 0.71073 Å). The APEX3 software was\nused to collect the data that were further reduced with\nSAINT/XPREP followed by an empirical absorption cor-\nrection using the SADABS program [ 20]. The phase pu-\nrity of the polycrystalline sample was confirmed from\npowder XRD (PANalytical Xpert-Pro, Cu Kαradia-\ntion with λav= 1.54182 Å). The temperature-dependent\npowder XRD measurement was performed in the temper-\nature range 15 K ≤T≤300 K with a low-temperature\n(Oxford Phenix) attachment to the diffractometer.\nMagnetization ( M) measurement was performed as a\nfunction of temperature ( T) and magnetic field ( H) using\na superconducting quantum interference device (SQUID)\n(MPMS-3, Quantum Design) magnetometer. The data\nwere collected in the temperature range 1.8 K ≤T≤\n350 K and in the magnetic field range 0 ≤H≤7 T. Heat\ncapacity ( Cp) as a function of T(0.5 K≤T≤300 K)\nandHwas measured on a small piece of sintered pellet\nusing the relaxation technique in the physical property\nmeasurement system (PPMS, Quantum Design). Mea-\nsurements below 2 K were carried out using an additional3He insert in the PPMS.\nHigh-field electron spin resonance (HF-ESR) spec-\ntroscopy was used to study the single crystals of LCPP.\nFor the measurements in a frequency range 75 - 330 GHz,\na vector network analyzer (PNA-X from Keysight Tech-\nnologies) was used and for frequencies up to 975 GHz\na modular Amplifier/Multiplier Chain (AMC from Vir-\nginia Diodes Inc.) was used for the generation of\nmicrowaves in combination with a hot electron InSb\nbolometer for detection. All measurements were per-\nformed at a given fixed frequency in the field-sweep mode\nup to 16 T, using a superconducting magnet system from\nOxford Inst. The sample was mounted onto a transmis-\nsion probe head which is then inserted in a4He variable\ntemperature insert (VTI) of the magnet cryostat to en-\nable measurements in a temperature range of 1.8−300K.\nDensity-functional (DFT) band-structure calculations\nwere performed in the FPLOcode [ 21] with the Perdew-\nBurke-Ernzerhof flavor of the exchange-correlation po-\ntential [ 22]. Correlation effects in the Cr 3dshell were\nincluded on the mean-field level within DFT+ Uusing the\non-site Coulomb repulsion parameter Ud= 2eV, Hund’s\ncoupling Jd= 1eV, and double-counting correction in\nthe atomic limit [ 23,24]. Exchange couplings Jijwere\nobtained by mapping [ 25] total energies of collinear mag-\nnetic configurations onto the spin Hamiltonian,\nH=/summationdisplay\n/angbracketleftij/angbracketrightJijSiSj (1)\nwhere the summation is over pairs, and S=3\n2. Energies\nwere converged on a kmesh with 64 points within the first\nBrillouin zone. Thermodynamic properties for the model\ndefined by Eq. ( 1) were obtained from quantum Monte-\nCarlo simulations performed with the loopalgorithm [ 26]\nof theALPSsimulation package [ 27]. Finite lattices with\nup to 752 sites and periodic boundary conditions were\nused.\nIII. RESULTS AND DISCUSSION\nA. X-ray Diffraction\nThe crystal structure of LCPP was solved from single-\ncrystal XRD data with direct methods using SHELXT-\n2018/2 [ 28] and refined by the full matrix least squares\nonF2using SHELXL-2018/3, respectively [ 29]. Details\nof the crystal structure and the refined parameters are\nsummarized in Table I. LCPP crystallizes in the trigonal\nspace group P¯3c1(No. 165). The refined atomic posi-\ntions at room temperature are listed in Table II. These\nstructural parameters are in good agreement with the\nprevious report [ 19].\nThe schematic view of the crystal structure of LCPP\nis presented in Fig. 1. It illustrates the corner sharing\nof CrO 6octahedra and PO 4tetrahedra forming equilat-\neral triangles with a geometrically deformed but regular3\nTABLE I. Crystallographic data for LCPP at room tempera-\nture, obtained from single-crystal XRD.\nEmpirical formula Cr 3Li9O29P8\nFormula weight(M r) 930.22 g mol−1\nTemperature 296(2) K\nCrystal system Trigonal\nSpace group P¯3c1\nLattice parameters a= 9.668(3) Åα= 90◦\nb= 9.668(3) Åβ= 90◦\nc= 13.610(6) Åγ= 120◦\nUnit cell volume 1101.7(8) Å3\nZ 2\nDensity (calculated) 2.804 g cm−3\nWavelength 0.71073 Å\nRadiation type MoK α1\nDiffractometer Bruker KAPPA APEX-II CCD\nCrystal size 0.049×0.035×0.027mm3\n2θrange 2.993 to 25.997◦\nIndex ranges −11≤h≤11\n−11≤k≤11\n−16≤l≤16\nF(000) 902\nReflections collected 6940\nIndependent reflections 735 [ Rint= 0.0454]\nData / restraints / parameters 735/0/76\nGoodness-of-fit on F21.098\nFinalRindices [I≥2σ(I)] R1= 0.0304,\nωR2= 0.0892\nRindices(all data) R1= 0.0351,\nωR2= 0.0919\nLargest diff. peak and hole +0.445/−0.970e.Å−3\nkagome lattice. Though all of the Cr3+– Cr3+distances\nin each hexagon are equal ( ∼4.949Å), the bond angles\nare different: three angles are about ∼146.3◦and the\nremaining three angles are about ∼93.7◦. The nearest-\nneighbour (NN) coupling between the Cr3+ions in the\nab-plane is denoted by J, while the shortest interplane\ndistance of ∼6.903Å leads to a weak coupling ( J⊥) be-\ntween the planes.\nIn order to confirm the phase purity and to scruti-\nnize the presence of any structural distortions, powder\nXRD data were collected at various temperatures. Le-\nBail analysis of the XRD patterns was performed us-\ningFullProf package [ 30] taking the initial structural\nparameters from the single crystal data (Table I). Fig-\nure2(a) and (b) present the powder XRD patterns at the\nhighest ( T= 300 K) and lowest ( T= 15 K) measured\ntemperatures, respectively, along with the Le-Beil fits.\nAll the peaks could be indexed based on the space group\nP¯3c1, suggesting phase purity of the polycrystalline sam-\nple. The obtained lattice parameters at room tempera-\nture are a=b= 9.6628(3) Å,c= 13.5769(3) Å, andTABLE II. Crystal structure of LCPP refined using single-\ncrystal XRD data. The atomic coordinates ( ×104) and the\nisotropic atomic displacement parameter Uiso(Å2×103), which\nis defined as one-third of the trace of the orthogonalized Uij\ntensor.\nAtomic\nsitesWyckoff\npositionsx y z U iso\nCr(1) 6f 5676(1) 0 2500 6(1)\nLi(1) 2b 0 0 5000 19(3)\nLi(2) 12g 3369(7) 2367(7) 4374(4) 16(1)\nLi(3) 4d 6667 3333 6178(7) 15(2)\nO(1) 12g 3758(2) −1053(2) 3331(1) 9(1)\nO(2) 6f 2120(3) 0 2500 9(1)\nO(3) 12g 792(2) −2535(2) 3440(2) 9(1)\nO(4) 12g 2299(3) 38(2) 4332(2) 11(1)\nO(5) 12g 6781(2) 1893(2) 3352(2) 10(1)\nO(6) 4d 6667 3333 4839(3) 23(1)\nP(1) 12g 2279(1) −894(1) 3440(1) 6(1)\nP(2) 4d 6667 3333 3736(1) 5(1)\nunit-cell volume Vcell≃1097.85(5) Å3, which are consis-\ntent with the single-crystal data. No extra peaks or fea-\ntures were observed in the XRD data corroborating the\nabsence of any structural transition or distortion down\nto 15 K.\nThe temperature variation of lattice parameters ( a,c,\nandVcell) is presented in Fig. 3. They are found to de-\ncrease monotonically upon cooling down to 15 K. Vcell(T)\nwas fitted by the equation [ 31]\nVcell(T) =γU(T)\nK0+V0, (2)\nwhereV0is the zero-temperature unit cell volume, K0\nis the bulk modulus, and γis the Grüneisen parameter.\nU(T)is the internal energy and it can be expressed in\nterms of the Debye approximation as\nU(T) = 9pkBT(T\nθD)3/integraldisplayθD/T\n0x3\nex−1dx. (3)\nHere,pis the number of atoms in the unit cell, kBis\nthe Boltzmann constant, and the Debye temperature is\nrepresented by θD. The fit (see Fig. 3) returns θD≃\n385K,γ/K0≃2.06×10−4Pa−1, andV0≃1094.2Å3.\nB. Magnetization\nMagnetic susceptibility χ(T) [≡M(T)/H]measured\nin an applied field of H= 0.5T perpendicular ( H⊥c)\nand parallel ( H∝bardblc) to the kagome plane is displayed\nin Fig. 4(a). With decreasing temperature, χ(T)in-\ncreases in a Curie-Weiss manner as expected in the high-\ntemperature paramagnetic (PM) regime, followed by a4\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s32\n/s32/s32/s73\n/s111/s98/s115\n/s32/s32 /s73\n/s99/s97/s108\n/s32/s32 /s73\n/s111/s98/s115/s45/s32 /s73\n/s99/s97/s108/s32\n/s32/s32/s66/s114/s97/s103/s103/s32/s112/s111/s115/s105/s116/s105/s111/s110/s115/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40/s97/s41\n/s84 /s32/s61/s32/s51/s48/s48/s32/s75\n/s32/s40/s100/s101/s103/s114/s101/s101/s41/s97\n/s32/s40/s98/s41\n/s84 /s32/s61/s32/s49/s53/s32/s75\n/s32/s32\n/s98\nFIG. 2. Powder XRD data measured at (a) T= 300 K and\n(b)T= 15K. The black solid line represents the Le-Bail fit\nof the data. Bragg positions are indicated by vertical bars\nand the solid green line at the bottom denotes the difference\nbetween experimental and calculated intensities. The inse t of\n(b) shows a representative single crystal.\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s57/s46/s54/s53/s50/s57/s46/s54/s53/s54/s57/s46/s54/s54/s48/s57/s46/s54/s54/s52/s49/s51/s46/s53/s53/s50/s49/s51/s46/s53/s54/s48/s49/s51/s46/s53/s54/s56/s49/s51/s46/s53/s55/s54\n/s32/s97\n/s32/s99/s99 /s32/s40 /s197 /s41/s32/s97 /s32/s40 /s197 /s41\n/s84 /s32/s40/s75/s41/s49/s48/s57/s51/s46/s53/s49/s48/s57/s53/s46/s48/s49/s48/s57/s54/s46/s53/s49/s48/s57/s56/s46/s48/s32/s86\n/s99/s101/s108/s108\n/s32/s102/s105/s116\n/s86\n/s99/s101/s108/s108/s32/s40 /s197/s51\n/s41\nFIG. 3. Variation of lattice parameters ( a,c, andVcell) with\ntemperature. The solid line denotes the fit of Vcell(T)by\nEq. (2).\nrapid enhancement at low temperatures. This rapid in-\ncrease suggests strong FM correlations below about 10 K.\nFurther, the susceptibility for H⊥candH∝bardblcshow only\na small difference even at low temperatures, which is an\nindication of weak magnetic anisotropy in the compound.\nFor a quantitative analysis, we have plotted the in-\nverse susceptibility ( 1/χ) as a function of temperature in/s49 /s49/s48 /s49/s48/s48/s48/s49/s50/s51\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51\n/s50 /s52 /s54 /s56/s49/s49/s48/s49/s48/s48/s32\n/s72 /s32/s61/s32/s48/s46/s53/s32/s84/s32/s91/s99/s109/s51\n/s32/s40/s109/s111/s108/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s84 /s32/s40/s75/s41/s32/s72 /s32 /s32/s99\n/s32/s72 /s32/s99/s40/s97/s41\n/s40/s99/s41\n/s32/s32/s49/s47 /s32/s40/s99/s109/s45/s51\n/s32/s109/s111/s108/s32/s67/s114/s51/s43\n/s41\n/s84 /s32/s40/s75/s41/s32/s67/s87/s32/s102/s105/s116\n/s32/s72 /s32 /s32/s99/s40/s98/s41/s32\n/s32/s84 /s32/s61/s32/s49/s46/s56/s32/s75\n/s32/s72 /s32 /s32/s99\n/s32/s72 /s32/s99/s77 /s32/s91\n/s66/s32/s40/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s72 /s32/s40/s84/s41\n/s40/s100/s41\n/s72 /s32 /s32/s32/s99/s32\n/s32/s48/s46/s48/s49/s32/s84\n/s32/s48/s46/s48/s50/s32/s84\n/s32/s48/s46/s48/s53/s32/s84\n/s32/s48/s46/s49/s32/s84\n/s32/s48/s46/s53/s32/s84\n/s32/s49/s32/s84/s84 /s32/s91/s99/s109/s51\n/s32/s75/s32/s40/s109/s111/s108/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s84 /s32/s40/s75/s41\nFIG. 4. (a) χ(T)measured in the field of H= 0.5T applied\nperpendicular ( H⊥c) and parallel ( H/bardblc) to the c-axis. (b)\nMagnetization at T= 1.8K as a function of applied field for\nbothH⊥candH/bardblcafter correcting for the demagnetiza-\ntion effect. (c) Inverse susceptibility ( 1/χ) as a function of\ntemperature for H⊥cand solid line is the CW fit. (d) χT\nvsTin different fields for H⊥cin low temperatures.\nFig.4(c) forH⊥c. ForT≥50 K, it exhibits a com-\npletely linear behaviour which was fitted by the modified\nCurie-Weiss (CW) law\nχ(T) =χ0+C\n(T−θCW). (4)\nHere,χ0is the temperature-independent susceptibility,\nCis the Curie constant, and θCWis the CW temper-\nature. The fit yields χ0≃ −3.6×10−4cm3mol−1,\nC≃1.92cm3K mol−1, andθCW≃6K. From the\nvalue of Cthe effective moment is calculated to be\nµeff≃3.92µBin agreement with the spin-only value\nof 3.87µBfor spin-3/2. The positive value of θCWsug-\ngests that the dominant exchange interactions between\nCr3+ions are FM in nature. Using the mean-field ex-\npression J/kB=−3|θCW|/zS(S+ 1)withz= 4neigh-\nbors on the kagome lattice, we estimate J/kB≃ −1.2K.\nMoreover, the small peak in χTin low magnetic fields\nrevealsTC≃2.6K [Fig. 4(d)].χ(T)measured in zero-\nfield cooled and field cooled conditions (not shown) in a\nsmall magnetic field of H= 0.01T forH⊥cshows no\ndifference, suggesting negligible hysteresis.\nThe magnetic isotherm ( MvsH) atT= 1.8K satu-\nrates in low fields of Hsat⊥c≃0.15T andHsat∝bardblc≃\n0.4T with the saturation magnetization of Msat∼\n3.2µB/Cr3+and2.8µB/Cr3+forH⊥candH∝bardblc,\nrespectively (not shown). The obtained Msatvalues are\nclose to the calculated Msat=gSµB≃2.952µBand\n2.937µB, taking the ESR values g≃1.968and 1.958 for\nH⊥candH∝bardblc, respectively. No visible hysteresis is\nobserved in any of the field directions. A slight difference5\nin the saturation field for H⊥candH∝bardblcmay be at-\ntributed to the anisotropic demagnetization field caused\nby the flat shape of the crystals [ 8]. The demagnetiza-\ntion factor is negligible when magnetic field is parallel\nto the crystal plates ( H⊥c). However, when field is\nperpendicular to crystal plate ( H∝bardblc), the demagne-\ntization effect is considerably amplified [ 32]. The data\nin Fig. 4(b) have been corrected for this demagnetizing\nfield asHeff=H0−4πNM whereHeffandH0are the\neffective and applied magnetic fields, respectively, Nis\nthe demagnetization factor, and Mis magnetic moment\nin emu cm−3. To calculate the demagnetization factor\nin the case of LCPP, we approximated the shape of the\nsample to a rectangular strip and did the calculation fol-\nlowing Ref. [ 33], which yields N= 0.84forH∝bardblc. The\nHsatin both directions after demagnetization correction\nis found to be almost same ( ∼0.15T).\nC. Heat Capacity\nTemperature-dependent heat capacity Cpmeasured on\nthe polycrystalline sample is shown in Fig. 5. TheCp\ndata exhibit a sharp λ-type anomaly at TC≃2.5K\ndemonstrating the transition to the magnetically ordered\nstate. Typically, in magnetic insulators, the major con-\ntributions to Cpare from magnetic ( Cmag) and phonon\n(Cph) parts. In high temperatures, Cphdominates over\nCmag, while at low temperatures it is reverse. One can\nestimate Cmagby subtracting Cphfrom the total heat\ncapacity. First, we approximate the phonon contribution\nby fitting the high- Tdata by a linear combination of one\nDebye and three Einstein terms as [ 34]\nCph(T) =fDCD(θD,T)+3/summationdisplay\ni=1giCEi(θEi,T).(5)\nThe first term in Eq. ( 5) is the Debye contribution to\nCph, which can be written as\nCD(θD,T) = 9nR/parenleftbiggT\nθD/parenrightbigg3/integraldisplayθD\nT\n0x4ex\n(ex−1)2dx. (6)\nHere,Ris the universal gas constant, θDis the character-\nistic Debye temperature, and nis the number of atoms\nin the formula unit. The second term in Eq. ( 5) gives the\nEinstein contribution to Cphthat has the form\nCE(θE,T) = 3nR/parenleftbiggθE\nT/parenrightbigg2eθE/T\n[eθE/T−1]2. (7)\nHere,θEis the characteristic Einstein temperature. The\ncoefficients fD,g1,g2, andg3represent the fraction of\natoms that contribute to their respective parts. These\nvalues are taken in such a way that their sum should be\nequal to 1 and are conditioned to satisfy the Dulong-\nPetit value ∼3nRat high temperatures. The high-\nTfit to the Cp(T)data was then extrapolated down/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s53/s49/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s49/s50/s51/s52\n/s48 /s49 /s50 /s51/s48/s51/s54/s57/s49/s50\n/s32/s32\n/s32/s67\n/s112\n/s32/s67\n/s112/s104\n/s32/s67\n/s109/s97 /s103/s67\n/s112/s32/s91/s74/s32/s40/s109/s111/s108/s32/s75/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93/s40/s97/s41\n/s32/s32/s67\n/s112/s32/s91/s74/s32/s40/s109/s111/s108/s32/s75/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s84 /s32/s40/s75/s41/s32/s48/s32/s84\n/s32/s53/s32/s84\n/s32/s55/s32/s84\n/s40/s98/s41/s67\n/s109/s97/s103/s47/s84 /s32/s91/s74/s32/s40/s109/s111/s108/s32/s75/s50\n/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93/s32\n/s84/s32 /s40/s75/s41/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s83\n/s109/s97/s103/s47/s82 /s108/s110/s52/s32/s32\n/s32 /s32/s84 /s32/s67\n/s112/s32/s91/s74/s32/s40/s109/s111/s108/s32/s75/s32/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s84/s32 /s40/s75/s41\nFIG. 5. (a) Heat capacity ( Cp) of LCPP measured in\nzero applied field. The solid line represents the simulated\nphonon contribution [ Cph(T)] and the dotted line represents\nthe magnetic contribution [ Cmag(T)]. Inset: Low tempera-\ntureCpmeasured in various applied fields. (b) Cmag/Tand\nSmag/Rln4in left and right y-axes, respectively, are plotted\nas a function of temperature. Inset: Cmag(T)vsT. Solid line\nis the power-law ( Cmag=aTα) fit in the low- Tregime.\nto low temperatures and subtracted from Cp(T). The\nobtained Cmag/Tis plotted as a function of tempera-\nture in the main panel of Fig. 5(b) and the correspond-\ning magnetic entropy is calculated to be Smag(T) =/integraltextT\n0KCmag(T′)\nT′dT′≃11.6J mol−1K−1at 12 K. This value\ncorresponds to the expected magnetic entropy for spin-3\n2:\nSmag=Rln4 = 11 .5J mol−1K−1. Unlike the conven-\ntional magnets which release the entire entropy near the\ntransition temperature, LCPP releases only ∼40% of\nthe total entropy at TCand the remaining entropy is re-\nleased only above 11 K, suggesting that TCis partially\nsuppressed as a result of low-dimensionality or magnetic\nfrustration [ 24].\nThe inset of Fig. 5(a) presents the Cp(T)data mea-\nsured in different applied fields. The influences of mag-\nnetic field is clearly reflected in the data. The zero-field\npeak broadens and shifts toward high temperatures with\nincreasing field, which is usual for ferromagnets. At low\ntemperatures, Cmag(T)in zero field could be well de-6\n/s48 /s50 /s52 /s54 /s56/s48/s49/s50/s51\n/s48 /s49 /s50 /s51/s48/s52/s56/s49/s50/s49/s48/s32/s75/s49/s46/s56/s32/s75\n/s32/s77 /s32/s91\n/s66/s32/s40/s67/s114/s51/s43\n/s41/s45/s49\n/s93\n/s72 /s32/s40/s84/s41/s72 /s32 /s32/s99/s40/s97/s41\n/s40/s98/s41\n/s32/s32/s77/s50\n/s32/s91\n/s66/s32/s40/s67/s114/s51/s43\n/s41/s45/s49\n/s93/s50\n/s72/s47/s77 /s32/s40/s84/s32/s67/s114/s51/s43\n/s66/s45/s49\n/s41/s49/s46/s56/s32/s75\n/s49/s48/s32/s75\nFIG. 6. (a) Isothermal magnetization ( MvsH) curves for\nH⊥cand (b) their corresponding Arrott plots ( M2vs\n(H/M)) for LCPP at different temperatures around TC.\nscribed using power-law ( Cmag∝Tα) behavior [inset of\nFig.5(b)] with an exponent α∼1.5that corresponds to\nFM spin-wave excitations [ 35].\nD. Magnetocaloric Effect\nMagnetocaloric effect (MCE) is an intrinsic property of\nmagnetic materials. Magnetic cooling is achieved by first\napplying magnetic field to the material isothermally and\nthen removing the field adiabatically. Therefore, MCE\nis generally quantified by the isothermal entropy change\n(∆Sm) and adiabatic temperature change ( ∆Tad) with\nrespect to the change in applied field. The ∆Smcan\nbe calculated from either magnetization isotherms ( M\nvsH) or heat capacity data measured in zero and non-\nzero magnetic fields. Figure 6(a) displays the magnetic\nisotherms measured in close temperature steps around/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s45/s49/s48/s45/s50/s48/s45/s51/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s45/s49/s48/s45/s50/s48/s45/s51/s48/s49/s32/s84/s55/s32/s84/s32\n/s32/s83\n/s109/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41/s40/s97/s41\n/s72 /s32 /s32/s99\n/s40/s98/s41\n/s32/s32/s83\n/s109/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84/s32 /s40/s75/s41/s32/s55/s32/s84\n/s32/s53/s32/s84\nFIG. 7. (a) Isothermal entropy change ( ∆Sm) vsTplotted\nupon sweeping the field to 0 T starting from different fields\nfrom 1 T to 7T, calculated using MvsHdata of single\ncrystals for H⊥cand employing Eq. ( 8). (b)∆SmvsTplot\nfor5T and7T, calculated using Cp(T)of polycrystalline\nsample in Eq. ( 9).\nTCforH⊥c. The first method utilizes Maxwell’s ther-\nmodynamic relation, (∂S/∂H)T= (∂M/∂T)H, and∆Sm\ncan be estimated using the MvsHdata as [ 36]\n∆Sm(H,T) =/integraldisplayHf\nHidM\ndTdH. (8)\nFigure 7(a) presents the plot of ∆Smas a function of\ntemperature ( T) in different values of ∆H=Hf−Hi.\n∆SmvsTexhibits a maximum entropy change around\n4.6 K, with a highest value of ∆Sm≃ −31J kg−1K−1\nfor the 7 T field change. As the magnetic anisotropy\nis negligibly small, no significant difference in ∆Smis\nexpected for H⊥candH∝bardblc[37].\nFurther, to cross check the large value of ∆Sm, we have\nalso estimated ∆Smfrom heat capacity data measured\nin zero field, 5 T, and 7 T. First, we calculated the total7\nentropy at a given field as\nS(T)H=/integraldisplayTf\nTiCp(T)H\nTdT, (9)\nwhereCp(T)His the heat capacity at a particular field\nHandTiandTfare the initial and final tempera-\ntures, respectively. We calculated ∆Smby taking the\ndifference of total entropy at non-zero and zero fields\nas∆Sm(T)∆H= [S(T)H−S(T)0]T. Here, S(T)Hand\nS(T)0are the total entropy in the presence of Hand\nin zero field, respectively. Figure 7(b) presents the esti-\nmated∆Smas a function of temperature in 5 T and 7 T\nmagnetic fields. The overall shape and peak position of\nthe∆Smcurves are identical with the curves [Fig. 7(a)]\nobtained from the magnetic isotherms but with a slight\nreduction in magnitude. This difference in magnitude\nat the peak position could be related to the polycrys-\ntalline sample used for heat capacity and single crystals\nfor magnetic measurements [ 38].\nSimilarly, the adiabatic temperature change ∆Tadcan\nbe estimated from either the combination of zero-field\nheat capacity and the magnetic entropy change obtained\nfrom magnetic isotherms or from the heat capacity alone\nmeasured in different magnetic fields. Using the heat\ncapacity in zero field and magnetization isotherm data,\nthe estimation of ∆Tadcan be done as [ 39]\n∆Tad=/integraldisplayHf\nHiT\nCpdM\ndTdH. (10)\nThe dependence of ∆TadonTfor different magnetic\nfields is shown in Fig. 8(a). The maximum value of ∆Tad\nis obtained to be ∼40K for∆H= 7T. However, as\nexplained in Ref. [ 40] the above expression overestimates\n∆TadsinceT/Cpis not constant over the range of applied\nfields as it was assumed. It is evident from the inset of\nFig.5(a) thatCpat low temperatures is changing drasti-\ncally as we apply magnetic field and this change should be\ntaken into account while calculating the entropy for that\nparticular field. Therefore, we tried to estimate ∆Tadby\ntaking the difference in temperatures corresponding to\ntwo different fields with constant (same) entropy as [ 40]\n∆Tad(T)∆H= [T(S)Hf−T(S)Hi]. (11)\n∆TadvsTfor∆H= 5T and7T calculated by this\nmethod is shown in Fig. 8(b). The maximum value of\n∆Tadat 7 T is around ∼9K which is significantly\nsmaller than the value obtained using the former method\n[Eq. (10)]. A similar difference has been reported earlier\nfor ErAl 2[40]. The latter method is considered to be\nmore reliable. It is expected to provide accurate results\nas the effect of magnetic field on Cpis accounted for.\nNote that the large values of ∆Smand∆Tadare not\nsufficient to characterize the potential of a material for\nthe magnetic refrigeration applications. Another im-\nportant parameter is the relative cooling power ( RCP)/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s50/s52/s54/s56/s49/s48/s72 /s32 /s32/s99\n/s32\n/s32/s32/s45 /s84\n/s97/s100/s32/s40/s75/s41\n/s49/s32/s84/s55/s32/s84/s40/s97/s41\n/s40/s98/s41\n/s32/s32/s45 /s84\n/s97/s100/s32/s40/s75/s41\n/s84 /s32/s40/s75/s41/s32/s32/s55/s32/s84\n/s32/s32/s53/s32/s84\nFIG. 8. (a) ∆TadvsTplotted for different field changes of\n∆H= 1T to7T calculated using Eq. ( 10). (b)∆TadvsT\nplotted for ∆H= 5T and7T calculated using Eq. ( 11).\nwhich is a measure of the amount of heat transferred be-\ntween the cold and hot reservoirs in a refrigeration cycle.\nMathematically, it can be expressed as\nRCP=/integraldisplayThot\nTcold∆Sm(T,H)dT, (12)\nwhere,TcoldandThotcorrespond to temperatures of cold\nand hot reservoirs, respectively. The formula for RCP\ncan be approximated as\n|RCP|approx= ∆Speak\nm×δTFWHM, (13)\nwhere∆Speak\nmandδTFWHM are the maximum value of\nentropy change (or the peak value) and full width at\nhalf maximum of the ∆Smcurve, respectively. RCP\nas a function of Hcalculated using the ∆Smdata from\nFig.7(a) is plotted in Fig. 9(a). The maximum value of\nRCP is calculated to about ∼284J kg−1at 7 T.\nThe application of a MCE material is also decided by\nthe nature of its magnetic phase transition. In materi-8\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s48 /s50 /s52 /s54 /s56/s49/s48/s50/s48/s51/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s50 /s52 /s54/s45/s50/s52/s45/s49/s54/s45/s56/s48\n/s32/s82/s67/s80 /s32/s40/s74/s32/s107/s103/s45/s49\n/s41\n/s72 /s32/s40/s84/s41/s40/s97/s41\n/s32/s32/s83\n/s109/s112/s101/s97/s107\n/s40/s74/s32/s107/s103/s45/s49\n/s75/s45/s49\n/s41\n/s72 /s32/s40/s84/s41\n/s40/s98/s41\n/s32/s32/s110\n/s84 /s32/s40/s75/s41\n/s32/s32/s83\n/s109/s32/s40/s74/s32/s107/s103/s45/s49\n/s32/s75/s45/s49\n/s41\n/s72 /s32/s40/s84/s41/s50/s50/s46/s53/s32/s75\n/s49/s48/s32/s75\n/s51/s46/s54/s32/s75\nFIG. 9. (a) Relative cooling power ( RCP =∆Speak\nm×\nδTFWHM ) as a function of magnetic field and the inset shows\nthe value of entropy change at the peak position ∆Speak\nm. Solid\nlines are the fits as described in the text. (b) The exponent n\nplotted as a function of temperature which is obtained from\nfitting power law to ∆Smvs field isotherms. Inset shows the\n∆Smisotherms for temperatures near and well above TC.\nals with a first-order phase transition, though the peak\nheight of the ∆Smand∆TadvsTcurves is large but\nthe curve width is not very broad, which limits the rele-\nvance of these materials in a cyclic operation. The second\nproblem with the first-order transitions is the energy loss\ndue to magnetic and thermal hysteresis [ 41]. Further,\nrelatively large magnetic fields are required to perturb\nthe first-order magneto-structural transitions and induce\nlarge MCE which is another drawback of these mate-\nrials. On the other hand, materials with second-order\nphase transition do not show very large peaks, but their\nRCP values are large due to the increased curve widthand the absence of thermal hysteresis, both effects mak-\ning them promising for practical applications. In order to\nanalyze the nature of the phase transition, we construct\nthe Arrott plot [ 42] in Fig. 6(b) by using the isother-\nmal magnetization data presented in Fig. 6(a). Clearly,\nthe slope of M2vsH/M curves is positive in the entire\nmeasured temperature range, well below and above TC.\nAccording to the Banerjee criterion [ 43], positive slope\nimplies the second-order phase transition. This confirms\nthe continuous second-order nature of the PM to FM\nphase transition in LCPP.\nFurthermore, MCE is also utilized to characterize the\nnature of a phase transition [ 39,44,45]. According to\nthe scaling hypothesis [ 41], the∆Sm(T)curves for dif-\nferent values of ∆Hshould collapse on a single universal\ncurve when the ∆Sm(T)is normalized to its peak value\n∆Speak\nm. However, due to the low transition temperature,\nthe universal curve construction is implausible with the\npresent data. Therefore, we performed only the power-\nlaw analysis of ∆SmandRCP. In Fig. 9(a) main panel\nand inset, we fitted the RCP and∆Speak\nmdata by power\nlaws of the form RCP∝HNand|∆Speak\nm| ∝Hn, re-\nspectively. The exponents Nandn, which are related\nto the critical exponents ( β,γ, andδ), are estimated to\nbeN≃0.7andn≃0.5. In order to perceive the tem-\nperature dependence of n, we fitted the field-dependent\nisothermal magnetic entropy change ∆Sm(H)at various\ntemperatures across the transition using the power law\n∆Sm∝Hn[see inset of Fig. 9(b)] [44]. The obtained\nnvsTdata are plotted in Fig. 9(b) and provide in-\nformation concerning the nature of the transition. For\ninstance, for a second-order magnetic transition the ex-\nponent should have the value n≃2in the paramagnetic\nregion (T >> T C) andn(T)typically exhibits a mini-\nmum near TC[44]. Indeed, our n(T)demonstrates the\nexpected behavior, further confirming the second-order\nmagnetic transition in LCPP.\nIn Table III, we compare the main parameters of LCPP\nwith those of well-studied magnets having low transition\ntemperatures and large MCE. Though the ∆Speak\nmvalue\nof LCPP is comparable to the values for most of the\npotential low-temperature magnetic refrigerant materi-\nals, the width of ∆SmvsTcurves is not very broad.\nDue to which LCPP has a slightly reduced value of RCP\ncompared to others. Nevertheless, the obtained value of\nRCP≃284J kg−1is still significantly large and LCPP\nmay have strong prerequisites for cryogenic applications\nin sub-Kelvin temperatures [ 9,18].\nE. Electron Spin Resonance\n1. Frequency Dependence at T= 1.8K\nThe ferromagnetic resonance (FMR) measurements on\nLCPP were performed at T= 1.8K (< TC) for the two\norientations of the applied magnetic field, H∝bardblcand\nH⊥c. The frequency ( ν) vs resonance field ( Hres) de-9\nTABLE III. Comparison of the adiabatic temperature change ( ∆Tad), maximum entropy change ( ∆Speak\nm), and relative cooling\npower (RCP) of LCPP with some known magnets having low transition tempe ratures ( TCorTN) and large MCE in a field\nchange of ∆H= 5to 8 T. More compounds with the similar behaviour are listed i n Refs. [ 38,46,47].\nSystem TC/TN |∆Tad| | ∆Speak\nm| RCP ∆H Ref.\n(K) (K) (J kg−1K−1) (J kg−1) (T)\nLCPP 2.6 9 31 284 7 This work\nHoMnO 3 5 6.5 13.1 320 7 [ 48]\nErMn 2Si2 4.5 12.9 25.2 365 5 [ 16]\nEuTi0.9V0.3O3 4.5 17.4 41.4 577 7 [ 15]\nGdCrTiO 5 0.9 15.5 36 – 7 [ 49]\nEdDy 2O4 5 16 25 415 8 [ 50]\nEuHo 2O4 5 12.7 30 540 8 [ 50]\nMn32 0.32 6.7 18.2 – 7 [ 51]\nHoB2 15 12 40.1 - 5 [ 52]\nEuTiO 3 5.6 21 49 500 7 [ 46]\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s32/s40/s71/s72/s122/s41\n/s72\n/s114/s101/s115/s32/s40/s84/s41/s32/s72 /s32/s124/s124/s32 /s99\n/s32/s72 /s32 /s32/s99\n/s32/s70/s77/s82/s32/s70/s105/s116/s44/s32 /s72 /s32/s124/s124/s32 /s99\n/s32/s70/s77/s82/s32/s70/s105/s116/s44/s32 /s72 /s32 /s32/s99/s84 /s32/s61/s32/s49/s46/s56/s32/s75\n/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\nFIG. 10. Left vertical scale: Frequency as a function of reso -\nnant field measured for both orientations at T= 1.8K. The\nhollow red and solid black squares represent the measured re s-\nonance fields for orientations H/bardblcandH⊥c, respectively.\nRight vertical scale: Selected HF-ESR spectra.\npendence of the FMR signal is shown in Fig. 10together\nwith few selected spectra (right axis). The spectra mea-\nsured at fields perpendicular to the c-axis are relatively\nbroader and more distorted when compared to the H∝bardblc\nmeasurements. The νvsHresdata were fitted with a\nspin-wave model for FMR, ν=h−1g/bardblµB(Hres−Ha)for\nH∝bardblcorientation and ν=h−1g⊥µB[Hres(Hres+Ha)]1/2\nforH⊥corientation to obtain the g-factors and the\nanisotropy field Ha. It was found that the g-factors in\nboth orientations are somewhat different amounting to\ng⊥= 1.968±0.003andg/bardbl= 1.958±0.003. Such a\nslightly anisotropic g-tensor is typical for a Cr3+ion (3d3,\nS= 3/2,L= 3) in a distorted octahedral ligand coordi-\nnation [ 53]. Further, the anisotropy field Hawas obtained\nas∼2864Oe from the fit. Generally, Haconsists of twocontributions [ 54]:\nHa= 4πM−2K/M. (14)\nThe first term accounts for the shape anisotropy and\nthe second term is the intrinsic magnetocrystalline\nanisotropy. In Eq. ( 14), the first term assumes the shape\nanisotropy of a thin plate with N= 1.\nUsing the saturation magnetization value, Ms=g/bardblS=\n2.94µB/Cr3+, the magnetocrystalline anisotropy con-\nstantK=−7.42×104erg cm−3was obtained. The neg-\native sign of Kimplies that LCPP is an easy-plane fer-\nromagnet with the hard magnetic axis normal to the ab-\nplane. We note that no other resonance excitations could\nbe found in a frequency range up to 975 GHz (4 meV).\n2. Temperature Dependence\nHF-ESR spectra of LCPP at various temperatures\nwere measured at a fixed excitation frequency of ν=\n141.310GHz for both field directions [see Figs. 11(a)\nand (b)]. The resonance fields Hreswere obtained from\nthe absorption minima of each spectrum and are plotted\nagainst temperature in Fig. 12(a). Interestingly, the Hres\nvsTdependence for both orientations does not converge\nto a paramagnetic line immediately at TC= 2.8K. Only\nabove 10 K, the resonance fields for H∝bardblcandH⊥cori-\nentations rapidly start to decrease and increase, respec-\ntively, toward the expected paramagnetic position and\nalmost merge around 100 K at a field corresponding to\ntheg-factor,g= 1.97. The lineshape distortions in mea-\nsurements at low temperatures for H⊥care accounted\nfor in the enlarged error bars.\nThe shift of the resonance position δH(T)from the\nparamagnetic one is shown in Fig. 12(b) (left y-axis).\nδH(T)is positive and larger when the external field is\nparallel to the magnetic hard axis, as compared to the\nsmaller negative shift for the in-plane field geometry, as10\n/s52/s46/s56 /s53/s46/s50 /s53/s46/s54 /s52/s46/s56 /s53/s46/s49 /s53/s46/s52/s40/s98/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s51/s48/s48/s32/s75\n/s50/s53/s48/s32/s75\n/s53/s48/s32/s75/s55/s53/s32/s75/s49/s48/s48/s32/s75/s50/s48/s48/s32/s75\n/s49/s53/s48/s32/s75\n/s52/s32/s75/s55/s32/s75/s49/s48/s32/s75/s50/s48/s32/s75\n/s50/s32/s75/s51/s48/s32/s75\n/s50/s46/s50/s53/s32/s75/s51/s32/s75\n/s50/s46/s55/s53/s32/s75\n/s50/s46/s53/s32/s75/s72 /s32 /s32/s99\n/s32/s32\n/s72 /s32/s40/s84/s41/s49/s46/s56/s32/s75/s32/s61/s32/s49/s52/s49/s46/s51/s49/s48/s32/s71/s72/s122\n/s32/s32\n/s72 /s32/s40/s84/s41/s40/s97/s41\n/s72 /s32 /s32/s99\nFIG. 11. Temperature dependence of the HF-ESR spectra at\nan excitation frequency of 141.310 GHz for (a) H⊥cand (b)\nH/bardblc. Line shape distortions at T <20K in the H⊥cfield\ngeometry are instrumental artifacts arising due to the stro ng\nmagnetization of the sample.\nexpected for an easy-plane ferromagnet. Such a shift can-\nnot be ascribed entirely to the shape anisotropy, which\nshould play a role also in the paramagnetic state if the\nsample’s magnetization Mis large. The M(T)curve\nplotted in Fig. 12(b) (right y-axis) for comparison de-\ncreases right above TCmore rapidly than the δH(T)de-\npendence. Therefore, the line shift observed for both\norientations well above the Curie temperature may be in-\ndicative of short-range FM spin correlations on the fast\nESR time scale, typical for low-dimensional magnets such\nas, e.g., the quasi-two-dimensional van der Waals com-\npound Cr 2Ge2Te6[55]. This is also the case for LCPP\nowing to its layered crystal structure.\nF. Microscopic Analysis\nOur DFT calculations return nearest-neighbor ex-\nchange coupling J/kB≃ −0.5K within the kagome\nplanes. This value is somewhat dependent on the\nchoice of the DFT+ Uparameters, but the negative\nsign is robust and suggests ferromagnetic nature of the\nkagome network in LCPP. The origin of this ferro-\nmagnetic coupling deserves some attention, as the sib-\nling Fe3+compound is clearly antiferromagnetic [ 17].\nSuperexchange theory stipulates that antiferromagnetic\ncouplings are mediated by hoppings between half-filled\norbitals, whereas ferromagnetic couplings arise from hop-\npings between the half-filled and empty orbitals. In/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32/s40/s98/s41/s32/s72 /s32/s124/s124/s32 /s99\n/s32/s72 /s32 /s32/s99/s72 /s32/s40/s84/s41/s61/s32/s49/s52/s49/s46/s51/s49/s48/s32/s71/s72/s122\n/s84 /s32/s40/s75/s41/s53/s46/s48/s53/s46/s50/s53/s46/s52/s72\n/s114/s101/s115/s32/s40/s84/s41/s103\n/s84 /s32/s61/s32/s51/s48/s48/s32/s75/s32/s61/s32/s49/s46/s57/s55/s40/s97/s41\n/s45/s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s48/s46/s48/s49/s53\n/s77\n/s72 /s32/s61/s32/s48/s46/s53/s32/s84/s32/s40/s101/s109/s117/s41\nFIG. 12. Temperature dependence of (a) the resonance field\nHresand (b) the resonance shift δH=Hres(T)−Hres(300 K)\n(lefty-axis) and the magnetization Min an applied field of\nH= 0.5T (right y-axis), for both H/bardblcandH⊥c.\nLCPP with the 3d3Cr3+magnetic ion, these orbitals\nhave the t2gandeσ\ngcharacters, respectively. Trigonal\nsymmetry of the crystal structure further splits the t2g\nlevels into a1gandeπ\ng.\n100\n0\n0 1 2a e1g g+π\n( )t2g\negσe eg gπ σ/c45\ne eg gπ π/c45200\n20 40a a1 1g g/c45a e1g g/c45σ\na e1g g/c45π\n60AFMFM\n80 0(b)(a)\nHopping amplitude (meV) Energy (eV)DOS (eV /f.u.)/c451\nFIG. 13. (a) PBE density of states for LCPP with the Fermi\nlevel placed at zero energy. (b) Nearest-neighbor Cr–Cr hop -\nping amplitudes within the kagome plane.\nDFT band structure of LCPP calculated on the PBE\nlevel features narrow t2g(a1g+eπ\ng) bands around the\nFermi level and almost equally narrow eσ\ngbands centered\nat around 2.0 eV (Fig. 13). This band structure is metal-\nlic because neither magnetism nor correlation effects have\nbeen taken into account. The small band width of 0.25 eV\nfor thet2gbands indicates that antiferromagnetic contri-11\n10/c45210/c451100\n10kagome planeexperiment\nkagome plane + interlayer\n100\nT(K)101102\nχ(cm mol Cr )3 3+( )/c451\n13\n069\n10Cp(J mol K )/c45 /c451 1\nT(K)\nFIG. 14. Magnetic susceptibility of LCPP measured in the\napplied field of 0.01 T upon field cooling and its fit using\nthe model of ferromagnetic kagome planes (dashed line) and\ncoupled ferromagnetic kagome planes ( J⊥/J= 0.01, solid\nline). The inset shows calculated specific heat for the cou-\npled kagome planes, with TC≃2.8K.\nbution to the exchange couplings should be minor. The\nt2g−eσ\nghoppings are small too, but somewhat larger than\nthet2g−t2ghoppings, as shown in Fig. 13(b). There-\nfore, ferromagnetic contribution to the exchange becomes\npredominant, and the overall coupling is ferromagnetic.\nMagnetic susceptibility of LCPP is well described by the\nmodel of nearest-neighbor ferromagnetic kagome planes\nwithJ/kB=−1.2K (g= 1.995). A minute interlayer\ncoupling J⊥/J= 0.01improves the fit below 3.5 K and\nleads to the Curie temperature TC= 2.8K in a perfect\nagreement with the experimental TC≃2.6K (Fig. 14).Our DFT calculations corroborate this result and reveal\na weakly ferromagnetic J⊥withJ⊥/J≃0.02.\nIV. SUMMARY\nWe synthesized single crystals of LCPP and confirmed\ntrigonalP¯3c1symmetry of this compound with the lat-\ntice constants a=b= 9.668(3) Å andc= 13.610(6) Å at\nroom temperature. Green-colored LCPP is a rare exam-\nple of an insulating kagome ferromagnet. Ferromagnetic\norder below TC≃2.6K is driven by the in-plane FM\ncoupling J/kB≃ −1.2K supplied with a minute inter-\nplane coupling J⊥/J= 0.02, which is also FM in nature.\nThe incomplete release of the magnetic entropy at TCand\nthe increased width of the ESR line above TCboth sug-\ngest quasi-2D magnetic behavior caused by the strong\nspatial anisotropy of FM couplings. The overall mag-\nnetic behavior of LCPP has striking resemblance with\nthat of other insulating kagome ferromagnets, such as α-\nMgCu 3(OD)6Cl2and Cu[1,3-bdc] [ 8,56]. Magnetization\nand ESR measurements on single crystals indicate a weak\neasy-plane anisotropy. The critical scaling of magnetiza-\ntion suggests a non-mean-field type second-order nature\nof the phase transition at TC. 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Kataev1\n1Leibniz IFW Dresden, D-01069 Dresden, Germany\n2Institute for Solid State and Materials Physics, TU Dresden, D-01062 Dresden, Germany\n3W urzburg-Dresden Cluster of Excellence ct.qmat, Germany\n4Department of Materials and Environmental Chemistry,\nStockholm University, SE-106 91, Stockholm, Sweden\n5Faculty of Chemistry and Food Chemistry, TU Dresden, D-01062 Dresden, Germany\n(Dated: June 1, 2020)\nWe report a comprehensive high-\feld/high-frequency electron spin resonance (ESR) study on sin-\ngle crystals of the van der Waals magnet CrCl 3. This material, although being known for quite a\nwhile, has received recent signi\fcant attention in a context of the use of van der Waals magnets in\nnovel spintronic devices. Temperature-dependent measurements of the resonance \felds were per-\nformed between 4 and 175 K and with the external magnetic \feld applied parallel and perpendicular\nto the honeycomb planes of the crystal structure. These investigations reveal that the resonance line\nshifts from the paramagnetic resonance position already at temperatures well above the transition\ninto a magnetically ordered state. Thereby the existence of ferromagnetic short-range correlations\nabove the transition is established and the intrinsically two-dimensional nature of the magnetism\nin the title compound is proven. To study details of the magnetic anisotropies in the \feld-induced\ne\u000bectively ferromagnetic state at low temperatures, frequency-dependent ferromagnetic resonance\n(FMR) measurements were conducted at 4 K. The observed anisotropy between the two magnetic-\n\feld orientations is analyzed by means of numerical simulations based on a phenomenological theory\nof FMR. These simulations are in excellent agreement with measured data if the shape anisotropy\nof the studied crystal is taken into account, while the magnetocrystalline anisotropy is found to be\nnegligible in CrCl 3. The absence of a signi\fcant intrinsic anisotropy thus renders this material as a\npractically ideal isotropic Heisenberg magnet.\nI. INTRODUCTION\nMagnetic van der Waals materials belong to a class of\nphysical systems that currently receive considerable at-\ntention in solid state and materials research [1{6]. As a\ncommon feature these materials crystallize in a layered\nstructure with the individual layers being separated by\nthe so-called van der Waals gap, see Fig. 1. The weak\nvan der Waals coupling between the adjacent layers re-\nsults in dominating magnetic interactions within the lay-\ners while magnetic couplings between the layers remain\nrelatively weak. Thus, van der Waals magnets can be\nconsidered as quasi-two-dimensional magnetic systems.\nThese systems enable experimental studies of the spe-\nci\fc magnetic properties arising from the interplay of\nan e\u000bectively reduced dimensionality and the respective\nsingle-ion anisotropies determined by the type of mag-\nnetic ions and their local environments, see for instance\n[4]. Moreover, the weak van der Waals bonds between the\nlayers allow mechanical exfoliation of bulk crystals down\nto the few-layer or even monolayer limit [2, 3], thereby\napproaching the experimental systems to the true two\ndimensional (2D) limit. In addition, the combination of\nvarious materials sharing similar layer structures and the\nweak van der Waals couplings between the layers open up\n\u0003These authors contributed equally to this work.numerous possibilities for the creation of (magnetic) van\nder Waals heterostructures [5{7]. These stacks of sev-\neral few-layer crystals of di\u000berent materials are a promis-\ning route towards electronic devices with speci\fcally tai-\nlored magnetic properties, see, e.g., Ref. [6] and refer-\nences therein. In both respects { the fundamental study\nof 2D magnetism and its application in the framework of\nmagnetic heterostructures { the characterization of mag-\nnetic anisotropies in van der Waals materials represents a\nkey task. Magnetic resonance spectroscopies are valuable\ntools to accomplish this task due to their high sensitivity\nwith respect to the presence of magnetic anisotropies in\na system. As an example, in a previous work [8] some\nof the authors quantitatively investigated the magnetic\nanisotropies in the van der Waals magnet Cr 2Ge2Te6by\nmeans of electron spin resonance (ESR) and ferromag-\nnetic resonance (FMR) studies. Here, we report a char-\nacterization of the (e\u000bective) magnetic anisotropy in the\nhigh-\feld phase of the related compound CrCl 3. The last\nis a member of the family of transition-metal trihalides\nwhose magnetic ions (here: Cr3+, 3d3,S= 3=2,L= 3)\nare situated in the center of an octahedron built by the\nhalogen ligands (here: Cl\u0000ions). These octahedra form\nedge-sharing networks in the crystallographic abplane\nwhich e\u000bectively results in a magnetic honeycomb lat-\ntice, see Fig. 1.\nIt is worthwhile to mention that the title compound\nbelongs to the \frst materials studied using the ESR tech-arXiv:2005.14559v1  [cond-mat.mtrl-sci]  29 May 20202\nFIG. 1. Crystal and magnetic structures of CrCl 3at low temperatures. (a) View along the b-axis of the unit cell (non-primitive\nhexagonal) of the rhombohedral low-temperature structure (space group R3 [9]). The magnetic Cr3+ions (red spheres) are\noctahedrally coordinated by the Cl ligands (green spheres). (b) These CrCl 6octahedra build an edge-sharing network in the\nabplane which e\u000bectively leads to a honeycomb-like arrangement of the magnetic ions (illustrated by solid red lines). Each\nof these honeycomb layers in the abplane is well separated from its neighbors along the caxis by the van der Waals gap,\nas shown in (a). At temperatures below the transition into a magnetically long-range ordered state, spins in the honeycomb\nlayers are coupled ferromagnetically and oriented within the ab-plane, while spins in neighboring layers are coupled by a weaker\nantiferromagnetic interaction [10, 11]. The resulting spin structure in the magnetically ordered state at zero magnetic \feld\n[10, 11] is schematically illustrated by the arrows at the Cr sites. Crystallographic data are taken from Ref. [9].\nnique by E. K. Zavoisky, the pioneer of this spectroscopy\n(see, for instance, [12, 13]). However, in the context of\nmagnetic van der Waals materials the magnetic prop-\nerties of CrCl 3came back to the focus of current re-\nsearch interest [11, 14{23]. Basic magnetic properties\nof bulk CrCl 3crystals have been reported in the past\ndecades, see, e.g., Refs. [11, 16, 24{28]. In particular,\na two-step transition into a magnetically long-range or-\ndered state was observed [11, 16, 28]. Below a temper-\natureT2D\nc\u001817 K, spins within the honeycomb layers\norder ferromagnetically and align parallel to the honey-\ncomb plane, i.e., the abplane [cf. Fig. 1(b)], while spins\nof individual Cr layers are not coupled to spins in the\nneighboring layers [11, 16, 28]. Consequently, the order-\ning at around 17 K is of an e\u000bectively 2D nature. The\nferromagnetic character of the dominant exchange inter-\nactions between spins in the abplane is also evidenced\nby the positive Curie-Weiss temperatures \u0002 CWbetween\n27 and 43 K derived from measurements of the static sus-\nceptibility [11, 16, 17, 25, 29, 30]. Below a temperature\nT3D\nNof about 14 K [11, 16] (15.5 K in Ref. [28]) CrCl 3en-\nters into a three-dimensional (3D) antiferromagnetically\nordered state at zero magnetic \feld. In this magnetic\nphase, ferromagnetically ordered spins in the honeycomb\nlayers are coupled by antiferromagnetic interactions be-\ntween neighboring layers [10], as illustrated in Fig. 1(a).\nHowever, the long range antiferromagnetic order can be\nsuppressed already in relatively small magnetic \felds of\nabout 0.6 T (external \feld applied perpendicular to thehoneycomb planes, i.e., Hkc) and 0.25 T (external \feld\napplied in the honeycomb planes, i.e., H?c) at 2 K [11],\nrespectively. Above these \felds, a saturation of the mag-\nnetization at around 3 \u0016B/Cr was observed [11, 16]. The\nlow values of the saturation \felds thus con\frm the (rel-\native) weakness of the antiferromagnetic interlayer cou-\nplings. Moreover, if demagnetization e\u000bects are taken\ninto account, the saturation \felds become almost identi-\ncal for both orientations of the external \feld with respect\nto the honeycomb layers [11, 16]. Therefore, the exper-\nimentally observed magnetic anisotropies appear to be\ndominated by dipole-dipole interactions which are at the\norigin of demagnetization \felds and the so-called shape\nanisotropy [11, 16]. The apparent size of the magnetic\nanisotropy thus depends strongly on the dimensions of\nthe studied samples whereas the intrinsic magnetocrys-\ntalline is expected to be much weaker, if it exists at all.\nIn order to disentangle and quantify the two contribu-\ntions to an anisotropic magnetic response, which could\nbe of relevance in the case of CrCl 3, we studied in this\nwork the details of the magnetic anisotropies in the \feld-\ninduced ferromagnetic-like phase of the title compound\nby means of high-\feld/high-frequency (HF) FMR over\na wide range of frequencies at magnetic \felds exceeding\nthe low-temperature saturation \felds of CrCl 3.3\nII. SAMPLES AND EXPERIMENTAL\nMETHODS\nSingle-crystalline CrCl 3samples used in this study\nwere synthesized by a chemical vapor transport reac-\ntion between Cr metal and Cl 2gas as described in detail\nin Ref. [29]. The results of single-crystal X-ray di\u000brac-\ntion studies at room temperature con\frming the mon-\noclinic (space group C2=m) high-temperature modi\fca-\ntion of the title compound are also reported there. More-\nover, magnetic properties of the samples were studied\nin Ref. [16] by means of speci\fc heat as well as static\nand dynamic magnetization measurements. These are\nconsistent with the \fndings reported in previous studies\n[11, 26, 28], in particular, they con\frm the presence of\ntwo successive magnetic phase transitions mentioned in\nthe previous section. This does not only demonstrate the\nhigh quality of the used CrCl 3crystals but also ensures\nthe comparability of the magnetic properties reported in\nthis work and in the literature [11, 23, 26, 28]. Additional\ncharacterization details can be found in the Appendix.\nThe compound CrCl 3is known to undergo a struc-\ntural phase transition at temperatures around 240 K from\nthe high-temperature monoclinic phase to an rhombo-\nhedral phase (space group R3) at lower temperatures\n[9, 11]. These two structural modi\fcations di\u000ber mainly\nin the stacking sequence of the honeycomb layers while\nthe structure within the layers is very similar in both\nphases [11]. The crystal structure of the rhombohedral\nmodi\fcation is shown in Fig. 1 as the focus of the present\nstudy lies on the magnetic properties at lower tempera-\ntures. In this phase, individual honeycomb layers are\nstacked along the caxis in an -ABC-sequence. Conse-\nquently, each unit cell contains three layers along the c\naxis, see Fig. 1(a). The strong chemical bonding within\ntheabplane and the comparatively weak van der Waals\ncouplings between the layers result in \rat, platelet-like\nsingle crystals allowing an easy identi\fcation of the caxis\nas the direction perpendicular to the platelet plane.\nThe ESR and FMR measurements were carried out us-\ning two di\u000berent setups. For continuous wave (cw) HF-\nESR/HF-FMR studies a homemade spectrometer was\nemployed. The spectrometer consists of a network vec-\ntor analyzer (PNA-X from Keysight Technologies) for\ngeneration and detection of microwaves in the frequency\nrange from 20 to 330 GHz, oversized waveguides, and a\nsuperconducting solenoid (Oxford Instruments) provid-\ning magnetic \felds up to 16 T. The magnetocryostat is\nequipped with a variable temperature insert that en-\nables measurements in the temperature range between\n1.8 and 300 K. All HF measurements presented in the\nfollowing section were carried out in transmission ge-\nometry employing the Faraday con\fguration. In addi-\ntion, cw ESR/FMR measurements at a \fxed microwave\nfrequency of about 9.6 GHz and in magnetic \felds up\nto 0.9 T were performed using a commercial X-band\nspectrometer (EMX from Bruker). This spectrome-\nter is equipped with a helium \row cryostat (ESR900from Oxford Instruments) and a goniometer, allowing\ntemperature-dependent measurements in the range 4 -\n300 K and angle-dependent studies, respectively.\nIII. RESULTS AND DISCUSSION\nIn the following, results of systematic ESR measure-\nments are presented and discussed. These measurements\nwere carried out with the external magnetic \feld Hap-\nplied parallel and perpendicular to the crystallographic\ncaxis, respectively. Since the focus of the present study\nlies on a detailed investigation of the (e\u000bective) magnetic\nanisotropies in the \feld-polarized ferromagnetic state of\nCrCl 3at low temperatures, this work is mainly con-\ncerned with the behavior of the resonance \feld Hresas a\nfunction of temperature and microwave frequency over a\nbroad frequency range. Further aspects of the magnetic\nproperties of CrCl 3derived from ESR measurements at\nlower microwave frequencies, such as the excitations of\nthe 3D antiferromagnetic state and the spin dynamics\nabove the magnetic phase transitions, were reported in\nRefs. [23, 31].\nA. Temperature dependence\nThe temperature dependence of the resonance shift\n\u000eH(T) =Hres(T)\u0000Hres(100K) measured at low and\nhigh microwave frequencies \u0017and in both magnetic \feld\ncon\fgurations is shown in Fig. 2. This quantity is a mea-\nsure of the deviation of the resonance \feld at a given\ntemperature Tfrom the ideal paramagnetic resonance\nposition. This position is determined by the standard\nresonance condition of a paramagnet [32]\n\u0017=g\u0016B\u00160Hres=h : (1)\nHere,gdenotes the gfactor of the resonating spins and\n\u0016B,\u00160, andhare Bohr's magneton, the vacuum per-\nmeability, and Planck's constant, respectively. In the\npresent case, the resonant shift was determined with re-\nspect to the expected resonance \feld at 100 K which was\ncalculated according to Eq. (1) using the gfactor de-\nrived from frequency-dependent measurements at 100 K,\nsee below. Upon lowering the temperature below \u001875 K,\nthe resonance position is shifted progressively to smaller\n\felds when the external magnetic \feld is applied perpen-\ndicular to the caxis, resulting in a negative shift \u000eH.\nForHkcthis trend is reversed yielding a positive res-\nonance shift. Thus, based on the qualitative behavior\nof the temperature-dependent resonance shift, it can be\nconcluded that the experimentally observable (e\u000bective)\nmagnetic easy direction lies within the abplane which is\nin agreement with the proposed spin structure [10] and\nprevious magnetization measurements [11, 16, 26]. More-\nover, the\u000eH(T) curves obtained for HkcandH?c\nare asymmetric with respect to the \u000eH= 0 line of an4\nideal paramagnet, see Fig. 2. This asymmetry is a direct\nconsequence of the di\u000berent frequency-\feld dependencies\n\u0017(Hres) expected for the two di\u000berent \feld orientations in\na ferromagnetically ordered system [33, 34]. In particu-\nlar, a shift of the resonance line should be larger when the\nexternal magnetic \feld is oriented parallel to the mag-\nnetic anisotropy axis which in the case of CrCl 3is the\nmagnetic hard axis parallel to the crystallographic caxis.\nThe onset of a \fnite resonance shift already at temper-\natures signi\fcantly above the ordering temperature T2D\nc\nprovides clear evidence for the low-dimensional character\nof the spin correlations in this material as it was discussed\nin the context of one-dimensional systems, for instance in\nRefs. [35{37]. Moreover, the 2D nature of magnetic cor-\nrelations above T2D\ncwas reported in Ref. [31] based on\nmeasurements of the resonance shift and the linewidth\nangular dependence at various temperatures and at low\nmicrowave frequencies of about 9.4 GHz. While our mea-\nsurements of \u000eH(T) at 9.6 GHz are consistent with this\nprevious study [31], we observed the onset of a \fnite\nresonance shift at higher temperatures when employing\nfrequencies of about 90 GHz. Thus, the higher exter-\nnal magnetic \felds associated with the higher microwave\nfrequencies strengthen the ferromagnetic correlations re-\nsponsible for the shift of the resonance line, similar to\nthe situation found in the related van der Waals magnet\nCr2Ge2Te6[8]. Finally, it is worthwhile mentioning that\nthe appearance of short-range correlations at tempera-\ntures far above the magnetic ordering temperature is con-\nsistent with the deviation of the temperature-dependent\nstatic susceptibility from a Curie-Weiss behavior already\nbelow\u0018125 K, which was reported in Ref. [29]. Taken to-\ngether, temperature-dependent measurements of the res-\nonance shift at two di\u000berent frequencies and \feld orien-\ntations demonstrate, \frst, the apparent easy-plane type\nmagnetic anisotropy and, second, the 2D character of dy-\nnamic spin correlations in CrCl 3far above the long-range\nordering temperatures.\nB. Frequency dependence\nTo shed light on the details of the magnetic\nanisotropies, frequency-dependent investigations were\nconducted at 4 and 100 K, i.e., at temperatures deep in\nthe magnetically ordered state and well above the mag-\nnetic phase transition, respectively. Exemplary spectra\nare presented in the insets of Fig. 3. At both tem-\nperatures, ESR/FMR spectra consist of a single nar-\nrow resonance line with typical linewidths (full width\nat half maximum) of about 25 mT at 100 K. The small\nlinewidths allow an easy and precise determination of\nthe resonance \felds which correspond to the positions\nof the minima in the microwave transmission. The re-\nsulting frequency-\feld diagrams are shown in the main\npanels of Fig. 3. At 100 K a linear frequency-\feld de-\npendence\u0017(Hres) is observed for both orientations of the\nexternal magnetic \feld. Such a behavior is expected in\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53 /s49/s53/s48 /s49/s55/s53/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48\n/s32/s72 /s32/s124 /s124 /s32 /s99 /s32/s58/s32 /s110 /s32/s61/s32/s57/s48/s46/s48/s32/s71/s72/s122\n/s32/s72 /s32/s124 /s124 /s32 /s99/s32 /s58/s32 /s110 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122\n/s32/s72 /s32/s94 /s32/s99 /s32/s58/s32 /s110 /s32/s61/s32/s57/s48/s46/s49/s32/s71/s72/s122\n/s32/s72 /s32/s94 /s32/s99/s32 /s58/s32 /s110 /s32/s61/s32/s57/s46/s54/s32/s71/s72/s122/s109\n/s48/s40/s72\n/s114/s101/s115/s40/s84 /s41/s32/s45/s32 /s72\n/s114/s101/s115/s40/s49/s48/s48/s32/s75/s41/s41/s32/s40/s109/s84/s41\n/s84 /s32/s40/s75/s41/s84/s32/s50/s68\n/s67/s84/s32/s51/s68\n/s78FIG. 2. Temperature dependence of the resonance shift\n\u000eH(T) =Hres(T)\u0000Hres(100K) at microwave frequencies\nof about 9.6 and 90 GHz (open and \flled symbols, respec-\ntively). In these measurements the external magnetic \feld\nwas oriented parallel (red circles) as well as perpendicular\n(blue squares) to the caxis. The dashed horizontal line rep-\nresents the zero shift \u000eH= 0 expected for an uncorrelated\nparamagnet. Vertical dashed lines indicate the zero-\feld tran-\nsition temperatures T2D\ncandT3D\nNof the transition into the 2D\nferromagnetically ordered phase and the 3D antiferromagnet-\nically ordered state [11], respectively.\nthe paramagnetic regime of CrCl 3and can be well de-\nscribed by the standard resonance condition of a param-\nagnet [Eq. (1)] [32]. Fits to the data according to Eq. (1)\nare shown in Fig. 3(a) as solid lines and yielded gfactors\ngk= 1:970\u00060:005 andg?= 1:990\u00060:005 forHkc\nandH?c, respectively. The experimentally determined\ngfactors are in good agreement with the values antici-\npated for Cr3+ions (3d3,S= 3=2,L= 3) in an octa-\nhedral crystal \feld [32]. The merely small deviation of g\nfrom the free-electron gfactor of 2 as well as the slight\nanisotropy observed in the measurements indicate that\nthe magnetism in CrCl 3is largely dominated by the spin\ndegrees of freedom while the orbital angular momentum\nis practically completely quenched in the \frst order. The\nmentioned small deviations from the ideal spin-only be-\nhavior result, most likely, from second-order spin-orbit\ncoupling e\u000bects [32]. The gfactors derived in this work\nfrom frequency-dependent measurements are, moreover,\nconsistent with the saturation magnetization of about\n3\u0016B/Cr which was experimentally observed [11, 16] and\nis theoretically expected for spins S= 3=2 and agfac-\ntor of 2. These observations already suggest that spin-\norbit coupling and, consequently, the intrinsic magne-\ntocrystalline anisotropies are very weak (or even negligi-\nble) in CrCl 3, as it was also mentioned in the literature\n[11, 16, 26].5\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s52/s48/s48/s110 /s32/s40/s71/s72/s122/s41\n/s109\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s32/s72 /s32/s124/s124/s32 /s99\n/s32/s72 /s32/s94 /s32/s99/s84 /s32/s61/s32/s52/s32/s75/s49/s46/s53 /s49/s46/s54 /s53/s46/s52 /s53/s46/s54 /s57/s46/s54 /s57/s46/s57 /s49/s48/s46/s50\n/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32\n/s115/s105/s103/s110/s97/s108/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s109\n/s48/s32/s72 /s32/s40/s84/s41/s51/s52/s32/s71/s72/s122 /s49/s52/s52/s32/s71/s72/s122 /s50/s54/s50/s32/s71/s72/s122\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50 /s49/s51/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48\n/s32/s72 /s32/s124/s124/s32 /s99/s32 /s58/s32 /s103 /s32/s61/s32/s49/s46/s57/s55/s48/s32 /s177 /s32/s48/s46/s48/s48/s53/s32\n/s32/s72 /s32/s94 /s32/s99/s32 /s58/s32 /s103 /s32/s61/s32/s49/s46/s57/s57/s48/s32 /s177 /s32/s48/s46/s48/s48/s53/s110 /s32/s40/s71/s72/s122/s41\n/s109\n/s48/s32/s72\n/s114/s101/s115/s32/s40/s84/s41/s84 /s32/s61/s32/s49/s48/s48/s32/s75\n/s51/s46/s48 /s51/s46/s53 /s55/s46/s48 /s55/s46/s53 /s49/s48/s46/s48 /s49/s48/s46/s53/s109\n/s48/s32/s72 /s32/s40/s84/s41/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32\n/s115/s105/s103/s110/s97/s108/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s57/s48/s32/s71/s72/s122\n/s49/s57/s55/s32/s71/s72/s122/s50/s55/s53/s32/s71/s72/s122/s40/s97 /s41 /s40/s98 /s41\nFIG. 3. Frequency dependence of the resonance \feld Hresat 100 K (a) and 4 K (b), respectively. To quantitatively investigate\nthe magnetic anisotropies in CrCl 3, measurements were carried out with the external magnetic \feld applied parallel (red circles)\nand perpendicular (blue squares) to the crystallographic caxis, respectively. Open symbols correspond to the resonance \felds\nmeasured at \u0017= 9:6 GHz. Solid lines in (a) are linear \fts to the \u0017(Hres) dependencies according to the standard resonance\ncondition of a paramagnet given in Eq. (1) [32]. At 4 K, the measured frequency-\feld dependencies were simulated using\nEq. (2) which describes the resonance frequency in the case of FMR. Corresponding simulations are shown in (b) as solid\nlines. Exemplary spectra recorded with Hkcare presented for both temperatures in the respective insets. For comparison, all\nspectra shown were normalized (note the di\u000berent breaks in the \feld axes). Arrows in the \u0017(Hres) denote the positions of the\nspectra given in the insets.\nFor a detailed analysis of the relevant anisotropies\npresent in the title compound, frequency-dependent mea-\nsurements were conducted at 4 K, i.e., in the magnetically\nordered state. We emphasize that the lowest frequency\nused in the systematic HF measurements corresponds to\na resonance \feld of about 0.75 T [cf. Fig. 3(b)] which\nexceeds the reported saturation \felds [26]. Therefore,\nthe \feld-polarized, e\u000bectively 2D ferromagnetic state of\nCrCl 3is probed by our HF magnetic resonance measure-\nments allowing us to describe the obtained \u0017(Hres) de-\npendencies in the framework of a theory of FMR that\nis applicable in a single-domain ferromagnetic material,\nas will be discussed in the following section. In contrast\nto the measurements in the paramagnetic state, the 4 K\ndata show a clear anisotropy regarding the two magnetic-\n\feld orientations which is consistent with the resonance\nshifts of opposite sign observed in the temperature-\ndependent studies (see Fig. 2). If the external magnetic\n\feld is applied within the abplane (H?c), the resonance\npositions are systematically shifted towards smaller mag-\nnetic \felds (with respect to the paramagnetic resonance\npositions) for all measured frequencies. For Hkca neg-\native intercept with the frequency axis of the \u0017(Hres)\ndiagram is observed due to the shift of the resonance po-\nsitions to higher magnetic \felds. Qualitatively, such a\nbehavior is expected in the case of a ferromagnetically\nordered system with an easy-plane anisotropy [34, 38].\nIn a semi-classical picture, these shifts of the resonanceposition result from anisotropic internal \felds in the mag-\nnetic crystal. These \felds are caused, in turn, by dipole-\ndipole interactions between the magnetic moments asso-\nciated with the spins or an intrinsic magnetocrystalline\nanisotropy, i.e., by spin-orbit coupling. In the following\nsection the resonance \felds in the ferromagnetic state\nwill be simulated employing a theory based on this semi-\nclassical description of magnetic resonance. This allows\nto determine quantitatively the contributions of the two\npossible sources of magnetic anisotropy in CrCl 3.\nC. Analysis of the magnetic anisotropies\nAs mentioned in the previous sections, it is possi-\nble to describe the experimentally observed frequency-\n\feld dependence in the ferromagnetic state at 4 K in the\nframework of a semi-classical, phenomenological theory\nof FMR [34, 38, 39]. Conceptually, the di\u000berence between\nESR and FMR lies in the fact, that ESR is the resonant\nexcitation of individual paramagnetic spins within a mag-\nnetic system, while FMR describes the resonance of the\ntotal magnetization Min a ferromagnetically ordered\nmaterial. Thus, the resonance frequencies expected for a\ncorrelated ferromagnetic spin system can be calculated\nby considering the classical vector of the macroscopic\nmagnetization and the appropriate free energy density\nF[34, 38, 39]. In this case, the resonance frequency is6\ngiven by the following expression [34, 38, 39]\n\u00172\nres=g2\u00162\nB\nh2M2ssin2\u0012\u0012@2F\n@\u00122@2F\n@'2\u0000\u0010@2F\n@\u0012@'\u00112\u0013\n;(2)\nwhereMsis the saturation magnetization and 'and\u0012\ndenote the spherical coordinates of the magnetization\nvectorM(Ms;';\u0012). Note that in the present case the\nspherical coordinate system is chosen such that the z\naxis coincides with the crystallographic caxis in the low-\ntemperature structure of CrCl 3. For a calculation of the\nresonance position under given experimental conditions,\ni.e., a speci\fc orientation of the external magnetic \feld\nrelative to the studied sample and a \fxed microwave fre-\nquency, Eq. (2) has to be evaluated at the equilibrium po-\nsition ('0;\u00120) of the macroscopic magnetization vector.\nThe orientation of Min the equilibrium state is found\nnumerically by minimizing Fwith respect to the spheri-\ncal coordinates, taking into account the particular exper-\nimental conditions. Since the qualitative considerations\nregarding the temperature dependence of the resonance\nshift as well as the \u0017(Hres) dependencies at 4 K suggest\nan easy-plane type anisotropy, a uniaxial magnetocrys-\ntalline anisotropy was included in the free energy density\nterm as an initial approach to describe the anisotropies\nin CrCl 3. The free energy density is then given (in SI\nunits) by\nF=\u0000\u00160H\u0001M\u0000KUcos2(\u0012)+\n1\n2\u00160M2\ns(Nxsin2(\u0012) cos2(')+\nNysin2(\u0012) sin2(') +Nzcos2(\u0012)):(3)\nThe \frst term is the Zeeman-energy density describing\nthe coupling between the magnetization vector Mand\nthe external magnetic \feld H. The second term repre-\nsents the uniaxial magnetocrystalline anisotropy whose\nstrength is parametrized by the energy density KU. Fi-\nnally, the third contribution to Fis the shape anisotropy\nenergy density which is characterized by the demagneti-\nzation factors Nx,Ny, andNz[40, 41]. These factors are\ndetermined by the dimensions of the crystals under study.\nIn the present case, the shape of the measured platelet-\nlike CrCl 3crystal was described by the demagnetization\nfactors of an extended \rat plate ( Nx=Ny= 0;Nz= 1\n[42]), whose xyplane corresponds to the crystallographic\nabplane and the zaxis lies parallel to the caxis. This\napproximation to the real sample shape is justi\fed by\nthe platelet-like shape of the studied CrCl 3crystal whose\nlateral dimensions in the abplane (of about 400 \u0016m\u0002\n450\u0016m) are much larger than the thickness of the crys-\ntal along the caxis. Due to experimental reasons, the\nsample thickness could not be measured precisely which\nhampered a determination of the demagnetization factors\nsolely based on the sample dimensions. However, the de-\nviations between the approximated and the true demag-\nnetization factors can be expected to be small. In addi-\ntion to the sample shape, the value of the saturation mag-\nnetizationMsenters into the simulation of the frequency-\n\feld dependence. For the simulations presented in thefollowing, the reported saturation magnetization of about\n3\u0016B/Cr at 1.8 K [16] was used which corresponds to\nMs\u0019314:97\u0002103J/Tm3. Furthermore, the gfactors\ndetermined independently from the frequency-dependent\nmeasurements at 100 K and for both \feld orientations\n(see Sec. III B) were set as \fxed parameters in the sim-\nulations. Thus, the only free parameter, which was ad-\njusted to match simulated with measured data, was the\nmagnetocrystalline anisotropy energy density KU.\nThe \fnal results of the simulations are presented in\nFig. 3(b) as solid lines and show an excellent agree-\nment between the simulated and the measured resonance\npositions. Most importantly, this agreement could be\naccomplished by solely taking into account the shape\nanisotropy, i.e., by setting KUto zero. Thus, it is ulti-\nmately proven that the anisotropy observed in dynamic\nand static magnetic investigations of CrCl 3[11, 16, 23,\n26] is, indeed, due to the shape anisotropy caused by\nlong-range dipole-dipole interactions, whereas the intrin-\nsic magnetocrystalline anisotropy can be neglected. The\npresent work therefore could verify by means of highly\nsensitive HF magnetic resonance investigations the con-\nclusions of these previous studies [11, 16, 23, 26] and\nsupports the results of the recent \frst-principle calcu-\nlations of the magnetic anisotropy of CrCl 3monolayers\n[43]. It can be concluded that, intrinsically, CrCl 3is\nmagnetically isotropic while the apparent anisotropy ob-\nserved in experiments can be tuned (within certain lim-\nits) by choosing a particular sample shape. Moreover,\nCrCl 3might serve as a valuable reference system for fu-\nture magnetic resonance studies of other van der Waals\nmagnets, since it illustrates the pure e\u000bect of the shape\nanisotropy on, e.g., the frequency-\feld dependence. As\nsimilar sample shapes can be expected for this large fam-\nily of layered materials, it follows that it is very important\nto take into account this source of magnetic anisotropy\nwhen aiming at a precise quanti\fcation of magnetocrys-\ntalline anisotropies in van der Waals magnets.\nIV. CONCLUSIONS\nIn summary, we studied the details of the mag-\nnetic anisotropies in the van der Waals magnet CrCl 3\nby means of systematic HF ESR and FMR measure-\nments. By extending the frequency range of previ-\nous works [23, 31], the \feld-polarized, e\u000bectively ferro-\nmagnetic low-temperature state of the title compound\nwas investigated. Numerical simulations of the mea-\nsured frequency-\feld dependence at 4 K revealed that\nthe anisotropy observed experimentally is governed by\nthe shape anisotropy of the studied CrCl 3crystal. In\ncontrast, the intrinsic magnetocrystalline anisotropy is\nfound to be negligible in this compound, thus supporting\nthe conclusions drawn in previous studies [11, 16, 23, 26].\nConsidering the large current scienti\fc interest in mag-\nnetic van der Waals materials, CrCl 3may serve as a\nreference for future quantitative analyses of magnetic7\nanisotropies in related layered magnets, since it provides\nan excellent example for the impact of the particular sam-\nple shape on the apparent magnetic anisotropy. Finally,\ntemperature-dependent measurements of the resonance\nshift showed the onset of a \fnite shift and, thus, the\nexistence of short-range spin correlations already at tem-\nperatures well above the magnetically ordered state. This\nobservation provides further evidence for the intrinsically\n2D nature of the magnetism in the van der Waals magnet\nCrCl 3.\nConsidering a continuously increasing number of quasi-\n2D van der Waals compounds exhibiting a rich variety\nof intriguing magnetic properties, it is appealing to ap-\nply systematically the frequency-tunable high-\feld ESR\nspectroscopy to investigate the spin dynamics and in par-\nticular the magnetic anisotropy of these materials. The\nlatter appears to be a key factor for determining the type\nof a magnetically ordered ground state. For example,\nthe members of the family of the van der Waals layered\nmetal phosphorous trichalcogenides MPS 3(M = Mn, Fe,\nNi) [44] feature di\u000berent types of magnetic anisotropy\n[45] and exhibit dissimilar antiferromagnetically ordered\nground states, as, e.g., was illustrated in Ref. [46]. Quan-\nti\fcation of the parameters of magnetic anisotropy with\nESR spectroscopy may shed more light on a possible rela-\ntionship between the anistropy and the type of magnetic\norder in these compounds.\nACKNOWLEDGMENTS\nThis work was \fnancially supported by the Deutsche\nForschungsgemeinschaft (DFG) within the Collabora-\ntive Research Center SFB 1143 \\Correlated Magnetism\nFrom Frustration to Topology (project-id 247310070)\nand the Dresden-Wrzburg Cluster of Excellence (EXC\n2147) \\ct.qmat - Complexity and Topology in Quantum\nMatter\" (project-id 39085490). K. M. acknowledges the\nHallwachsRntgen Postdoc Program of ct.qmat for \fnan-\ncial support. A. A. acknowledges \fnancial support by\nthe DFG through Grant No. AL 1771/4-1.\nAppendix: Details on the sample characterization\nA typical x-ray di\u000braction (XRD) pattern of a powder\nsample of CrCl 3is shown in Fig. 4. It was collected\nusing a PANalytical XPert Pro MPD di\u000bractometer with\nCu-K\u000b1radiation (\u0015= 1:54056 \u0017A) in the BraggBrentano\ngeometry at room temperature in the 2 \u0012range between\n10 and 90\u000e. The CrCl 3sample displays strongly preferred\norientation leading to only (00 l) peaks visible.\nA typical scanning electron microscopy (SEM) image\nof CrCl 3crystallites is shown in Fig. 5. It was made with\na Hitachi SU8020 microscope equipped with an Oxford\nSilicon Drift X-MaxN energy dispersive x-ray spectrome-ter (EDX) at an acceleration voltage of 20 kV. Results of\nthe EDX analysis at selected areas indicated in Fig. 5 are\nIntensity (arb. units)\n2q (°)\nFIG. 4. XRD powder pattern of a sample of CrCl 3. Red ticks\ndisplay the peak positions of the reference CrCl 3from the\nInorganic Crystal Structure Database (22080-ICSD, SCXRD,\n298 K). 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Lett. 124,\n027601 (2020)."    },    {        "title": "2204.10596v2.A_short_circuited_coplanar_waveguide_for_low_temperature_single_port_ferromagnetic_resonance_spectroscopy_set_up_to_probe_the_magnetic_properties_of_ferromagnetic_thin_films.pdf",        "content": "arXiv:2204.10596v2  [cond-mat.mtrl-sci]  19 Jul 2022A short-circuited coplanar waveguide for low-temperature single-port ferromagnetic\nresonance spectroscopy set-up to probe the magnetic proper ties of ferromagnetic thin\nfilms\nSayani Pal, Soumik Aon, Subhadip Manna and Chiranjib Mitra∗\nIndian Institute of Science Education and Research Kolkata ,\nWest Bengal, India\nA coplanar waveguide shorted in one end is proposed, designe d, and implemented successfully to\nmeasure the properties of magnetic thin films as a part of the v ector network analyser ferromag-\nnetic resonance (VNA-FMR) spectroscopy set-up. Its simple structure, potential applications and\neasy installation inside the cryostat chamber made it advan tageous especially for low-temperature\nmeasurements. It provides a wide band of frequencies in the g igahertz range essential for FMR\nmeasurements. Our spectroscopy set-up with short-circuit ed coplanar waveguide has been used to\nextract Gilbert damping coefficient and effective magnetizat ion values for standard ferromagnetic\nthin films like Py and Co. The thickness and temperature depen dent studies of those magnetic\nparameters have also been done here for the afore mentioned m agnetic samples.\nINTRODUCTION\nIn recent years, extensive research on microwave mag-\nnetization dynamics in magnetic thin films[1–3], planar\nnanostructures[4–6] and multi-layers[7–9] havebeen per-\nformedduetotheirpotentialapplicationsinvariousfields\nof science and technology. Spintronics is one such emerg-\ning discipline that encompasses the interplay between\nmagnetization dynamics and spin transport. It also in-\ncludes fields like spin-transfer torque [10–13], direct and\ninversespin hall effect [14–18], spin pumping [19, 20] etc.,\nwhich are crucial in industrial applications for develop-\ning devices like magnetic recording head[21], magnetic\ntunnel junction(MTJ) sensors [22, 23], magnetic memory\ndevices[24, 25] andspin-torquedevices[26, 27]. Thus ex-\nploring more about the static and dynamic properties of\nmagnetic materials in itself is an interesting subject. Fer-\nromagnetic resonance spectroscopy(FMR) is a very ba-\nsic and well-understood technique that is used to study\nthe magnetization dynamics of ferromagnets[28, 29, 31].\nNowadays, most advanced FMR spectroscopy methods\nuse a vector network analyzer (VNA)[30, 31] as the mi-\ncrowave source and detector. We have used VNA in our\nset-up too.\nTo determine the magnetic parameters of the ferromag-\nnetic materials using the VNA-FMR spectroscopy, one\nneeds to carry out the measurements at a wide range of\nfrequencies. Since the microwave magnetic field in the\ncoplanar waveguide (CPW) is parallel to the plane, it\nservesthepurposeofexploringthemagneticpropertiesof\nthe concernedsystem overabroadfrequencyrangein the\nGHz region. The advantage of using CPW in the spec-\ntroscopy system lies in the fact that we no longer need\nto remount samples at different waveguides or cavities\nforeveryotherfrequency measurements, which consumes\n∗Corresponding author:chiranjib@iiserkol.ac.ina lot of time and effort in an experiment[32, 33]. Re-\nsearchers design and use different types of CPW for vari-\nous other purposes like micron-sized CPW in microwave-\nassisted magnetic recording; two-port CPW in antenna;\nshorted CPW in ultra-wideband bandpass-filter and per-\nmeability measurements [34–36]. However, in broadband\nFMR spectroscopy two-port CPW jigs have most com-\nmonly been used till date. Using two-port CPW in FMR\nspectroscopy, one gets absorption spectra in terms of\nthe transmissioncoefficient of scatteringparameters, and\nfrom there magnetic parameters of the samples can be\ndetermined. The use of two-port CPW in VNA-FMR\ncan be replaced by one-port CPW where the reflection\ncoefficient of scattering parameters of the FMR spectra\ncan be used to determine the magnetic parameters of\nthe sample. One port reflection geometry is a lot more\nconvenient in terms of easy design, calibration, installa-\ntion, and sample loading. This is especially true when\nthe whole CPW arrangement is kept inside the cryostat\nchamber for low-temperature measurements and the sys-\ntem becomes very sensitive to vibration and any kind\nof magnetic contacts, one port CPW seems very con-\nvenient to operate rather than the two-port one. Previ-\nously, manyhavedesignedandusedshort-circuitedCPW\njigs for other purposes but to the best our knowledge it\nhas not been used for low-temperature VNA-FMR spec-\ntroscopy measurements before.\nIn this work, we report the development of short-\ncircuited CPW based low-temperature broadband VNA-\nFMR spectroscopy set-up to study the magnetic param-\neters of standard ferromagnetic samples. For measure-\nments, we chose the permalloy(Py) thin films as ferro-\nmagnetic (FM) material which has greatly been used in\nresearchfields like spintronics and industrial applications\ndue to its interesting magnetic properties like high per-\nmeability, large anisotropy magnetoresistance, low coer-\ncivity, and low magnetic anisotropy. We have also con-\nsidered another standard magnetic thin film, Co of thick-\nness 30nm as a standard for ascertaining the measure-2\nment accuracy. In our system, we swept the magnetic\nfield keeping the frequencies constant, and got the FMR\nspectra for several frequencies. From there we found the\nvariation of resonance fields and field linewidths with\nthe resonance frequencies. We have used the linear fit\nfor resonance frequencies vs field line-widths data to\ncalculate the Gilbert damping coefficient( α). We fit-\nted the set of resonance frequencies vs resonance fields\ndata to the Kittel formula [59] to obtain the effec-\ntive magnetization(4 πMeff). Subsequently, we investi-\ngated the thickness and temperature-dependent studies\nof 4πMeffandαfor FM thin films of different thickness\ninthetemperaturerangeof7.5Kto300K.Tocharacterise\nthe measurement set-up using short-circuited CPW, we\ncompared the previous measurements in the literature\nwith ourresults and there wasa good agreementbetween\nthe two[36, 41].\nEXPERIMENTAL DETAILS\nA short-circuited CPW has been designed and fab-\nricated as a part of our low-temperature VNA-FMR\nspectroscopy set-up. To make the CPW we have used\nRogers AD1000, a laminated PCB substrate with copper\ncladding on both sides of the dielectric. The thickness of\nthe dielectric and the copper layer are 1.5 mm and 17.5\nmicrons respectively and the dielectric constant of the\nsubstrate is 10.7. The main concern about the design of\nthe CPW is to match its characteristic impedance with\nthe impedance of the microwave transmission line con-\nnected to it. We haveused the line calculatorto calculate\nthe dimensions of CPW. For a CPW with a characteris-\ntic impedance of 50 ohms, the line calculator calculated\nthe width of the signal line and the gap to be 900 mi-\ncrons and 500 microns respectively. The fabrication is\ndone using optical lithography which is described in de-\ntail in the literature[49]. Other components of our mea-\nCryostatVNA\nElectromagnetSample\nCPWCoaxial Transmission Line\nFIG. 1. The schematic diagram of measurement system and\nthe arrangement inside the cryostat with the sample on top\nof the CPW\nsurement system are a)Vector Network Analyser(VNA),\nwhich is a microwave source as well as a detector, b)theelectromagnet that generates the external magnetic field,\ni.e., Zeemanfieldand, c)optistatdrycryogen-freecooling\nsystem from Oxford instruments which is used for low-\ntemperature measurements. One end of the CPW signal\nline is shorted to the ground, and the other end is con-\nnected to the VNA through a SMA connector and coax-\nial cable (fig 3b). On top of the CPW, thin-film samples\nhave been placed face down after wrapping them with\nan insulating tape to electrically isolate them. For low-\ntemperature measurements, the sample has been glued\nto the CPW using a low-temperature adhesive to ensure\ncontact of sample and resonator at all times, in spite of\nthe vibration caused by the cryostat unit. This whole ar-\nrangementis then placed inside the twopole pieces of the\nelectromagnet as we can see from the diagram in fig 1.\nTherearetwostandardmethods ofgettingFMR spectra:\nsweeping the frequency keeping the field constant and\nsweeping the magnetic field while keeping the frequency\nconstant. We have adopted the second method. We have\nworked in the frequency range from 2.5GHz to 5.5GHz\nand in the magnetic field range from 0 Oe to roughly\naround 500 Oe. We have used 1mW of microwave power\nthroughout the experiment. From the FMR spectra, we\nhavedeterminedeffectivemagnetizationanddampingco-\nefficient of FM thin films and studied their variation with\ntemperature and sample thickness.\nSAMPLE PREPARATION AND\nCHARACTERIZATION\nPy (Ni80Fe20) and Co thin films were fabricated by\nthermal evaporation technique on Si/SiO 2substrates,\nfrom commercially available pellets (99 .995%pure) at\nroom temperature. The substrates were cleaned with\nacetone, IPA and DI water respectively in ultrasonica-\ntor and dried with a nitrogen gun. The chamber was\npumped down to 1 ×10−7torr using a combination of\na scroll pump and turbo pump. During the deposition,\npressure reached upto 1 ×10−6torr. Thin films were fab-\nricated at a rate of 1 .2˚A/swhere thickness can be con-\ntrolled by Inficon SQM 160 crystal monitor. For our\nexperiments a series of Py thin films of different thick-\nnesses were fabricated by keeping the other parameters\nlike base pressure, deposition pressure and growth rate\nconstant. Film thickness and morphology was measured\nby using atomic force microscopy technique as shown in\nfig 2(a). We have used Py films with thicknesses 10nm,\n15nm, 34nm, 50nm, and 90nm with a surface roughness\nof around 1nm and one Co film of thickness 30nm. X-ray\ndiffraction experiment confirms the polycrystalline struc-\nture of the samples as shown in fig 2b and fig 2c for Py\nand Co respectively.3\n2µm\n2µm\n(a)\n/s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53 /s54/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s52/s52/s46/s51/s54/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41\n/s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s80/s121/s32/s40/s49/s53/s110/s109/s41\n(b)\n/s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41\n/s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s67/s111/s32/s40/s51/s48/s110/s109/s41\n(c)\nFIG. 2. (a)Atomic force microscope (AFM) image of 30 nm\nthick Py thin film with a surface roughness of 1 nm . X-ray\ndiffraction peak of (b)15nm thick Py film and (c)30nm Co\nprepared by thermal evaporation.\nRESULTS AND DISCUSSION\nWe have calculated the dimensions of the short-\ncircuited CPW using the line calculator of the CST Stu-\ndio Suite software as mentioned in the experimental de-\ntails section. Using those dimensions we have also done\nthe full-waveelectromagneticsimulation in CST software\nto get the electric and magnetic field distribution of the\nCPW. One can see from the simulation result displayed\nin figure 3a that the farther it is from the gap, the weaker\nthe intensity of the magnetic field, and the magnitude of\nthe field in the gap area is one order of greater than that\non the signal line. When placing the thin film sample\non top of the CPW, the dimension of the sample shouldDielectricSampleSignal  Line\nGap\nMagnetic field lines\nElectric field lines\na) b)\nc)\nFIG. 3. (a) Schematic diagram of the cross-sectional view of\nCPW. (b) Top view of the short-circuited CPW after fabri-\ncation. (c)Intensity distribution of microwave magnetic fi eld\nin the one end shorted CPW at 5GHz (top view)\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s83\n/s49/s49/s40/s100/s66/s41\n/s72/s32/s40/s79/s101/s41/s32/s32/s32 /s102/s114/s101/s113/s117/s101/s110/s99/s121\n/s32/s50/s46/s53/s71/s72/s122\n/s32/s51/s46/s53/s71/s72/s122\n/s32/s52/s46/s53/s71/s72/s122\n/s32/s53/s46/s53/s71/s72/s122/s49/s53/s110/s109/s32/s80/s121\n/s84/s61/s51/s48/s48/s75\nFIG. 4. Ferromagnetic Resonance spectra of absorption at\nfrequencies 2.5 GHz, 3.5 GHz, 4.5 GHz, 5.5 GHz for 15nm Py\nthin films at room temperature after background subtraction\nbe such that it can cover the gap area on both sides of\nthe signal line of the CPW because the magnetic field is\nmost intense in that area. This microwave magnetic field\ncirculatingthe signal line ofthe CPW is perpendicular to4\n/s50 /s51 /s52 /s53 /s54 /s55/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s51/s48/s48/s75\n/s32/s67/s111/s32/s40/s51/s48/s110/s109/s41\n/s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s68 /s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41\n(a)/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s97\n/s116/s32\n/s80/s121 /s32/s40/s110/s109/s41/s32/s84/s61/s51/s48/s48/s75\n(b)\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s51/s52/s53/s54/s55\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s32/s51/s48/s48/s75\n/s32/s67/s111/s32/s40/s51/s48/s110/s109/s41\n/s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s79/s101/s41\n(c)/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s56/s57/s49/s48/s49/s49/s52 /s112 /s77\n/s101/s102/s102/s32/s40/s107/s71/s41\n/s116/s32/s45/s49\n/s32/s80/s121/s32/s40/s110/s109/s45/s49\n/s41/s32/s84/s61/s51/s48/s48/s75\n(d)\nFIG. 5. a)Field linewidth variation with resonance frequen cies at 300K for 34nm Py and 30nm Co thin films. Equation 1 has\nbeen used for fitting the curve and to determine the Gilbert da mping coefficient; b)thickness dependence of Gilbert dampin g\ncoefficient at room temperature for Py thin films; c)resonance field variation with resonance frequencies at 300K for 34 nm P y\nand 30 nm Co thin films. Kittel formula (eqn-3)has been used fo r fitting the curve and to determine the effective magnetizati on;\nd)thickness dependence of effective magnetization for Py th in films at room temperature.\nthe external magnetic field and both the magnetic fields\nare parallel to the film surface as can be seen from fig\n3a and 3b. On account of the static magnetic field, the\nmagnetic moment will undergo a precession around the\nstatic magnetic field at a frequency called the Larmor\nprecession frequency. Absorption of electromagnetic en-\nergy happens when the frequency of the transverse mag-\nnetic field (microwave) is equal to the Larmor frequency.\nFig4exhibitsthe absorptionspectrafor15nmbarePy\nfilm after subtraction of a constant background for four\ndifferent frequencies, 2.5 GHz, 3.5 GHz, 4.5 GHz and 5.5\nGHz at room temperature in terms of S-parameter re-\nflection coefficient ( S11) vs. external magnetic field. We\nfitted these experimental results to the Lorentz equation\n[56]. We extracted the field linewidth at half maxima\nfrom the FMR spectra at different frequencies and fitted\nthem using equation 1 to obtain αas one can see from\nfig 5a and fig 6a. The experimental values of the absorp-tion linewidth (∆ H) contains both the effect of intrinsic\nGilbert damping and the extrinsic contribution to the\ndamping. Linewidth due to Gilbert damping is directly\nproportional to the resonance frequency and follows the\nequation:\n∆H= (2π\nγ)αf+∆H0 (1)\nwhereγis the gyromagneticratio, αis the Gilbert damp-\ning coefficient and ∆ H0is the inhomogeneous linewidth.\nA number of extrinsic contributions to the damping coef-\nficient like magnetic inhomogeneities, surface roughness,\ndefects of the thin films bring about the inhomogeneous\nlinewidth broadening [55]. αhas been determined using\nthe above equation only. Damping coefficient values ob-\ntainedhereareintherangeofabout0 .005to0.009forPy\nsamplesofthicknessescoveringthe whole thin film region\ni.e., 10nm to 90nm at room temperature. These values5\n/s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48\n/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121/s68 /s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s53/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121\n(a)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50\n/s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s116 /s32\n/s80/s121 \n/s32/s49/s53/s110/s109\n/s32/s49/s48/s110/s109\n(b)\n/s54/s48 /s49/s50/s48 /s49/s56/s48 /s50/s52/s48 /s51/s48/s48 /s51/s54/s48 /s52/s50/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48\n/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s53/s110/s109/s32/s84/s61/s32/s52/s53/s75/s80/s121/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s79/s101/s41/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121\n(c)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s56/s46/s48/s56/s46/s50/s56/s46/s52/s56/s46/s54/s56/s46/s56/s57/s46/s48/s57/s46/s50/s57/s46/s52/s57/s46/s54/s52 /s77\n/s101/s102/s102/s32/s40/s107/s71/s41\n/s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s116/s32\n/s80/s121 \n/s32/s49/s53/s110/s109\n/s32/s49/s48/s110/s109\n(d)\nFIG. 6. a)Field linewidth variation with resonance frequen cies at 300K and 45K for 10 nm and 15nm Py films. Equation 1\nhas been used for fitting the curve and to determine the Gilber t damping coefficient; b)temperature dependence of damping\ncoefficient for 10nm and 15nm Py thin films; c)resonance field va riation with resonance frequencies at 300K and 45K for 10nm\nand 15nm Py thin films. Kittel formula (eqn-3) has been used fo r fitting the curve and to determine the 4 πMeff; d)temperature\ndependence of 4 πMefffor 10nm and 15nm Py thin films.\nare pretty close to the values previously reported in the\nliterature [39–41, 43, 44]. For the Co film of thickness 30\nnm we have obtained the value of αto be 0.008 ±0.0004.\nBaratiet al.measured the damping value of 30nm Co\nfilm to be 0.004 [37, 38]. There are other literature also\nwhere Co multilayers have been studied where damping\ncoefficient value increasesbecause ofspin pumping effect.\nαis a veryinterestingparameterto investigatebecause it\nis used in the phenomenological LLG equation [57], [58]\nto describe magnetization relaxation:\nd/vectorM\ndt=−γ/vectorM×/vectorHeff+α\nMS/vectorM×d/vectorM\ndt(2)\nwhere,µBisBohrmagneton, /vectorMisthemagnetizationvec-\ntor,MSis the saturation magnetization and Heffis the\neffectve magnetic field which includes the external field,\ndemagnetization and crystalline anisotropy field. The in-troduction of the Damping coefficient in LLG equation is\nphenomenological in nature and the question of whether\nit has a physical origin or not has not been fully under-\nstood till date. We have measured 4 πMeffalso from\nthe absorption spectra. We have fitted the Kittel for-\nmula (equation 3) into resonance field vs. the resonance\nfrequency ( fres) data as shown in fig 5c and fig 6c.\nfres= (γ\n2π)[(H+4πMeff)H]1\n2 (3)\nwhere,His the applied magnetic field, and Meffis the\neffective magnetization which contains saturation mag-\nnetization and other anisotropic contributions. We ob-\ntained the 4 πMeffvalue for 30nm thick Co and 34nm Py\nto be 17.4 ±0.2kG and 9.6 ±0.09kG respectively at room\ntemperature. These values also agree quite well with the\nliterature. For a 10nm Co film, Beaujour et al.measured\nthe value to be around 16 kG[45] and for a 30nm Py the6\nvalue is 10 .4kGas measured by Zhao et al[41].\nWe tried to address here the thickness and tempera-\nture dependence of αand 4πMeffusing our measure-\nment set-up. The variation of the αwith thickness is\nshown here in figure 5b. It increases smoothly as film\nthickness decreases and then shows a sudden jump below\n15nm. Increased surface scattering could be the reason\nbehind this enhanced damping for thinner films. It has\nbeen previously observed [60] that damping coefficient\nand electrical resistivity follows a linear relation at room\ntemperature for Py thin film. It suggests a strong corre-\nlation between magnetization relaxation( α) and electron\nscattering. Magnetization relaxation could be explained\nby electron scattering by phonons and magnons. In the\nformer case, αis proportional to the electron scatter-\ning rate, τ−1and in the later case, α∼τ. Theoretical\npredictions by Kambersky [61] suggests that at higher\ntemperature α∼τ−1as electron scattering by phonons\nare predominant there. So, here in our case we can elim-\ninate the possibility of electron scattering by magnons as\nthickness dependent study has only been done at room\ntemperature where phonon scattering is prevalent. Ing-\nvasson et.al in their paper[60] also suggests that the re-\nlaxation of magnetization is similar to bulk relaxation\nwhere phonon scattering in bulk is replaced by surface\nand defect scattering in thin films.\nThicknessdependent studyof4 πMeffalsohasbeen done\nfor Py thin films at room temperature. As we can see\nfrom fig 5d, Meffis linear for thinner films and becomes\nalmost independent of thickness for thicker films. The\nchange in Meffwith thickness mainly comes from the\nsurface anisotropy,\nµ0Meff=µ0Ms−2Ks\nMsd(4)\nwhereMsis the saturation magnetization and2Ks\nMsdis\nthe surface anisotropy field. Surface anisotropy is higher\nfor thinner films and the anisotropy reduces as one in-\ncreases the film thickness. We have obtained saturation\nmagnetization(4 πMs) value of Py to be 10 .86kGusing\nthe linear fit (equation 4). Previously Chen et al.has re-\nported the 4 πMeffvalue for a 30nm Py film to be 12 kG\n[54] which includes both 4 πMsand anisotropy field.\nTemperature dependence of αfor 15nm and 10nm Py\nfilm is represented in figure 6b. The αvalue decreases\nmonotonically from room temperature value and reaches\na minimum value at around 100K and then starts to in-\ncrease with further decrease of temperature and reaches\na maximum value at 45K. Zhao et al.have seen this\nkind of damping enhancement at around50Kin their low\ntemperature experiment with Py thin films with differ-\nent types of capping layers and Rio et al.observed the\ndamping anomaly at temperature 25K when they have\nusedPtas a capping layer on Py thin film.[39, 41]. We\ndid not use any capping layer on Py film in our mea-\nsurement. So there is no question of interface effect for\nthe enhanced damping at 45K. A possible reason for the\nstrong enhancement of damping at 45K could be the/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48\n/s32/s57/s71/s72/s122\n/s32/s52/s71/s72/s122/s51/s48/s110/s109/s32/s67/s111/s68 /s72 /s32/s40/s79/s101/s41\n/s84/s32/s40/s75/s41\nFIG. 7. Temperature induced linewidth variation of 30nm Co\nthin film at two different frequencies 4GHz and 9GHz\nspin reorientation transition(SRT) on the Py surface at\nthat particular temperature [41, 42]. Previously it has\nbeen established that the competition between different\nanisotropy energies: magnetocrystalline anisotropy, sur-\nface anisotropy, shape anisotropy decides the magnetiza-\ntion direction in magnetic films. For thin films, the vari-\nation of temperature, film thickness, strain can alter the\ncompetition between shape and surface anisotropy. In\nour case, temperature variation could be the reason for\nthe spin reorientation transition on Py surface at around\n45K.Foradeeperunderstandingofthespinreorientation\nwe investigated the temperature dependence of 4 πMeff\nfor 15nm and 10nm Py film as shown in fig 6d. There\nMeffis showing an anomaly at around 45K, otherwise it\nis increasing smoothly with the decrease of temperature.\nSince there is no reason of sudden change in saturation\nmagnetization at this temperature, the possible reason\nfor the anomaly in Meffshould come from any change\ninmagneticanisotropy. Thatchangeofanisotropycanbe\nrelated to a spin reorientation at that particular temper-\nature value. Sierra et.al., [42], have also argued that in\nthe temperature dependent spin re-orientation (T-SRT),\nthe central effect of temperature on the magnetic prop-\nerties of Py films was to increase the in-plane uniax-\nial anisotropy and to induce a surface anisotropy which\norients the magnetization out of plane in the Py sur-\nface. They have verified this using X-Ray diffraction\nexperiments and high resolution transmission electron\nmicroscopy images. This establishes reasonably enough\nthat it is a spin re-orientation transition around 45K.\nLastly, for a 30nm Co thin film we have studied the\ntemperature variation of FMR linewidth(∆ H) at mi-\ncrowave frequencies 9GHz and 4GHz. One can see from\nfig7 that the linewidth does not change much in the tem-\nperature range 100 <T<300 but below 100K, ∆ Hstarts\nto increase significantly. This behaviour of ∆ Hhas been7\nobserved previously by Bhagat et al.[62]. They sug-\ngested that the increase of relaxation frequency at low\ntemp might be related to the rapid variation in the elec-\ntronicmeanfreepath. Thiscomparisonsuggeststhatour\nset-up with a single port CPW has the same sensitivity\nof the previously reported set-up.\nCONCLUSIONS\nShort-circuited coplanar waveguide can successfully be\nused for low-temperature single-port broadband FMR\nspectroscopy measurements. The magnetic parameters\nobtained here for the standard ferromagnetic materials,\nPy and Co are in good agreement with other experimen-\ntal works and theoretical predictions. Temperature de-\npendentstudiesof αand4πMeffforPyfilmshereexhibit\nspin reorientation phenomenon at low temperatures. We\nbelieve that the findings with Py sample will help in bet-\nter understanding of magnetic phenomena for other fer-\nromagnetic materials at low temperature. Though this\nsetup has limitations for angle-dependent FMR measure-\nments, itcanreadilybe usedforstudiesonmagnetization\ndynamics for multi-layered films and planar nanostruc-\ntures and is currently under way. In future we aim to in-\ntegrate electrical measurement facilities with the existing\nset-up thereby extending the possibilities of inverse spinhall(ISHE) measurements and spin-torque ferromagnetic\nresonance(ST-FMR)spectroscopyexperimentswhich are\nvery relevant for current scientific interests. We believe\nthe short-circuited CPW will serve its purpose conve-\nniently there too.\nACKNOWLEDGEMENTS\nTheauthorssincerelyacknowledgesMinistryofEduca-\ntion, Government of India and Science and Engineering\nResearch Board (SERB) (grant no:EMR/2016/007950)\nand Department of Science and Technology (grant\nno. DST/ICPS/Quest/2019/22) for financial sup-\nport. S.P. acknowledges Department of Science and\nTechnology(DST)-INSPIRE fellowship India, S. A. ac-\nknowledges Ministry of Education of Government of In-\ndia and S.M. acknowledges Council Of Scientific and\nIndustrial Research(CSIR),India for research fellowship.\nThe authors would like to thank Dr. Partha Mitra of the\nDepartment of Physical Sciences, Indian Institute of Sci-\nence Education and Research Kolkata for providing the\nlab facilities for sample deposition. The authors would\nalso like to thank Mr. Subhadip Roy, IISER Kolkata for\nhis help in fabricating the CPW structure using home-\nbuilt optical lithography set-up.\n[1]Barman, A. and Haldar, A., 2014. Time-domain study\nof magnetization dynamics in magnetic thin films and\nmicro-and nanostructures. Solid State Physics, 65, pp.1-\n108.\n[2]Silva, T.J., Lee, C.S., Crawford, T.M. and Rogers, C.T.,\n1999. Inductive measurement of ultrafast magnetization\ndynamics in thin-film Py. 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J. \nFerguson1 \n1Microelectronics Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, \nCambridge CB3 0HE, United Kingdom . \n2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom.  \nWe use ferromagnetic resonance to study the c urrent -induced torques in  YIG/heavy  metal \nbilayers. YIG samples with thi ckness varying from 14.8  nm to 80 nm, with Pt or Ta  thin film on top, \nare measured by applying a microwave current into the heavy  metals and measuring the longitudinal  \nDC voltage  gener ated by both spin rectification and spin pumping . From a symmetry analysis of the \nFMR lineshape  and its dependence on YIG thickness , we deduce that the  Oersted field domi nates \nover spin -transfer torque in driving magnetization dynamics . \n \nIntroduction  — Insulating  magnetic materials  have recently  played an important role  in \nspintronics , since they allow pure spin currents  to flow without associated charge transport . Within \nthe family of ferromagnetic insulator s, yttrium iron garnet (YIG) holds a special place  owing  to \nseveral favourable  properties, including ultra -low damping, high Curie temperature and  chemical \nstability  [1–3]. By g rowing a n over layer of heavy  metal  (HM), such as platinum or tantalum , several  \nimportant  spintronic  phenomena have been explored in the YIG/ HM bilayer system, including  the \nmagnetic proximity effect  [4,5] , spin pumping  [6,7] , spin Hall magnetoresistance (SMR)  [8,9] , spin \nSeebeck effect  [10,11]  and so on.  Furthermore, the spin Hall effect in HM can convert  a charge \ncurrent into a transverse  pure spin current, making it  possible to manipulate the ferromagnetic \ninsulator by spin transfer torque (STT) . Recently, several groups have  reported  control ling the \ndamping  in YIG by applying  a DC charge current in a Pt cap ping layer  [12], by which  spin-Hall auto-\noscillation  can be realized  [13,14] . Replacing the DC  current with  a microwave  current , the electric al \nsignal  in Pt  can also be transmit ted via spin wave s in YIG  [3]. In order to further explore the \napplication of the YIG/HM  system, it is necessary to  understand  the torque  on YIG induced by  the \ncharge current in HM. \nCurrent -induced  ferromagnetic resonance (CI -FMR)  is an effective  method to  characterize  \nferromagne tic samples at  micrometre -scale and quantify the current -induced magnetic torques  in \nferromagnetic/ HM bilayer systems  [15]. As shown in  FIG. 1(a), an oscillating  charge current in the \nHM layer generates  a perpendicular  pure spin-current  oscillating at the same frequency via the spin \nHall effect . This oscillating  spin current  flows into the ferromagnetic  layer, exert ing an oscillating  \nSTT, which can drive magnetization  precession when  the FMR condition is satisfied  [15–19]. Since \nno charge current is required to fl ow inside the ferromagnetic  layer in this process,  it is possible  to \nextend this method to  ferromagnetic  insulator /HM bilayers. Instead of penetrating  into the \nferromagnetic  layer,  the electrons undergo  spin-dependent scattering  at the interface  between  the \nferromagnetic  insulator  and the HM, transferring angular momentum to the ferromagnetic insulator . \nThe accompanied Oersted field generated by the charge current in Pt can also drive the magnetization \n                                                      \n* zf231@cam.ac.uk  precession. Although from the symmetry point of view , the torques induced by the Oersted field and \nthe field -like component of STT are indistinguishable from  each other , in our work  we confirm that \nthe driving field  is dominated by  the Oersted contribution  by repeating the measurement with Pt and \nTa.  \n \n \n  \nFIG. 1. (a) Scheme of CI -FMR  in YIG/Pt and the experiment setup. (b) -(c) Results from  a \nYIG(14.8)/Pt(4.2)  sample:  (b) spectra of CI -FMR at  4-8 GHz ; (c) resonance frequency f as a function \nof the resonant field μ0Hres, fitted with the in -plane Kittel ’s formula in dash ed line. Inset: f requency \ndependence of the FMR linewidth μ0ΔH. (d) Plot of the effective damping αeff as a function of tYIG-active. \nRed dashed line represents the fitting result using EQ. (4). \nRecently , several groups have studied  the CI-FMR in YIG/Pt  both theoretically  [20,21]  and \nexperimentally  [22–24]. Using  the theoretical model built by Chiba  et al.  [21], Schreier  et al.  did the \nfirst experiment on in-plane CI-FMR  in YIG/Pt and identified the current -induced torque  by the \nsymmetry  and the lineshape of the  signal s [22]. Sklenar  et al.  then repeat ed the experiment for an out -\nof-plane  extern al magnetic field  [23]. Very recently, Jungfleisch  et al.  imaged the  current -driven  \nmagnetization precession  in YIG/Pt  at resonance condition  with Brillouin light scattering \nspectroscopy and argued that uniform precession is  no longer applicable at high microwave \npower  [24]. The behaviour of CI -FMR in YIG/H M, however,  should depend  on the thickness of the \nfilms  [20], which is on e aspect that remains under explored to the best of our knowledge.  \nHere, we study  the current -induced torque  in YIG/Pt (or Ta)  bilayer structur es with different YIG \nthick ness using CI-FMR . By applying a microwave current into the HM and sweep ing the external \nmagnetic field  in the plane of the devices , a DC voltage is observed at the resonance condition . This \nDC voltage is generated  simultaneously  by two mechanisms  [20]: spin rectification  and spin pumping . \nThe nature of the torque can be understood from the lineshape  and the symmetry  of the DC voltage \nobtained from different samples.  \n(a) (b) \n(c) (d) Sample s and experimental methods — YIG film s with different thickness (listed in  TAB. 1) were \ngrown using RF sputtering on substrates of (111) gadolinium gallium garnet at a pressure of 2.4 \nmTorr.  Since the deposited YIG was nonmagnetic, t he film was annealed  ex situ at 850˚C for 2 hours . \nA layer of 4.2 ± 0.1 nm Pt (or 5.0 ± 0.1 nm Ta) was then deposited via  DC magnetron sputtering . \nBoth YIG and Pt thicknesses were measured by x -ray reflectivity . The samples  were  patterned into \n5×50  μm2 bars by using optical lithography and argon ion milling . After a  second  round of optical \nlithography , a layer of 5  nm Cr/50  nm Au were evapo rated as the contact electrodes . Each bar was \nmounted on a low -loss dielectric circuit board and connected to a microstr ip transmission line  via wire \nbonding . By using  a bias -tee, the DC voltage  across the bar  was measured at the same time  as \nmicrowave power was applied . A magnetic field Hext was swept  in the film plane at an angle θ with \nrespect to the bar, as defined in FIG. 1(a).  \nTAB. 1. Summary of sample characteristics.  The HM cap is Pt unless specified.  \n \n \n \nSignal symmetry — The lineshape and the symmetry of the  resonance in the  DC voltage that we \nmeasure depends both on how the magnetization  precession  is driven  and how the DC voltage is \ngenerated . As for the driving  mechanism, w hen a microwave current  I0ejωt flows though the HM, two \ntypes  of tor ques are expected to act on YIG : a field -like torque τOe = M  hOe induced  by the Oersted \nfield hOe // y, where y is the unit vector along the  y axis , and an antidamping -like STT τST = M  hST \ninduced by an effective field hST // y  M. If M is in the x-y plane, both torques  reach  their maximum \nwhen M is along  the x -axis, and be come zero when  M is perpendicular to the current direction .  \nAs for  the generation of the longitudinal voltage , two mechanisms are  mainly  invol ved: spin \nrectification  and spin pumping . At the FMR condition , the oscillating magnetization leads to a time -\ndependent  SMR in the HM at the same frequency : R = R0 + ΔRcos2θ(t) [8,25] , which rectifies the \nmicrowave current , induc ing a DC voltage along the bar . We have characterised the SMR for each \nYIG thickness by measuring t he resistance of the Pt bar as an  external in -plane magnetic field is \nrotated. The results are reported in TAB. 1. The  spin-rectification  DC voltage consist s of a symmetric \n(Vsym-SR) and an antisymmetric (Vasy-SR) Lorentzian  component s [26,27] : \n \n\n2\next res\nDC sym-SR asy-SR 22 22\next res ext resH H H HV V V\nH H H H H H \n      (1) \n \n\n res res eff 0\nsym-SR\nres effsin 222STH H M IRVhH H M \n  (2) \n \n\n res eff 0\nasy-SR\nres effsin 2 cos22OeHM IRVhH H M \n (3) \nHere, Hext, Hres and Δ H are external applied magnetic field, the resonant field and the linewidth \n(half width at h alf maximum ) respectively ; Meff = Ms  2K2/μ0Ms is the effective magnetization , tYIG (nm)  SMR (10-5) γeff/2π  (GHz/T)  Meff (kA/m)  αeff (10-3) K2 (kJ/m3) \n14.8 4.8 ± 0.6  30.0 ± 0.1  69 ± 3 1.41 ±  0.02 12.6 ± 1.3  \n22 5.9 ± 1.0 29.9 ± 0.1 81 ± 4 1.15 ± 0.03 11.2 ± 1.1  \n36 2.9 ± 0.3  29.8 ± 0.1  82 ± 3  1.00 ± 0.02  11.1 ± 1.0  \n49.5 5.1 ± 0.1  29.8 ± 0.1  82 ± 3  0.97 ± 0.02  11.1 ± 1.0  \n62 3.1 ± 0.5  29.9 ± 0.1  77 ± 4  0.93 ± 0.04  11.6 ± 1.2  \n80 (Ta)  1.2 ± 0.5  28.1 ± 0.9  90 ± 7  1.36 ± 0.31  10.2 ± 0.9 where Ms, K2 and μ0 refer to the saturation magnetization,  the interface anisotropy energy density and \nthe vacuum permeability  respectively . EQ. (2)-(3) show that  Vsym-SR is induced by the out -of-plane  \nfield hST while Vasy-SR is induced by  the Oersted field hOe. \n The other mechanism that could lead to a DC longitudinal  voltage is spin pumping . The DC \nvoltage from spin pumping , irrespective of the driving mechanism, is described by  a symmetric \nLorentzian component Vsym-SP, being independent of the phase between the microwave current  in the \nHM and the precessing magnetization . TAB. 2 summarizes  the lineshape and the angle dependence of \nthe DC voltage for each  of the  driving and detecting mechanism s discussed above . Here , the effective  \ndamping factor  αeff is a function of tYIG and includes the spin pumping term αSP [28]: \n \nB\neff YIG 0 SP 0 eff\ns YIG 4gtgMt      \n (4) \nwhere  α0 is the intrinsic  Gilbert damping coefficient  of YIG  without HM cap respectively ; g is the g -\nfactor ; μB is the Bohr magneton ; g↑↓ \neff is the  interface  effective spin mixing conductance  taking into \naccount the backflow . Assuming that α0 does not change with tYIG, g↑↓ \neff can be determined by \nmeasuring αeff for samples of different YIG thickness.  \nTAB. 2. Summary  of resonance DC signal components involved, with their Lorentzian lineshape and \ndependence on θ, spin Hall angle ϑSH, tYIG and αeff. Ci are the positive coefficients independent from \nthe parameters listed above.  \nDriving  Detecting  lineshape  Dependence on θ, ϑSH, tYIG, and αeff \nhST SR Symmetric  \n3\nST-SR SH eff YIG sin 2 cos Ct     \nSP Symmetric  \n2 3\nST-SP SH eff YIG sin 2 cos Ct     \nhOe SR Anti-symmetric  \n2\nOe-SR SH eff sin 2 cos C     \nSP Symmetric  \n2\nOe-SP SH eff sin 2 cos C     \n \nResults and Discussion  — FIG. 1(b) shows an example of  CI-FMR signals measured at f = 4-8 \nGHz and θ = 45o for a sample YIG(14.8)/Pt (4.2) . The resonance s are well described by  a Lorentzian \nlineshape consisting of symmetric and antisymmetric component s. FIG. 1(c) plots the frequency \ndependence of resona nce field and linewidth (inset) , which are well fitted by  the in-plane Kittel \nformula f = (μ0γeff/2π)[ Hres(Hres + Meff)]1/2 and the  linear  linewidth  function μ0ΔH = μ0ΔH0 + 2πfαeff/γeff \nrespectively . Here, γeff is the effective gyromagnetic ratio  and ΔH0 is the inhomogeneous linewidth  \nbroadening . From the fitting we calculate t he parameters,  γeff, Meff and αeff, as summarized in  TAB. 1, \ntogether with those obtained from each  of the samples  considered in this study . By using a  vibrating \nsample magnetometer  (VSM) , we measured the values of Ms to be 180 ± 20 kA/m for all the samples \nat room temperature,  and K2 can now be calculated ( TAB. 1). From VSM measurement, w e also find \nthat there is a 5.1 ± 0.1 nm -thick  non-magnetic  dead layer in our YIG films . Therefore, we define the \nthickness of active YIG layer as tYIG-active = tYIG  5.1 in nm , and this value should be use d in our \ncalculation . As shown in FIG. 1(d), the value of  αeff for different tYIG-active is well fitted by  EQ. (4) and \nwe find the values of  α0 and g↑↓ \neff to be  (8.1 ± 0.1)  10-4 and (7.1  ± 0.2)  1017 m-2 respectively , in \nreasonably good agreement with the literature  [29].  To characterize the current -induced torque , we now analy se the angle  dependence of the \nsymmetric and the antisymmetric components of the resonance signal. FIG. 2(a) shows the result \nobtained from the YIG(14.8)/Pt (4.2)  sample . The Vasy is fitted well with a –sin2θcosθ function alone \n(red dash), in agreement with a resonance driven by the Oersted field and detected by spin -\nrectification ( TAB. 2).) In contrast, Vsym is fitted by the sum of a sin2 θcosθ term (orange  dash) and a \nsinθ (green dash ) term, noted as Vsym-sin2θcosθ  and Vsym-sinθ respectively. All components are linear in \npower  (FIG. 2(b)), indica ting that the small -angle precess ion approximation is satisfied.  By carrying a \nquantitative analysis  of Vasy based on EQ. (3), we extract the value of the effective field that generates \nthe torq ue for each sample  (FIG. 2(c)), normalized to a unit current density of jc = 1010 A/m2. This can \nbe compared with the value of t he Oersted field calculated from  Ampere’s law as μ0hOe = μ0jctPt/2  26 \nμT (red dash in FIG. 2(c)), where tPt is the thickness of Pt . The good agreement between th e two \nvalues  confirms that the field -like torque is mainly attributed to the  Oersted field.  \n   \n  \nFIG. 2. (a) Angle dependence of the symmetric part Vsym (blue) and anti -symmetric part Vasy (red) \nfrom  YIG(14.8)/Pt (4.2)  at 8 GHz. Dash ed lines are fitting results, where Vasy is fitted by sin2 θcosθ, \nwhile Vsym needs a sin θ term (green) in addition to the sin2 θcosθ term (orange).  (b) Power dependence \nof the three  resonance component s at 8 GHz. ( c) Oersted field μ0hy calculated from Ampere ’s law  (red \ndashed line)  and Vasy using EQ. (3) (black dot)  for each sample , normalized to jc = 1010 A/m.  (d) \nAngle dependence measurement from a  YIG(80)/Ta(5.0)  sample .  \nThe analysis o f the sin2θcosθ term (orange dash)  in symmetric  component  is richer , since it  \ncontains three different terms  as shown in TAB. 2. Despite this, we can still identify the main driving \nmechanism by comparing  Vsym-sin2cos to Vasy. FIG. 3 plots the ratio Vsym-sin2cos/Vasy in each sample \nwith respect to 1/ αeff, showing a linear dependence . Referring to TAB. 2, only  |VOe-SP/VOe-SR| ∝ 1/αeff, \nwhile the ratios between other terms have a more complicated relation  to αeff, as αeff also depends on \ntYIG-active (EQ. (4)). From this we conclude that Vsym-sin2cos can be mainly attributed to the spin \npumping driven by the Oersted field.  In addition to this, we carried  out an experiment in which we \n(a) \n (b) \n(c) \n (d) replaced the Pt with Ta . FIG. 2(d) shows  the angle dependence for a  YIG(80)/Ta(5.0 ) sample . While \nVsym change s its sign compared with the YIG/Pt case, the sign of Vasy stays t he same. The change in \nthe sign of  Vsym is explained with the opposite sign  of the spin -pumping, which  results from the \nopposite value of the sp in-Hall angle of Ta compared with Pt  [15,16,30] . The fact that the sign of Vasy \ndoes not depend on the sign of the spin -Hall angle of the metal layer further confirms that the Oersted \nfield dominates over the field-like STT in driving the magnetization dynamics in our sample s [31].  \n \nFIG. 3. Plot of the ratio Vsym-sin2θcosθ/Vasy as a function of  1/αeff, measured at 8 GHz.  The dashed line \nrepresents the linear fitting.  \nWe also briefly comment on an additional  sinθ (green dash ) term  that appears in the fitting of \nVsym. We note that when measuring other material systems in our setup, e.g. Co/Pt  [27] or Py/Pt  [32], \nthis sinθ component  is absent, indicating its origin in the sample.  This term was previously attributed \nto an  on-resonance  contribution from the longitudinal  spin Seebeck effect  [24]. However , neither STT \nnor Oersted field can drive FMR at θ = 90°, where this term is maximum.  \nConclusion  — In conclusion, we have used CI -FMR to  investigate  the charge -current -induced \ntorque on  YIG magnetization in a series of YIG/ HM samples with different YIG thickness  between \n14.8 nm and 80 nm . Our measurements show that the Oersted field gives the dominant contribution to \ndriving the magnetisation precession and shoul d therefore be taken into account when carryin g out \nCI-FMR studies in YIG/HM systems.  \nACKNOWLEDGMENTS  \nWe thank L. Abdurakhimov  for the experiment help, and  T. Jungwirth, M. Jungfleisch &  A. \nHoffmann for  the valuable discussion s. Z.F. thanks the  Cambridge T rusts for financial support . B.J.H. \nthanks D.  Williams and Hitachi Cambridge for support and advice.  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Chen, A. D. Kent, J. Z. Sun, M. J. Rooks, and R. H. Koch, \nPhys. Rev. B - Condens. Matter Mater. Phys. 70, 1 (2004).  \n[32] S. Langenfeld, Control of Magnetisation Dynamics in Permalloy Microstructures by Spin -Hall \nEffect, University of Cambridge, 2015.  \n "    },    {        "title": "1603.03207v1.Resonant_excitation_of_the_spin_wave_current_in_hybrid_nanostructures.pdf",        "content": "arXiv:1603.03207v1  [cond-mat.mes-hall]  10 Mar 20161\nResonant excitation of the spin-wave current in\nhybrid nanostructures\nI. I. Lyapilin, M. S. Okorokov, V. V. Ustinov\nInstitute of Metal Physics, UD RAS, Ekaterinburg, Russia\nemail: Okorokovmike@gmail.com\nUsing the non-equilibrium statistical operator method (NS O), we have investigated the\nspin transport through the interface in a semiconductor/fe rromagnetic insulator hybrid\nstructure. We have analyzed the approximation of effective p arameters, when each of the\nconsidered subsystems (conduction electrons, magnons, an d phonons) is characterized by its\neffective temperature. We have constructed the macroscopic equations, describing the spin-\nwave current caused by both resonantly excited spin system o f conduction electrons and by\nan inhomogeneous thermal field in the ferromagnetic insulat or.\nIntroduction\nNon-equilibrium statistical mechanics covers many proble ms of the interaction\nbetween a macroscopic system consisting of several sub-sys tems and external fields.\nIn this case, the system’s non-equilibrium state depends on both the external field\nenergy absorbed by the subsystems per unit time and the energ y exchange rate\nbetween the subsystems, and on the energy leakage rate from t he non-equilibrium\nsubsystems toward a thermostat. The typical examples of the se are the well-known\nOverhauser [1] and Feher [2] effects. The latter are exhibite d through observable\ndeviations in interacting subsystems of a crystal as one of w hich (nuclear or electron)\npreviously disturbed. In spintronics, the spin pumping effe ct has been implemented\nin a paramagnetic/ferromagnetic insulator system [3, 4, 5, 6]. The effect mentioned\nabove is an analog of the Overhauser effect resulting to elect ron polarization. The\nsaturation of ferromagnetic resonance in a localized spin s ubsystem causes the electrons\nto heat up. Otherwise speaking, their excess energy produce s a deviation of the other\nsubsystems from the equilibrium: the electron spins of the p aramagnetic, phonons,\netc. At the same time, the magnetic subsystem (localized spi n subsystem of the\nferromagnetic) transfers the spin angular momentum to the e lectronic subsystem of\nthe paramagnetic material. It has been found that two types o f spin pumping are\npossible to exist. The first of these is resonance or \"coheren t\". It involves the excitation\nof ferromagnetic (spin-wave) resonance in the localized sp in subsystem. The paper\n[7] demonstrates such a way of pumping in an experiment. Anot her type is a non-\ncoherent method of unbalancing either one or both the spin su bsystems (subsystems\nof conduction electrons and localized spins) by external pe rturbations, for example,\nby thermal. Wherein, each of the spin subsystems can be descr ibed by its effective\ntemperature. The different temperature relaxation ratios m ake it possible to realize\nboth the spin pumping and the spin torque effect: the transfer of the spin angular\nmomentum from the paramagnetic to the magnetic subsystem. T he spin Seebeck effect\n(SSE) observed in Ni81Fe19conducting crystals can serve as an example of the non-\ncoherent spin pumping. Afterwards, the SSE could be observe d in various materials,2\nboth semiconductors and metallic ferromagnets [8, 9, 10]. B esides, later the spin Nerst\neffect (or the thermal spin Hall effect), the spin Peltier effec t, and others have been\ndiscovered [11, 12, 13, 14, 15, 16, 17]. The spin current is po ssible to be generated by\nmeans of surface plasmons [18].\nStudying the SSE in a non-conducting magnet in the system of a non-magnetic\nconductor/magnetic insulator (N/F) LaY2Fe5O12[19] has shown that, against\nconducting crystals where the transfer of the spin angular m omentum is due to band\ncharge carriers, in non-conducting magnetic materials the spin Seebeck effect can be\nrealized by exciting a localized spin system. For the SSE, th e angular momentum\ntransfer is driven by the spin-wave current (spin wave) unde rlain by excitations of\nthe localized spin subsystem (magnons). Thus, unlike the co nducting crystals, in the\nnon-conducting magnet another spin current type - a spin-wa ve current can emerge.\nSince spin waves relax weakly enough, the spin-wave current propagates far greater\ndistances than the electron spin current. This circumstanc e promises possible practical\napplications of the effect [20, 21].\nThe work [6] investigates experimentally the spin-wave cur rent. The detection as\nwell as the generation of the spin current is a daunting task. The first detections\nof spin currents have been performed by indirect methods, by measuring the effects\naccompanied with the spin current generation. So, the work [ 22] uses optical methods\nto measure the spin accumulation occurring on the lateral su rfaces of the sample when\ngenerating the spin current in spin-Hall systems. Subseque ntly, the inverse spin Hall\neffect has been proposed as an electrical method of detecting spin currents [5, 23, 24].\nThe essence of this method is based on inducing a voltage by th e spin current against\nthe background of spin-orbit interaction [25, 26]. For the fi rst time such a spin current\ndetection method has been demonstrated in the work [27] and h as actually become the\nmain method of detecting spin currents [7, 28, 29].\nThe spin-wave current generation requires producing a non- equilibrium distribution\nof magnons, i.e. unbalancing the localized spin system. As h as been noted above,\nthis is possible to be done a variety of ways. Accompanied by t he creation (or\nannihilation) of magnons, the interaction between the spin -polarized electrons and\nthe localized spins at the interface (N/F) disturbs the magn etic subsystem. In the\ncase of electron spin pumping, the effect is achieved by means of an alternating\nmagnetic field under ferromagnetic resonance conditions. T his way provides the spin\ncurrent without transferring spin-polarized charge carri ers through the interface in the\nhybrid structures. As such, the approach avoids the mismatc h problem [30, 31, 32, 33]\nthat prevents to obtain high spin polarization values by inj ection. The discovery of\nthe spin Seebeck effect has shown that the spin current can be i nduced by thermal\ngradients. Thus, magnetic, electric, thermoelectric and q uantum-relativistic (spin-\norbit) interactions underlying various physical effects ma ke the spin current generation\npossible.\nBoth the spin Hall effect (SHE) and its inverse owe their origi n mainly to the\nspin-orbit interaction (SOI). The latter couples the kinet ic (translational) and spin\nsubsystems of conduction electrons. Thus, the SOI is one of t he possible channels\nto act on one of the subsystems via another, for example, on th e spin subsystem of\nconduction electrons via the kinetic subsystem and vice ver sa. Due to the translational\nand spin motion locking, the quantum transitions cannot be c onventionally divided3\ninto pure configurational (orbital) and pure spin ones. We ca n only talk about either\npredominantly configurational or predominantly spin trans itions. But this circumstance\nsignificantly changes the conditions for the excitation of d ifferent transitions. Namely,\nthe electrical component of an electromagnetic field initia tes the spin transitions,\nand the magnetic component – the orbital ones. The spin-orbi t interaction gives rise\nto the resonant electron transitions at frequencies being l inear combinations of the\ncyclotron and Zeeman frequencies. Besides, such transitio ns can exist at the antinode\nof both electric and magnetic fields. Such resonance is known as the combined Rashba\nresonance [34, 35, 36, 37]. The powers absorbed by the electr ons under saturation of\nthe combined resonance (CR) and the paramagnetic resonance (PR) at the antinode\nof the electric field differ dramatically in magnitude. The fo rmer is much larger (by\nseveral orders of magnitude) than the latter. As to the combi ned resonance, a change\nin the average energy of the electron thermal motion (or thei r kinetic temperature) is\nchiefly explained by the fact that the kinetic degrees of free dom directly absorb the\nalternating electric field energy. However, as to the parama gnetic resonance, the kinetic\ndegrees of freedom derive energy from the spin subsystem onl y indirectly, through the\nspin-orbital interaction.\nIn this work we look at how the electric-dipole excitation of the electron\nspin subsystem of a semiconductor affects the spin-wave curr ent generation in a\nnon-conducting magnetic ferromagnetic material in the sem iconductor/ferromagnetic\ninsulator structure. The spin relaxation of conduction ele ctrons is assumed to be due\nto the exchange interaction between the electrons and the lo calized spins located at the\ninterface. Accompanied by the creation (or annihilation) o f magnons, the inelastic spin-\nflip electron scattering when the electrons resonantly abso rb energy from external fields\n(spin resonance saturation), causes the localized spin sub system to non-monotonically\ndeviate from its equilibrium state. Qualitatively, the effe ct can be explained as follows.\nThe system is characterized by some average angular momentu m having both electronic\nand magnetic components of an external magnetic field. The sp in resonance saturation\n(the change in the spin temperature of the electron subsyste m) changes the electronic\ncomponent of the angular momentum. By virtue of the conserva tion law of angular\nmomentum of the entire system, the magnetization of the magn etic subsystem must\nalso be changed. This can be interpreted as a change in the spi n temperature of the\nlocalized spin subsystem.\nThe paper is organized as follows. The first part formulates t he model at\nhand, contains the Hamiltonian of the system, and enters bas ic operators and their\nmicroscopic equations of motion. The second section involv es constructing both the\nnon-equilibrium entropy operator, accounting for the pert urbed system, and the Non-\nequilibrium Statistical Operator (NSO). The third section of the work covers analysis\nof macroscopic equations.\nThe Hamiltonian\nConsider a structure comprising two subsystems: a semicond uctor and a\nferromagnetic insulator (S/FI). Let them interact between each other and the lattice\nin crossed alternating electrical and magnetic fields. The e xternal impact is assumed4\nto give rise to combined (spin) resonance in the conduction e lectron subsystem.\nAccompanied by the emission and absorption of magnons, the i nelastic scattering\nof the conduction electrons by localized spins located near the interface, unbalances\nthe localized spin system. As an additional mechanism of mag non scattering, the\nmagnon-phonon interaction is concerned. The system of cond uction electrons in the\nsemiconductor (S) should be viewed as a system consisting of kinetic and spin\nsubsystems. The Hamiltonian of the system can be represente d as:\nH=HSC+HFI+HL+HF.\nHere\nHSC=/integraldisplay\ndxHe(x),\nHe(x) =/summationdisplay\nj{He,j,δ(x−xj)}, H e,j=Hk,j+Hs,j+Hks,j,\nHk=/summationdisplay\njp2\nj\n2m, H s=−/planckover2pi1ωs/summationdisplay\njsz\nj. (1)\nHksdescribes the interaction of the kinetic and spin degrees of freedom of the electrons.\nAn expression for Hkscan be written in the general form\nHks=/summationdisplay\njf(pj)Sj=R/summationdisplay\njpα1\njpα2\nj...pαs\niSβ\nj, (2)\nwhereRis a constant depending on the spin-orbital interaction int ensity.f(pj)is\na pseudo-vector whose components are a formula of order s of t he kinetic impulse\ncomponents pα\nj. The integration is over the volume occupied by (S). sz\njandpγ\njare\nthe spin and momentum operator components of the j-th electr on, respectively. ωs=\ngsµ0H//planckover2pi1is the Zeeman precession frequency of free electrons in an ex ternal magnetic\nfield directed along the z-axis ( gs, µ0are the effective electron spectroscopic splitting\nfactor and the Bohr magneton, respectively); {A,B}= (AB+BA)/2\nHFI=/integraldisplay\ndx(Hm(x)+Hsm(x)) (3)\nis the Hamiltonian of the localized spin subsystem. Hm(x)is the energy density operator\nfor the magnetic subsystem; it is of a sum of the exchange HSS(x)(over the nearest\nneighbors) and Zeeman energies HS(x).\nHSS=−J/summationdisplay\njδSjSj+δ, HS=−/planckover2pi1ωm/summationdisplay\njSz\nj. (4)\nJ is the exchange integral, ωm=gmµ0H//planckover2pi1.Hms(x)is the energy density operator of\ninteraction with the conduction electrons at the interface .\nHsm=−J0/summationdisplay\nj/integraldisplay\ndxs(x)S(Rj)δ(x−Rj), (5)5\nwhereJ0is the exchange integral, S(Rj)being the operator of the localized spin with\nthe coordinate Rjat the interface. The integration in (2) is over the volume oc cupied\nby(FI).HLis the lattice Hamiltonian\nHL=/integraldisplay\ndx(Hp(x)+Hpm(x)), (6)\nwhereHp(x)is the energy density operator for the phonon subsystem. Hpm(x)is the\nenergy density operator of interaction between the localiz ed spins and phonons. HFis\nthe interaction between the conduction electrons and the ex ternal alternating electrical\nfieldE(t).\nAs a rule, the therms of the Hamiltonian Hksin a certain sense are small\n[34]. In this case, to eliminate the interaction of the spin a nd kinetic degrees of\nfreedom of the electrons in the linear approximation, we can perform an momentum-\ndependent canonical transformation of the Hamiltonian. Th is also modifies the rest\nof the Hamiltonian’s therms describing the interaction of t he electrons with the\nlattice, and the electromagnetic field. As a result, we obtai n a new Hamiltonian\nwith autonomous subsystems (k) and (s) , the electronic syst em effectively interacting\nwith the electromagnetic field, which determines the resona nt energy absorption. Such\na transformation corresponds to gauge-invariant equation s of motion for physical\nquantities under conditions, typical for combined resonan ce (CR). Suppose the\ncanonical transformation of the Hamiltonian to be already m ade and the renormalized\ninteraction with the alternating electric field has the form [37]\nHef(t) = [r, T(p)]eE(t), T (p) =R/summationdisplayTα1,...αs;β\n/planckover2pi1Ωα1,...αs;β(7)\nwhereT(p)is the operator of the canonical transformation; Ωα1,...αs;βis a linear\ncombination of the cyclotron ω0and the Zeeman ωsfrequencies of electrons, this\ncombination depends on the particular structure of the oper atorTα1,...αs;βand the\nspin-orbital interaction constant.\nTo describe the non-equilibrium state of our system, we need calculate average\nenergies of the subsystems (s) and (m) or their thermodynami cally conjugate inverse\neffective temperatures βs,βm. Such a description corresponds to the case of establishing\nequilibrium inside each of the subsystems at a rate greater a s compared to the energy\nexchange rate between them. We have earlier used it to analyz e spin-thermal effects (the\nspin Seebeck effect) in hybrid structures [38]. In general ca se, CR is possible to occur\nat frequencies being linear combinations of the cyclotron a nd the Zeeman frequencies.\nThen, both the kinetic and spin subsystems of conduction ele ctrons are expected to be\nunbalanced. In the case of CR at the spin frequency, only the s pin subsystem absorbs\nenergy from the external alternating field, thereby the kine tic subsystem remains\nequilibrium. Thus, under combined (spin) resonance condit ions, it is sufficient to discuss\nthe evolution only two spin subsystem: the conduction elect rons spin subsystem and the\nlocalized spin subsystem. In the effective parameter approx imation, the temperature\nTscharacterizes the spin electron subsystem, the temperatur eTm– the localized spin\nsubsystem of the ferromagnetic insulator. The equilibrium temperature Tcorresponds\nto the phonon subsystem.6\nThe entropy operator\nTo construct balance equations for average energies (or effe ctive temperatures) of\nthe subsystems, we employ the method of the non-equilibrium statistical operator\n(NSO) [39, 40, 38]. The entropy operator corresponding to a n on-equilibrium state\nin terms of average density values can be written as\nS(t) = Φ(t)+/integraldisplay\ndx{β(Hk(x)−µ(x,t)N(x,t))+βs(x,t)(Hs(x,t)+(1/2)Hsm(x,t))\n+��m(x,t)[Hm(x,t)+(1/2)(Hsm(x,t)+Hpm(x,t))]+β(Hp(x,t)+(1/2)Hpm(x,t))}\n=S0+δS(t). (8)\nΦ(t)is the Massieu-Plank functional. βi(x,t), i=s,mare local-equilibrium values of\nthe inverse temperatures of the subsystems (s) and (m), resp ectively. µ(x)is a local-\nequilibrium value of the chemical potential of electrons. N(x) =/summationtext\niδ(x−xi)is the\nelectron number density operator. S0is the entropy of the equilibrium system with the\nHamiltonian H\nS0= Φ0+β(He−µN)+β(Hm+Hp+Hsm+Hmp),\nwhereβ−1=Tis the equilibrium temperature of the system. The operator δS(t) =/integraltext\ndxδS(x,t)describes the system deviation from its equilibrium state. Provided that\nthe exchange interaction is the main mechanism of inelastic spin-electron scattering,\nwe arrive at\nδS(t) = ∆/integraldisplay\ndx{δβs(x,t)(Hs(x,t)+(1/2)Hsm(x,t))−βδµ(x,t)N(x)\n+δβm(x,t)[Hm(x,t)+(1/2)(Hsm(x,t)+Hmp(x,t))]}, (9)\n∆A=A−/an}b∇acketle{tA/an}b∇acket∇i}ht0,/an}b∇acketle{t.../an}b∇acket∇i}ht0=Sp(...ρ0).\n/an}b∇acketle{t.../an}b∇acket∇i}htt=Sp(...ρ(t)).\nThe non-equilibrium statistical operator ρ(t)(NSO or the density matrix) in the linear\napproximation in deviation from equilibrium can be written in the form [40]:\nρ(t) =ρq(t)−0/integraldisplay\n−∞dt1eǫt11/integraldisplay\n0dτ ρτ\n0˙S(t+t1,t1)ρ1−τ\n0. (10)\nHereρq(t) = exp {−S(t)}is the quasi-equilibrium statistical operator, ˙S(t)is the\nentropy production operator\n˙S(t) =δ˙S(t) =∂S(t)\n∂t+1\ni/planckover2pi1[S(t),H].\nA further algorithm for constructing the operator ρ(t)reduces to finding the entropy\nproduction operator ˙S(t). Commuting the operators Hs(x),N(x),Hm(x), with the\nHamiltonian (1), we come up with the operator equations of mo tion\n˙Hs(x) =−∇IHs(x)+(1/2)˙Hs(sm)(x)+˙Hs(eF)(x),\n˙N(x) =−∇IN(x). (11)7\nHere\nIN(x) =1\nm/summationdisplay\nj{pj,δ(x−xj)}, I Hs(x) =−/planckover2pi1ωs1\nm/summationdisplay\njsz\nj{pj,δ(x−xα\nj)}(12)\nare the particle flux and the Zeeman energy densities; ˙Hs(sm)(x)is the rate of change in\nlocal electron energy due to the interaction Hsm(x).˙Hs(eF)(x)determines the change\nin electron energy due to the interaction with the electrica l field. Here ˙Aλ(λγ)(x) =\n(i/planckover2pi1)−1[Aλ(x),Hλγ].\nLet us turn to the examination of the magnetic subsystem. Usi ng the Holstein-\nPrimakov method [41], the Hamiltonian of the localized spin subsystem can be\nrepresented through spin-wave (magnon) variables (using t he creation b+\nkand\nannihilation bkoperators). Treating the magnon gas as free, we have\nHm=/summationdisplay\nkε(k)b+\nkbk,гдеε(k) =/planckover2pi12k2\n2m∗.\nThis expression can be interpreted as a sum of the energies of the quasiparticles-\nmagnons having the quasi-momentum Pwith their own effective mass m∗and the\nmagnetic moment [42]. The equations of motion for the magnet ic subsystem can be\nwritten in the form:\n˙Hm(x) =−∇IHm(x)+(1/2)[˙Hm(sm)(x)+˙Hm(pm)(x)]. (13)\nHere\nIHm(x) =−/planckover2pi1ωmISz(x) (14)\nis the magnon energy flux density. The rest of the terms in the r ight-hand side of the\nequation are responsible for the magnon scattering process es at the interface and by\nphonons. Then, the inelastic part of the exchange interacti on can be written via the\ncreation and annihilation operators for electrons and magn ons:\nHsm=−J∗/summationdisplay\nk,k′,q{b+\nqa+\nk↑ak′↓+bqa+\nk′↓ak↑}δk′,k+q, (15)\nwherea+\nkα(akα)are the creation (annihilation) operators for electrons wi th a certain\nspin value α=↑,↓.\nUltimately, the equation of motion for the lattice subsyste m has the form:\n˙Hp(x) =−∇IHp(x)+(1/2)˙Hp(pm)(x). (16)\nSubstituting the equations of motion found into the entropy production operator yields:\n˙S(t) = ∆/integraldisplay\ndx{−βδµ(x,t)∇IN(x)+β˙Hs(sF)(x)+δβs(x,t)[−∇IHs(x)+˙Hs(sm)(x)\n+δβm(x,t)[−∇IHm(x)+˙Hm(sm)(x)+˙Hm(mp)(x)]}.(17)\nIntegrating by parts the terms containing the divergence of fluxes and discarding the\nsurface integrals, we represent the entropy production ope rator as follows:\n˙S(t) = ∆/integraldisplay\ndx{IN(x)β∇µ(x,t)+IHs(x)∇βs(x,t) +IHm(x)∇βm(x,t)+β˙Hs(F)(x,t)\n+δβs(x,t)(1/2)˙Hs(sm)(x) +δβm(x,t)(1/2)[˙Hm(sm)(x)+˙Hm(pm)(x)]}.(18)8\nBy expanding the quasi-equilibrium operator ρq(t) = exp{−S(t)}in powers of δS(t),\nwe establish the linear relationship between the deviation s of the thermodynamic\ncoordinates and thermodynamic forces from their equilibri um values. Then, we have\nδ/an}b∇acketle{tHm(x)/an}b∇acket∇i}htt=−/integraldisplay\ndx′δβm(x′,t)(Hm(x);Hm(x′))0,\nδ/an}b∇acketle{tHs(x)/an}b∇acket∇i}htt=−/integraldisplay\ndx′{δβs(x′,t)(Hs(x);Hs(x′))0−βδµ(x′,t)(Hs(x);N(x′))0},\nδ/an}b∇acketle{tN(x)/an}b∇acket∇i}htt=−/integraldisplay\ndx′{δβs(x′,t)(N(x);Hs(x′))0−βδµ(x′,t)(N(x);N(x′))0},(19)\nwhere\nδ/an}b∇acketle{tA/an}b∇acket∇i}htt=/an}b∇acketle{tA/an}b∇acket∇i}htt−/an}b∇acketle{tA/an}b∇acket∇i}ht0,\n(A,B)0=1/integraldisplay\n0dλSp{Aρλ\n0∆Bρ1−λ\n0}.\nGoing over to the Fourier components of the spatial coordina tes and taking into account\nthat/an}b∇acketle{tN(x)/an}b∇acket∇i}htt=/an}b∇acketle{tN(x)/an}b∇acket∇i}ht0, we get\nβδµ(q,t) =−(N(q),Hs(−q))0\n(N(q),N(−q))0δβs(q,t),\nδ/an}b∇acketle{tHs(q)/an}b∇acket∇i}htt=−δβs(q,t)(/planckover2pi1ωs)2Czz(q), (20)\nwhere\nCzz(q) = (sz(q);sz(−q))0−(sz(q);N(−q))0(N(q);sz(−q))0\n(N(q);N(−q))0.\nδ/an}b∇acketle{tHm(q)/an}b∇acket∇i}htt=−δβm(q,t)(/planckover2pi1ωm)2(Sz(q);Sz(−q))0. (21)\nNote that with help of (19), (20), the entropy production ope rator (18) can appear\nas a functional of the Fourier components of the deviations o f the thermodynamic\ncoordinates from their equilibrium values.\nMacroscopic equations\nAveraging the operator equations (12), (16) by the NSO (10), we can construct the\nmacroscopic equations for the density of the spin magnetiza tion of conduction electrons\nand localized spins: δmz(x,t) =gsµ0δ/an}b∇acketle{tsz(x)/an}b∇acket∇i}httandδMz(x,t) =gmµ0δ/an}b∇acketle{tSz(x)/an}b∇acket∇i}htt,\nrespectively. We have\n∂\n∂tδmz(x,t) =−∇/an}b∇acketle{tIHs(x,t)/an}b∇acket∇i}htt+/an}b∇acketle{t˙Hs(sm)(x,t)/an}b∇acket∇i}htt+/an}b∇acketle{t˙Hs(ef)(x,t)/an}b∇acket∇i}htt,\n∂\n∂tδMz(x,t) =−∇/an}b∇acketle{tIHm(x,t)/an}b∇acket∇i}htt+/an}b∇acketle{t˙Hm(sm)(x,t)/an}b∇acket∇i}htt+/an}b∇acketle{t˙Hm(pm)(x,t)/an}b∇acket∇i}htt(22)\nEqs. (22) describe the change of the density of the spin magne tization of the electronic\nand magnetic subsystems due to the following processes: diff usion (the first summands9\nin the right-hand sides of the equations), relaxation as a re sult of the exchange\ninteraction between electrons with localized moments at th e interface (the summands\n/an}b∇acketle{t˙Hi(sm)(x,t)/an}b∇acket∇i}htt,i=s,m), and energy absorption from an external electrical field by the\nspin subsystem of conduction electrons /an}b∇acketle{t˙Hs(ef)(x,t)/an}b∇acket∇i}htt≡Qs(x,t).\nThe summands /an}b∇acketle{t˙Hm(pm)(x,t)/an}b∇acket∇i}httgovern the magnon-phonon relaxation. In addition,\nthe time and spatial dispersion effects should be also taken i nto account. These\ncoefficients are given by the explicit expressions below\n/an}b∇acketle{tIα\nHs(x,t)/an}b∇acket∇i}htt=/integraldisplay\ndx′0/integraldisplay\n−∞dt1eǫt1{(Iα\nHs(x);Iγ\nHs(x′,t1))∇βγ\ns(x′,/tildewidet)\n+(Iα\nHs(x);Iγ\nN(x′,t1))β∇µγ(x′,/tildewidet)+(Iα\nHs(x);Iγ\nHm(x′,t1))∇βγ\nm(x′,/tildewidet)}, (23)\n/an}b∇acketle{tIα\nHm(x,t)/an}b∇acket∇i}htt=/integraldisplay\ndx′0/integraldisplay\n−∞dt1eǫt1{(Iα\nHm(x);Iγ\nHm(x′t1))∇βγ\nm(x′,/tildewidet)\n+(Iα\nHm(x);Iγ\nHs(x′,t1))∇βγ\ns(x′,/tildewidet)+(Iα\nHm(x);Iλ\nHp(x′,t1))∇βλ\nm(x′,/tildewidet)}, (24)\n/an}b∇acketle{t˙Hi(nm)(x,t)/an}b∇acket∇i}htt=/integraldisplay\ndx′0/integraldisplay\n−∞dt1eǫt1(˙Hi(in);˙Hi(in)(x′,t1))δβj(x′,/tildewidet). (25)\nQ(x,t) =β/integraldisplay\ndx′0/integraldisplay\n−∞dt1eǫt1(˙Hs(eF)(x);˙Hs(eF)(x′,/tildewidet)). (26)\n/tildewidet=t+t1;i,j=s,m, n=m,p. Eqs. (22) and the formulas for the kinetic coefficients\n(23) - (26) solve the problem of macroscopic description of t he non-equilibrium spin\nsubsystems in terms of average magnetization densities.\nNext, we consider the case of stationary flow of energy betwee n the subsystems. For\nthis purpose, we average the equations (22) over time tby the rule\nA(t) =ǫ/integraldisplay0\n−∞dtA(t)eǫt, ǫ→+0,\nwhere∂tA(t) = 0,δβi(x,t+t1) =δβi(x), i=s,m,p . Finally, the set of equations (22)\nacquires the form\nδβsm(x)Ls(sm)(x)+Qs= 0, (27)\nδβms(x)Lm(sm)(x)+δβmp(x)Lm(mp)(x) = 0.\n(28)\nδβik=βi−βkand\nLi(jk)(x)=/integraldisplay\ndx′/integraldisplay0\n−∞dt1eǫt1(˙Hi(ik)(x);˙Hi(ik)(x′,t1)).10\nFurther, we proceed to the Fourier-representation in corre lation functions and put\nthat\nHs(x) =/summationdisplay\nqHs(q)eiqx, Hs(q) =/summationdisplay\njHsje−iqxj,\nI(x) =/summationdisplay\nqI(q)eiqx, I(q) =/summationdisplay\nj{pj/m, e−iqxj},...и т. д.\nNow, we write down an expression for average power absorbed b y the spin subsystem\nof conduction electrons under CR Qs=/integraltext\ndxQ(x,t). According to [37], we have\nQs=βω2\nsR2/summationdisplay\nq,ω|E−(ω)|2Γ(q,ω)ω2c±(q)\n(ω−ωs)2+Γ2(q,ω). (29)\nHerec±(q) = (s±(q);s∓q))0,\nΓ(q,ω) =ν(q,ω)+qαqγD±\nα,γ(q,ω), (30)\nwhereν(q,ω)is the known formula for the frequency of the transverse elec tron spin\nrelaxation; it defines, for example, the line width of parama gnetic resonance [43]\nν(q,ω)=1\nc±(q)Re0/integraldisplay\n−∞dt1e(ǫ−iω)t1(˙s+\n(sm)(q); ˙s−\n(sm)(−q,t1)), (31)\nandD±\nα,γ(q,ω)is the diffusion tensor for the transverse spin magnetizatio n components-\nD±\nα,γ(q,ω) =1\nc±(q)Re0/integraldisplay\n−∞dt1e(ǫ−iω)t1(Iα\ns+(q);Iγ\ns−(−q,t1)), (32)\nThe expressions for the kinetic coefficients ν(q,ω), D±\nα,γ(q,ω)account for time and\nspatial dispersion and are suitable for both quantizing and classical magnetic fields.\nPreviously, papers [37, 43] have derived similar expressio ns for absorbed power in the\nhomogeneous case.\nLet us now look at the set of equations (22) in the stationary c ase. We have\n(βm−βs)Lm(sm)(q,ω)+(βm−β)Lm(mp)(q,ω)−qαqγDzz\nα,γ(q,ω) = 0,\nQs(q,ω)+(βs−β)Ls(sm)(q,ω) = 0 (33)\nwhere\nDzz\nα,γ(q,ω) =1\nCzz(q)Re0/integraldisplay\n−∞dt1e(ǫ−iω)t1(Iα\nSz(q);Iγ\nSz(−q,t1)), (34)\nandCzz(q) = (Sz(q),Sz(−q))0.\nFrom (27) it follows that the expression for spin-wave curre nt in the ferromagnetic\ninsulator is due to nonequilibrium magnon system and can be w ritten as\nδMz(q,ω) =χ0[qαqγDzz\nα,γ(q,ω)+Os(q,ω)(ωm/ωs)2]\nLm(sm)(q,ω)+Lm(mp)(q,ω)(35)11\nwhereχ0=β(gmµ0)2(Sz(q);Sz(−q))is the static susceptibility of localized spins. As\ncan be seen from the formula (35), the spin-wave current depe nds on the frequency of an\nexternal field in a resonance manner. As for the correlation f unctions Li(jk)describing\nthe relaxation processes, the papers [43, 37] have calculat ed them.\nConclusions\nUsing the non-equilibrium statistical operator method (NS O), we have investigated\nthe spin transport through the interface in a semiconductor /ferromagnetic insulator\nhybrid structure. We have analyzed the approximation of effe ctive parameters, when\neach of the considered subsystems (conduction electrons, m agnons, and phonons)\nis characterized by its effective temperature. We have const ructed the macroscopic\nequations describing the spin-wave current caused by both t he resonantly exciting\nspin subsystem of conduction electrons and an inhomogeneou s temperature field in\nthe ferromagnetic insulator. Also, we have derived the gene ralized Bloch equations\ndescribing the spin-wave current propagation in the insula tor and taking into account\nthe resonant-diffusion nature of the propagation of magnons and their relaxation\nprocesses. 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Physics 7, 79 (1971)."    },    {        "title": "0806.4104v1.Experimental_Observation_of_the_Inverse_Proximity_Effect_in_Superconductor_Ferromagnet_Layered_Structures.pdf",        "content": "arXiv:0806.4104v1  [cond-mat.supr-con]  25 Jun 2008\n/BX/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /C7/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C1/D2 /DA /CT/D6/D7/CT /C8/D6/D3 /DC/CX/D1/CX/D8 /DD /BX/AR/CT\r/D8 /CX/D2/CB/D9/D4 /CT/D6\r/D3/D2/CS/D9\r/D8/D3/D6/BB/BY /CT/D6/D6/D3/D1/CP/CV/D2/CT/D8 /C4/CP /DD /CT/D6/CT/CS /CB/D8/D6/D9\r/D8/D9/D6/CT/D7/CA/BA /C1/BA /CB/CP/D0/CX/CZ/CW/D3 /DA/B8 /C1/BA /BT/BA /BZ/CP/D6/CX/CU/D9/D0/D0/CX/D2∗/CP/D2/CS /C6/BA /C6/BA /BZ/CP/D6/CX/CU /B3/DD /CP/D2/D3 /DA/CI/CP/DA/D3/CX/D7/CZ/DD /C8/CW/DD/D7/CX\r /CP/D0/B9/CC /CT \r/CW/D2/CX\r /CP/D0 /C1/D2/D7/D8/CX/D8/D9/D8/CT/B8 /C3/CP/DE/CP/D2 /CB\r/CX/CT/D2/D8/CX/AS\r /BV/CT/D2/D8/CT/D6 /D3/CU /CA /D9/D7/D7/CX/CP/D2 /BT \r /CP/CS/CT/D1/DD /D3/CU /CB\r/CX/CT/D2\r /CT/D7/B8 /BG/BE/BC/BC/BE/BL /C3/CP/DE/CP/D2/B8 /CA 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/BF/BH/BL /B4/BD/BL/BH/BG/B5/BA/CJ/BD/BI℄ /C4/BA /C6/CT/DA /D3/D8 /CP/D2/CS /C8 /BA /BV/D6/D3 \r/CT/B8 /CA/CT/DA/BA /C8/CW /DD/D7/BA /BT/D4/D4/D0/BA /BD/BH /B8 /BJ/BI/BD /B4/BD/BL/BK/BC/B5/BA/CJ/BD/BJ℄ /C4/BA /C4/CP/DE/CP/D6/B8 /C3/BA /CF /CT/D7/D8/CT/D6/CW/D3/D0/D8/B8 /C0/BA /CI/CP/CQ /CT/D0 /CT/D8 /CP/D0/BA /B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /BU/BI/BD /B8 /BF/BJ/BD/BD /B4/BE/BC/BC/BC/B5/BA/CJ/BD/BK℄ /BT/BA /BU/BA /C8/CX/D4/D4/CP/D6/CS/B8 /CA/CT/D4/BA /C8/D6/D3/CV/BA /C8/CW /DD/D7/BA /BE/BF /B8 /BD/BJ/BI /B4/BD/BL/BI/BC/B5/BA/CJ/BD/BL℄ /C2/BA /B9/C5/BA /BW/CT/D0/D6/CX/CT/DB/B8 /CB/D3/D0/CX/CS /CB/D8/CP/D8/CT /BV/D3/D1/D1 /D9/D2/BA /BK /B8 /BI/BD /B4/BD/BL/BJ/BC/B5/BA/CJ/BE/BC℄ /C4/BA /BW/D3/CQ/D6/D3/D7/CP /DA/D0/CY/CT/DA/CX\r/B8 /BV/BA /CA/BA /BT \r/CP/CS/BA /CB\r/CX/BA /C8 /CP/D6/CX/D7 /BE/BI/BF /B8 /BH/BC/BE /B4/BD/BL/BI/BI/B5/BA/CC /BT/BU/C4/BX /BD/BA /BX/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /D8/CW/CT /D7/D8/D9/CS/CX/CT/CS/D7/CP/D1/D4/D0/CT/D7/BA /BZ/CX/DA /CT/D2 /CP/D6/CT /D8/CW/CT /D8/CW/CX\r /CZ/D2/CT/D7/D7 /D3/CU /D8/CW/CT /CE /D0/CP /DD /CT/D6dV\n/B8 /D8/CW/CT\n/D6/D3/D9/CV/CW/D2/CT/D7/D7 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /CA/D3/D9/CV/CW /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT /AS/D8 /D3/CU /D8/CW/CT/D7/D1/CP/D0/D0 /CP/D2/CV/D0/CT /DC/B9/D6/CP /DD /D6/CT/AT/CT\r/D8/CX/DA/CX/D8 /DD /B8 /D8/CW/CT /D7/D9/D4 /CT/D6\r/D3/D2/CS/D9\r/D8/CX/D2/CV /D8/D6/CP/D2/B9/D7/CX/D8/CX/D3/D2 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT Tc\n/B8 /D8/CW/CT /D6/CT/D7/CX/CS/D9/CP/D0 /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /D6/CP/D8/CX/D3RRR /B8/D8/CW/CT /CT/D0/CT\r/D8/D6/D3/D2 /D1/CT/CP/D2 /CU/D6/CT/CT /D4/CP/D8/CW /CX/D2 /D8/CW/CT /CE /D0/CP /DD /CT/D6l /B8 /CP/D2/CS /D8/CW/CT /D7/D9/B9/D4 /CT/D6\r/D3/D2/CS/D9\r/D8/CX/D2/CV \r/D3/CW/CT/D6/CT/D2\r/CT /D0/CT/D2/CV/D8/CWξs\n/BA /CC/CW/CT /D8/CW/CX\r /CZ/D2/CT/D7/D7 /D3/CU /D8/CW/CT/D1/CP/CV/D2/CT/D8/CX\r /D0/CP /DD /CT/D6/D7 /CX/D7 /CP/CQ /D3/D9/D8 /BF /D2/D1 /CU/D3/D6 /CP/D0/D0 /D8/D6/CX/D0/CP /DD /CT/D6 /D7/CP/D1/D4/D0/CT/D7/BA/CB/CP/D1/D4/D0/CT dV\n/CA/D3/D9/CV/CWTcRRRlξs/B4/D2/D1/B5 /B4/D2/D1/B5 /B4/C3/B5 /B4/D2/D1/B5 /B4/D2/D1/B5/CE /BF/BC /BC/BA/BF /BG/BA/BI/BH /BD/BD /BD/BH /BD/BG/C8/CS0.98\n/BY /CT0.02\n/BB/CE/BB/C8/CS0.98\n/BY /CT0.02\n/BF/BI /BD/BA/BF /BF/BA/BC/BE /BG/BA/BI /BH /BK/C8/CS0.97\n/BY /CT0.03\n/BB/CE/BB/C8/CS0.97\n/BY /CT0.03\n/BG/BE /BD/BA/BF /BF/BA/BH/BH /BI /BJ /BD/BC/C6/CX/BB/CE/BB/C6/CX /BG/BG /BD/BA/BI /BG/BA/BC/BH /BG/BA/BG /BH /BK/BH\n0 1 2 3 4 52468\n  H (kOe) \nT (K) /BY/CX/CV/D9/D6/CT /BD/BM /CC/CW/CT /D9/D4/D4 /CT/D6 \r/D6/CX/D8/CX\r/CP/D0 /AS/CT/D0/CS /DA/D7 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CU/D3/D6 /D7/D8/D9/CS/B9/CX/CT/CS /D7/CP/D1/D4/D0/CT/D7 /CX/D2 /D8/CW/CT /D4 /CT/D6/D4 /CT/D2/CS/CX\r/D9/D0/CP/D6 /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/DC/B9/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3 /D8/CW/CT /AS/D0/D1 /D4/D0/CP/D2/CT/BA /BY /D9/D0/D0/D7/D5/D9/CP/D6/CT/D7 /D6/CT/D0/CP/D8/CT /D8/D3 /D8/CW/CT /D7/CX/D2/CV/D0/CT /CE/B9/D0/CP /DD /CT/D6/B8 /D3/D4 /CT/D2 /D8/D6/CX/CP/D2/CV/D0/CT/D7 /B9/D8/D3 /C8/CS0.98\n/BY /CT0.02\n/BB/CE/BB/C8/CS 0.98\n/BY /CT0.02\n/B8 \r/D0/D3/D7/CT/CS /D8/D6/CX/CP/D2/CV/D9/D0/CP/D6/CT/D7 /B9 /D8/D3/C8/CS0.97\n/BY /CT0.03\n/BB/CE/BB/C8/CS 0.97\n/BY /CT0.03\n/B8 /CP/D2/CS \r/D0/D3/D7/CT/CS \r/CX/D6\r/D0/CT/D7 /B9 /D8/D3 /C6/CX/BB/CE/BB/C6/CX/D8/D6/CX/D0/CP /DD /CT/D6/D7/BA\n4900 4920 4940 4960 4980 51 V\n(e) 1,8K \nH (Oe) dP/dH (arb. units) \n \n  1,4K \n \n  1,4K \n \n  (d) (c) (b) 1,4K \n \n  3K \n \n   (a) /BY/CX/CV/D9/D6/CT /BE/BM /C6/C5/CA /D7/D4 /CT\r/D8/D6/CP /CU/D3/D6 /D8/CW/CT /D7/CX/D2/CV/D0/CT /CE/B9/D0/CP /DD /CT/D6 /CX/D2 /D8/CW/CT /D2/D3/D6/D1/CP/D0 /B4T/BP /BF /C3/B5 /B4/CP/B5 /CP/D2/CS /D7/D9/D4 /CT/D6\r/D3/D2/CS/D9\r/D8/CX/D2/CV /B4T /BP/BD/BA/BG /C3/B5 /B4/CQ/B5 /D7/D8/CP/D8/CT/D7 /CP/D2/CS /CU/D3/D6/C8/CS0.98\n/BY /CT0.02\n/BB/CE/BB/C8/CS 0.98\n/BY /CT0.02\n/B4\r/B5/B8 /C8/CS0.97\n/BY /CT0.03\n/BB/CE/BB/C8/CS 0.97\n/BY /CT0.03/B4/CS/B5 /CP/D2/CS /C6/CX/BB/C8/CS/BB/C6/CX /B4/CT/B5 /D8/D6/CX/D0/CP /DD /CT/D6/D7 /CX/D2 /D8/CW/CT /D7/D9/D4 /CT/D6\r/D3/D2/CS/D9\r/D8/CX/D2/CV /D7/D8/CP/D8/CT/BA/BT/D0/D0 /CS/CP/D8/CP /CP/D6/CT /CV/CX/DA /CT/D2 /CU/D3/D6 /D8/CW/CT /CT/DC/D8/CT/D6/D2/CP/D0 /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D4 /CT/D6/D4 /CT/D2/CS/CX\r/B9/D9/D0/CP/D6 /D8/D3 /D8/CW/CT /D7/CP/D1/D4/D0/CT /D4/D0/CP/D2/CT/BA /CC/CW/CT /C6/C5/CA /D7/D4 /CT\r/D8/D6/CP /CU/D3/D6 /D8/CW/CT /D7/CX/D2/CV/D0/CT /CE/D0/CP /DD /CT/D6 /CP/D6/CT /D7/CX/D1 /D9/D0/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /BZ/CP/D9/D7/D7/CX/CP/D2 /D0/CX/D2/CT/D7/CW/CP/D4 /CT /D3/CU /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/D4 /CT/CP/CZ/B9/D8/D3/B9/D4 /CT/CP/CZ /D0/CX/D2/CT/DB/CX/CS/D8/CW /B4/CS/CP/D7/CW/CT/CS \r/D9/D6/DA /CT/D7/B5/BA/BI\n0.1 0.2 0.3 0.4 -1 01\n(b)  \n dP/dH (arb. units) \n(H-HN)/Hm\n  -1 012\n \n   \n(a) /BY/CX/CV/D9/D6/CT /BF/BM /B4/CP/B5 /CC/CW/CT /D4/D9/D6/CT /BZ/CP/D9/D7/D7/CX/CP/D2 /D0/CX/D2/CT/D7/CW/CP/D4 /CT /CS/CT/D6/CX/DA /CP/D8/CX/DA /CT /DB/CX/D8/CW/D4 /CT/CP/CZ/B9/D8/D3/B9/D4 /CT/CP/CZ /DB/CX/CS/D8/CW /CP/D2/CS /D4 /D3/D7/CX/D8/CX/D3/D2 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8 /D8/D3 /D8/CW/CT /D1/D3 /CS/CT/D0 /D7/D4 /CT\r/B9/D8/D6/D9/D1 /CQ /CT/D0/D3 /DB/BA /B4/CQ/B5 /C5/D3 /CS/CT/D0 \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /C6/C5/CA /D0/CX/D2/CT /D7/CW/CP/D4 /CT/CX/D2 /CP/D2 /BY/BB/CB/BB/BY /D8/D6/CX/D0/CP /DD /CT/D6 /DB/CX/D8/CWξs/d /BP/BC/BA/BE /CP/D2/CS /D8/CW/CT /BZ/CP/D9/D7/D7/CX/CP/D2 /CQ/D6/D3/CP/CS/B9/CT/D2/CX/D2/CV /D4/CP/D6/CP/D1/CT/D8/CT/D6 σ/Hm\n/BP/BC/BA/BC/BI /B4/CQ/B5/BAHN\n/CX/D7 /D8/CW/CT /D6/CT/D7/D3/D2/CP/D2\r/CT /AS/CT/D0/CS/CX/D2 /D8/CW/CT /D2/D3/D6/D1/CP/D0 /D7/D8/CP/D8/CT/BA /C7/D2/D0/DD /D8/CW/CT /D0/CX/D2/CT/D7/CW/CP/D4 /CT /CS/CX/D7/D8/D3/D6/D8/CX/D3/D2 /CP/D2/CS /D8/CW/CT/D0/CX/D2/CT /D7/CW/CX/CU/D8 /CS/D9/CT /D8/D3 /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D4/D6/D3 /DC/CX/D1/CX/D8 /DD /CT/AR/CT\r/D8 /DB /CT/D6/CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS/CX/D2 \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4 /CT\r/D8/D6/D9/D1 /B4/CQ/B5/BA"    },    {        "title": "0707.1627v1.Soft_X_ray_Circular_Reflectivity_from_Ferromagnetic_Transition_Metal_Films_Near_the_Brewster_s_Angle__Theoretical_and_Numerical_X_ray_Resonant_Magnetic_Scattering_Study.pdf",        "content": " 1Soft X-ray Circular Reflectivity fr om Ferromagnetic Transition-Metal \nFilms Near the Brewster’s Angle:  Theoretical and Numerical X-ray \nResonant Magnetic Scattering Study \n \nDae-Eun Jeong and Sang-Koog Kim * \nResearch Center for Spin Dynamics & Spin-W ave Devices, Seoul National University, Seoul \n151-744, Republic of Korea \nNanospintronics Laboratory, Department of Mate rials Science and Engineering, College of \nEngineering, Seoul National University, Seoul 151-744, Republic of Korea \nAbstract  \nWe first report a novel phenomenon that manifests itself in a colossal difference in soft x-ray \nreflectivity from ferromagnetic transition-metal fi lms between the left- and right-handed circular \npolarization (LCP and RCP) modes at a resonance near  the normal Brewster’s angle.  Theoretical and \nnumerical studies of soft x-ray resonant magnetic scattering using the circular-polarization-mode basis reveal that this effect arises fro m a totally destructive interference of photons scattered individually from \ncharge, orbital, and spin degrees of freedom in magnetized thin films that selectively occurs only for one helicity of the opposite circular modes when the re quired criteria are fulfilled.  Across the normal \nBrewster’s angle, the polarization state of scattered so ft x rays is continuously variable from the RCP to \nthe LCP mode (or vice versa) through the linear s polarization mode by changing the incidence angle of  \nlinearly p-polarized x rays at the resonance. \n \nPACS: 78.70.Ck, 78.20.Ls  2Soft or hard x-ray resonant magnetic scattering (XRMS) measurement techniques have been \nwidely used to investigate charge, orbital, and spin degrees of freedom in multi-component magnetic materials because those techniques offer exceedingl y enhanced, element-speci fic sensitivity to such \ndifferent orderings at energies close to the absorption edges of a selected element [1-6].  Due to a variety of microscopic interactions between incident photons an d each of those orderings, as well as their angular \nand polarization dependence in the XRMS process,  the initial polarization state of incident photons is converted to various polarization st ates of the scattered photons, which in turn makes it possible to \ndetermine element-specific charge, orbital, and spin orderings by analyzing the changed polarization \nstates of the scattered soft x rays and their angular and polarization dependence. \nAny arbitrary polarized states of photons can in principle be described in terms of the \northogonal right- and left-handed circular polarization (RCP and LCP) modes (or opposite photon helicities). Since the RCP and LCP modes are not only the basis of an irreducible representation of rotational symmetries in atomic transition processes [7], but are also the eigenmodes of photon beams interacting with the different kinds of orderings in broken symmetries, such circular polarizations are useful in the determination of the fundamental atomic transition spectra in the x-ray resonant region for magnetic materials [8]. For example, material syst ems with broken symmetries can yield circular-mode-\ndependent dichroism, such as natural circular dichroism in chiral materials [9, 10], magnetic circular dichroism in magnetic materials [11, 12], and magnetochiral dichroism in parity-nonconserving magnetic  3materials [13]. Thus, in order to obtain better or deeper insight into not only the interactions of incident \nphotons with different scattering sources of charge, orbital, and spins, but also their polarization and angular dependences, theoretical interpretations of  XRMS in the framework of the RCP-and-LCP-modes \nbasis is more fundamental than the linear-polarization-modes basis.  \nIn this Letter, we report the first prediction of differential soft x-ray circular reflectivity from \nferromagnetic transition-metal thin films at the reso nance energies in a wide angular region across the \nnormal Brewster’s angle \n45n\nBφ≅° .  It was found that this novel effect arises from a totally destructive \ninterference occurring selectively for either circular polarization mode of photons scattered individually \nfrom charge, orbital, and spin degrees of freedom. This can be verified not only by theoretical derivations of the XRMS amplitudes with respect to the circular-mode basis, but also by numerical calculations of the individual intensities of the RCP and LCP component s using circular-mode-bas ed magneto-optical Kerr \nmatrices. Also, the continuously variable polarization state of reflected soft x rays from the RCP to the \nLCP mode (or vice versa) through the linear s polarization mode is observable in a wide angular region \nacross the \nn\nBφ, and is thus important in that it can be implemented into polarizers or analyzers that enable \nthe control or determination of the polarization states of incoming photons. \nFirst, we derive the XRMS amplitudes with respect to the circular-mode basis by considering a \ntotal coherent elastic scattering amplitude in the pure electric dipole ( 1E) transition for the case of 3d \ntransition-metal ferromagnetic materials. The amplitude can be expressed by tot c xres mf ff f≅+ + ,  4where cf, mf, and xresf are the charge, non-resonant magnetic, and resonant scattering contributions, \nrespectively[1, 14, 15].  In general, mf is noticeably weaker than cf by a factor of \n20.002 mcω≅=  at 1ω== keV , but xresf is comparable to cf and is much stronger than mf in \nthe vicinity of the absorption edges [1]. In fact, xresfconsists of radial and angular parts. The former is \nrepresented by the reduced resonant scattering amplitude, \n() () ( ) ()( )2\n11 2 11 2,; 1 ; ,; 1 ; 1 R l jl R l jl x i′ =− Δ Γ −  in a fast collision approximation, and the latter \nby  k-th rank spin-orbital coupled moments, ()()11 2,; 1 ;kM lj l  as well as the polarization tensors, \n()()ˆˆ,; ,k\nrr i i Te k e k∗. Here, ()ˆrieand ()ˆ\nrik represent the polarization and propagation unit vectors of a \nreflected (incident) phot on beam, respectively. \nSince the magnetic linear dichroism term ()2M  is negligible in the magnitude of the \nscattering amplitude for 3d transition metals [11, 12, 14-17], only the ()0M  and ()1M  terms are \nconsidered in determining the resonant scattering amp litudes with respect to the circular polarization basis, \nwhich is given in matrix form by [15, 18] \n    ()()()()()\n()()()()()2\n11 2 11 2\n0\n01\n11 2 11 2 11 23ˆˆ 4, ; 1 ; , ; , , ; 1 ;21\n3,; 1 ; ,; 1 ; ,; 1 ; ,2 2R k\nkk xres\nqr r i i q L\nkq k xres\nR\ni\nm L\nifRlj l T eke kM lj lk f\nN iR l jl M l jl Ml jl\nNπ∗∗\n== −⎛⎞′=⎜⎟+ ⎝⎠\n⎛⎞ ⎛⎞′≅− ⎜⎟ ⎜⎟⎝⎠ ⎝⎠∑∑\nAB\u001a\n\u001a      \n(1) \nwhere ()() 11ˆmM Mm=⋅  with a unit vector of magnetization, 11 2 2 3 3ˆˆ ˆ ˆmm u m u m u=++ . ()RL\niN  \nimplies the complex amplitude factor of the RCP(LCP ) mode with respect to the circular polarization  5basis, ˆˆ ˆRR LL\nii i i ieN e N e=+ . The matrices of  A  and B  are given as \n() ()†ˆˆ ˆˆsps p\nri R L ri R LRL spee ee∗∗=⋅ = ⋅AU U  and ()()†ˆˆ ˆˆsps p\nri R L ri R LRL spee ee∗∗=× = ×BU U . The unitary \noperation in the matrices transforms a linear to circul ar polarization basis or vice versa, which leads to \n() ()\n() ()111c o s 2 1c o s 222\n111c o s 2 1c o s 222φ φ\nφ φ⎛⎞+−⎜⎟\n=⎜⎟\n⎜⎟−+⎜⎟⎝⎠A and 12 13\n13 1211sin 2 cos sin 2 sin22,11sin 2 sin sin 2 cos22mi m mi m\nmi m mi mφφ φφ\nφ φφ φ⎛⎞−− −⎜⎟\n=⎜⎟\n⎜⎟+− + ⎜⎟⎝⎠B  where \nφ is the angle of incidence from the reflection surface as defined in Fig. 1. These matrices represent the \npolarization dependence of the total amplitude in the scattering geometry; in particular,  B shows the \nrelation of φ and ˆm for each resonant scattering  source. For the longitudinal \n()21 3 1, 0 mm m=± = =  and polar magnetizations ( )31 2 1, 0 mm m=±= = , the corresponding B  \nare transformed into 2\n2cos 0\n0c o sim\nimφ\nφ±−⎛⎞=⎜⎟⎝⎠lonB and  3\n30s i n\nsin 0im\nimφ\nφ±− ⎛⎞=⎜⎟⎝⎠polB.  The operation of B   \nleads to ()()ˆˆRL RL\niree→  for the longitudinal case and ()()ˆˆRL LR\niree→  for the polar case, originating \nfrom the conservation of each helicity in the presence of an axial symmetry with respect to ˆm [7]. \n For the specific case of 1E resonant scattering at the 3L line from a 3d transition metal, \nusing the relations of () () ()( ) []1 2 2 2\n3232 913 2 225 2Ib Re x i d r p i i F εε ω−⎧⎫⎡⎤′≅− Δ Γ − − − − Γ ≡ ⎨⎬⎢⎥⎣⎦⎩⎭\u001a\u001a =, \n()() ( )() 00222299hm M NS x i M≅− Δ Γ − ≡⎡⎤⎣⎦, ()() ( )() 1 1 1132 2\n92 92mm m hM SLN x i M≅− + −Δ Γ − ≡−⎡⎤⎣⎦, the total scattering \namplitudes of the RCP and LCP modes are finally obtained as \n()()()1\n0\n0() .4RR\ntot i\nc LL\ntot ifN FMrF K i F M\nfN⎛⎞⎛⎞ ⎛⎞=− + −⎜⎟⎜⎟ ⎜⎟⎜⎟⎝⎠ ⎝⎠⎝⎠ABG\n               ( 2 )   \nEq. (2) implies that each of the charge () 0crF KG\n, resonant non-magnetic ()0FM , and resonant  6magnetic ()1FM  scattering terms can contribute to R\ntotf and L\ntotf, by way of the correspondence to \nthe scattering geometry factors represented by the matrices of A and B.  To obtain the individual \nvalues of R\ntotf andL\ntotf, one should consider the individual phase factors of the amplitudes of photons \nscattered from different scattering sources for each circular mode.   \nThe relations between the phases of photons s cattered from the three scattering sources and \nincident photons for the RCP and LCP modes are described in Table I for the case of an incident p \npolarization and the longitudinal magnetization of 2ˆˆmu=+. The phase factors of 1+, 1−,  i+, \nand i− represent the phase changes in the reflected ph otons with respect to those of the incident \nphotons, which correspond to /2π− , 2π , 0, and π, respectively, for  \nˆˆ ˆ\n22pR L\nii iiiee e=− + .  The asymmetry in these phase factors arises from the phase difference of \nπ between the RCP- and LCP-mode components of the incident p polarization. Using the relations \nshown in Table I, the total scattering amplitudes are given as  \n            ()()()1\n0\n0cos 2 cos 1() .cos 2 cos 4 2R\ntot\nc L\ntotf FMir F K FM\nfφ φ\nφ φ⎛⎞ ⎛⎞ ⎛⎞⎛ ⎞=+ +⎜⎟ ⎜⎟ ⎜⎟⎜ ⎟ ⎜⎟ −⎝⎠⎝ ⎠ ⎝⎠ ⎝⎠G\n        ( 3 )  \nHere, ()RL\ntotf  consists of three different contributions such that() () () ()\n:0 :1RL RL RL RL\ntot c xres xresf fff=++ . The \nphase factors of ()RL\ncf  and ()\n:0RL\nxresf  are determined by multiplying the cos 2φ(cos 2φ− ) angular \ndependence.  In the case of 4φπ>  , the phase factor of ()RL\ncf  is ()ii−+, and that of ()\n:0RL\nxresf  is \nof the multiple states of ()11−+  and ()ii+−.  On the other angular side, /4φπ< ,  the phase  7factors of ,: 0R\ncx r e sf and ,: 0L\ncx r e sf have signs opposite to those for 4φπ> .  However, the phase factor \nof ()\n:1RL\nxresf  does not vary with φ across /4φπ= , but retains the allowed multiple states of 1+ and \ni−.  Furthermore, due to the different angular dependence, that is, cos 2φ for both ()RL\ncf  and \n()\n:0RL\nxresf , and cosφ for ()\n:1RL\nxresf , each value of ()RL\ncf  and ()\n:0RL\nxresf  could become comparable in \nmagnitude to ()\n:1RL\nxresf  at certain angles close to /4φπ= . Thus, the asymmetry of the polarization \ndependence between the RCP and LCP modes gives rise to the differential scattering amplitude, that is, \nRL\ntot totf f≠ . This inequality can yield differential circular reflectivity for the RCP and LCP modes under \nspecific conditions, wher e either condition, 0R\ntotf≠ and 0L\ntotf≅ or 0L\ntotf≠and 0R\ntotf≅, can be \nfulfilled.  The illustration of an example of the complete destructive interference for the RCP mode and a \nnon-vanishing interference for the LCP mode are give n in Fig. 2. The total amplitude is nearly zero \nthrough the vector sum of the imaginary phase terms (0)\n:0R\nxresf , ()\n:1R\nxresfπ, and ()R\ncfπ, and the real phase \nterms   ()2\n:0R\nxresfπ and ()2\n:1R\nxresfπ− for the RCP mode, but is of a finite value for the counterpart LCP mode.  \nConsequently, we can obtain ()0RL\ntotf≈  and ()0LR\ntotf≠  at certain circular-mode-dependent \nBrewster's angles (denoted by ,R\nBl o nφ+ and ,L\nBl o nφ+) . The estimate of ()\n,RL\nBl o nφ+ can be made by solving \n()0RL\ntotf= at () ()\n,, 4RL RL\nBl o n Bl o nφπ δ φ++=+  for a strong resonant case of 0crF F<  in a first-order \napproximation of ()\n,RL\nBl o nδφ+, \n             ()\n() ()2\n()\n,23 3() .16 2 16 2mm RL mm\nBl o n\nhhSL SL\nNNδφ+⎛⎞+ +≅+− − ⎜⎟⎜⎟⎝⎠                    ( 4 )  \nFrom Eq. (4), ,, ,RL\nBlon B lon B lonφφφ+++Δ=− is given as approximately ()( ) 23 / 8 2mm hSLN+ . By inserting  8into Eq. (4) the numerical values of () 2 8cFK Z≅=G\n, 050Fr≅ , 1.56mS≅ , 0.13mL≅ , \n()22 . 5hN≅ , 5eVΓ≅ , and 1.2eVΔ≅ for Co [17, 19, 20],  we predicted the numerical values \nof ()()\n, 3.81 4.09RL\nBl o nδφ+=° −°  and , 7.9Bl o nφΔ= ° . This angular difference is somewhat large for 3 d-\ntransition metals. This allo ws us to select either the LCP- or RCP-mode component of scattered soft x \nrays by changing the incidence angle across /4n\nBφπ≅ . \n  To confirm the theoretical XRMS prediction of  a colossal difference in soft x-ray circular \nreflectivity between the opposite photon helicities near /4n\nBφπ≅ , we also numerically calculated the \nintensities of the individual RCP and LCP components as well as the linear s and p components of \nphotons reflected from a model thin film consisting of a Co ( 10nm) layer on an Si substrate for both \ncases of 2ˆˆmu=±  with a linearly p-polarized incident photon beam at the Co 3L edge. In the \ncalculations, we used the circular-mode-based magneto- optical Kerr matrix [21].  Figure 3(a) shows the \nangular variations of the individual reflectivities of the s- and p- and RCP- and LCP-mode components in \nspecular reflection geometry for a de magnetized state of the Co film.  In the reflectivity profiles, there \nexists a n\nBφ where the reflectivities are extremely low for the linear p polarization, RCP, and LCP modes \nof the scattered soft x rays, while the reflectivity of the linear s polarization mode is zero in the whole \nangular range due to no net magnetization.  In comparison with the non-magnetized case, contrasting \nreflectivity profiles between the s and p linear modes, and the RCP and LCP modes are observed near n\nBφ \nfrom the longitudinally magnetized Co film for the incident p polarization, as shown in Fig. 3(b).  The  9numerical values of , 39.5L\nBl o nφ+=° , , 45.2p\nBl o nφ±=°, and , 48.4R\nBl o nφ+=°are observed, as predicted \nby the circular-mode-based XRMS theory. \n At ()\n,RL\nBl o nφ±, the degree of circular polarization represented by the Stokes parameter 3S \nreaches 1+ or 1−, indicating that almost pure circular pola rizations can be obtained from the incident \np polarization and that the opposite photon helicities can be readily switchable either by changing the \nincident angle from ,L\nBl o nφ± to ,R\nBl o nφ± or by reversing the longitudinal magnetization.  Also, at \n,p\nBl o nφ±  the degree of linear polarization represented by 1S reaches 1+,  indicating the pure s \npolarization. The values of ()()\n, 3.4 5.5RL\nBl o nδφ+=° −°  obtained from the nume rical calculation are in \ngood agreement with those values of ()()\n, 3.81 4.09RL\nBl o nδφ+=°− °  predicted using Eq. (4).  In the above \nnumerical calculation, it was also found that continuously variable polarization states can be obtained by \nchanging φ across n\nBφ in the specular reflection.  The variation from the RCP to the LCP mode \nthrough the linear s polarization mode can be simply obtained by tuning to , 39.5L\nBl o nφ+=° , \n, 45.2p\nBl o nφ±=° , and , 48.4R\nBl o nφ+=°   in a wide angular range, or vice versa for the opposite \nmagnetization orientation, as shown in Fig. 3(d).  This novel phenomenon can be implemented into \npolarizers or analysers that enable the control or determination of the polarization states of incoming \nphotons simply and at very low cost, as multilayer-thin-film linear polarizers [22].  \nIn conclusion, we made a theoretical derivation of the XRMS amplitudes for the individual \nRCP and LCP modes of soft x rays scattered individually from the charge, orbital and spin degrees of  10freedom for linearly p–polarized incident photons at the resonance.  From this derivation, we found a \ncolossal difference in the soft x-ray circular reflectiv ity from ferromagnetic transition-metal films over a \nwide incidence-angle range near the normal Brewster’s angle.  This difference originates from a totally \ndestructive interference effect occu rring selectively for either the RCP  or LCP mode at certain angles \nacross the normal Brewster's  angle. The XRMS theory in the framework of the circular-mode basis offers \na more fundamental understanding of the polarization and angular-dependent scattering of soft x rays from different scattering sources such as charge, orbital, and spin degrees of freedom in a magnetized material.  Also, the findings on the wide angular difference of the opposite circular-mode-dependent \nBrewster’s angles, as much as \n, 8.9Bl o nφΔ= ° , and on the continuously variable polarization state with \nslight changes in the incident angle of a linearly p-polarized photon beam, might provide practical \napplications for an optical production (or determination) of the linear and circular components using polarizing elements (or analyzers).  This work was supported by Creative Resear ch Initiatives (ReC-SDSW) of MOST/KOSEF.   11References  \n[1] M. Blume, J. Appl. Phys. 57, 3615 (1985). \n[2] D. Gibbs, G. Gr übel, D. R. Harshman, E. D. Isaacs, D. B. McWhan, D. Mills and C. Vettier, Phys. Rev. \nB 43, 5663 (1991). \n[3] J. B. Kortright, D. D. Awschalom, J. St öhr, S. D. Bader, Y . U. Idzerda, S. S. P. Parkin, Ivan K. Schuller \nand H. -C. Siegmann, J. Magn. Magn. Mater. 207, 7 (1999). \n[4] J. B. Kortright and S. -K. Kim, Phys. Rev. B 62, 12216 (2000). \n[5] J. B. Kortright, S. -K. Kim, G. P. Denbeaux, G. Zeltzer, K. Takano, and  E. E. Fullerton Phys. Rev. B \n64, 092401 (2001). \n[6] G . Srajer , L. H. Lewis, S. D. Bader, A. J. Epstein, C. S. Fadley , E. E. Fullerton, A. Hoffmann, J. B. \nKortright, K. M. Krishnan, S. A. Majetich, T. S. Rahman, C. A. Ross, M. B. Salamon, I. K. Schuller, \nT. C. Schulthess, J. Z. Sun, J. Magn. Magn. Mater. 307, 1 (2006). \n[7] V . B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynamics  (Pergamon Press, \nOxford 1982). \n[8] G. van der Laan, Phys. Rev. B 57, 112 (1998). \n[9] F. Gel'mukhanov and H. Ågren, Phys. Rep. 312 87 (1999). \n[10] L. Alagna, T. Prosperi, S. Turchini, J. Goulon, A. Rogalev, C. Goulon-Ginet, C. R. Natoli, R. D. \nPeacock, and B. Stewart, Phy. Rev. Lett. 80, 4799 (1998). \n[11] B. T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev. Lett. 68, 1943 (1992). \n[12] P. Carra, B. T. Thole, M. A ltarelli, and X. Wang, Phys. Rev. Lett, 70, 694 (1993). \n[13] J. Goulon, A. Rogalev, F. Wilhelm, C. Goulon-Ginet, and P. Carra, Phys Rev. Lett. 88, 237401 (2002). \n[14] J. P. Hannon, G. T. Trammell, M.  Blume, and D. Gibbs, Phys. Rev. Lett. 61, 1245 (1988); ibid 62, \n2644(E) (1989). \n[15] J. Luo, G. T. Trammell, and J. P. Hannon, Phys. Rev. Lett. 71, 287 (1993). \n[16] G. van der Laan, Phys. Rev. Lett. 82, 640 (1999). \n[17] C. Kao, J. B. Hastings, E. D. Johnson, D. P . Siddons, G . C. Smith, and G . A. Prinz, Phys. Rev. Lett. \n65, 373 (1990). \n[18] This is a case where the symmetry is 32SO SO⊃  with a magnetic ordering and a negligible \ncrystal-field effect. \n[19] R. Wu and A. J. Freeman, Phys. Rev. Lett. 73, 1994 (1994). \n[20] S. -K. Kim and J. B. Kortright, Phys. Rev. Lett. 86, 1347 (2001). \n[21] D. -E. Jeong, K. -S. Lee, and S. -K. Kim, Appl. Phys. Lett. 88, 181109 (2006). \n[22] J.B. Kortright, M. Rice, S.-K. Kim, C.C.  Walton, T. Warwick J. Magn. Magn. Mater. 191 79 (1999).  12Table Ⅰ. Phase factors of the RCP- and LCP-mode components of photons scattered \nindividually from three different scattering sour ces, which indicate the phase change of the \nscattered photons with respect to those of the RCP- and LCP-mode components of a linearly p-\npolarized incident photon beam.  In the notation of ()\nnγαβ, α and β indicate the circular \npolarization components in the incident and reflected photons, respectively. The γ and n \nvariables represent the phase difference a nd the scattering source, respectively.  \n \n \n              ˆˆRR\niree→  ˆˆLR\niree→  ˆˆRL\niree→  ˆˆLL\niree→  \nCharge scattering (c)  ()0:c iR R+  ():c iL Rπ−  ()0:c iR L+  ():c iL Lπ−  \nResonant non-ma gnetic scattering \n(xres : 0) ()\n()2\n:0\n:01:\n:xres\nxresRR\niR Rπ\nπ−+\n−()\n()2\n:0\n0\n:01:\n:xres\nxresLR\niL Rπ−+()\n()2\n:0\n:01:\n:xres\nxresRL\niR Lπ\nπ−+\n− ()\n()2\n:0\n0\n:01:\n:xres\nxresLL\niL Lπ−+\nResonant magnetic scattering \n(xres : 1) ()\n()2\n:1\n:11:\n:xres\nxresRR\niR Rπ\nπ−+\n−Not allowed Not allowed ()\n()2\n:1\n:11:\n:xres\nxresLL\niL Lπ\nπ−+\n− 13Figure captions  \nFIG. 1. (color online) Definition and coordinate system used in the text for a specular reflection geometry \nwith the linear s- and p- as well as RCP- and LCP-mode bases in the incident and reflected photons. \n \nFIG. 2. (color online) Illustration of interferences in each of the RCP- and LCP-mode components of \nphotons scattered individually from the charge cf, resonant non-magnetic :0xresf , and resonant \nmagnetic :1xresf  scattering sources. The radius of each circle represents the magnitude of the \ncorresponding scattering amplitudes. The direction of the arrow inside each circle indicates the \ncorresponding phase factor, as noted in the inset, which is also described by γ in ,()RLfγ.  \n \nFIG. 3. ( color online) Calculations of the intensities of the linear s, p polarization and of the RCP- and \nLCP-mode components at the Co 3L edge for incident p-polarized x rays for (a) the demagnetization \nstate of Co and (b) the longitudinally magnetized states of 2ˆˆmu=±. The superscripts and subscripts in \n,,,\n,spRLI+− denote the corresponding polarization component in the scattered photons and either state of \n2ˆˆmu=± , respectively.  (c) Calculations of the Stokes parameters, 1S, 2S and 3S as a function of \nφ for 2ˆˆmu=+ . The relations of the Stokes parameters and the degree of linear PL or circular PC \npolarization are expressed by 3 C SP= and 22 2\n12 L SSP+= .  (d) Continuously variable polarization \nstates at the indicated angles in a wide angular  region across the normal Brewster’s angle for 2ˆˆmu=+ .   14Figure 1.   \nReflected photon\nφφ\n1ˆu3ˆu2ˆuIncident photon\nˆp\nie\nˆs\nieˆR\nie\nˆL\nie\n \n \n                         15Figure 2.   \n \n \n \n                       \n 16Figure 3.   \n \n "    },    {        "title": "2108.09132v1.Probing_anisotropy_in_epitaxial_Fe_Pt_bilayers_by_spin_orbit_torque_ferromagnetic_resonance.pdf",        "content": "Probing anisotropy in epitaxial Fe/Pt bilayers by spin-orbit torque\nferromagnetic resonance\nMohammad Tomal Hossain,1Sergi Lendinez,1Laura Scheuer,2Evangelos Papaioannou,3and M. Benjamin\nJungfleisch1,a)\n1)Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716,\nUSA\n2)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern,\n67663 Kaiserslautern, Germany\n3)Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, Halle, Germany\n(Dated: August 23, 2021)\nWe report the generation and detection of spin-orbit torque ferromagnetic resonance (STFMR) in micropat-\nternedepitaxialFe/Ptbilayersgrownbymolecularbeamepitaxy. Themagneticfielddependentmeasurements\nat an in-plane magnetic field angle of 45\u000ewith respect to the microwave-current direction reveal the presence\nof two distinct voltage peaks indicative of a strong magnetic anisotropy. We show that STFMR can be em-\nployed to probe the underlying magnetic properties including the anisotropies in the Fe layer. We compare\nour STFMR results with broadband ferromagnetic resonance spectroscopy of the unpatterned bilayer thin\nfilms. The experimental STFMR measurements are interpreted using an analytical formalism and further\nconfirmed using micromagnetic modeling, which shed light on the field-dependent magnetization alignment in\nthe microstructures responsible for the STFMR rectification. Our results demonstrate a simple and efficient\nmethod for determining magnetic anisotropies in microstructures by means of rfspectroscopy.\nExploring spin-orbit torques (SOTs) in novel mate-\nrial systems is a prosperous field in spintronics that\nhas attracted enormous attention in the past decade.\nStudies on SOTs enable both the realization of highly\nenergy efficient storage applications and an improve-\nment of our understanding of fundamental spin physics\nat interfaces. In this regard, spin-torque or spin-orbit\ntorque ferromagnetic resonance (STFMR) is a prominent\nchoice for studying spin-orbit toques in multilayers1,2.\nIt has been demonstrated that this mechanism can be\nobserved in a large range of material systems includ-\ning metallic1and insulating ferromagnetic bilayers3,4,\nantiferromagnets5, heterostructures made of topological\ninsulators6and transition metal dichalcogenides7, etc.\nA particular emphasis of these efforts has been on de-\ntermining the spin-to-charge conversion efficiency and\nthe spin-Hall angle. Moreover, most of these works re-\nlied on a macro-spin approach without considering any\nanisotropies1,8.\nAn overlooked aspect of STFMR has been the dy-\nnamicresponsedrivenbyoscillatorySOT.Othermagnon\nspintronic effects such as spin pumping, spin-Hall effect\nand auto-oscillation studies have extensively been used\nto determine what role the underlying magnetization dy-\nnamics plays. For instance, Sandweg et al. reported\nan enhanced contribution of surface spin-wave modes to\nthe spin pumping signal9. Meanwhile, Papaioannou et\nal. observed strong magnetic anisotropies giving rise to\ntwo distinct inverse spin-Hall effect voltage peaks driven\nby spin pumping in Fe/Pt10. Corresponding reports on\nSTFMR driven dynamics are scarce11,12and successful\na)mbj@udel.edudemonstration of probing magnetic anisotopy directions\nby STFMR remained elusive until now.\nHere, we report the generation and detection of\nSTFMR in micropatterned epitaxial Fe/Pt bilayers\ngrown by molecular beam epitaxy. We compare our\nSTFMR results with standard ferromagnetic resonance\nspectroscopy and show that in-plane magnetic field de-\npendentSTFMRcanbeemployedtoprobeallunderlying\nmagnetic properties including the anisotropies in the epi-\ntaxial Fe layer. The experimental results are interpreted\nusing an analytical formalism and further confirmed us-\ning micromagnetic modeling. Our results demonstrate\na simple and efficient method for determining magnetic\nanisotropies in microstructures by means of rfspec-\ntroscopy.\nThe bilayers were grown epitaxially on MgO ( 100) sub-\nstrates by electron-beam evaporation under a base pres-\nsure of 5\u000210\u00009mbar. The substrate’s surface was ini-\ntially cleaned in organic solvents, followed by annealing\nat 650\u000eC for 1 hour. The bilayers were then deposited\nwith a rate of 0.01 nm/s at a substrate temperatures\n300\u000eC, followed by 1 hour total annealing time at the\ngrowthtemperature, resultinginaFe( 10nm)/Pt( 5nm)\nbilayer configuration.\nThe epitaxial films were then etched to produce the\nSTFMR devices: first, a negative-tone photoresist was\nused to cover 80 \u0016m\u0002130\u0016m rectangular sections\nof the film. The rectangular sections are rotated 15\u000e\nto cover relative angles between \u0012STFMR = 45\u000eand\n\u0012STFMR =\u0000120\u000e. The surrounding film was then etched\nusing Ar ion milling. The shorted coplanar waveguides\nwere patterned on the top of the rectangular epitaxial\nfilm sections using a direct laser writer and a positive-\ntone photoresist. 10 nm of Ti and 100 nm of Au were\ne-beam evaporated and lifted off to finalize the waveg-arXiv:2108.09132v1  [cond-mat.mtrl-sci]  20 Aug 20212\nFigure 1. Illustration of experimental setup and measurement\nconfiguration. (a) Schematic illustration of flip-chip FMR\nmeasurement configuration of the unpatterned Fe/Pt films\nusing a coplanar waveguide (CPW). The films were placed on\nthetopofaCPWparalleltotheexternalmagneticfield. Mea-\nsurements for three different in-plane field directions ( \u0012FMR =\n45\u000e;0\u000e;\u000045\u000e) were collected. (b) Illustration of the STFMR\nmeasurement configuration with the patterned devices. The\nFe/Pt microstructures have lateral dimensions of 80 \u0016m x 130\n\u0016m. A series of samples with different orientations in 15\u000e-\nsteps with respect to crystallographic directions ( \u0012STFMR =\n45\u000e;30\u000e;15\u000e;0\u000e:::) are patterned. The solid straight arrows\nshow the magnetic anisotropy and the in-plane magnetic field\ndirections. Schematic of the coordinate systems used for (c)\nFMR and (d) STFMR. The red arrows indicate the orienta-\ntions of the corresponding in-plane fields which rotates clock-\nwise by 45\u000e(FMR) or 15\u000e(STFMR) (please note that for\nthe FMR measurements 'H;FMR =\u0012FMRand for the STFMR\nmeasurements 'H;STFMR = 45\u000e+\u0012STFMR).'uistheharduni-\naxial anisotropy direction (easy axis direction perpendicular\nto'u) which remains fixed at 0\u000ein our coordinate setup. All\nangles are measured with respect to the horizontal direction.\nuide contacts on the devices; see Fig. 1 for a schematic\nWeintroducetwoangularcoordinatesetsforoursetup:\n\u0012, the sample/device orientation angle, and ', the field\nangle; both defined with respect to the anisotropy axes\n[see Fig. 1]. For both FMR and STFMR measurements,\nthe coordinate system is defined with respect to the\nhard axis of the uniaxial anisotropy of the sample. In\nour Fe/Pt cubic system, an in-plane fourfold magnetic\nanisotropy is expected due to the cubic lattice symme-\ntry of Fe, together with an additional uniaxial magnetic\nanisotropy, which is superimposed on top of the four-\nfold anisotropy13,14. The FMR measurements were per-\nformed prior to the microstructuring in STFMR devices\nusing the identical samples. For the FMR measurements,\nthe entire substrate is rotated anti-clockwise in steps of\n45\u000efor successive measurements. In our coordinate setup(where hard axis of the uniaxial anisotropy is fixed along\nx-axis), the magnetic field is effectively rotated clockwise\nby45\u000e. We set the sample angle \u0012FMR = 0\u000ewhen the\nexternal magnetic field is parallel to the hard axis [see\nFig. 1(a)]. The external field angle 'H;FMRis rotated\nclockwise with respect to the hard axis of the uniaxial\nanisotropy [see Fig. 1(c)], so that 'H;FMR =\u0012FMR. The\nbilayers were then patterned into microstrips for STFMR\nexperiments at orientations ( \u0012STFMR) clockwise with re-\nspect to the hard axis in steps of 15\u000eas illustrated in\nFig. 1(b). We set \u0012STFMR = 0\u000ewhen the pattern\nis parallel to the hard axis of the uniaxial anisotropy\nfield and assign the device angles \u0012STFMRwith respect\nto this position [see Fig. 1(b)]. The external magnetic\nfield [shown as a red arrow in Fig. 1(b)] for all mea-\nsurements is applied at 45\u000ewith respect to the device\nedge so that a maximum STFMR signal strength can be\nachieved in saturation1,2,15. Hence, the external field an-\ngle is given by 'H;STFMR = 45\u000e+\u0012STFMR. The fourfold\ncubic anisotropy direction is always along the substrate\nedges [blue arrows in Figs. 1(a,b)]. In this coordinate\nsetup it is easy to realize that the FMR device oriented\nat\u0012FMRis equivalent to the STFMR configuration ori-\nented at\u0012STFMR =\u0012FMR\u000045\u000e. All measurements were\nperformed at room temperature.\nFor the FMR measurements [Fig. 1(a)], the magneti-\nzation dynamics is excited by the microwave magnetic\nfield of a coplanar wave guide (CPW) in transmission.\nThe transmitted S 12parameter is measured using a vec-\ntor network analyzer. The external magnetic field is ap-\nplied in-plane and swept between \u00002000Oe and 2000Oe.\nUponachievingtheFMRresonancecondition[Eq.(2)be-\nlow], the microwave magnetic field induces a precession\nof the magnetic moments in the Fe layer. This leads to\na resonant absorption of the microwave signal and thus\nresults in a characteristic symmetric Lorentzian absorp-\ntion when approaching resonance16. Compared to the\nbare Fe film, the Gilbert damping of the Fe/Pt bilayer is\nenhanced due to spin pumping from the Fe layer into the\nadjacent Pt layer10,17.\nFor the STFMR measurements a bias tee is used to si-\nmultaneously apply a microwave current and to measure\nthe rectified dcvoltage using a lock-in amplifier. A mi-\ncrowave frequency signal of 22dBm power is supplied by\nan Agilent E8257D signal generator. The external mag-\nnetic fieldHextapplied in the sample plane and swept\nbetween \u00002000Oe and 2000Oe for each device.\nThe micromagnetic simulations are carried out us-\ning the graphics processor unit (GPU)-accelerated\nprogram Mumax318. The device is modelled into\n1024 \u00021024 \u00021cells with an individual cell size of\n3:0nm\u00023:0nm\u00025:0nm with periodic boundary con-\nditions in two dimensions. The material parameters em-\nployed in simulations were obtained from the experimen-\ntal data [fits to Eqs. (1) and (2)]. The Gilbert damp-\ning constant19\u000b= 0:0081and the exchange stiffness\nconstant20A= 2\u000210\u00006erg/cm were used as simula-\ntion parameters. An acsincpulse driving magnetic field3\nFigure 2. (a) Absorption spectrum obtained by broadband FMR at \u0012FMR = 45\u000e. (b) Corresponding results of the STFMR\nmeasurements of the microstructure at an orientation of \u0012STFMR = 0\u000e(note that due to the chosen coordinate system \u0012STFMR =\n\u0012FMR\u000045\u000e). (c) Resonant frequency vs. magnetic field plot extracted from the FMR and STFMR results shown in (a) and\n(b). Open circles represent results obtained from FMR, filled circles represent results obtained from STFMR measurements.\nof amplitude 5 Oe and 50GHz cut-off frequency is ap-\nplied at an angle 90\u000e+\u0012simwith respect to the hard axis\nof the uniaxial anisotropy. A sweeping external field var-\nied between 0 Oe and 2000 Oe is applied in the plane\nat\u000045\u000e(45\u000ein clockwise orientation) angle from the\nacdriving field, hence 'H;sim= 45\u000e+\u0012sim. Thus,\u0012sim\nis equivalent to \u0012STFMR. The magnetization is relaxed\nand simulated for a total duration of 3 ns without an ac\ndriving field to find the ground state configuration of the\nmagnetization. The simulation is then run for another 4\nns for the dynamics simulation. The resonance frequency\nis then found from the Fourier transformation of the time\ndependence of the magnetization.\nFigure 2(a) shows the FMR spectra ( S12parameter)\nat\u0012FMR = 45\u000efor different frequencies. The external\nfield is swept between \u00002000Oe to 2000Oe at each fre-\nquency. Upon achieving the resonance condition, a min-\nimum transmission ( S12) is observed, corresponding to a\nmaximum absorption. The spectrum is symmetric with\nFigure 3. Fit to analytical model for an in-plane geometry,\nEq. (2). The solid and dotted lines show the fitted curves\nfor different devices and the solid circles are the experimen-\ntal results for STFMR measurement. The global minimum\nof the least square residue is numerically found and used as\nfitting method to fit all experimental STFMR data points for\ndifferent device angles \u0012STFMRsimultaneously. The obtained\nparameters are summarized in Tab. I.respect to zero field. At low frequencies (below 5 GHz),\nonly one mode is detectable. As the frequency increases,\na second peak emerges, see Fig. 2(a). This second peak\ndecreases in field as the frequency is increased [Figs. 2(a)\nand (b)]. At even higher frequency ( f >9.5 GHz), this\nsecond mode disappears and the remaining mode shows\na typical Kittel-like increase of the resonance field with\nfrequency.\nFigure 2(b) shows the frequency-dependent STFMR\nresultsforadevicealignedat \u0012STFMR = 0\u000e[seeFig.1(b)].\nAn oscillatory pure spin current is generated in the Pt\nlayer by means of the spin-Hall effect21,22. Upon injec-\ntion of this pure spin current in the Fe layer, it interacts\nwith the magnetic moments in the ferromagnet by exert-\ning a SOT, which results in the onset of a precession of\nthe moments around an equilibrium axis. In addition,\nthe Oersted field created by the alternating charge cur-\nrent contributes to the precession. Due to magnetore-\nsistance effects, this results in an oscillatory resistance\nchange that mixes with the microwave current leading to\na rectifieddcsignal. This rectified dcvoltage is then de-\ntected by a lock-in amplifier through a bias tee1. As\nis shown in Fig. 2(b), we observe two peaks for each\nfrequency for positive and negative biasing field. The\npeaks separate from each other for increasing frequency,\nin agreement with the FMR results shown in Fig. 2(a).\nThe observed FMR and STFMR lineshapes are different\nas fundamentally different physical mechanisms lead to\nthe signal observed on resonance2.\nTo compare the FMR and STFMR measurements\n[Fig. 2(a) and Fig. 2(b)], we plot the frequency (f)/ field\n(H0)relationships obtained from both techniques at dif-\nferent angles, see Fig. 2(c). The resonance conditions\nwere determined from fits to Lorentzian lines in the FMR\nand STFMR spectra [Fig. 2(a,b)]. For any given fre-\nquency/ field combination the results of a STFMR mea-\nsurement for a device at \u0012STFMRqualitatively matches\nthat of the FMR measurement at \u0012STFMR =\u0012FMR\u000045\u000e\nas expected from the definition of our coordinate sys-\ntem [see Fig. 1]. This result shows that STFMR can be\nused to determine magnetic anisotropies in microstruc-4\ntures. For example, two distinct resonances are found\nwhen the external field is parallel to the cubic hard axis\n(i.e.,\u0012STFMR = 0\u000eand\u000090\u000eand\u0012FMR = 45\u000eand\u000045\u000e).\nThephysicalreasonbehindthiscanbeexplainedthrough\nresonance conditions as detailed in the following.\nWe analyze the STFMR results considering the free\nenergy expansion and finding the resonance condition for\nour in-plane geometry as follows23:\nH0sin('\u0000'H) +1\n2H1sin4'\u0000Husin2'= 0(1)\n\u0012!0\n\r\u00132\n= [H0cos('\u0000'H) +1\n2H1(3 +cos4')\n+ 4\u0019Me\u000b\u00002Hucos2'][H0cos('\u0000'H)\n+ 2H1cos4'\u00002Hucos2'](2)\nHere,H0is the external resonance field, H1andHuare\nthe cubic and uniaxial anisotropies, and 'and'Hare\nthe direction of the magnetization and the direction of\nthe external field, respectively [see Fig. 1]. The appear-\nance of two resonance peaks can be understood by con-\nsidering that the cubic hard direction is determined by\nan energy unstable equilibrium state superimposed with\nan energy gradient from the uniaxial anisotropy. When\nmagnetic field is applied in that direction, it lowers the\nenergy in that direction. As a result, the orientation of\nthe magnetization is locked in the direction of external\nfield [see Fig. 4(b)] and the resonance frequency drops\nwith reduced magnetic field. When the magnitude of the\nenergy associated with the external field is comparable\nto the energy gradient from the uniaxial anisotropy, the\nmagnetization direction starts rotating to the easy di-\nrection. As the magnetization rotates towards the easy\naxis the effective field He\u000bincreases, making it possible\nto meet the resonance condition at two different external\nfield values10[see Fig. 4(a) and (b)].\nThe FMR condition is modeled based on the set of\nequations, Eqs. (1) and (2). In the following, we describe\nthe fitting procedure to experimental STFMR data we\nuse to extract the magnetic parameters including satura-\ntion magnetization Msand anisotropy fields HuandH1.\nThe equations have two independent variables, the exter-\nnal fieldH0, and the external field angle 'H, while the\nstatic equilibrium magnetization direction 'is a hidden\nvariable that cannot be eliminated by analytically solv-\ning Eq. (1). Therefore, we numerically solve Eq. (1) for\neach pair of Hand'Hto obtain the equilibrium orien-\ntation'. This result is then used in Eq. (2) to find the\nresonance frequency. An optimization process is imple-\nmented to find the global minimum of the least square\nresidue24for the fitting parameters: saturation magne-\ntizationMs= 2226 \u00065Oe, crystalline anisotropy fields\nH1= 215 \u00061Oe andHu= 4:4\u00060:1Oe. The solid lines\nin Fig. 3 show the fits to the experimental data STFMR\ndata (solid dots).\nFigure 4. (a) Frequency vs. resonating field for different \u0012sim\nfrom micromagnetic modeling using Mumax3. The magnetic\nparameters were extracted from experimental data shown in\nFig. 3. (b) Polar plot of the magnetization direction ', ,\nwhere the color of the curves represents the simulated device\nangle\u0012simasintroducedin(a). Wenoticethatthemagnetiza-\ntionalignswiththeexternalfielddirection 'H;sim= 45\u000e+\u0012sim\nat high field and rotates towards the nearby cubic anisotropy\ndirection ( 90\u000e,0\u000e, and \u000090\u000e) as the external field is lowered.\nWhentheexternalfield 'H;simisinthedirectionofcubichard\naxis(\u0012sim= 0\u000e;\u000090\u000e), themagnetizationdirection 'islocked\nin that direction (i.e., '='H;sim) until the lowest frequency\nis reached, after which 'rapidly changes to the nearest easy-\naxis direction ( \u000690\u000e, not 0\u000eas uniaxial easy axis makes the\nformer favorable than the later for \u0012sim= 0\u000eor\u000090\u000e). (c)\nMagnetization in the simulated sample for \u0012sim= 0\u000efor ex-\nternal field H= 260Oe. The magnetization direction 'along\nwith other related directions is illustrated at the bottom.\nA comparison of our magnetic parameters and the lit-\nerature is presented in Tab. I. The saturation magnetiza-\ntion we find is higher than the bulk value of Fe20, while\nthe anisotropy constants are slightly lower than the liter-\nature values25. Enhancement of magnetization in 3d/5d\nmultilayershavebeenreportedpreviously, e.g.,26–29. The\neffect is usually attributed to the narrowed d bands and\nlocalized electronic states. The latter emanates from the\nchanges in the symmetry and the coordination number\nof ferromagnetic atoms located at or near a surface or a\nmetal-metal interface. Particularly in the Fe/Pt system,\nthe interplanar distance and the Fe-Pt hybridization of\nthe electronic wave functions are considered as the key\nfactors for this enhancement26,27.\nWe further verified our result through micromagnetic\nsimulation using Mumax318. For the magnetization pa-\nrameters we relied on the values obtained from the fits5\nParameter Value (This work) (Oe) Value (Literature) (Oe)\nHs 2226\u00065 170020\nH1 215\u00061 26025\nHu 4.4\u00060.1 Negligible25\nTable I. Magnetization parameters obtained by fitting experi-\nmental results to Eq. (2) and comparison to literature values.\nto the experimental data as summarized in Tab. I. Fig-\nure 4(a) shows the simulated resonance frequency vs.\nfield plot, where an excellent agreement with the exper-\nimental FMR and STFMR results is found. Moreover,\nthe micromagnetic simulations enable us to determine\nand visualize the magnetization direction with respect to\nthe hard axis of the uniaxial anisotropy (as per our coor-\ndinate system) as a function of the external field for each\nsimulated field angle, \u0012sim. As is apparent from Fig. 4(b)\nthe magnetization tries to align with the external field\nat a high external field ( 'H;sim= 45\u000e+\u0012sim) and ro-\ntates continuously towards the nearby easy anisotropy\ndirection ( 90\u000e,0\u000e, or\u000090\u000e) as the external field reduces\n(note that the curves are slightly “bent” as the field is\nlowered). By comparing Fig. 4(a) and (b), we see that\nthe resonance frequency increases as the orientation of\nthe magnetization moves away from the direction of the\nexternal field. However, when the external field is along\nthe cubic hard direction, for \u0012sim= 0\u000eor\u0012sim=\u000090\u000e,\nthe magnetization stays locked in the direction of the ex-\nternal field ( 'H;sim= 45\u000eand\u000045\u000e, respectively) while\ntheresonancefrequencysteadilyreduces. Afteritreaches\nthe lowest resonance frequency at about 450Oe [see Fig.\n4(a)], the magnetization attempts to quickly align in the\ndirectionoftheeasyaxis( 'H;sim= 45\u000eand\u000045\u000e, respec-\ntively). This magnetization re-alignment away from the\nexternal field direction is accompanied by a fast increase\nin frequency [see Fig. 4(a)].\nIn summary, we demonstrated the generation and de-\ntection of STFMR in micropatterned epitaxial Fe/Pt bi-\nlayers grown by molecular beam epitaxy. Using an an-\nalytical formalism we extract the material parameters\nincluding saturation magnetization, uniaxial, and cubic\nanisotropies. It is found that saturation magnetization\nis larger than the literature value for bare Fe thin films,\nwhich is likely due to induced magnetic moments medi-\nated by the presence of the Pt capping layer and the mi-\ncrostructuring of the devices. Micromagnetic modeling\nusing Mumax3 revealed that the magnetization rotates\nin the direction of the nearby cubic anisotropy direction\nas the field is lowered to minimize the total energy. Thus,\nthe condition for maximum STFMR is no longer fulfilled\nand the signal intensity decreases in that field range.\nOur results show that STFMR can reveal these magnetic\nanisotropiesinindividualmicrostructuredevices–achal-\nlenging task for conventional rfspectroscopy techniques\nsuch as broadband FMR due to their relatively signal\nstrength for microstructures.ACKNOWLEDGMENTS\nWe thank Dr. Hang Chen and Dr. John Xiao for as-\nsistance with the broadband FMR measurements. This\nwork was supported by NSF through the University of\nDelaware Materials Research Science and Engineering\nCenter DMR-2011824.\nThe data that support the findings of this study are\navailable upon reasonable request.\nREFERENCES\n1L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys.\nRev. Lett. 106, 036601 (2011).\n2M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M. Hu,\nPhys. Rev. B 84, 054423 (2011).\n3M. B. Jungfleisch, W. Zhang, J. Sklenar, J. Ding, W. Jiang,\nH. Chang, F. Y. Fradin, J. E. Pearson, J. B. Ketterson,\nV. Novosad, M. Wu, and A. Hoffmann, Phys. Rev. Lett. 116,\n057601 (2016).\n4J. Sklenar, W. Zhang, M. B. Jungfleisch, W. Jiang, H. Chang,\nJ. E. Pearson, M. Wu, J. B. Ketterson, and A. Hoffmann, Phys.\nRev. B 92, 174406 (2015).\n5W. Zhang, M. B. Jungfleisch, F. Freimuth, W. Jiang, J. Sklenar,\nJ. E. Pearson, J. B. Ketterson, Y. Mokrousov, and A. Hoffmann,\nPhys. Rev. B 92, 144405 (2015).\n6A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. 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Lett. 85, 413 (2000)."    },    {        "title": "0704.3139v2.Element_resolved_x_ray_ferrimagnetic_and_ferromagnetic_resonance_spectroscopy.pdf",        "content": "Element-resolved x-ray ferrimagnetic and\nferromagnetic resonance spectroscopy\nG Boero, S Mouaziz, S Rusponi\nEcole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\nP Bencok\nEuropean Synchrotron Radiation Facility (ESRF), F-38043 Grenoble, France\nF Nolting\nSwiss Light Source (SLS), Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland\nS Stepanow\nCentre d'Investigacions en Nanoci\u0012 encia i Nanotecnologia (CIN2-ICN), UAB Campus,\nE-08193 Bellaterra, Barcelona, Spain\nP Gambardella\nInstituci\u0013 o Catalana de Recerca i Estudis Avan\u0018 cats (ICREA)\nand Centre d'Investigacions en Nanoci\u0012 encia i Nanotecnologia (CIN2-ICN), UAB\nCampus, E-08193 Bellaterra, Barcelona, Spain\nE-mail: pietro.gambardella@icrea.es\nAbstract. We report on the measurement of element-speci\fc magnetic resonance\nspectra at gigahertz frequencies using x-ray magnetic circular dichroism (XMCD). We\ninvestigate the ferrimagnetic precession of Gd and Fe ions in Gd-substituted Yttrium\nIron Garnet, showing that the resonant \feld and linewidth of Gd precisely coincide\nwith Fe up to the nonlinear regime of parametric excitations. The opposite sign of\nthe Gd x-ray magnetic resonance signal with respect to Fe is consistent with dynamic\nantiferromagnetic alignment of the two ionic species. Further, we investigate a bilayer\nmetal \flm, Ni 80Fe20(5 nm)/Ni(50 nm), where the coupled resonance modes of Ni and\nNi80Fe20are separately resolved, revealing shifts in the resonance \felds of individual\nlayers but no mutual driving e\u000bects. Energy-dependent dynamic XMCD measurements\nare introduced, combining x-ray absorption and magnetic resonance spectroscopies.\nPACS numbers: 76.50.+g, 78.70.Dm, 78.20.Ls, 76.30.DaarXiv:0704.3139v2  [cond-mat.mtrl-sci]  22 Jan 2008Element-resolved XFMR 2\n1. Introduction\nRecent interest in magnetization dynamics has been fostered by progress in fast\nmagnetic recording and microwave technologies [1, 2]. Despite considerable e\u000borts,\nhowever, the description of magnetodynamics remains essentially phenomenological.\nInductive, magnetoresistive, and magneto-optical techniques solely measure the\nintegrated magnetic response of complex heterogeneous materials, typically magnetic\nalloys and multilayer structures, whose functionality depends on the interplay of several\nelements. The development of methods capable of elemental analysis constitutes\nan obvious advantage for investigating fundamental problems related to time- or\nfrequency-dependent magnetization phenomena. Examples include the dynamic\ncoupling of elemental moments in ferrites [3, 4, 5, 6], metallic alloys [7], and spin-valve\nheterostructures [8, 9], as well as spin-orbit induced damping e\u000bects attributed to the\npresence of high [5, 10, 11] and low [12] Z elements. Advances in this direction are\nmostly based on stroboscopic pump-probe experiments exploiting the element-resolving\npower of x-ray magnetic circular dichroism (XMCD) and the sub-ns bunch structure of\nsynchrotron radiation beams. Pulsed magnetic \felds in synchrony with x-ray photon\nbunches are usually employed to excite the reversal [8, 13] or the precessional motion\n[7] of the magnetization. More recently, continuous wave rf \felds have been applied to\nexcite resonant modes in trilayer metal \flms [14, 15] and microstructures [16, 17].\nWith respect to time-resolved measurements, techniques such as ferromagnetic\nresonance spectroscopy (FMR) o\u000ber an alternative and powerful way to gain insight\ninto the energy scales that govern magnetization dynamics. Frequency-domain methods\nthat allow to detect magnetic resonance using the core level absorption of circularly\npolarized x-rays have been developed independently by our group in the soft x-ray\nenergy range [18] and by Goulon et al. in the hard x-ray regime [19, 20]. These methods\nexploit the XMCD dependence on the scalar product M\u0001Pof the magnetization vector\nMand photon helicity Pto measure the time-invariant changes of the longitudinal\nmagnetization component \u0001 Mzas a function of microwave (MW) \feld B1and bias\n\feldB0. Microstrip resonators [18] and tunable cavities [21] have been employed to\ngenerate MW excitations together with di\u000berent detection schemes. In the hard x-ray\nregime, XMCD at the Kedge of transition metals relates purely to orbital magnetization\ncomponents; measurements at the Fe K-edge and Y L2;3edges by Goulon et al. provided\nevidence for the precession of the Fe orbital moments as well as induced Y spin moments\nin yttrium iron garnet (YIG) [19, 20].\nIn this article, we report on di\u000berent applications of soft x-ray MCD to FMR\nmeasurements and on a novel way to combine FMR and XMCD spectroscopy. Element-\nspeci\fc magnetic resonance spectra are measured on both magnetic oxides and metallic\nmultilayers. We show that ferrimagnetic resonance measurements of Gd-substituted\nYIG are consistent with the antiferromagnetic (AFM) alignment of Gd and Fe ions in the\nferromagnetic resonance mode of YIG in the non-linear regime, above the threshold for\nparametric spin wave excitations. Further, FMR spectra of coupled thin metal bilayersElement-resolved XFMR 3\nFigure 1. (a) Diagram of the experimental setup. (b) Close-up view of the resonator\nand photodiode situated between the poles of the electromagnet. Note that one of the\nmagnet poles and the photodiode have an opening to allow for the passage of x-rays.\nare separately resolved, allowing the investigation of interlayer dynamics in stacks of\nmagnetic layers. Finally, we show that the x-ray FMR (XFMR) signal measured at\nresonance as a function of photon energy yields dynamic XMCD spectra, which relate\nto the magnetic state of the atoms undergoing microwave absorption. The latter can\nbe combined with static XMCD spectra to derive information on the dynamics of the\norbital and spin magnetization components.\n2. Experimental\nA schematic diagram of the experimental setup is given in Fig. 1. A coplanar waveguide\n\u0015=2-resonator is used to generate a MW \feld B1\u00190:01 to 0.5 mT parallel to the sample\nsurface with input power 0 to 34 dBm at frequency !=2\u0019= 2:21 GHz. The resonator-\nsample assembly is positioned between the pole expansions of an electromagnet, which\nproduces a \feld 0 \u0014B0\u00140:8 T aligned perpendicular to the sample surface and parallel\nto the photon propagation direction. In the absence of MW \feld, Maligns with B0\nparallel to P, which is the geometry commonly employed in static XMCD measurements.\nIfB1is turned on, as B0matches the resonance \feld of the sample ( Br) the precessional\nmotion of Minduces a reduction of the longitudinal magnetization component Mzthat\ncan be measured as a steady-state e\u000bect in the frequency domain, i.e., without requiring\nsub-ns time resolution. Here, x-ray absorption spectra (XAS) corresponding to positive\n(P+) and negative (P\u0000) helicity are measured by recording the dc \ruorescence yield\n(FY) of the sample as a function of photon energy using a Si photodiode (Eurisys-\nCanberra, Ref. [22]). XMCD is de\fned as the di\u000berence spectrum P+-P\u0000(Fig. 2). The\nXFMR signal, either P+or P\u0000, is obtained by square-modulating the MW power source\nat relatively low frequency ( <100 kHz) and by measuring the corresponding amplitude\nof the ac FY photocurrent by means of a lock-in ampli\fer, as shown in Fig. 1 (a). We\nintroduce two methods to measure magnetic resonance using XMCD: the \frst, in analogy\nwith FMR spectroscopy, consists in recording the XFMR intensity during a sweep of B0\nacrossBr, \fxing the photon energy in correspondence of a static XMCD peak [18]. We\ndenote this type of measurements as XFMR B-scan , which e\u000bectively generate element-Element-resolved XFMR 4\nFigure 2. (a) One octant portion of the unit cell of GdIG, showing the AFM\nspin alignment of octahedral Fe (black circles), tetrahedral Fe (gray circles), and\ndodecahedral Gd sites (empty blue circles), from Ref. [23]. Oxygen ions have been\nomitted. (b) FY XAS spectra and corresponding XMCD of Fe and (c) Gd sites\nmeasured at room-temperature with B0= 0:21 T.\nspeci\fc longitudinal magnetic resonance spectra. The second method consists in taking\nthe sample at resonance by setting B0=Brand recording the XFMR as a function\nof photon energy. This, denoted as XFMR E-scan , is analogous to recording XAS and\nXMCD spectra, but corresponding to the precessional motion of Mrather than to a\nstatic situation. Examples of either type of measurements will be given later.\nTwo di\u000berent type of samples are employed in the present study: a\nrare earth substituted iron oxide and a metallic heterostructure, which were\nchosen in order to highlight the broad spectrum of materials where new insight\ncan be obtained by XFMR. A polished 30 \u0016m-thick slab of polycristalline\nGd 1Y2Fe5O12(Gd:YIG) with lateral dimensions 1 \u00022 mm2was selected to investigate\nferrimagnetic resonance in garnet systems composed of di\u000berent magnetic ions. An\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) multilayer deposited on glass by e-\nbeam evaporation in high vacuum (1 \u000210\u00006mbar) was fabricated in order to address\nlayer-speci\fc resonance modes in metallic heterostructures. The x-ray spot size at the\nsample position was 0.1 mm long and 1 mm wide at full width half maximum, while\nthe coplanar resonator had a central conductor with a width of 1.5 mm and a length\nof 44 mm, thus ensuring that the MW excitation covers the whole area sampled by the\nx-ray beam. XAS and XFMR spectra were recorded at the L2;3edges of Fe and Ni,\nand at the M4;5edges of Gd. XAS spectra are normalized to the incident photon \rux\nmeasured by the photocurrent of an Au grid upstream from the sample, and are givenElement-resolved XFMR 5\nFigure 3. (a) Magnetization of a 30 \u0016m thick, 1\u00022 mm2wide Gd 1Y2Fe5O12slab\nmeasured by SQUID with applied \feld perpendicular to the sample plane at 300 K.\n(b) Magnetization vs. temperature of a 100 \u0016m thick Gd 1Y2Fe5O12slab \feld-cooled\nin a 3 mT \feld.\nin arbitrary units. Apart from normalization, the spectra are raw data; in particular, no\nenergy-dependent correction for self-absorption has been applied. As the signal-to-noise\nratio is proportional to the square root of the photocurrent [18], energy resolution has\nbeen sacri\fced to intensity by opening the exit slits of the beamline monochromator.\nThe e\u000bective energy resolution corresponds to about 1.2 and 3 eV at 700 and 1200 eV,\nrespectively, which results in signi\fcant broadening of the multiplet features of Fe and\nGd spectra in Gd:YIG, as shown in Fig. 2. This is not an essential problem for XFMR\nB-scans, but may limit the spectral resolution of E-scans; in the latter case, however,\nhigher resolution can be achieved simple by reducing the slit apertures while increasing\nthe averaging time to maintain a constant signal-to-noise ratio. Throughout the paper\nXFMRB-scans are given in pA, as measured by the FY photodiode. Simultaneously\nwith XFMR, the transverse part of the imaginary susceptibility \u001f00was measured, as in\nconventional FMR, by monitoring the power re\rected o\u000b the \u0015=2-resonator via a MW\nbridge and diode detector, as schematized in Fig. 1 (a). XFMR B-scans were measured\nat the ID08 beamline of the European Synchrotron Radiation Facility, while E-scans\nwere recorded at the SIM beamline of the Swiss Light Source; two undulators were\noperated in series with 99 \u00061 % circularly polarized beams in both type of measurements.\n3. Element-resolved XFMR spectra of Gd:YIG\nThe structure of Gd 1Y2Fe5O12(Gd:YIG) consists of three sublattices [Fig. 2 (a)]. Two\nof them, the octahedral and tetrahedral sites, contain Fe ions which are strongly AFM\ncoupled by superexchange. The third lattice, the dodecahedral sites, contains Gd and\ndiamagnetic Y ions [23]. While their mutual interaction is very weak, Gd ions couple\nAFM to tetrahedral Fe ions with a moderate exchange \feld of the order of 24 T (16 K)\n[24]. Such a system thus e\u000bectively behaves as a two-sublattice ferrimagnet, where\nthe Gd moments order spontaneously only at low temperature ( <50 K). Figure 3\n(a) shows the out-of-plane magnetization of Gd:YIG measured by superconducting\nquantum interference device magnetometry (SQUID) at room temperature. The curveElement-resolved XFMR 6\nFigure 4. FMR spectra of Gd:YIG measured by the re\rected power from the \u0015=2-\nresonator at 0 dBm using \feld and MW amplitude modulation (bottom and middle\ntraces, respectively). The top trace shows the high power (31 dBm) FMR for MW\namplitude modulation. B0is oriented perpendicular to the sample surface in all cases.\nis composed by a hard-axis ferromagnetic loop that saturates above 0.1 T, as expected\nfrom shape anisotropy considerations, and a linear term proportional to the applied \feld.\nThe latter is a common feature of rare-earth garnets and ascribed to the continuous\nrotation of MGdtowards MFewith increasing \feld, in accordance with N\u0012 eel's theory\nof ferrimagnetism. The temperature behavior of the magnetization, shown in Fig. 3\n(b), is characteristic of two AFM-coupled lattices with inequivalent magnetization.\nWhile for all rare-earth garnets the Curie temperature is associated to the pairing of Fe\nmoments and nearly independent on rare-earth composition [23, 25], the compensation\ntemperature depends sensibly on the rare-earth content. In Gd 3Fe5O12compensation\noccurs at 290 K [23]. Figure 3 (b) shows that the total magnetization of Gd 1Y2Fe5O12is\napproximately constant from 300 to 150 K; below this temperature magnetic order sets\nin throughout the Gd lattice, compensating the Fe magnetization at about 45 K. The\nXAS and XMCD spectra of Fe and Gd in Gd 1Y2Fe5O12recorded at room temperature\nwith applied \feld B0= 0:21 T are shown in Figs. 2 (b) and (c). The opposite sign of\ntheM5vsL3andM4vsL2intensity re\rects the static alignment of the resultant MGd\nagainst MFe.\nLinearization of the coupled equations of motion shows that two resonances can be\nexcited in a ferrimagnetic compound: the ferromagnetic mode, which is independent\nof the exchange \feld since the angle between MFeandMGddoes not vary during the\nprecession, and the high-frequency exchange mode, where the two sublattices precess\nout-of-phase but phase-locked to each other with non collinear magnetization vectors\n[3, 4, 26]. The \frst mode is the one accessible at relatively low \felds in usual FMR\nexperiments, as in our case, while the second one is situated at \felds of several\ntens of Teslas for frequencies in the MW range [27]. Neglecting magnetocrystalline\nanisotropy, the resonant \feld for uniform precession in the ferromagnetic mode is\ngiven byBr=!\n\r+\u00160Nz(MFe\u0000MGd) = 190 mT, where \ris the gyromagneticElement-resolved XFMR 7\nratio,Nz= 0:935 is the demagnetizing factor calculated for our geometry [28], and\n\u00160(MFe\u0000MGd) = 120\u00066 mT. Figure 4 shows the conventional FMR spectra of\nGd:YIG. Owing to the sample \fnite dimensions, the low power FMR shows a series of\nmagnetostatic modes with the principal one close to Br. The longer wavelength modes\nare resolved in the \feld-modulated spectrum (bottom trace) and appear as shoulders\nof the main peak in the MW-modulated spectrum (middle trace). For a sample 30 \u0016m\nthick with lateral dimensions of the order of 1 mm their separation corresponds to that\nexpected for magnetostatic forward volume wave modes with the excitation geometry\nof Fig. 1 [29, 30]. At high MW power (top trace) the FMR shifts to a lower \feld due to\nheating of the sample and related decrease of the resultant magnetization MFe\u0000MGd.\nMoreover, the FMR lineshape is signi\fcantly distorted due to e\u000bects such as foldover\nand nonlinear spin wave instabilities [31]. In such a regime, nonlinear terms in the\nLandau-Lifschitz equation of motion transfer energy from the uniform precession mode\ndriven by the external MW \feld to nonuniform magnon modes, which become unstable\nabove a critical \feld threshold [32]. These phenomena lead to saturation of the main\nresonance and precession angle together with excitation of spin waves above thermal\nvalues. Of relevance to the present discussion is the fact that nonlinear coupling terms\nescape conventional treatments of ferrimagnetic resonance, which reduce the dynamics\nof individual sublattices to that of a single macrospin (e.g., of amplitude MFe\u0000MGd\nfor Gd:YIG) [3, 4, 5, 6]. Moreover, the assumed equivalency of the equations of motion\nfor di\u000berent sublattices might not hold true when nonlinear phenomena are taken into\naccount. For example, substitution of foreign ions in a material where all equivalent\nlattice sites are occupied by identical ions, as in Gd:YIG, provides a site-dependent\nadditional scattering channel leading to spin wave excitations [33]. Element-resolved\nFMR spectra can thus put the macrospin concept to test, speci\fcally in the nonlinear\nregime where relatively large deviations \u0001 Mzmake the XFMR intensity easier to detect.\nFigure 5 compares the inductive FMR spectrum of Gd:YIG (a) with the XFMR\nP+-P\u0000intensity recorded at the Fe L2edge (b) and Gd M4edge (c) as a function of\nB0. Several comments are in order. First, we note that conventional FMR and XFMR\nspectra di\u000ber for obvious reasons, namely: (i) XFMR is a measure of \u0001 Mz, while\nFMR is proportional to the transverse dynamic magnetization component. Only if jMj\nis conserved the two measurements can be considered to be equivalent. (ii) XFMR is\nsurface-sensitive, with the same probing depth as FY XAS ( \u001820 nm at the Fe L2;3edges\n[34]) and probes a limited portion of the sample, while FMR averages over the whole\nsample volume. In Fig. 5 (a) the FMR lineshape is asymmetric and heavily saturated due\nto nonlinear e\u000bects that limit the FMR precession cone amplitude. The XFMR signal\nin (b), on the other hand, is composed of a broad resonant feature and a sharp peak\nlocated at about B0= 165 mT with linewidth \u0001 B= 1 mT. It may be observed that the\nintensity of both features is centered around the low-\feld rising edge of the FMR peak\nand does not follow the FMR intensity distribution. The origin of such di\u000berences lies\nin (i) and (ii); a detailed understanding of the XFMR vs. FMR lineshape, however, isElement-resolved XFMR 8\nFigure 5. (a) FMR spectrum of Gd:YIG measured simultaneously with the XFMR\ndata. (b) XFMR P+\u0000P\u0000intensity measured at the L2edge of Fe (723.8 eV) and (c)\nat theM4edge of Gd (1222 eV). The MW power is 31 dBm. The data are averaged\nover 40 sweeps of B0in the positive direction, with a sweep time of 80 s and lock-in\ntime constant of 100 ms.\npresently missing. To appreciate this point, we o\u000ber a number of consideration based on\nprevious FMR and XFMR studies of YIG. The sharp peak observed by XFMR denotes\na sudden increase of \u0001 Mz, whereMzis proportional to the total number of magnons\nin the system. De Loubens et al. , using magnetic resonance force microscopy on a\nsingle crystal YIG \flm, observed a dramatic increase of \u0001 Mzat the onset of the second\norder Suhl's instability threshold, which was attributed to the parametric excitation of\nlongitudinal spin waves with a low spin-lattice relaxation rate compared to the uniform\nmode [35, 36]. In this model, the total number of magnons is considered to be constant,\nwhile changes of Mzare attributed to a redistribution of their occupation number from\nmodes with relatively high to low relaxation rate, favoring larger precession angles [37].\nGoulon et al. , using XFMR on a single crystal Y 1:3La0:47Lu1:3Fe4:84O12\flm, also observed\na sharp decrease of Mzmeasured at the Fe K edge, taking place in correspondence with\nthe foldover critical \feld of the FMR spectrum [21]. They explained this e\u000bect by\nthe degeneracy of the uniform mode with long-wavelength longitudinal magnetostatic\nwaves caused by foldover in perpendicular FMR. In this regime, parametric excitation\nof coupled magnetostatic-magnetoelastic waves becomes possible [21], which may lead\nto an e\u000bective transfer of angular momentum to the lattice and therefore to a decrease\nofMz. This is substantially di\u000berent from the model proposed by De Loubens et al. ,Element-resolved XFMR 9\nFigure 6. Restricted range of (a) FMR and (b) XFMR spectra of Gd:YIG at the L2\nedge of Fe (723.8 eV) and M5edge of Gd (1191 eV) recorded with the parameters of\nFig. 5.\nas the total number of magnons needs not be conserved. The validity of either of these\nexplanations for the present measurements may be questioned due to the inhomogeneous\ncharacter of local magnetic \felds in polycrystalline samples, e.g., owing to magnetic\nanisotropy \ructuations or microstructure \raws, which results in broadened FMR lines.\nSpeci\fcally, if individual crystal grains went through resonance individually according\nto their orientation in the applied \feld and one would have to worry about strongly\ninhomogeneous resonance conditions; however, as the magnetocrystalline anisotropy\n\feld is more than a factor 10 smaller compared to the saturation magnetization in\nGd:YIG, dipolar coupling between di\u000berent grains predominates and resonance occurs\nas a collective phenomenon [38, 39]. The observation of di\u000berent magnetostatic modes\nin Fig. 4 supports this view, although a much smaller number of modes are resolved\ncompared to single crystal YIG \flms [21, 35]. The granular structure of the material\nand related local changes of the anisotropy \feld have also a well-known e\u000bect on the\ncritical \feld for parametric spin wave excitations, raising it up to 0.1-1 mT in YIG\n[40], and leading to a smooth onset of this e\u000bect rather than an abrupt threshold [41].\nThe saturation as well as the distorted shape of the FMR spectrum indicate that the\nconditions for foldover and parametric spin wave ampli\fcations are met at high power\nin Gd:YIG and likely contribute to the observed XFMR features. In general, however,\nwe cannot identify a unique origin for the XFMR peak nor exclude it to be related to\na mode localized at the vacuum-Gd:YIG interface, which would be selectively probed\nby XFMR and only weakly observed in the bulk FMR signal [see Fig. 6 (a)]. More\nmeasurements shall be performed to clarify this point.\nWe proceed now to compare the XFMR spectra of Fe and Gd, discussing whatElement-resolved XFMR 10\ntype of information may be derived on the relative motion and relaxation of dissimilar\nmagnetic moments in a bulk compound at resonance. Apart from the noise and a\nscaling factor, the Gd M4spectrum in Fig. 5 re\rects specularly the one measured at\nthe FeL2edge. The resonant \feld and linewidth derived from the Gd B-scan XFMR\nprecisely match those of Fe, but the XFMR intensity has opposite sign. This is even\nmore evident in the restricted range B-scan in Fig. 6 (b), where the Fe L2and GdM5\nspectra are reported; note that the relative sign of the Fe and Gd intensity depends on\nthe absorption edge, as for XMCD. Sign inversion of the XFMR at the Fe L2(L3) and Gd\nM4(M5) edges, consistent with that observed in the static XMCD [Figs. 2 (b) and (c)],\nreveals the coupled AFM dynamics of the Fe and Gd magnetic moments. Their relative\n\u0001Mz=Mdeviations can be quanti\fed in terms of the XFMR cross section, de\fned as\nthe ratio between the dynamic and static dichroism FY photocurrents \u001b=XFMR (E)\nXMCD (E),\nwhich depends on the x-ray photon energy Eas well as on the spin and orbital magnetic\nmoment precession in a way dictated by the XMCD sum rules [42]. At 31 dBm MW\npower, we have \u001bL2(Fe) = (2:0\u00060:2)\u000210\u00003and\u001bM4(Gd) = (1:7\u00060:2)\u000210\u00003. These\ndata, together with the above observations, are consistent with Fe and Gd maintaining\nrigid AFM alignment in nonlinear excitation modes (diagram in Fig. 6). We note that,\nin principle, the same result can be obtained for noncollinear MFeandMGdvectors\nprecessing on the cone shown in Fig. 6; however, in the noncollinear case, di\u000berent \rexing\nangles (\u001b) would be expected for Fe and Gd, given that the local exchange \felds acting\non the two ionic species are strongly dissimilar [3, 24, 27]. Full con\frmation of the type\nof AFM coupling would in any case require to measure the phase of the precessing Fe and\nGd moments, which may be retrieved only by time-resolved detection of the transverse\nmagnetization components [14, 15, 21]. Within the experimental error, XFMR data thus\nshow that the resonating longitudinal components of MFeandMGdhave opposite sign\nand equal relative deviations from static equilibrium up to the nonlinear regime of high-\npower MW excitations. This is consistent with collinear dynamic AFM alignment of\nMFeandMGdpredicted by the theory of ferrimagnetic resonance for uniform precession\nat low \felds, but extends into the nonlinear regime beyond the approximations usually\nmade in theoretical models [3, 4, 26] and at temperatures where thermal \ructuations\nstrongly a\u000bect magnetic order in the Gd lattice (Fig. 3). Further, the observation of\nequal Fe and Gd linewidths, within the experimental accuracy of the results reported\nin Fig. 6 (b), implies that the relaxation mechanisms of the Fe and Gd lattice can be\ndescribed by a common e\u000bective damping parameter, as also predicted by theory [4].\nEven though \u001b, and therefore \u0001 Mz, cannot be uniquely related to precessing\nmagnetic moments in the uniform mode due to the presence of nonlinear excitations,\nit is interesting to de\fne an e\u000bective precession angle related to \u0001 Mz=Mmeasured by\nXFMR. In doing so, one must take into account that \u001bis a photon energy-dependent\nparameter. In other words, considering that XAS involves 2 p!3d(3d!4f)\ntransitions for the Fe L2;3(GdM4;5) edges,\u001bdepends on the precession of both spin\nand orbital magnetic components of the d- (f-) projected density of states probed by\nphotons of energy E. This point has been discussed in detail by Goulon et al. inElement-resolved XFMR 11\nRef. [42], who have shown that the precession angles of the spin and orbital magnetic\ncomponents may be derived by combining \u001bL2and\u001bL3measurements and applying the\ndi\u000berential form of the XMCD sum rules. By assuming spin-only magnetic moments,\nthe relationship between \u001band the e\u000bective precession angle becomes extremely simple,\n\u001b= (1\u0000cos\u0012eff), yielding \u0012eff(Fe) = 3:6\u000e\u00060:2\u000eand\u0012eff(Gd) = 3:4\u000e\u00060:2\u000efor the\nmeasurements reported above. Even if the orbital magnetization of Gd and trivalent\nFe ions is usually very small, the extent to which orbital precession contributes to \u001b,\nin particular for Fe, remains to be determined. This matter touches on the interesting\nquestion of separately measuring the spin and orbital moment precession angles, which\nrequires either a comparison between Kedge andL2;3edges measurements recorded\nusing identical experimental conditions [42] or full XMFR E-scans over the entire L2;3\nregion. The latter possibility is further discussed in Sect. 5.\n4. Element-resolved XFMR spectra of metallic bilayers\nWe consider now the extension of XFMR to thin metallic \flms, and show that\nlayer-speci\fc magnetic resonance spectra of multilayer magnetic structures can be\nseparately resolved. This is of interest, e.g., to investigate interlayer coupling e\u000bects,\ndistinguish superposed spectra of layers with similar resonance \felds, and investigate\ncurrent induced precessional dynamics in spin-torque devices. Here we study a\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) multilayer, where the thickness of the\ntwo magnetic \flms was adjusted so as to reduce Brof Ni 80Fe20to within range of our\nelectromagnet for perpendicular FMR.\nFigure 7 (a) shows the inductive FMR of the magnetic bilayer, where two resonances\nare observed at 530 and 740 mT. These are close but not equal to the resonances\nof individual Ni and Ni 80Fe20\flms, respectively, that were prepared with the same\nprocedure. The high \feld resonance peak, in particular, appears to be shifted by an\namount \u0001B=\u0000170 mT with respect to the resonance of an individual Ni 80Fe20layer,\nwhich is indicative of ferromagnetic exchange coupling at the Ni - Ni 80Fe20interface.\nThe elemental components of the two resonance peaks are straightforwardly resolved by\nXFMR, as shown in Fig. 7 (b). We observe that the low-\feld resonance originates from\nthe Ni layer alone, while the high-\feld one comprises both Ni and Fe components. In\nthe high-\feld resonance, the scaled Ni and Fe XFMR intensities coincide, implying\na common g-value and relaxation channel for the two elements, as expected for a\nferromagnetic alloy such as Ni 80Fe20[7]. We therefore conclude that, despite the\npresence of exchange coupling at the interface, mutual resonance-driving e\u000bects between\nperpendicularly-magnetized Ni and Ni 80Fe20layers are not signi\fcant. This result can be\nrationalized within the theoretical model developed by Cochran et al. for a thin overlayer\ncoupled to a thick magnetic substrate [43]. The model assumes that two ferromagnetic\nlayersAandBdeposited on top of each other are exchange coupled at their interface by\na surface energy per unit area of the form Eexc=\u0000JMA\u0001MB, whereJis the interface\ncoupling constant [44, 45]. In the two extreme limits of strong and zero coupling, theElement-resolved XFMR 12\nFigure 7. (a) FMR of Ni 80Fe20(5 nm)/Ni(50 nm) measured simultaneoulsy with (b)\nL2XFMR spectra of Fe and Ni at E= 722:2 and 871.7 eV, respectively. The MW\npower is 34 dBm.\nmagnetizations of the two layers precess locked together or independently of each other,\nrespectively. For small but \fnite J, mutual driving terms in the equations of motion\nbecome unimportant, with the overlayer responding to the driving MW radiation as if\nit were an isolated \flm subject to an e\u000bective anisotropy \feld of magnitude JMB=tA,\nwheretAdenotes the overlayer thickness and MBthe thick \flm magnetization [43]. This\nbehavior corresponds to the data reported in Fig. 7. From the shift \u0001 Bwe estimate\nJ= 2:1\u000210\u000015Vs/A andEexc\u00196\u000210\u00004J/m2. According to theory [43, 45], also\nthe resonance position of the thicker Ni layer should be down-shifted in the presence of\nferromagnetic interface coupling, namely by the amount JMA=tB. Indeed, with respect\nto a single 50 nm thick Ni layer in a Al(10 nm)/Ni(50 nm)/Cr(5 nm) stack, a shift\n\u0001B=\u000030 mT is observed, which yields J= 1:9\u000210\u000015Vs/A, consistently with the\nvalue reported above.\nCompared to the exchange energy of ferromagnetic metals, Eexcestimated from the\nresonance shifts turns out to be rather small for metallic \flms in direct contact with\neach other. Although this explains the absence of Ni 80Fe20(Ni) response upon excitation\nof the Ni (Ni 80Fe20) resonance, its origin could not be uniquely determined during the\npresent study. The magnitude of Eexcis known to be extremely sensitive to the quality of\nthe interface between magnetic materials. Roughness, as well as adsorption of impurities\nsigni\fcantly diminish the coupling strength. In high vacuum, the few seconds intervened\nbetween evaporation of the Ni and Ni 80Fe20\flms are su\u000ecient to deposit a monolayer-\nlike quantity of contaminants, which may strongly decrease the magnetization of the\ninterface metal layers. In vacuum conditions similar to ours, Ho\u000bmann et al. found\nEexc= 1:2\u000210\u00003J/m2for a double Ni/Ni 80Fe20/Ni interface [44]. Fully oxidizedElement-resolved XFMR 13\nFigure 8. Static Fe XMCD (solid line) of Ni 80Fe20(5 nm) and Fe XFMR E-scan\nmeasured at B0= 0:74 T (squares) and 0.70 T (dashed line). The MW power is\n34 dBm.\nNiO/Ni 80Fe20interfaces, on the other hand, have interfacial coupling energies as small\nas 2\u000210\u00005J/m2[46].\nFinally, we note that the smallest XFMR cross-section measured for Ni 80Fe20(5 nm)\ncorresponds to \u001bFe= 5\u000210\u00004, representing a very remarkable dichroism sensitivity in\nthe soft x-ray range, still susceptible of further improvements.\n5. Dynamic XMCD spectra\nSo far we have dealt with the information contained in XFMR B-scans. One of the main\npoints of XFMR, however, is that the measured intensity contains all the information\nderived from the x-ray absorption process, in particular that related to the unoccupied\n\fnal density of states of a given chemical species together with its spin and orbital\nmagnetization components. In other words, two powerful spectroscopical methods, x-\nray absorption and magnetic resonance, are combined together in XFMR. Here we show\nhow the information related to the electronic state of the atoms whose magnetization is\nprecessing can be practically retrieved by XFMR E-scans, i.e., by recording the XFMR\nintensity as a function of photon energy at B0=Br. Figure 8 shows the XFMR\nenergy-dependent intensity of Fe in the Ni 80Fe20layer measured on- and o\u000b-resonance,\ncompared with the static XMCD signal measured at the same \feld value. One can\nsee that, while the on-resonance XFMR displays a strong energy dependent intensity,\nthe XFMR measured o\u000b-resonance is zero within the noise, emphasizing the dynamic\norigin of the XFMR E-scan. Indeed, the latter can be considered as a dynamic XMCD\nspectrum, where the probed magnetization corresponds to that resonantly excited by\nthe MW \feld into uniform precession or other resonant modes selected by the choice of\nB0. Here, although the signal-to-noise ratio needs to be improved to reach quantitative\nconclusions, the overall similarity between the static and dynamic XMCD lineshape\nsuggests a similar orbital-to-spin ratio for the static and precessing magnetic moments\nof Fe.\nThis method eliminates the need to resort to the di\u000berential form of the XMCD\nsum rules to extract information on the precession dynamics of the spin and orbitalElement-resolved XFMR 14\nmagnetization components of the d-density of states introduced in Ref. [42]. By\nintegrating XFMR E-scans and XMCD spectra simultaneously measured, the standard\nXMCD sum rules [47, 48] can be applied, deriving information on the dynamic vs. static\ntotal orbital and spin magnetic moments. Assumptions made in applying the XMCD\nsum rules regarding integration cut o\u000bs, magnitude of the spin dipole moment, and\nisotropic absorption intensity [47, 48, 49] shall hold equally well (or badly) for XFMR\nE-scans and XMCD spectra, thus making their relative comparison most relevant. Two\ncaveats should be mentioned concerning this type of measurements. The \frst is the\nquantitative accuracy of the XMCD sum rules for soft x-ray absorption spectra measured\nin the FY mode, as discussed, e.g., in Ref. [50]. The second is the presence of strong self-\nabsorption e\u000bects for thick \flms and bulk samples, which alter the measured intensity\nof the most prominent XAS and XMCD features. Di\u000berent methods may be used to\nretrieve the true XAS absorption coe\u000ecients from FY data [51, 52]; a relative, qualitative\ncomparison of static and dynamic XMCD measurements is nonetheless always possible\nsince self-absorption a\u000bects them in the same way. Moreover, such e\u000bects may be\nneglected in ultrathin \flms and dilute samples, and entirely bypassed by measuring\nXFMR in a transmission geometry, with a signi\fcant additional gain of XAS intensity.\nRecently, XAS and XMCD spectra have been measured also by time-resolved pump-\nprobe methods, addressing the transfer of angular momentum from the spin and orbital\nmagnetic moments to the lattice in Fe/Gd multilayers [53] and polycrystalline Ni \flms\n[54]. Ultrafast heat transients produced by fs-laser pulses are used to pump electronic\nexcitations, inducing strong demagnetization e\u000bects and consequent transfer of angular\nmomentum from the magnetic system to the lattice. XMCD spectra recorded at \fxed\ndelay times allow to monitor the spin and orbital magnetic moments during this process.\nTime resolution is achieved either by temporally dispersing the intensity of x-ray photon\nbunches transmitted by the sample using a streak camera [53] or by employing fs x-\nray probe pulses produced by femtoslicing techniques [54], achieving resolutions of the\norder of 2 ps and 100 fs, respectively. \"Slower\" time-resolved schemes based on pulsed\nmagnetic \felds [7, 13] or continuous wave excitations [14, 15] as pump and x-ray photon\nbunches of\u001850\u0000100 ps duration as probe may also be employed to measure full XMCD\nspectra, although this, to our knowledge, has not yet been reported. With respect to\ntime-resolved methods, XFMR E-scans appear particularly suited to study stationary\nprecessional dynamics. The averaging time required to measure the Fe spectrum in\nFig. 8 amounts to about 1 hour. Improving the detection e\u000eciency using transmission\nrather than FY is expected to reduce this time further while leading to a better XFMR\nsignal-to-noise.\n6. Conclusions\nIn summary, we have shown that time-invariant x-ray magnetic dichroism and magnetic\nresonance spectroscopy at GHz frequency can be combined to yield element-resolved\nmagnetic resonance spectra as well as dynamic XMCD spectra, depending on whetherElement-resolved XFMR 15\nthe photon energy is kept constant while the applied magnetic \feld is varied or\nviceversa. We reported two case studies concerning a Gd 1Y2Fe5O12garnet and an\nAl(10 nm)/Ni 80Fe20(5 nm)/Ni(50 nm)/Cr(5 nm) metallic \flm. Antiferromagnetic\ncoupling at resonance between Fe and Gd sublattices in Gd:YIG has been resolved\nand shown to hold also in the nonlinear regime where the FMR response is heavily\nsaturated. The Fe and Gd XFMR linewidths coincide to within the experimental\naccuracy, supporting the notion of a common e\u000bective damping parameter for the two\nsublattices introduced in early theoretical treatments of ferrimagnetic resonance [4].\nThe Ni 80Fe20(5 nm)/Ni(50 nm) bilayer presents two resonance modes whose elemental\ncomponents have been separately identi\fed by XFMR. It was shown that while one\nlayer is excited the other is at rest, i.e., that interlayer driving e\u000bects are negligible\nfor moderate values of the interface exchange energy, as predicted by theory [43].\nFinally, the comparison between static and dynamic Fe XMCD lineshape in Ni 80Fe20\nsuggests a constant orbital-to-spin magnetic moment ratio for the steady and precessing\nmagnetization.\n7. 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The linear response to an oscillating external magnetic field is calculated\nwithin the framework of the inertial Landau–Lifshitz–Gilbert equation using analytical theory and\ncomputersimulations. Precessionandnutationresonancepeaksareidentified,anditisdemonstrated\nthat the precession frequencies are reduced by the spin inertia, while the lifetime of the excitations\nis enhanced. The interplay between precession and nutation is found to be the most prominent in\nantiferromagnets, where the timescale of the exchange-driven sublattice dynamics is comparable to\ninertial relaxation times. Consequently, antiferromagnetic resonance techniques should be better\nsuited for the search for intrinsic inertial spin dynamics on ultrafast timescales than ferromagnetic\nresonance.\nI. INTRODUCTION\nDeterministic spin switching at ultrashort timescales\nbuilds the fundament for future spin-based memory tech-\nnology [1–5]. At femtosecond timescales inertial switch-\ning becomes particularly relevant, where the reversal is\nachieved with a linear momentum gained by the interac-\ntion of an ultrashort pulse and spin inertia [6, 7]. The\nunderstanding of magnetic inertia has been pursued along\ntwo different directions so far.\nOn the one hand, spin dynamics in antiferromagnets\n(AFMs) and ferrimagnets (FiMs) has successfully been de-\nscribedbytheLandau–Lifshitz–Gilbert(LLG)equation[8–\n10] for two sublattices coupled by the exchange interaction.\nThe exchange energy created by tilting the sublattice mag-\nnetization directions away from the antiferromagnetic ori-\nentation is dynamically transformed into anisotropy energy\nby collectively rotating the sublattices away from the easy\nmagnetic direction [11], analogously to the transition be-\ntween kinetic and potential energy terms in a harmonic\noscillator. While the LLG equation for the two sublat-\ntices is of first order in time, this effect gives rise to an ef-\nfectively inertial second-order differential equation for the\norder parameter in AFMs [12, 13]. The interaction be-\ntween exchange and anisotropy degrees of freedom causes\nan exchange enhancement of AFM resonance frequencies\nand linewidths [14].\nOn the other hand, an intrinsic inertia also arises in mag-\nnetic systems, if it is assumed that the directions of spin\nangular and magnetic moments become separated in the\nultrafast dynamical regime [15, 16]. The inertia gives rise\nto spin nutation, a rotation of the magnetization around\nthe angular momentum direction [17], caused by the en-\nergytransferbetweenmagnetickineticandpotentialenergy\nterms. The emergence of spin inertia has been explained\nbased on an extension of the breathing Fermi surface model\n[18, 19], calculated from a s\u0000dlike interaction between\nthe magnetization density and electron spin [20] and de-\nrived from a fundamental relativistic Dirac theory [21, 22].\nMagnetic inertia can be associated with a torque term\ncontaining a second-order time derivative of the magnetic\nmoment appearing in the inertial LLG (ILLG) dynamical\n\u0003ritwik.mondal@uni-konstanz.deequation. The characteristic inertial relaxation time, using\nits definition in Eq. (1) below, is expected to range from\n1 fs [15, 20, 23, 24] to a few hundred fs [25].\nLinear-response theory predicted the emergence of a nu-\ntation resonance besides the conventional precession reso-\nnance in ferromagnets (FMs) [26–28], providing a possible\nway of detecting inertial dynamics by applying oscillating\nexternal fields. An indirect evidence of the inertial dynam-\nics was found in NiFe and Co samples [23] by following\nthe field dependence of the ferromagnetic precession reso-\nnance (FMR) peaks. The experimental observation of the\nnutation resonance has only been achieved very recently\nin NiFe and CoFeB using intense terahertz magnetic field\ntransients [25].\nWhile the notion of inertial dynamics has been applied\nboth in the context of the LLG equation for AFMs as well\nas in the ILLG equation for FMs, the linear response of\nthese two examples is fundamentally different. While in\nboth cases a pair of resonances is found in contrast to the\nsingle FMR peak, the excitation frequencies in an AFM are\ndegenerate in the absence of a static external field, while\ntheydifferbyseveralordersofmagnitudeintheILLGequa-\ntion. The effective damping parameter of the precession,\ndefined as the half-width of the peak at half-maximum, is\nconsiderably higher in AFMs than in FMs, where it corre-\nsponds to the Gilbert damping. In contrast, it was demon-\nstrated that the effective damping decreases in the ILLG\nequation applied to FMs [27], particularly at the nutation\nresonance [29]. However, the ILLG has not been applied\nto AFMs so far.\nHere, we explore the effects of the ILLG equation in\ntwo-sublattice AFMs and FiMs using linear-response the-\nory and computer simulations. It is shown that a pair of\nnutation resonance peaks emerges, and that the inertial re-\nlaxation time influences the precessional resonance signifi-\ncantly stronger in AFMs than in FMs due to the exchange\ncoupling between the sublattices. The effective damping\nparameter is found to decrease in AFMs, reaching consid-\nerably lower values than the Gilbert damping at the nuta-\ntion peak, thereby enhancing the lifetime of these excita-\ntions. The inertial effects in FiMs are found to interpolate\nbetween those in AFMs and FMs.\nII. METHODS\nAsderivedinearlierworks[15,21,22],theILLGequationarXiv:2012.02790v3  [cond-mat.mtrl-sci]  15 Apr 20212\nreads\n_Mi=\u0000\riMi\u0002Hi+\u000bi\nMi0Mi\u0002_Mi+\u0011i\nMi0Mi\u0002Mi;\n(1)\ngeneralized here to multiple sublattices indexed by i. The\nfirst, second and third terms in Eq. (1) describe spin pre-\ncession with gyromagnetic ratio \ri, transverse relaxation\nwith Gilbert damping \u000bi, and inertial dynamics with re-\nlaxation time \u0011i. Note that an alternative notation for\nthe inertial term with \u0011i=\u000bi\u001ciis also used in the liter-\nature [15, 23, 25]; where comparison with earlier works is\nmentioned in the following, the relaxation time is converted\nto the formulation of Eq. (1). The equation of motion was\ntreated analytically as described in the following sections,\nand also solved numerically using an algorithm presented\nin detail in Appendix A.\nIII. INERTIAL EFFECTS IN FERROMAGNETS\nFirst, we summarize the effects of the inertial term on\nFM resonance. The FM is described by the free energy\nF(M) =\u0000H0Mz\u0000KM2\nz=M2\n0, modeling a single sublat-\ntice where spatial modulations of the magnetization are ne-\nglected.M0is the magnitude of the magnetic moment, H0\nistheappliedexternalfieldand Kistheuniaxialanisotropy\nenergy, also considered to include demagnetization effects\nin the form of a shape anisotropy. The effective field can\nbe written as H=\u0000@F=@M= (H0+ 2KMz=M2\n0)^ez, and\nthe magnetic moment is oriented along the zdirection in\nequilibrium.\nThe linear response to a small transversal external field\ncomponent h(t)is calculated considering M=M0^ez+\nm(t)and expanding Eq. (1) up to first order in h(t)and\nm(t). The exciting field is assumed to be circularly polar-\nized,h\u0006=hx\u0006ihy=he\u0006i!t, with a similar time depen-\ndence for the response, m\u0006=mx\u0006imy=me\u0006i!t. The\ncalculated susceptibility reads (see Appendix B for details)\nm\u0006=\u001f\u0006h\u0006=\rM0\n\n0\u0000!\u0000\u0011!2\u0006i\u000b!h\u0006;(2)\nwith \n0=\r(H0M0+ 2K)=M0. It is found that the\nGilbert damping is associated with the imaginary part of\nthe susceptibility, while the inertial term contributes to the\nreal part of the susceptibility, which is consistent with the\nprevious calculation in Ref. [21]. The dissipated power is\ncalculated as P=_m\u0001h=!Im(\u001f+)jhj2. We note that a\nlinearly polarized exciting field can be described as a linear\ncombination of circularly polarized fields with !and\u0000!\nfrequencies.\nThe dissipated power with and without the inertial term\nis shown in Fig. 1. The data points denoted by symbols\nin Fig. 1 denote the results of the atomistic spin simula-\ntions (see Appendix A for details). The relaxation time is\nchosen to range from \u0011= 10\u000015s to\u0011= 10\u000012s. This\ncovers the fs timescales described in Refs. [20, 23, 24] andthe values of around 300 fs in Ref. [25]. It can be observed\nthat the inertial dynamics reduces the precession resonance\nfrequency. The resonance peak position is well approxi-\nmated as!p=\u0000p1 + 4\fFM\u00001\u0001\n=(2\u0011)\u0019\n0(1\u0000\fFM),\nwith\fFM=\u0011\n0. The associated shift in the resonance\nfieldHpwas investigated in Ref. [23]. However, note that\nthe relative value of this shift is very low since \fFM\u001c1,\nmeaning that it can only be observed if \n0is shifted to high\nvalues, for example by a strong external field H0.\nThe most profound effect of the inertial dynamics is the\nemergence of a second resonance peak, associated with\nthe spin nutation. Its frequency is approximately !n=\n\u0000\u0000p1 + 4\fFM+ 1\u0001\n=(2\u0011)\u0019\u00001=\u0011\u0000\n0(1\u0000\fFM). Simi-\nlarly to the precession frequency, the subleading corrections\n\fFM\n0are small. The negative sign of the frequency im-\nplies an opposite rotational sense [30]: while the precession\nis excited by a circularly polarized field rotating counter-\nclockwise, the nutation resonance reveals an opposite po-\nlarization.\nThe effective damping parameter is defined as the ra-\ntio of the imaginary and the real parts of the frequency\nwhere Eq. (2) has a node, and is approximately expressed\nas\u000beff,p=\u000beff,n\u0019\u000b(1\u00002\fFM), see Appendix B for the\nderivation. Since the imaginary part characterizes the half-\nwidth of the resonance peak at half maximum, the latter\nsuggests that the linewidth of FMR decreases due to the\ninertia, in agreement with the numerical results in Ref. [27].\nThe relative value of the reduction is once again governed\nby the factor \fFM.\nIV. INERTIAL EFFECTS IN\nANTIFERROMAGNETS AND FERRIMAGNETS\nNext, we consider AFMs and FiMs with two sublattices\nAandB. Assuming once again homogeneous sublattice\nmagnetizations, the free energy is expressed as\nF(MA;MB) =\u0000H0(MAz+MBz)\n\u0000KA\nM2\nA0M2\nAz\u0000KB\nM2\nB0M2\nBz+J\nMA0MB0MA\u0001MB;(3)\nwith the external field applied along the zdirec-\ntion, H0=H0^ez, uniaxial easy-axis anisotropy con-\nstantsKA;KBand intersublattice exchange coupling J.\nFrom the free energy, the associated fields entering\nthe sublattice ILLG equations (1) can be determined\nusing HA=B =\u0000@F(MA;MB)=@MA=B =H0^ez+\n2KA=BMA=Bz=M2\nA=B 0^ez\u0000JMB=A=(MA0MB0). Inequilib-\nrium, the sublattice magnetizations are aligned antiparallel\nalong thezdirection. Linear response to the transverse ho-\nmogeneous external field hA(t) =hB(t)may be calculated\nsimilarly to the FM case, using the expansions MA(r;t) =\nMA0^ez+mA(t)andMB(r;t) =\u0000MB0^ez+mB(t).\nThe two-sublattice susceptibility tensor is expressed as\nfollows (see Appendix C for details):\n\u0012\nmA\u0006\nmB\u0006\u0013\n=\u001fAB\n\u0006\u0012\nhA\u0006\nhB\u0006\u0013\n=1\n\u0001\u0006\u00121\n\rBMB0\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u00001\nMA0MB0J\n\u00001\nMA0MB0J1\n\rAMA0\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001\u0013\u0012\nhA\u0006\nhB\u0006\u0013\n;(4)3\n-200-150-1000.00.51.0Dissipated power£Æ∞M0|h|2(a)\n0.0250.050.0750.1!/2º(THz)¥=0¥= 10°15ssimulations-20-100.00.51.0Dissipated power£Æ∞M0|h|2(b)\n0.0250.050.0750.1!/2º(THz)¥=0¥= 10°14ssimulations\n-2-10.00.51.0Dissipated power£Æ∞M0|h|2(c)\n0.0250.050.0750.1!/2º(THz)¥=0¥= 10°13ssimulations-0.3-0.2-0.10.00.51.0Dissipated power£Æ∞M0|h|2(d)\n0.0250.050.0750.1!/2º(THz)¥=0¥= 10°12ssimulations\nFigure 1. The rate of energy dissipation in the ferromagnet as a function of frequency for several values of the inertial relaxation\ntime, (a)\u0011= 1fs, (b)\u0011= 10fs, (c)\u0011= 100fs, and (d) \u0011= 1ps. The lines denote the results of the analytical calculations\nand the symbols of the atomistic simulations for a single macrospin. All curves are compared to the analytical expression obtained\nwithout the inertial term. The other parameters are \r= 1:76\u00021011T\u00001s\u00001,M0= 2\u0016B,H0= 1T,K= 10\u000023J,\u000b= 0:05, and\njhj= 0:001T.\nHereweusethedefinitions \u0001\u0006= (\rAMA0\rBMB0)\u00001\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u0000J2=\u0000\nM2\nA0M2\nB0\u0001\nas well as \nA=\rA=MA0(J+ 2KA+H0MA0)and\nB=\rB=MB0(J+ 2KB\u0000H0MB0).\nTo compare with FMR, we compute the dissipated power\nfor AFMR, P=_mA\u0001hA+_mB\u0001hB, with the explicit for-\nmula given in Appendix C. The result is shown in Fig. 2,\nusing the same parameters for both sublattices as for the\nFM in Fig. 1. The insets of Fig. 2 show that without the\ninertial term the AFM precession resonance peaks are sup-\npressed with respect to the FM one by a factor of about\nJ=(2K) = 50. This is caused by the fact that the magne-\ntization in the two sublattices rotates around the equilib-\nrium direction with a phase shift of \u0019, meaning that the\nhomogeneous exciting field only couples to the difference of\nthe sublattice precession amplitudes [14] in the dissipated\npower. Also, the inertial term shifts the precession reso-\nnancepeakstolowerfrequenciesconsiderablystrongerthan\nin the FM, and further reduces their magnitude. At higher\nfrequency, two additional nutation resonance peaks can be\nobserved. Remarkably, their height is significantly larger\nthan that of the precession resonances, even exceeding the\nintensity of the FMR peaks (cf. Fig. 1 where the same\nnormalization was used). The latter suggests that probing\nthe AFM nutation resonance peak is experimentally moresuitable than in the FM case. Most of these effects can be\nexplained by the fact that the precession and nutation res-\nonance frequencies lie much closer in AFMs than in FMs,\nas will be discussed in detail below.\nTo obtain the AFM resonance frequencies, we calculate\nthe nodes of the susceptibility tensor in Eq. (4), obtaining\n\u0001\u0006=a\u0006!4+b\u0006!3+c\u0006!2+d\u0006!+e\u0006= 0:(5)\nwith the following definitions:\na\u0006=\u0011A\u0011B; (6)\nb\u0006=\u0007i(\u000bA\u0011B+\u000bB\u0011A)\u0000(\u0011A\u0000\u0011B); (7)\nc\u0006=\u00001\u0006i(\u000bA\u0000\u000bB)\u0000(\nA\u0011B+ \nB\u0011A)\n\u0000\u000bA\u000bB; (8)\nd\u0006= (\nA\u0000\nB)\u0006i(\u000bB\nA+\u000bA\nB); (9)\ne\u0006=\u0000\rA\nMA0\rB\nMB0J2+ \nA\nB: (10)\nNote that inertial effects enter via a;b, andc, terms which\nare of higher order in frequency. Setting the inertial re-4\n-200-100010020000.51.0Dissipated power£ÆA∞AMA0|hA|2\n!/2º(THz)¥=0¥= 10°15ssimulations\n-0.6-0.4-0.200.20.40.600.010.02\n-20-100102000.51.0Dissipated power£ÆA∞AMA0|hA|2\n!/2º(THz)¥=0¥= 10°14ssimulations\n-0.6-0.4-0.200.20.40.600.010.02\n-0.8-0.6-0.4-0.200.20.40.60.800.51.01.52.0Dissipated power£ÆA∞AMA0|hA|2\n!/2º(THz)¥=0¥= 10°12ssimulations\n-0.6-0.4-0.200.20.40.600.010.02-3-2-1012300.51.01.5Dissipated power£ÆA∞AMA0|hA|2\n!/2º(THz)¥=0¥= 10°13ssimulations\n-0.6-0.4-0.200.20.40.600.010.02\nFigure2. Therateofenergydissipationfor theantiferromagnetasafunctionof frequencyforseveralvaluesofthe inertialrelaxation\ntime\u0011A=\u0011B=\u0011, (a)\u0011= 1fs, (b)\u0011= 10fs, (c)\u0011= 100fs and (d)\u0011= 1ps. The lines denote the results of the analytical\ncalculations and the symbols of the atomistic spin simulations for two coupled macrospins. All curves are compared to the analytical\nexpression obtained without the inertial term. The other parameters are MA0=MB0= 2\u0016B,\rA=\rB= 1:76\u00021011T\u00001s\u00001,\n\u000bA=\u000bB= 0:05,KA=KB= 10\u000023J,J= 10\u000021J,H0= 1T, andjhAj=jhBj= 0:001T. The insets show the precession\nresonances on a smaller frequency and power scale.\nlaxation times to zero, we obtain a second-order equa-\ntion that results in well-known antiferromagnetic reso-\nnance frequencies [31–33]. For equivalent sublattices and\nassuming\u000b\u001c1andK\u0019H0M0\u001cJ, these read\n!p\u0006\u0019\u0010\n1\u0006i\u000bp\nJ=(4K)\u0011\u0010\n\rH0\u0006\r=Mp\n4KJ\u0011\n. Com-\npared to the FM case, two resonance frequencies are found,\nand they are exchange enhanced by about a factor ofp\nJ=K. However, the lifetime of the excitations is reduced\nsince the effective damping is also higher by a factor ofp\nJ=(4K).\nIn the presence of the inertial term, the resonance fre-\nquencies are found as a solution of a fourth-order equation.\nThe real and imaginary parts of the calculated frequencies\nare denoted by Re(!p;n\u0006)andIm(!p;n\u0006)for precession and\nnutation resonances, respectively. These have been calcu-\nlated for an AFM and a FiM as a function of the relaxation\ntime\u0011A=\u0011B=\u0011in Fig. 3. In the absence of external field\nand damping, Eq. (5) simplifies to a second-order equa-\ntion in!2. The precession resonance frequencies are given\nby!p\u0006\u0019\u0006\r=Mp\n4KJ(1 + 2\fAFM)\u00001\n2forK\u001cJ. It is\nimportant to note here that the relative strength of the in-\nertial corrections is defined by the dimensionless parameter\n\fAFM =(\u0011\r=M 0)J, which is enhanced by a factor of J=Kas\ncompared to \fFM. The characteristic time scale of the ex-change interactions typically falls into the fs range in AFMs\nwhich are ordered at room temperature ( \rJ=M\u00191013s\u00001\nwiththeparametersusedhere), whichissimilartothetypi-\ncal values of the inverse inertial relaxation time [20, 23, 25].\nThis explains the considerable decrease of the AFMR pre-\ncession frequencies in Fig. 2, while Fig. 3(a) demonstrates\nthat deviations from the non-inertial case already become\nobservable for \u0011\u00191fs. This more pronounced inertial ef-\nfect should also be observable if the resonance is measured\nby sweeping the external field, as in Ref. [23]. The strongly\nasymmetric ( MA0= 5MB0) FiM in Fig. 3(b) is charac-\nterized by a high-frequency exchange mode, strongly influ-\nencedbyinertialeffectsasintheAFM,andalow-frequency\nmode which is less affected like in the FM.\nThe nutation resonance frequencies in the AFM can be\nexpressed as !n\u0006\u0019\u0006p1 + 2\fAFM=\u0011. Just as for the pre-\ncession resonance, the correction factor arising due to the\ninterplaybetweeninertiaandmagneticinteractionsisgiven\nby\fAFM, which is exchange enhanced compared to the FM\ncase. This gives rise to an increase of the nutation frequen-\ncies, as demonstrated in Fig. 2. For the FiM in Fig. 3(b),\nthe nutation frequency Re(!n+)belonging to the exchange\nmode Re(!p\u0000)starts deviating from the low-inertia \u0011\u00001\nasymptote at considerably lower frequencies than the FM-\nlike nutation Re(!n\u0000).5\n10−1610−1510−1410−1310−12\nη(s)10−1100101102103ωAFM\n±/2π(THz)\n(a)\n1/η\nRe(ωp+)\nRe(ωp−)\nRe(ωn+)\nRe(ωn−)\n10−1610−1510−1410−1310−12\nη(s)10−1100101102103ωFiM\n±/2π(THz)\n(b)\n1/η\nRe(ωp+)\nRe(ωp−)\nRe(ωn+)\nRe(ωn−)\nFigure 3. (Color Online) Real part of the precession resonance frequencies as a function of inertial relaxation time \u0011, (a) for AFMs\nwithMA0=MB0= 2\u0016Band (b) for FiMs with MA0= 5MB0= 10\u0016B. The other parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,\n\u000bA=\u000bB= 0:05,KA=KB= 10\u000023J,J= 10\u000021J,H0= 1T.\nThe effective damping parameters of the excitation\nmodes, defined as the ratio of the imaginary to the real part\nof the frequencies, are shown in Fig. 4. They no longer co-\nincide between precession and nutation as in the FM case,\nsince the exchange enhancement discussed above does not\naffect the nutation resonance. A reduction of the effec-\ntive damping is observed with increasing inertial relaxation\ntimes, which becomes noticeable for \fAFM =O\u0000\n10\u00002\u0001\n, just\nas in the case of the resonance frequencies. The consid-\nerable reduction of the effective damping compared to the\nGilbert damping leads to sharper nutation resonance peaks\nas demonstrated in Fig. 2, with higher intensities than for\nthe FM. In the FiM, the exchange modes !n+and!p\u0000\nstarttobecomeinfluencedatlowerinertialrelaxationtimes\nthan the FM modes !n\u0000and!p+[34]. The difference be-\ntween the effective damping parameters vanishes between\nexchange and FM modes for higher \u0011, but it remains to be\nobservable between precession and nutation modes.\nV. CONCLUSIONS\nTo conclude, we applied the ILLG equation to FMs and\ntotwo-sublatticeAFMsandFiMs, andinvestigatedtheres-\nonance frequencies using linear-response theory and com-\nputer simulations. The precession frequencies are found to\ndecrease with increasing inertial relaxation time and addi-\ntional high-frequency nutation peaks become observable.\nFurthermore, the calculation of the resonance linewidth\nshows that the effect of inertia reduces the effective damp-\ning parameter. While in FMs these corrections scale with\n\fFM=\u0011\n0, in AFMs the dimensionless coupling between\nprecession and nutation is given by \fAFM = (\u0011\r=M 0)J,\nwhich is typically several orders of magnitude higher.\nTherefore, an antiferromagnetic system with higher ex-\nchange to anisotropy energy ratio and higher \u0011will be suit-\nable to observe inertial effects. Such antiferromagnetic sys-\ntems include NiO [35] and CrPt [36, 37], even though the\ncharacteristic inertial relaxation time \u0011is unknown. The\nFiM is observed to interpolate between the FM and AFM\nlimits. The reduced effective damping gives rise to particu-\nlarly sharp and high-intensity nutation resonance peaks in\nAFMs, with frequencies comparable to the values alreadyobserved in FMs [23, 25]. These findings are expected to\nmotivatethesearchforthesignsofintrinsicallyinertialspin\ndynamics on ultrafast timescales using AFMR techniques.\nACKNOWLEDGMENTS\nWe acknowledge financial support from the Alexander\nvon Humboldt-Stiftung, the Deutsche Forschungsgemein-\nschaft via Project No. NO 290/5-1, and the National Re-\nsearch, Development, and Innovation Office of Hungary via\nProject No. K131938.\nAppendix A: Atomistic simulations of the ILLG\nequation\nThe inertial Landau-Lifshitz-Gilbert (ILLG) equation of\nmotion, given in Eq. (1) in the main text, can be rewritten\nfor the normalized spin si(t) =Mi(t)=Mi0as [21]\n@tsi=\u0000\risi\u0002Hi+\u000bisi\u0002@tsi+\u0011isi\u0002@ttsi:(A1)\nThe first term denotes precession of the spins around an\neffective field Hi, the second term corresponds to a trans-\nverse relaxation of the spins, and the last term defines the\ninertial dynamics [15]. The ILLG equation can be rewrit-\nten from the implicit form of Eq. (A1) to an explicit dif-\nferential equation which can be solved numerically without\niterations. By taking a scalar product of Eq. (A1) with siit\nis easy to see that the length of the spin remains conserved\nin the ILLG equation, i.e., @tjsij2= 0andsi\u0001@tsi= 0.\nFurthermore, we use\nsi\u0002(si\u0002@ttsi) =si(si\u0001@ttsi)\u0000@ttsi;(A2)\n@t(si\u0001@tsi)|{z}\n=0= (@tsi)2+si\u0001@ttsi: (A3)\nBy multiplying Eq. (A1) by si\u0002and using the conditions\nEqs. (A2) and (A3), we obtain the explicit equation of mo-\ntion (cf. Ref. [30])\n@ttsi=\u0000\ri\n\u0011isi\u0002(si\u0002Hi)\u0000\u000bi\n\u0011i@tsi\u00001\n\u0011isi\u0002@tsi\n\u0000si(@tsi)2=Fi(s;@ts;t): (A4)6\n10−1610−1510−1410−1310−12\nη(s)0.000.050.100.150.200.25Im(ω±)/Re(ω±)|AFM\n(a)\nη= 0\nIm(ωp+)/Re(ωp+)\nIm(ωp−)/Re(ωp−)\nIm(ωn+)/Re(ωn+)\nIm(ωn−)/Re(ωn−)\n10−1610−1510−1410−1310−12\nη(s)0.000.010.020.030.040.050.060.07Im(ω±)/Re(ω±)|FiM\n(b)\nη= 0\nIm(ωp+)/Re(ωp+)\nIm(ωp−)/Re(ωp−)\nIm(ωn+)/Re(ωn+)\nIm(ωn−)/Re(ωn−)\nFigure 4. (Color Online) Effective damping parameters of the resonance modes as a function of inertial relaxation time \u0011, for (a)\nAFMs with MA0=MB0= 2\u0016Band (b) FiMs with MA0= 5MB0= 10\u0016B. The other parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,\n\u000bA=\u000bB= 0:05,KA=KB= 10\u000023J,J= 10\u000021J,H0= 1T.\nNote that a second-order explicit differential equation is\nobtained because of the inertial term, while the LLG equa-\ntion is of first order. With the definition pi=@tsi, we can\nconvert the second-order differential equation into a system\nof first-order differential equations as follows:\n@ttsi=@tpi=Fi(s;p;t); (A5)\n@tsi=pi=Gi(s;p;t): (A6)\nIt is obvious that one has to solve six coupled differential\nequations of first order per lattice site i. We numerically\nsolve these equations with Heun’s method [38], where the\npredictor steps are\n\u0016si=si(t) + \u0001tGi(s;p;t); (A7)\n\u0016pi=pi(t) + \u0001tFi(s;p;t); (A8)\nand the corrector steps are implemented as\nsi(t+ \u0001t) =si(t) +\u0001t\n2[Gi(s;p;t) +Gi(\u0016s;\u0016p;t+ \u0001t)];\n(A9)\npi(t+ \u0001t) =pi(t) +\u0001t\n2[Fi(s;p;t) +Fi(\u0016s;\u0016p;t+ \u0001t)]:\n(A10)\nIn order to calculate the resonance curves, we employed a\ncircularly polarized field h(t)\u0018ei!tin thexyplane in ad-\ndition to the static magnetic field H0along thezdirection,and solved the equations of motion for one and two spins by\nstarting from the equilibrium state along the zdirection.\nBy multiplying Eq. (A4) by Mi0\u0011i@tsi=\ri, summing over\nthe sublattices, and rearranging the terms, one arrives at\n@t X\niMi0\u0011i\n2\ri(@tsi)2+F!\n=X\ni@tMi0@tsihi\n\u0000X\ni\u000biMi0\n\ri(@tsi)2: (A11)\nThe left-hand side of Eq. (A11) describes the change of rate\nof the energy of the system, consisting of a kinetic part and\na potential partF. The former sheds light on the meaning\nof\u0011ias an inertial parameter. The right-hand side con-\nsists of the power loss due to damping processes, which\nis compensated by the external driving force in a steady\nstate. Accordingly, we computed the dissipated power us-\ningP=P\niMi0@tsi\u0001hi.\nAppendix B: Calculation of the linear response in\nferromagnets\nIn ferromagnets, we consider that the initial magnetiza-\ntion points towards the zdirection, such that the magne-\ntization is expanded as M=M0^ez+m(t)in linear order.\nThe considered dynamical field is denoted by h(t). Using\nthe effective field in the main text, the linearized ILLG\nequation can be written in the following way:\n@tm=\u0000\r2\n664M0^ez\u0002H0^ez|{z}\n= 0+M0^ez\u00022K\nM0^ez\n|{z}\n= 0+M0^ez\u0002h(t) +m(t)\u0002H0^ez+m(t)\u00022K\nM0^ez+m(t)\u0002h(t)|{z}\nnegligible3\n775\n+\u000b\nM02\n664M0^ez\u0002@m\n@t+m\u0002@m\n@t|{z}\nnegligible3\n775+\u0011\nM02\n664M0^ez\u0002@2m\n@t2+m\u0002@2m\n@t2|{z}\nnegligible3\n775: (B1)7\nThus, we obtain the following two equations for the\ntransversal components:\n@tmx=\rM0hy\u0000\rH0my\u00002\rK\nM0my\u0000\u000b@tmy\u0000\u0011@ttmy;\n(B2)\n@tmy=\u0000\rM0hx+\rH0mx+2\rK\nM0mx+\u000b@tmx+\u0011@ttmx:\n(B3)\nWe define \n0=\r=M 0(H0M0+ 2K)as in the main text.\nTherefore, Eqs. (B2) and (B3) can be recast as\nhx=1\n\rM0[\n0mx+\u000b@tmx+\u0011@ttmx\u0000@tmy];(B4)\nhy=1\n\rM0[\n0my+\u000b@tmy+\u0011@ttmy+@tmx]:(B5)\nIn matrix form we write\n\u0012\nhx\nhy\u0013\n=1\n\rM0\u0012\n\n0+\u000b@t+\u0011@tt\u0000@t\n@t \n0+\u000b@t+\u0011@tt\u0013\u0012\nmx\nmy\u0013\n:\n(B6)\nWe switch to the circularly polarized basis, m\u0006=mx\u0006imy\nandh\u0006=hx\u0006ihy, where the equations decouple,\n\rM0\u0012\nh+\nh\u0000\u0013\n=\n\u0012\n\n0+\u000b@t+\u0011@tt+i@t 0\n0 \n 0+\u000b@t+\u0011@tt\u0000i@t\u0013\u0012\nm+\nm\u0000\u0013\n:\n(B7)\nFor the time dependence we consider h\u0006=he\u0006i!t, describ-\ningtwotypesofpolarizationwithoppositehandedness. We\nassumem\u0006=me\u0006i!t. Thus, we have\nhei!t=1\n\rM0\u0000\n\n0+i\u000b!\u0000\u0011!2\u0000!\u0001\nmei!t\n)m+=\rM0\n\n0+i\u000b!\u0000\u0011!2\u0000!hei!t; (B8)\nhe\u0000i!t=1\n\rM0\u0000\n\n0\u0000i\u000b!\u0000\u0011!2\u0000!\u0001\nme\u0000i!t\n)m\u0000=\rM0\n\n0\u0000i\u000b!\u0000\u0011!2\u0000!he\u0000i!t: (B9)\nThis leads to the susceptibility given in Eq. (2). Its real\nand imaginary parts are derived as\nRe(\u001f\u0006) =\rM0\n0\u0000!\u0000\u0011!2\n(\n0\u0000!\u0000\u0011!2)2+\u000b2!2;(B10)\nIm(\u001f\u0006) =\u0006\rM0\u000b!\n(\n0\u0000!\u0000\u0011!2)2+\u000b2!2:(B11)The dissipated power can be calculated according to its\ndefinition based on Eq. (A11),\nP=@tm\u0001h\n= (@tmxhx+@tmyhy)\n=1\n2(@tm+h\u0000+@tm\u0000h+)\n=i!\n2(\u001f+\u0000\u001f\u0000)jhj2\n=i!\n2\u0012\u00002i\u000b!\rM 0\n(\n0\u0000!\u0000\u0011!2)2+\u000b2!2\u0013\njhj2\n=!Im(\u001f+)jhj2: (B12)\nThe positions and the linewidths of the resonance peaks\nmay be analyzed by finding the poles of the susceptibility\nin Eq. (B8),\n!=1\n2\u0011\u0014\n\u0000(1\u0000i\u000b)\u0006q\n(1\u0000i\u000b)2+ 4\fFM\u0015\n=1\n2\u0011\u0002\n\u00001\u0006a+i\u000b\u0000\n1\u0007a\u00001\u0001\u0003\n; (B13)\nwhere\fFM=\u0011\n0andais the single positive real solution\nof the fourth-order equation\na4\u0000\u0000\n1\u0000\u000b2+ 4\fFM\u0001\na2\u0000\u000b2= 0:(B14)\nFor\fFM\u001c1, one hasa= 1 + 2\fFM+O\u0000\n\f2\nFM\u0001\n. For\nthe real parts of the frequencies, corresponding to the\npeak positions, one obtains !p\u0019\n0(1\u0000\fFM)and!n\u0019\n\u00001=\u0011\u0000\n0(1\u0000\fFM), as described in the main text. Note\nthat the latter expression agrees with Eq. (14) in Ref. [28],\nbut the correction terms are different from Ref. [27], where\n!n=\u0000p1 +\fFM=\u0011\u0019\u00001=\u0011\u0000\n0=2\u0000\n1\u0000\fFM=4\u0001\nwas sug-\ngested. It is apparent from Eq. (B13) that effective damp-\ning parameter, i.e. the ratio of the imaginary and the real\nparts of the frequency, is \u000ba\u00001\u0019\u000b(1\u00002\fFM)both for\nthe precession and the nutation peaks. The full width of\nthe resonance peaks at half maximum can be expressed as\n\u0001!=!1\u0000!2, which frequencies satisfy\n\n0\u0000!1\u0000\u0011!2\n1=\u0000\u000b!1; (B15)\n\n0\u0000!2\u0000\u0011!2\n2=\u000b!2: (B16)8\nThe ratio of the linewidth and the peak position is given by\n\u0001!\n!p=(\n0+\u000b\n0(1\u0000\fFM))\u0000\u0011(\n0+\u000b\n0)2\u0000(\n0\u0000\u000b\n0(1\u0000\fFM)) +\u0011(\n0\u0000\u000b\n0)2\n\n0(1\u0000\fFM)\n=2\u000b\n0\u00006\u000b\fFM\n0\n\n0(1\u0000\fFM)= 2\u000b1\u00003\fFM\n1\u0000\fFM\u00192\u000b(1\u00002\fFM) (B17)\nfor the precession resonance, confirming that dividing the\nhalf-width at half maximum by the resonance frequency\nis approximately equal to the effective damping parameter\ndescribed above.Appendix C: Calculation of the linear response in\ntwo-sublattice antiferromagnets and ferrimagnets\nWe expand the magnetization around the equilibrium\ndirection in small deviations, MA=MA0^ez+mAand\nMB=\u0000MB0^ez+mB, which are induced by the trans-\nverse external field hA=B(t). The linearized ILLG equation\nfor the two sublattices reads\n@tmA=\u0000\rA\nMA0[\u0000(H0MA0+ 2KA+J)mAx^ey+ (H0MA0+ 2KA+J)mAy^ex] +\rA\nMB0[JmBx^ey\u0000JmBy^ex]\n\u0000\rAMA0(hAx^ey\u0000hAy^ex) +\u000bA(@tmAx^ey\u0000@tmAy^ex) +\u0011A(@ttmAx^ey\u0000@ttmAy^ex); (C1)\n@tmB=\u0000\rB\nMB0[\u0000(H0MB0\u00002KB\u0000J)mBx^ey+ (H0MB0\u00002KB\u0000J)mBy^ex]\u0000\rB\nMA0[JmAx^ey\u0000JmAy^ex]\n+\rBMB0(hBx^ey\u0000hBy^ex)\u0000\u000bB(@tmBx^ey\u0000@tmBy^ex)\u0000\u0011B(@ttmBx^ey\u0000@ttmBy^ex): (C2)\nFor thexandycomponents we obtain\n\rAMA0hAy=\rA\nMA0(H0MA0+ 2KA+J)mAy+\rA\nMB0JmBy+\u000bA@tmAy+\u0011A@ttmAy+@tmAx; (C3)\n\rAMA0hAx=\rA\nMA0(H0MA0+ 2KA+J)mAx+\rA\nMB0JmBx+\u000bA@tmAx+\u0011A@ttmAx\u0000@tmAy; (C4)\n\rBMB0hBy=\rB\nMB0(\u0000H0MB0+ 2KB+J)mBy+\rB\nMA0JmAy+\u000bB@tmBy+\u0011B@ttmBy\u0000@tmBx;(C5)\n\rBMB0hBx=\rB\nMB0(\u0000H0MB0+ 2KB+J)mBx+\rB\nMA0JmAx+\u000bB@tmBx+\u0011B@ttmBx+@tmBy:(C6)\nIn the circularly polarized basis with mA=B\u0006=mA=Bx\u0006imA=By;hA=B\u0006=hA=Bx\u0006ihA=Byand defining \nA=\n\rA=MA0(H0MA0+ 2KA+J);\nB=\rB=MB0(J+ 2KB\u0000H0MB0), we obtain\n\rAMA0hA\u0006= (\nA+\u000bA@t+\u0011A@tt\u0006i@t)mA\u0006+\rA\nMB0JmB\u0006; (C7)\n\rBMB0hB\u0006= (\nB+\u000bB@t+\u0011B@tt\u0007i@t)mB\u0006+\rB\nMA0JmA\u0006: (C8)\nThe four equations of motion are separated into two pairs of coupled equations for the +and\u0000components. In matrix\nformalism we have\n\u0012\nhA\u0006\nhB\u0006\u0013\n=0\nB@1\n\rAMA0(\nA+\u000bA@t+\u0011A@tt\u0006i@t)1\nMA0MB0J\n1\nMA0MB0J1\n\rBMB0(\nB+\u000bB@t+\u0011B@tt\u0007i@t)1\nCA\u0012\nmA\u0006\nmB\u0006\u0013\n: (C9)\nBy substituting the time dependence hA=B\u0006;mA=B\u0006/e\u0006i!twe have\n\u0012\nhA\u0006\nhB\u0006\u0013\n=0\nB@1\n\rAMA0\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001 1\nMA0MB0J\n1\nMA0MB0J1\n\rBMB0\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u00011\nCA\u0012\nmA\u0006\nmB\u0006\u0013\n:(C10)9\nWe introduce the definition \u0001\u0006 = (\rAMA0\rBMB0)\u00001\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u0000\nJ2=\u0000\nM2\nA0M2\nB0\u0001\nfor the determinant of the matrix above. The susceptibility tensor is obtained by matrix inversion,\n\u0012\nmA\u0006\nmB\u0006\u0013\n=1\n\u0001\u00060\nB@1\n\rBMB0\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u00001\nMA0MB0J\n\u00001\nMA0MB0J1\n\rAMA0\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u00011\nCA\u0012\nhA\u0006\nhB\u0006\u0013\n=\u001fAB\n\u0006\u0012\nhA\u0006\nhB\u0006\u0013\n;\n(C11)\nas also given in Eq. (4).\nSimilarly to the ferromagnet, we calculate the dissipated power from Eq. (A11) as\nPAB=@tmA\u0001hA+@tmB\u0001hB\n=1\n2h\n@tmA+hA\u0000+@tmA\u0000hA++@tmB+hB\u0000+@tmB\u0000hB+i\n=i!\n2h1\n\u0001+\u00121\n\rBMB0\u0000\n\nB+i!\u000bB\u0000\u0011B!2+!\u0001\nhA+\u00001\nMA0MB0JhB+\u0013\nhA\u0000\n\u00001\n\u0001\u0000\u00121\n\rBMB0\u0000\n\nB\u0000i!\u000bB\u0000\u0011B!2+!\u0001\nhA\u0000\u00001\nMA0MB0JhB\u0000\u0013\nhA+\n+1\n\u0001+\u0012\n\u00001\nMA0MB0JhA++1\n\rAMA0\u0000\n\nA+i!\u000bA\u0000\u0011A!2\u0000!\u0001\nhB+\u0013\nhB\u0000\n\u00001\n\u0001\u0000\u0012\n\u00001\nMA0MB0JhA\u0000+1\n\rAMA0\u0000\n\nA\u0000i!\u000bA\u0000\u0011A!2\u0000!\u0001\nhB\u0000\u0013\nhB+i\n=!2jhAj2\n\rBMB02\n4(\rAMA0\rBMB0)\u00001\u000bAh\u0000\n\nB\u0000\u0011B!2+!\u00012+!2\u000b2\nBi\n+J2=\u0000\nM2\nA0M2\nB0\u0001\n\u000bB\n\u0001+\u0001\u00003\n5\n+!2jhBj2\n\rAMA02\n4(\rAMA0\rBMB0)\u00001\u000bBh\u0000\n\nA\u0000\u0011A!2\u0000!\u00012+!2\u000b2\nAi\n+J2=\u0000\nM2\nA0MB0\u00012\u000bA\n\u0001+\u0001\u00003\n5\n\u00002!2JjhAhBj\n\rAM2\nA0\rBM2\nB0\u0014(\nA\u000bB+ \nB\u000bA) + (\u000bA\u0000\u000bB)!\u0000(\u0011A\u000bB+\u0011B\u000bA)!2\n\u0001+\u0001\u0000\u0015\n: (C12)\nAs discussed in the main text, the peak positions and the linewidths may be understood by finding the nodes of the\ndeterminant \u0001\u0006,\n\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u0001\n\u0000\rA\rB\nMA0MB0J2= 0\n)\u0011A\u0011B|{z}\n=a\u0006!4+ [\u0007i(\u000bA\u0011B+\u000bB\u0011A)\u0000(\u0011A\u0000\u0011B)]| {z }\n=b\u0006!3\n+ [\u00001\u0006i(\u000bA\u0000\u000bB)\u0000(\nA\u0011B+ \nB\u0011A)\u0000\u000bA\u000bB]| {z }\n=c\u0006!2\n+ [(\nA\u0000\nB)\u0006i(\u000bB\nA+\u000bA\nB)]| {z }\n=d\u0006!+ \nA\nB\u0000\rA\rB\nMA0MB0J2\n|{z}\n=e\u0006= 0: (C13)\nThe fourth-order equation (C13) may be solved in a\nclosed form. However, in order to arrive at solutions which\nhave a simpler form, we consider the antiferromagnet with\nidentical sublattices, MA0=MB0=M0,\u000bA=\u000bB=\u000b,\n\u0011A=\u0011B=\u0011, andKA=KB=K. Furthermore, we\nassume\u000b\u001c1andM0H0;K\u001cJ, as is typical in most\nsystems. Consequently, we will treat the terms propor-\ntional to the damping and the external field in first-orderperturbation theory, leading to\n\u00112!4\u0000\u0012\n1 + 2\u0011\r\nM0(J+ 2K)\u0013\n!2\u0000i2\u000b\u0011!3\n(0)+ 2\rH0!(0)\n+i2\u000b\r\nM0(J+ 2K)!(0)+\r2\nM2\n0(J+ 2K)2\u0000\r2(H0)2\n\u0000\r2\nM2\n0J2= 0; (C14)\nwhere!(0)is the solution for \u000b= 0andH0= 0, and we10\nonly treat \u0001+for simplicity since \u0001\u0000may be obtained\nby complex conjugation. Equation (C14) is a second-order\nequation in !2, the solutions of which are simple to express.Expanding them up to first order in \u000bandH0for consis-\ntency with the order of the perturbation, and also in first\norder inK=J\u001c1, one obtains\n!p\u0006\u0019\u0006\r\nM0p\n4K(J+K)r\n1 + 2\u0011\r\nM0(J+ 2K)+1r\n1 + 2\u0011\r\nM0(J+ 2K)\n\u0002\f\f!(0)\f\f\n\r\nM0p\n4K(J+K)\u0014\n\rH0+i\u000b\u0012\r\nM0(J+ 2K)\u0000\u0011!2\n(0)\u0013\u0015\n; (C15)\n!n\u0006\u0019\u00061\n\u0011r\n1 + 2\u0011\r\nM0(J+ 2K)0\nBBB@1\u0000\u00112\r2\nM2\n04K(J+K)\n2\u0014\n1 + 2\u0011\r\nM0(J+ 2K)\u001521\nCCCA\n\u0000\u0011\f\f!(0)\f\f\n\u0014\n1 + 2\u0011\r\nM0(J+ 2K)\u00153\n2\u0014\n\rH0+i\u000b\u0012\r\nM0(J+ 2K)\u0000\u0011!2\n(0)\u0013\u0015\n; (C16)\nfor the precession and the nutation frequencies, respectively. Substituting in\f\f!(0)\f\ffrom the leading term in the expression\ninto the perturbative terms, one arrives at\n!p\u0006\u0019\u0006\r\nM0p\n4K(J+K)r\n1 + 2\u0011\r\nM0(J+ 2K)\n+1\n1 + 2\u0011\r\nM0(J+ 2K)\u0014\n\rH0+i\u000b\r\nM0(J+ 2K)\u0015\n; (C17)\n!n\u0006\u0019\u00061\n\u0011r\n1 + 2\u0011\r\nM0(J+ 2K)0\nBBB@1\u0000\u00112\r2\nM2\n04K(J+K)\n2\u0014\n1 + 2\u0011\r\nM0(J+ 2K)\u001521\nCCCA\n\u00001\n1 + 2\u0011\r\nM0(J+ 2K)\u0014\n\rH0\u0000i\u000b\u00121\n\u0011+\r\nM0(J+ 2K)\u0013\u0015\n: (C18)\nThe leading-order terms for H0;\u000b= 0and usingJ+K\u0019\nJare also reported in the main text. As discussed there,\nin the antiferromagnet the corrections caused by the in-\nertial dynamics surpass in magnitude those in the ferro-\nmagnet, since the characteristic dimensionless parameter\n\fFM=\u0011\n0is replaced by \fAFM =\u0011\r=M 0(J+ 2K)\u0019\n\u0011\r=M 0J+2K. Thisdifferenceisalsomanifestinthedepen-\ndence of the excitation frequencies on the static magnetic\nfieldH0: while in the ferromagnet the Larmor frequency\nis renormalized as (1\u0000\fFM)\rH0, in the antiferromagnet\nthecorrespondingfactoris (1 + 2\fAFM)\u00001\rH0forboththe\nprecession and the nutation frequencies, causing an appar-\nent decrease in the gyromagnetic factor.\nFrom Eqs. (C17) and (C18), the effective damping pa-rameters in the antiferromagnet may be expressed as\nIm(!p)\nRe(!p)\u0019\u000bs\n(J+ 2K)2\n4K(J+K)1r\n1 + 2\u0011\r\nM0(J+ 2K);\n(C19)\nIm(!n)\nRe(!n)\u0019\u000b1 +\u0011\r\nM0(J+ 2K)\n\u0014\n1 + 2\u0011\r\nM0(J+ 2K)\u00153\n2: (C20)\nWhile the inertial dynamics decrease the resonance\nlinewidth of the antiferromagnet by a larger factor\n(1 + 2\fAFM)\u00001=2compared to the ferromagnet (1\u00002\fFM),\nthis is compensated by the exchange enhancement ex-\npressed in the factorp\nJ=4K. Remarkably, the effective\ndamping of the nutation resonance is not exchange en-11\nhanced, while it is still reduced compared to the Gilbert damping due to the inertial motion, giving rise to the par-\nticularly sharp peaks in Fig. 2.\n[1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett.\n99, 047601 (2007).\n[2] K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D. Hinzke,\nU. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kiri-\nlyuk, and T. Rasing, Phys. Rev. Lett. 103, 117201 (2009).\n[3] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,\nH. A. Dürr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W.\nChantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing,\nand A. V. Kimel, Nature 472, 205 (2011).\n[4] K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, S. Ger-\nlach, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto,\nA. Itoh, A. Kirilyuk, and Th. Rasing, Phys. Rev. B 85,\n104402 (2012).\n[5] A. Hassdenteufel, B. Hebler, C. 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Samuelsen, Phys. Rev. B 6, 3447\n(1972).\n[36] M. J. Besnus and A. J. P. Meyer, phys. stat. sol. (b) 58,\n533 (1973).\n[37] R. Zhang, R. Skomski, X. Li, Z. Li, P. Manchanda,\nA. Kashyap, R. D. Kirby, S.-H. Liou, and D. J. Sellmyer,\nJ. Appl. Phys. 111, 07D720 (2012).\n[38] U. Nowak, Handbook of Magnetism and Advanced Mag-\nnetic Materials (2007)."    },    {        "title": "2306.08043v1.Circuit_QED_detection_of_induced_two_fold_anisotropic_pairing_in_a_hybrid_superconductor_ferromagnet_bilayer.pdf",        "content": "Circuit QED detection of induced two-fold anisotropic pairing in a hybrid\nsuperconductor-ferromagnet bilayer\nC. G. L. Bøttcher,1,∗N. R. Poniatowski,1,∗A. Grankin,2\nM. E. Wesson,3Z. Yan,1U. Vool,1, 4V. M. Galitski,2, 5and A. Yacoby1, 3\n1Department of Physics, Harvard University, Cambridge, MA 02138, USA\n2Joint Quantum Institute, Department of Physics,\nUniversity of Maryland, College Park, MD 20742, USA\n3Harvard John A. Paulson School of Engineering and Applied Sciences,\nHarvard University, MA 02138, Cambridge, USA\n4Max Planck Institute for Chemical Physics of Solids, Dresden 01187, Germany\n5Center for Computational Quantum Physics, The Flatiron Institute, New York, NY 10010, United States\n(Dated: June 16, 2023)\nHybrid systems represent one of the frontiers in the study of unconventional superconductivity\nand are a promising platform to realize topological superconducting states. Owing to their meso-\nscopic dimensions, these materials are challenging to probe using many conventional measurement\ntechniques, and require new experimental probes to successfully characterize. In this work, we\ndevelop a probe that enables us to measure the superfluid density of micron-size superconductors\nusing microwave techniques drawn from circuit quantum electrodynamics (cQED). We apply this\ntechnique to a paradigmatic hybrid system, the superconductor/ferromagnet bilayer, and find that\nthe proximity-induced superfluid density is two-fold anisotropic within the plane of the sample and\nexhibits power law temperature-scaling which is indicative of a nodal superconducting state. These\nexperimental results are consistent with the theoretically predicted signatures of induced triplet\npairing with a nodal p-wave order parameter. Moreover, we unexpectedly observe drastic modi-\nfications to the microwave response at frequencies near the ferromagnetic resonance, suggesting a\ncoupling between the spin dynamics and induced superconducting order in the ferromagnetic layer.\nOur results offer new insights into the unconventional superconducting states induced in super-\nconductor/ferromagnet heterostructures and simultaneously establish a new avenue for the study\nof fragile unconventional superconductivity in low-dimensional materials such as van der Waals\nheterostructures.\nHeterostructures constructed from superconductors\nand other materials (e.g. semiconductors, ferromagnets,\nand topological materials) offer a rich platform to realize\nunconventional superconducting states via proximity ef-\nfects. In these hybrid systems, the coupling between dis-\ntinct materials leads to the formation of emergent phases\nthat feature new physical properties that are otherwise\nabsent in the isolated constituents. These include topo-\nlogical superconducting phases [1] hosting non-Abelian\nexcitations, as well as states supporting spin-triplet pair-\ning [2], both of which have potential applications for\nquantum computing technology [3]. Given the extreme\nscarcity of naturally occurring topological [4–7] or spin-\ntriplet superconductors [8–11], hybrid systems are an in-\nvaluable resource to realize these exotic superconducting\nstates [12–15].\nThe archetypal superconducting hybrid system is the\nsuperconductor-ferromagnet (S/F) bilayer, where spin-\ntriplet superconductivity can be induced in the ferromag-\nnet due to the combined effect of the exchange field and\nsuperconducting proximity effect [16, 17]. These systems\nare well-studied for their numerous applications for su-\nperconducting spintronics [2, 18], as well as their poten-\ntial use as a platform for topological quantum computing.\nHowever, while S/F heterostructures have been exten-\nsively studied using transport techniques, direct probesof the induced superconducting order have been lacking.\nThis disparity lies in the fact that the mesoscopic nature\nof the induced superconducting state, which is tightly\nconfined to the S/F interface and exists over nanometer-\nscale distances, renders most well-developed techniques\nin the study of bulk unconventional superconductors (e.g.\nheat capacity, penetration depth, or neutron scattering\nmeasurements) challenging to apply. Although there has\nbeen promising recent work applying conventional tech-\nniques to superconductor heterostructures [19–21], new\nexperimental probes are required to enable the direct\nstudy of induced unconventional superconducting states\nin hybrid superconducting systems [22, 23].\nIn this work, we employ an on-chip superconducting\nmicrowave resonator as a sensitive probe of a micron-\nscale S/F bilayer. Resonator circuits allow for the cre-\nation and control of highly localized electromagnetic\nfields, enabling one to attain strong coupling even to\nmicron-scale samples [24–28]. By integrating a S/F bi-\nlayer into our resonator circuit, one can probe the induc-\ntive response of the bilayer which, as will be discussed\nat length below, is a direct manifestation of the induced\nsuperfluid density in the hybrid system. The temper-\nature dependence of the superfluid density directly re-\nflects the underlying pairing symmetry of the supercon-\nducting state, and has been employed with great successarXiv:2306.08043v1  [cond-mat.supr-con]  13 Jun 20232\nto gain insight into bulk superconductors [29]. However,\ntemperature-dependent superfluid density measurements\nhave yet to be applied to the study of superconductivity\nin mesoscopic systems.\nWhen a metallic ferromagnet is placed into contact\nwith a conventional s-wave superconductor, the strong\nexchange field in the ferromagnet depairs the spin-singlet\nCooper pairs inherited from the superconductor and sup-\npresses induced singlet superconductivity [30]. How-\never, interfacial spin-orbit coupling (which is generically\npresent) or magnetic inhomogeneities can flip the spin of\nan electron as it tunnels across the S/F interface and con-\nvert singlet pairs into spin-triplet Cooper pairs which can\nsurvive in the ferromagnet, leading to the formation of\na mini-gap in the majority-spin band in the ferromagnet\n[31, 32]. To satisfy fermionic antisymmetry, these triplet\npairs must either have an odd-parity (e.g. p-wave) orbital\nstructure or an “odd-frequency” pairing structure, with\nthe pairing correlations being antisymmetric with respect\nto time [17, 33]. Although the presence of triplet pairs\nhas been indirectly inferred from the persistence of long-\nrange supercurrents in long S/F/S Josephson junctions\n[34–36], the detailed symmetry of the induced pairing is\nnot yet well understood.\nTo directly address the induced superconducting state\nin an S/F bilayer requires a probe that is amenable to\nthe small spatial size of typical devices (with nanometer\nto micron scale dimensions), as well as the ability to se-\nlectively address the weak induced superconducting state\nthat exists in parallel with the intrinsic bulk supercon-\nductivity of the superconducting layer. To achieve both\nof these requirements, we employ an on-chip supercon-\nducting coplanar waveguide resonator, which has been\nextensively developed as a part of the circuit quantum\nelectrodynamics (cQED) architecture for superconduct-\ning quantum information devices [37]. The resonator is\nfabricated from a 25 nm-thick Nb film perforated with\nflux pinning holes to maximize performance in external\nmagnetic fields [38], in a quarter-wavelength configura-\ntion with one end of the resonator shorted to ground and\nthe other open (Fig. 1a,b). These Nb resonators are de-\nsigned to have resonance frequencies ωr/2π= 4−7 GHz\nand attain quality factors of Q≈350,000 at our base\noperating temperature of T≈55 mK, enabling high sen-\nsitivity in our measurements.\nTo study the superconducting state of an S/F bilayer,\nwe deposit a 30 nm thick permalloy (Py) stripe directly\non top of the Nb center conductor where the resonator is\nshorted to ground, forming a Nb/Py bilayer that is situ-\nated at a current antinode of the circuit as shown in Fig.\n1a. Because the current is concentrated at the location\nof the bilayer, the resonator response is dominated by\nthe properties of the S/F subsystem. One manifestation\nof the strong coupling between the Nb resonator and Py\nstripe is the drastic reduction in the quality factor of the\nPy-loaded resonator to Q≈7,000.Moreover, spectroscopy of the hanger style resonator,\ninterrogated via the transmission S21across a capaci-\ntively coupled transmission line (Fig. 1b), allows one to\ndirectly probe the microwave dynamics of the ferromag-\nnet. In particular, by applying an in-plane magnetic field\nH∥along the length of the Py stripe, we may tune the\nferromagnetic resonance (FMR) frequency which follows\nthe Kittel law ωm(H∥) =γp\n(H∥+Ms)H∥, where γis\nthe gyromagnetic constant and µ0Ms≈1.2 T is the sat-\nuration magnetization of Py [39, 40]. When the FMR\nfrequency is brought to coincide with the frequency of\nthe resonator mode (which is only weakly affected by\nsmall in-plane fields due to pair-breaking effects), we ob-\nserve clear anti-crossings in the resonator spectrum asso-\nciated with the formation of magnon-polaritons. These\nanti-crossings are observed at both low fields µ0H∥≈7\nmT when the FMR intersects the fundamental frequency\nof the resonator, as well at higher fields µ0H∥≈110\nmT when the FMR crosses the third-harmonic of the\nresonator (a quarter-wavelength resonator exhibits only\nodd harmonics), as shown in Fig. 1d,e. In both cases,\nwe can fit the resonator spectrum to the spectral func-\ntion of two coupled harmonic oscillators, and extract an\neffective coupling strength g/2π≈100 MHz between the\nresonator and ferromagnetic resonance mode (see Sup-\nplemental Information for details). From the dimensions\nof the Py stripe, this corresponds to a coupling strength\nof 150 Hz/spin, which drastically exceeds the coupling\nstrengths reported in prior works [27, 28] on Py/Nb hy-\nbrid circuits, where the Py stripe was separated from the\nsuperconductor with an insulating layer to prevent any\ndegradation in Qfrom the inverse proximity effect. In\ncontrast, our devices feature a direct interface between\nthe superconductor and ferromagnet, enabling proximity\neffects and the possibility of new dynamics generated by\nthe interplay between the order parameters in each layer\nof the hybrid S/F system.\nAt microwave frequencies, a superconductor behaves as\nan inductive element characterized by a “kinetic induc-\ntance” arising from the superfluid response [41]. This\ninductance is fundamentally related to the density nsof\nsuperfluid carriers as LK= (m/n se2)(ℓ/s) for a super-\nconducting wire of length ℓand cross-section s. Notably,\nthe kinetic inductance is large for fragile or dilute su-\nperconductors with a small superfluid density, as well\nas for thin systems with small cross-sections. Both of\nthese features make kinetic inductance measurements es-\npecially favorable for probing weak and low-dimensional\nsuperconductors, such as the proximity-induced super-\nconducting state in an S/F bilayer. The kinetic in-\nductance is reflected in the resonant frequency 2 πf=\n1/p\n(Lg+LK)C, where LgandCare the geometric\ninductance and capacitance of the circuit, respectively.\nWhen the system is weakly perturbed by changing an\nexternal parameter such as the temperature or applied\nmagnetic field (under the reasonable assumption that Lg3\nFigure 1 \nS21(dB)\n1st harmonicmodelPhotonMagnon\nH∥H∥(mT)ω/2π(GHz)\nPhotonH∥MagnonS21(dB)deω/2π(GHz)  =120 MHz\n<latexit sha1_base64=\"Ek2JhIh2FH+QquZu37us8DrxtGI=\">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqe6KqMeiF48V7Ae0S8mm2TY2myxJVihL/4MXD4p49f9489+YtnvQ1gcDj/dmmJkXJoIb63nfqLCyura+UdwsbW3v7O6V9w+aRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuHoduq3npg2XMkHO05YEJOB5BGnxDqpOTg77ya8V654VW8GvEz8nFQgR71X/ur2FU1jJi0VxJiO7yU2yIi2nAo2KXVTwxJCR2TAOo5KEjMTZLNrJ/jEKX0cKe1KWjxTf09kJDZmHIeuMyZ2aBa9qfif10ltdB1kXCapZZLOF0WpwFbh6eu4zzWjVowdIVRzdyumQ6IJtS6gkgvBX3x5mTTPq/5l1bu/qNRu8jiKcATHcAo+XEEN7qAODaDwCM/wCm9IoRf0jj7mrQWUzxzCH6DPH/Kljrg=</latexit>g/2⇡\n3rd harmonicmodel  =100 MHz\n<latexit sha1_base64=\"Ek2JhIh2FH+QquZu37us8DrxtGI=\">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqe6KqMeiF48V7Ae0S8mm2TY2myxJVihL/4MXD4p49f9489+YtnvQ1gcDj/dmmJkXJoIb63nfqLCyura+UdwsbW3v7O6V9w+aRqWasgZVQul2SAwTXLKG5VawdqIZiUPBWuHoduq3npg2XMkHO05YEJOB5BGnxDqpOTg77ya8V654VW8GvEz8nFQgR71X/ur2FU1jJi0VxJiO7yU2yIi2nAo2KXVTwxJCR2TAOo5KEjMTZLNrJ/jEKX0cKe1KWjxTf09kJDZmHIeuMyZ2aBa9qfif10ltdB1kXCapZZLOF0WpwFbh6eu4zzWjVowdIVRzdyumQ6IJtS6gkgvBX3x5mTTPq/5l1bu/qNRu8jiKcATHcAo+XEEN7qAODaDwCM/wCm9IoRf0jj7mrQWUzxzCH6DPH/Kljrg=</latexit>g/2⇡LkLindPyLNbC\nPyNbkEk↑↑ΔindPyΔNbaNbPy nm25 nm30 m40μm5μ\nTransmission linem100μλ/4 resonatorhrf\nNb\nm10μSiH∥H⊥Py\nm1μbc\nH∥(mT)\nFIG. 1. Device geometry and ferromagnetic resonance .a.False-colored scanning electron micrograph of the supercon-\nductor/ferromagnet (S/F) bilayer composed of a 30 nm permalloy film deposited directly on top of a 25 nm thick Nb film, as\nshown in the cross-section in panel b. The bilayer is integrated into a quarter-wavelength coplanar resonator patterned into\nthe Nb film, shown in an optical micrograph in panel b. The resonator is capacitively coupled to a transmission line and is\nperforated with artificial flux-pinning holes (inset of panel a.) to improve the resonator performance in magnetic fields. c.Top:\nAt microwave frequencies, the bilayer response can be treated as a circuit of two parallel inductors, corresponding to the kinetic\ninductances associated with the bulk Nb superfluid density ( LNb∼1/nNb\ns) and the induced superfluid density in the bilayer.\nBottom: As a result of their direct contact, the Nb is able to proximity induce superconductivity in the Py strip, leading to\nthe formation of a mini-gap ∆ Pyin the majority spin band. d.Transmission S21across the circuit as a function of in-plane\nmagnetic field µ0H∥oriented along the length of the Py stripe. When the resonator frequency is tuned to the ferromagnetic\nresonance frequency of the Kittel magnons in the Py, an anti-crossing is observed in the resonator spectrum, where the cavity\nphotons hybridize with the FMR mode to form cavity magnon-polaritons. Note that given the low field µ0H∥≈7 mT associ-\nated with the FMR at this frequency, the magnon-photon hybridization leads to substantial damping of the photon mode even\nat zero field. The black lines is an overlay of the modelled spectral function of coupled harmonic oscillators (see Supplemental\nInformation), which allow us to extract an effective coupling strength g/2π= 120 MHz between the resonator photons and Py\nmagnons. e.Transmission spectrum at the third harmonic of the resonator. Anti-crossings associated with magnon-polariton\nmodes are again observed, now at a higher field µ0H∥≈120 mT, at which the FMR crosses the third harmonic frequency ≈11\nGHz. Fitting the transmission spectrum (black lines) yields a similar coupling g/2π= 100 MHz to that observed at the first\nharmonic. The broad shoulder on the left-hand side of both sweeps is due to hysteretic effects related to trapped flux in the\nsuperconducting resonator.\nandCare constant), the frequency shift of the resonator\nis directly proportional to the change in the kinetic in-\nductance or, equivalently, in the superfluid density\nδf\nf0≈ −1\n2δLK\nLK,0≈κ\n2δns\nns,0, (1)\nwhere we have assumed that the frequency shift is small\ncompared to the resonance frequency f0at our base op-\nerating temperature of 55 mK such that δf/f 0≪1,\nand have introduced the kinetic inductance fraction κ=\nLK,0/(Lg+LK,0). Thus, by studying the evolution of the\nresonator frequency with temperature or magnetic field,\nwe are able to sensitively measure the changes in the su-\nperfluid density of the S/F bilayer, offering a direct probe\nof the induced superconducting order.\nIn fact, superfluid density measurements have provento be an essential tool in the study of unconventional\nsuperconductivity [29, 42]. In conventional fully gapped\nsuperconductors, the superfluid density exhibits a ther-\nmally activated temperature dependence δns(T)/ns,0≡\n[ns(T)−ns(0)]/ns(0)∼e−∆/T/√\nTwhere ∆ is the su-\nperconducting energy gap. In contrast, unconventional\nsuperconductors with nodal order parameters host low-\nlying quasiparticles residing at the gap nodes, leading\nto a power law dependence of the superfluid density\nδns(T)/ns,0∼Tn, where the exponent ndepends on\nspatial dimensionality, the dimensionality of the nodes,\nand the degree of disorder in the system [29, 42]. As\na consequence of this distinction between activated and\npower law temperature dependence, superfluid density\nmeasurements have become a standard, powerful tool in\nassessing the gap topology of bulk superconductors.4\nFigure 3 \n0.20.40.60.8-25-20-15-10-50\n0.20.40.60.8-15-10-50\n0.20.40.60.8-10-50\n0.20.40.60.8-10-50\nμ0H⊥=25mTExp.T(K)μ0H⊥=300mTExp. (w Py)\nμ0H∥=20mTδf/f0(×10−5)T(K)Fit to αTnμ0H∥=300mTab\ncdFit to αTnH⊥H∥n=2.3\nn=1.5n=1.3δf/f0(×10−5)H⊥H∥T(K)T(K)Exp.Fit to αTnExp.Fit to αTnn=2.65Exp. (w/o Py)\nTcut−off(K)en0.10.20.30.40.50.60.70.8123\nFIG. 2. Anisotropic temperature dependence of induc-\ntance .a.Shift in resonance frequency δf/f 0= [f(T, H)−\nf(55 mK , H)]/f(55 mK , H) in an in-plane field µ0H∥= 300\nmT oriented along the length of the Py stripe, as illustrated\nin the inset. The comparatively negligible temperature de-\npendence of the resonance frequency of a bare Nb resonator\n(without a Py stripe) is shown for comparison. b.Shift in\nthe resonance frequency in an in-plane field µ0H⊥= 300 mT\noriented perpendicular to the length of the Py stripe. c.Fre-\nquency shift in an in-plane field µ0H∥= 20 mT; d.Frequency\nshift in an in-plane field µ0H⊥= 25 mT. In all plots, the grey\nline is a fit of the data over the full temperature range to the\npower-law dependence δf/f 0=αTn, with αandnfitting\nparameters. e.Extracted temperature-scaling exponent nas\na function of the upper cutoff of the temperature range over\nwhich the data is fit, for the data in each panel b-d. Irrespec-\ntive of the details of the fit procedure, the scaling exponents\nfor fields parallel and perpendicular to the stripe are distinct.\nWe may simplistically imagine that the microwave re-\nsponse of the bilayer can be described as that of two\nparallel inductors, as illustrated in Fig. 1c: one corre-\nsponding to the kinetic inductance of the induced super-\nconducting state in the Py, and the other corresponding\nto the bulk superfluid density of the Nb film below. We\nwill focus on the low-temperature regime T≲800 mK\nin our measurements, well below the critical tempera-tureTNb\nc≈8 K, such that the kinetic inductance of the\nNb film is effectively frozen out and equal to its zero-\ntemperature value. Experimentally, as shown in Fig. 2a,\nthe resonance frequency of bare Nb resonators exhibits\nvery little temperature dependence in this range, with\nδf/f 0∼10−6, consistent with this assumption. More-\nover, we have further validated this technique by measur-\ning the superfluid density of a small micron-scale Al film\ninserted at the end of the resonator (see Supplemental\nInformation), which leads to an activated temperature-\ndependence of the resonance frequency with a rate con-\nsistent with the gap of Al. Thus, we can attribute the\ntemperature-dependent changes studied below to the mi-\ncrowave response of the S/F bilayer.\nWe may begin by studying the response of the hybrid\nresonator in an applied in-plane magnetic field so that\nthe ferromagnetic resonance is detuned to be far above\nour operating frequency, which in this case is ωr/2π≈7\nGHz. The system’s behavior when the ferromagnetic res-\nonance is near the resonator frequency, and the cavity\nmode takes on the character of a magnon-polariton, will\nbe discussed later. In Fig. 2a, we present the tempera-\nture dependence of the fundamental resonant frequency\nof the hybrid superconductor-ferromagnet circuit in an\nin-plane magnetic field of µ0H∥= 300 mT oriented along\nthe length of the Py stripe. Notably, the temperature\ndependence is manifestly non-exponential, in contrast to\nthe expectation for a conventional fully gapped super-\nconductor. Fitting the temperature-dependent frequency\nshift to a simple power law, δf/f 0=α Tn, we find an\nexponent of n= 2.3. The overall magnitude αof the\nfrequency shift is determined by several non-universal\nfactors, and we will primarily focus on the exponent n\nthroughout this work (see the Supplemental Information\nfor further discussion). In this measurement configura-\ntion, the external magnetic field µ0H∥is applied paral-\nlel to the microwave current flowing in the resonator, as\nshown in the inset.\nIn Fig. 2b, we instead apply the field perpendicular to\nthe current, and present the temperature dependence of\nδf/f 0atµ0H⊥= 300 mT. We again find a power-law,\nrather than exponential, temperature dependence with a\ndifferent, faster exponent n= 1.3 compared to the H∥\nconfiguration. That is, we observe a two-fold anisotropy\nin the temperature scaling of the hybrid resonator fre-\nquency, and by extension of the kinetic inductance of the\nS/F bilayer.\nWe may also perform measurements at lower mag-\nnetic fields, and probe the temperature dependence of\nthe resonance below the FMR frequency. In Fig. 2c,d we\npresent δf/f 0traces for µ0H∥= 20 mT and µ0H⊥= 25\nmT, where we again see power-law temperature depen-\ndences in both cases. Further, we again find a two-fold\nanisotropy in the exponent n, with n= 2.65 in the paral-\nlel configuration and n= 1.5 in the perpendicular config-\nuration. Thus, we find that the temperature-dependent5\nresponse of the S/F bilayer is qualitatively unchanged by\nthe magnitude of the applied magnetic field.\nThe temperature in our dilution refrigerator is only\nstable below ∼800 mK, constraining the accessible tem-\nperature range for our measurements. To ensure that\nour results are independent of this upper limit, we may\nrestrict the fits of the temperature-dependent resonance\nfrequencies to progressively lower temperatures, and ex-\ntract the scaling exponents nfor each field orientation for\ndifferent values of the upper cutoff of the fitting range,\nTcut-off . We plot the extracted exponents nas a function\nofTcut-off in Fig. 2e, where we see that the scaling expo-\nnents for parallel vs. perpendicular field orientations are\nclearly distinct independent of the fitting range, empha-\nsizing the robustness of the observed scaling anisotropy.\nIt is natural to attribute the temperature dependence\nof the induced superfluid density, manifested in the shift\nof the hybrid S/F resonance, to the thermal excitations\nabove the proximity-induced mini-gap. In particular, we\nnote that the features we observe occur on temperature\nscales on the order of tens to hundreds of milliKelvin,\nwhich is substantially smaller than the energy scales as-\nsociated with either the superconductor (with a critical\ntemperature ≈8 K) or ferromagnet (with a Curie tem-\nperature ≈500 K) independently. This strongly suggests\nthat the physics underlying the observed temperature-\ndependent response arises due to the low-energy coupling\nbetween the two states, e.g. from a proximity-induced\nsuperconducting state.\nIn general, a variety of superconducting correlations\nwith different spin and orbital symmetries are generated\nat the S/F interface [16–18]. Typically, however, only the\ncorrelations which can persist over long distances (such\nas the odd-frequency triplet state) into the ferromagnet\ncontribute meaningfully in traditional transport experi-\nments, and hence have been the principal focus of theo-\nretical study. Nonetheless, other superconducting corre-\nlations are always present, albeit potentially confined to\nthe interface over atomic-scale distances and thus chal-\nlenging to detect using conventional probes.\nOur observation of a power-law, rather than acti-\nvated, temperature-dependence of the superfluid density\nsuggests that we are coupling to a nodal, rather than\nfully-gapped, induced superconducting state. Such an\nanisotropic state would not be protected by Anderson’s\ntheorem and thus susceptible to pair-breaking from im-\npurity scattering, and consequently would be confined to\nwithin a mean free path of the S/F interface. The possi-\nbility of our experiment to detect such a weak state lies\nin the fact that we measure changes in the kinetic in-\nductance, and thus are primarily sensitive to the lowest-\nlying thermally excited quasiparticles and the most frag-\nile superconducting states, as opposed to being immedi-\nately shunted by the fully-gapped superconducting state.\nMoreover, the lateral geometry of the bilayer integrated\ninto our superconducting circuit enables even states lo-calized to the S/F interface to contribute to the inductive\nresponse.\nIn general, the superfluid density is a tensor quantity\nthat can have two distinct components in a (quasi-) two-\ndimensional system [29, 43, 44]. Moreover, the superfluid\ndensity in a nodal superconductor can display different\ntemperature scalings depending on the relative orienta-\ntion between the current and nodal direction [44]. In-\ntuitively, one can imagine that the gapless quasiparticle\nstates residing near the gap nodes are most efficiently\nexcited when the current is aligned along the nodal di-\nrection, leading to a temperature scaling δns∼Tthat re-\nflects the linear dispersion of the nodal quasiparticles. In\ncontrast, when the current is aligned along the anti-nodal\ndirection, nodal quasiparticles are less efficiently excited,\nleading to a slower temperature dependence δns∼Tn\nwith n >1. In this case, the precise power law depen-\ndence of the superfluid density is determined by the mi-\ncroscopic details of the system (e.g. spatial dimensional-\nity, co-dimension of the gap nodes, disorder, etc.).\nThus, our finding of a two-fold anisotropic power-law\nscaling of the superfluid density strongly constrains the\npossible superconducting states detected in our experi-\nment. In particular, the two-fold anisotropy is only con-\nsistent with an induced order parameter with a p-wave\norbital symmetry. Moreover, the power-law dependence\nof the superfluid density implies that the induced state is\nnodal, and that by applying the dc magnetic field parallel\nor perpendicular to the microwave current, we are able\nto selectively address a nodal and anti-nodal orientation\nof the p-wave order parameter.\nTo inform our experimental findings, we now construct\na phenomenological model for the induced superfluid den-\nsity in the S/F bilayer. We consider a bilayer system\nconsisting of an s-wave superconductor and a ferromag-\nnet with an in-plane magnetization oriented along Hex.\nThe inter-layer tunneling is assumed to have a spin-\nindependent component, t, as well a component with the\nRashba spin-orbit texture ∝tsoc(k×σ)zwhere σis the\nelectron spin and kis the in-plane electron momentum.\nThe spin-orbit coupling at the interface arises due to the\ninversion symmetry breaking, and, as was shown in [32],\ncan generate chiral px+ipysuperconducting correlations\nin the ferromagnet when the magnetization is oriented\nout of the plane of the sample. Here, we take the Zee-\nman field to lie in an in-plane orientation, which gives\nrise to a nodal p-wave order parameter. In the Supple-\nmental Information, we use this model to derive the effec-\ntivep-wave order parameter for the majority spin com-\nponent of the ferromagnet, which is shown to have the\nform ∆ k= ∆ tcosθ, where θis the angle between kand\nHex, and ∆ tis the amplitude of the triplet order param-\neter. Within the mean-field approximation, the Meissner6\nH∥H⊥<latexit sha1_base64=\"bsrVFKbx6fYQwHYJvyb/JM30/Lw=\">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</latexit>⇠`mfp\nat+tSOC(k×σ)zFigure 4 \n10110210310-5100δns(T)Tτb\n∼T∼T2∼T3Hex\nFIG. 3. Superfluid density for a disorder nodal p-\nwave state. a. Illustration of the S/F bilayer with\nthe in-plane field directions H∥,H⊥indicated. The cross-\nsection schematically depicts how interfacial spin-orbit cou-\npling can convert isotropic spin-singlet pairs in the Nb layer\ninto spin-triplet p-wave pairs in the ferromagnet, which can\nsurvive into the ferromagnet over lengths scales on the or-\nder of the electronic mean free path. b.Superfluid density\nδns(T) =ns(T)−ns(T= 0) as a function of temperature for\n∆τ≈5×102, ∆τ≈103, ∆τ≈3×103, ∆τ≈6×103, where\n∆ is the triplet gap. Solid and dashed curves correspond to\nthe response along and transverse to the nodes of the super-\nconducting gap, respectively, where darker colors correspond\nto higher τ. Lines corresponding to temperature scalings of\nT, T2,andT3are included in purple/pink as guides to the\neye. The blue-shaded region indicates the range of parameter\nspace compatible with the experimental results.\nkernel at temperature Tcan be found to be [45]\nδKi,j=−2e2\nc∞Z\n0dϵnF(ϵ)*\nℜvivj∆2\nkh\n(ϵ−Σ)2−∆2\nki3/2+\nFS,\n(2)\nwhere ⟨. . .⟩FSdenotes a Fermi surface average, viis\nthe Fermi velocity, nF(ϵ) is the Fermi distribution,\nδKi,j=Ki,j(T)−Ki,j(0), Σ ( ϵ) is the diagonal com-\nponent of the self-energy which we evaluate within the\nstrong-scattering self-consistent T−matrix approxima-\ntion ˆΣ (ϵ) =τ−1/P\nkˆGk, where τis the scattering time\nand ˆGis the Nambu electron Green’s function. We\nnote that vertex corrections must be included due to the\nanisotropy of the order parameter, as discussed in the\nSupplemental Information. At finite disorder, the low-\ntemperature scaling of the Meissner kernel is quadratic\nδKi,i∼T2. At higher temperatures the response scales\nasTandT3when probed along the nodal and antin-odal directions of the superconducting order parameter,\nrespectively. For a general temperature and disorder\nscattering time τ, the Meissner response can be eval-\nuated numerically. The result for the anisotropic su-\nperfluid density defined as ns=Ki,jc/e2is shown in\nFig. 3, where we find that the temperature scaling con-\ntinuously evolves from a quasi-isotropic ∼T2dependence\nat strong disorder to a strongly anisotropic T/T3depen-\ndence along the nodal/antinodal direction in the clean\nor high-temperature limit. Notably, the experimentally\nobserved temperature scaling along the two directions is\ncompatible with this theory over a wide region of param-\neter space, highlighted in blue in Fig. 3.\nTo intuitively understand the origin of these power\nlaws, we recall that in a clean superconductor with line\nnodes, one expects that the component of the superfluid\ndensity along the nodal direction scales linearly with tem-\nperature, reflecting the linear dispersion of the low-lying\nquasiparticles as discussed above. However, the introduc-\ntion of weak non-magnetic disorder gives rise to low-lying\nimpurity states which “fill in” the node, leading to a finite\nquasiparticle density of states at low energies, manifested\nas a quadratic temperature dependence of the superfluid\ndensity at low temperatures [45]. Above the energy scale\nT⋆of these impurity states (which is set by the super-\nconducting gap and impurity scattering rate), the usual\nlinear-in-temperature scaling is recovered. In fact, such\na quadratic-to-linear crossover has been extensively used\nto successfully describe superfluid density measurements\nof cuprate superconductors with varying degrees of dis-\norder. In the language of temperature-scaling exponents,\nthis quadratic-to-linear crossover translates to intermedi-\nate scaling exponents 1 < n < 2 in the nodal direction\n(as observed experimentally), where the precise value of\nnvaries continuously with the degree of disorder. Simi-\nlarly, one expects 2 < n < 3 in the antinodal direction,\nwhich is again consistent with the experimental results.\nSo far, we have focused on the temperature-dependent\nresponse of the hybrid S/F resonator subjected to in-\nplane magnetic fields such that the resonator is far de-\ntuned from the ferromagnetic resonance frequency. If\nwe perform the same measurements at fields where the\nresonator frequency is near the FMR frequency, we ob-\nserve strikingly different behavior as illustrated in Fig.\n4. Namely, we observe a sharp “upturn” in the reso-\nnance frequency as the temperature is lowered, which\ncan be described as a nearly divergent power law scal-\ningδf/f 0∼Tnwith n < 1 at low temperatures. By\ncomparing the response of the first and third harmon-\nics, which intersect the FMR at different magnetic fields,\nwe can confirm that these upturn features track with the\nproximity to the FMR field (i.e. the field HFMR such\nthat ωm(HFMR) =ωr) as opposed to the magnitude of\nthe in-plane magnetic field itself. These upturns become\nincreasingly sharp as the FMR field is approached, and\nweaker upturns persist over a relatively wide field range,7\n00.5-3-2-10\n00.5-2-1.5-1-0.50\n00.5-6-4-20\n00.5-2-1.5-1-0.50\n00.5-4-3-2-10\n00.5-8-6-4-20\nFigure 6 \nKittel modeH∥ω1ω3aω/2πωm(H∥)\n30mT65mT\n−50mT=−20mTδHFMR1stδf/f0(×10−5)δf/f0(×10−5)T(K)T(K)T(K)\nT(K)T(K)T(K)b\nc−70mTFMR1stFMR3rd\nδHFMR3rd=15mT\nFIG. 4. Temperature scaling near the ferromagnetic\nresonance .a.Schematic illustration of the first and third\nharmonic modes of the resonator and the evolution of the fer-\nromagnetic resonance (Kittel) mode frequency with in-plane\nmagnetic field. b.Temperature dependence of the first har-\nmonic resonator frequency at fields above the ferromagnetic\nresonance field, i.e. δHFMR,1st =H−HFMR,1st >0. Pro-\ngressively steeper upturns in the temperature dependence are\nobserved as the ferromagnetic resonance field is approached.\nc.Temperature dependence of the third harmonic resonator\nfrequency at fields below the ferromagnetic resonance field,\nδHFMR,3rd <0. The steepness of the upturns again scales\nwith the proximity to the ferromagnetic resonance field.\nDashed lines are guides the eye that mark the approximate\ntemperature at which the upturns onset.\non the order of 100 mT, away from the FMR field. The\nexact field range over which the upturns persist is device-\ndependent, but in all cases the upturns track with the\nFMR frequency.\nThe appearance of these low-temperature upturns,\nwhich manifest on temperature scales far lower than the\nrelevant scales in either the superconductor or ferromag-\nnet independently, are again indicative of strong inter-\nactions and hybridization between the two subsystems.\nHowever, on account of the unusual, seemingly diver-\ngent, temperature-scaling exponent nin this regime, itis unclear whether the temperature dependence of the\nresonance frequency near the FMR can be simply at-\ntributed to changes in the superfluid density of the S/F\nbilayer. We also note that qualitatively similar upturns,\nand history-dependent artifacts presumably related to\ntrapped magnetic flux, are occasionally observed in the\ntemperature-dependence of bare Nb resonators after re-\npeated magnetic field cycling (as elaborated on in the\nSupplemental Information). In contrast, the upturns ob-\nserved in the S/F devices near the FMR are a repro-\nducible feature of the phenomenology of these devices.\nRegardless of whether the low-temperature upturns\ncan be associated with the induced superfluid density,\nthese unusual features clearly reflect a non-trivial low-\nenergy coupling between the superconductor and ferro-\nmagnet subsystems. The origin of these upturns is yet\nto be theoretically understood and necessitates further\nstudy of the coupled dynamics of S/F heterostructures.\nAltogether, our kinetic inductance measurement tech-\nnique has enabled us to access previously inaccessible as-\npects of the physics of S/F heterostructures. We are\nable to sensitively couple to fragile sub-dominant in-\nduced superconducting orders, beyond the usual long-\nrange triplet states that typically dominate the transport\nresponse of S/F systems. Our work thus establishes ki-\nnetic inductance techniques as a complementary probe to\nconventional transport experiments in the study of super-\nconductor heterostructures, and enables a deeper under-\nstanding of induced unconventional superconductivity in\nthese systems.\nMore broadly, our technique is applicable to a wide\nvariety of mesoscopic superconducting systems, as\nrealized in other proximity-coupled systems such as\nsuperconductor/semiconductor heterostructures [22],\ninterfacial superconductivity in oxide heterostructures,\nand two-dimensional superconducting materials such as\ngraphene heterostructures or transition metal dichalco-\ngenides [46]. Consequently, our technique can be\nleveraged as a novel means to directly probe (potentially\nunconventional) superconductivity in a wide array of\nexotic low-dimensional systems that have thus far been\nchallenging to probe via conventional techniques.\nAcknowledgements: The authors thank Eugene Dem-\nler, Bertrand Halperin, Jonathan Curtis, and Leonid\nGlazman for fruitful discussions relating to this work.\nThe experimental work is supported by the Quantum\nScience Center (QSC), a National Quantum Information\nScience Research Center of the U.S. Department of En-\nergy (DOE). Device fabrication was performed at the\nCenter for Nanoscale Systems at Harvard, supported in\npart by an NSF NNIN award ECS-00335765. N.R.P.\nand M.E.W. are supported by the Department of De-\nfense through the NDSEG fellowship program. A.G. and\nV.M.G. are supported by the National Science Founda-\ntion under Grant No. DMR-2037158, the U.S. Army8\nResearch Office under Contract No. W911NF1310172,\nand the Simons Foundation. A.Y. is partly supported by\nthe Gordon and Betty Moore Foundation through Grant\nGBMF 9468 and by the National Science Foundation un-\nder Grant No. DMR-1708688.\nTechnical details of the calculation of the superfluid\ndensity\nWe now discuss the technical details of calculating\nthe superfluid density in the simplified phenomenologi-\ncal model. The clean-limit of this theory agrees with the\nexperimental results presented in the main text, while\nthe dirty limit of the theory leads to different behavior.\nThis limit, studied using the Usadel equation, will be\npresented elsewhere.\nWe begin by discussing the generation of the nodal p-\nwave condensate component inside the ferromagnet. We\nnote that while a detailed calculation can be found in\n[32], here we only study a simplified model described by\nthe 2x2 Bogolyubov-de-Gennes Hamiltonian:\nHBdG= \nξ(s)\nkˆτ3+ ∆ˆτ2ˆσ2 tˆτ3+tsoc(kxˆσ2ˆτ3−kyˆσ1)\ntˆτ3+tsoc(kxˆσ2ˆτ3−kyˆσ1) ξ(f)\nkˆτ3+hxˆτ3ˆσ1!\n,\nwhere ξ(s/f)\nkare the electronic dispersions, hxis the Zee-\nman field value, ∆ is the bulk gap inside the supercon-\nductor, tis the regular tunneling and tsocis the tun-\nneling with spin-orbit interaction, the Nambu and spin\nPauli matrices are denoted as ˆ τiand ˆσi. Using projec-\ntor formalism, we now integrate the superconductor and\nthe minority-spin component degree of freedom. The in-\nduced imaginary-frequency self-energy reads:\nˆΣk(iϵn) =−PHBdG(iϵn−QHBdGQ)−1HBdGPwhere Pis the projector onto majority-spin component,\nQ=I− P andϵn= (2n+ 1)π/β. The resulting ex-\npression can be found analytically by expanding up to\nthe second order in the tunneling but it is still too cum-\nbersome to be reproduced here. Importantly, depending\non the Fermi surface geometry and scattering properties,\nthere are two kinds of terms: induced spin-orbit interac-\ntion∼2kyttsocˆτ0\n∆2+δE2δEand the nodal p-wave triplet compo-\nnent∼2kxttsoc∆\n∆2+δE2ˆτ2, where δEis the difference of Fermi\nenergies of the majority-spin and the superconductor. In\nthe following, we focus only on the triplet component\nassuming ˆΣk≈kx∆tˆτ2with ∆ tbeing a free parameter.\nSelf-energy and vertex corrections\nWe now consider the disorder averaging and vertex\ncorrections to the superfluid density. Within the self-\nconsistent T-matrix approximation, the self-energy due\nto disorder scattering reads [47]:\nˆΣ (iϵn) =niˆT(iϵn),\nwhere niis the impurity concentration. The ˆT-matrix is\ngiven by the sum of ladder diagrams and is equal to:\nˆT(iϵn) =v0\u0010\n1−v0D\nˆGk(iϵn)E\u0011−1\n,\nwhere ⟨. . .⟩=L−2P\nkandv0is the disorder scattering\nstrength and Gkbeing full Green’s function. In the main\ntext, we take the limit v0→ ∞ and denote the scattering\nrate as τ−1=ni/ν0, where ν0is the electronic density of\nstates. In this case, the T-matrix is ˆT(iϵn) =T(iϵn) ˆτ3,\nand the remaining equation for T(iϵn) can be solved self-\nconsistently to find the self-energy.\nWe also need to consider the proper vertex correc-\ntions to compute the Meissner response.Within the self-\nconsistent T-matrix approximation [47] the correction to\nthe current vertex Γ µare given by the Bethe-Salpeter\nequation:\nΓµ(iϵn, iϵn+iΩm,k) =γµ(k) +niZd2k′\n(2π)2T(iϵn)T(iϵn+iΩm)τ3ˆGk′(iϵn+iΩm) Γµ(iϵn, iϵn+iΩm,k)ˆGk′(iϵn)τ3.\nFollowing [47], we first analytically continue this equa-\ntion to real frequencies and then solve it numerically.\nDevice fabrication\nThe devices studied in this work are fabricated by\nfirst thermally evaporating gold bond pads and align-\nment marks onto a high-resistivity silicon chip. Next,\nthe chip is dipped in hydroflouric (HF) acid and a 25 nmthick Nb film is immediately sputtered onto the cleaned\nchip. The resonator structure is defined using electron-\nbeam lithography and the unwanted Nb is removed via\nreactive ion etching with CF 4. To fabricate the S/F hy-\nbrid resonators, the S/F bilayer region at the end of the\nresonator is defined in another electron-beam lithogra-\nphy step. The exposed Nb region is cleaned in HF to\nensure a transparent interface, after which a 30 nm thick\npermalloy film is immediately thermally evaporated.9\nExperimental setup\nThe experiments described in this work are performed\nin a dilution refrigerator (Oxford Instruments Kelvinox\nMX50) with a base temperature of 55 mK equipped with\na three-axis vector magnet. Microwave signals generated\nby a vector network analyzer (Keysight PNA Microwave\nNetwork Analyzer N5227B) are sent down a stainless\nsteel coaxial line which is thermally anchored via at-\ntenuators to each plate of the cryostat as illustrated in\nFig. 5a. The sample is mounted on the mixing chamber\nin a copper sample holder designed by IBM Research,\nand is connected to the measurement circuit via non-\nsuperconducting gold wirebonds. The signal transmitted\nacross the device is routed through a circulator (Quinstar\nQCY-G0400801) via Nb superconducting coaxial lines to\na cold amplifier (Low Noise Factory LNF-LNC03-14SA)\non the 4K plate. The amplified signal then leaves the\ncryostat via stainless steel coaxial lines and is further am-\nplified at room temperature (MITEQ LNA-40-04000800-\n07-10P) before being read out into one of the ports of the\nnetwork analyzer.\nThe microwave transmission S21is recorded and fit to\nthe standard form for a hanging resonator [48] to extract\nthe resonance frequency. Representative traces of S21\nversus frequency featuring the resonator mode for devices\nterminated both with and without the S/F bilayer are\nshown in Fig. 5b, along with the corresponding fits used\nto extract the resonance frequency.\nMeasuring the superconducting gap of aluminum\nAs a proof of concept for our kinetic inductance mea-\nsurement technique, we studied a device in which the Nb\nresonator is shorted to ground through a small Al strip,\nrather than an S/F bilayer. The termination of the res-\nonator is shown in Fig. 6a: the center conductor of the\nNb resonator is “cut” and replaced with a 40 µm long,\n20 nm thick Al film with a width of either 2.5 µm or\n5µm. As discussed in the main text, the resonator de-\nsign localizes most of the current to the Al strip, making\nthe resonator response particularly sensitive to the Al\nregion. Moreover, since the critical temperature and su-\nperconducting gap of Al are much lower than that of Nb,\nthe temperature dependence of the resonator frequency\nwill be almost exclusively due to the temperature de-\npendence of the superfluid density in the Al strip. In\nFig. 6b,c we show the temperature dependence of the\nresonance frequency for the two resonators with different\nwidths of the Al strips. In both cases, the curves are\nactivated with temperature, as one would expect for a\nfully-gapped conventional superconductor. Moreover, we\nmay fit these data to the low-temperature limit of the\nFigure 1S\nPCB with sampleωr\nS21PNA Network analyzer (N5227B)Port 1Port 2\nAu bond wireCu4 K1 K700 mK100 mK50 mK20 dB10 dB3 dB 300 K∼Stainless steelNba\n3.75733.75743.757501233.5033.50463.23.43.6\nω/2π(GHz)FitExp. Q0=366,000\nLinearS21b\n3.75733.75743.757501233.5033.50463.23.43.6FitExp. Q0=7,079ω/2π(GHz)\nFIG. 5. Experimental setup .a.Schematic wiring diagram\nfor the microwave measurement setup. All lines are coaxial\ncables, with the materials for each segment indicated in the\nfigure. Grey boxes represent attenuators thermally anchored\nto each plate of the dilution refrigerator. b.Traces of the\nmicrowave transmission S21for bare Nb resonators and hybrid\nresonators terminated with an S/F bilayer, respectively. The\nfit used to extract the resonance frequency is shown along\nwith the raw data, along with the quality factor estimated\nfrom the fit.\nstandard BCS form for the superfluid density [29],\nδf\nf0=Ar\n2π∆0\nTe−∆0/T. (3)\nThe amplitude of the frequency shift, A, and the zero-\ntemperature superconducting gap ∆ 0are treated as fit\nparameters. The fits to each curve are superimposed\non the data in Fig. 6b,c, and yield values of the gap\n∆0= 240 µeV (280 µeV) for the resonator with a 2.5\nµm (5 µm) Al strip. These values are consistent with10\nT(K)00.20.40.60.81T (K)-2.5-2-1.5-1-0.500.5f/f010-45μmAl\n<latexit sha1_base64=\"ypwokR+zEEiqZPlaQwlGkc6SiXE=\">AAACC3icdVDJSgNBEO1xN25Rj16aBMFT6BmNy0EQ9eAxglkgE0JPp6KNPQvdNWIYcvfir3jxoIhXf8Cbf2NnEVT0QcHjvSqq6gWJkgYZ+3AmJqemZ2bn5nMLi0vLK/nVtZqJUy2gKmIV60bADSgZQRUlKmgkGngYKKgH1ycDv34D2sg4usBeAq2QX0ayKwVHK7XzBf8UFPI2o4fU22c+wi1mtO+H6YhCrd/OF1nJ29k7cBllJTaEJV65vF32qDtWimSMSjv/7ndikYYQoVDcmKbLEmxlXKMUCvo5PzWQcHHNL6FpacRDMK1s+EufblqlQ7uxthUhHarfJzIeGtMLA9sZcrwyv72B+JfXTLG738pklKQIkRgt6qaKYkwHwdCO1CBQ9SzhQkt7KxVXXHOBNr6cDeHrU/o/qXkld7fEzneKR8fjOObIBimQLeKSPXJEzkiFVIkgd+SBPJFn5955dF6c11HrhDOeWSc/4Lx9AvFlmmg=</latexit>\u00000= 280µeVcb\n00.20.40.60.81T (K)-8-6-4-20f/f010-42.5μmAl\n<latexit sha1_base64=\"JJ18bGRBftJpjlT/kiMwkyMwKTc=\">AAACC3icdVDJSgNBEO1xjXGLevTSJAiewkyMmosQ1IPHCGaBTAg9nUrSpGehu0YMQ+5e/BUvHhTx6g9482/sLEIUfVDweK+KqnpeJIVG2/60FhaXlldWU2vp9Y3Nre3Mzm5Nh7HiUOWhDFXDYxqkCKCKAiU0IgXM9yTUvcHF2K/fgtIiDG5wGEHLZ71AdAVnaKR2JutegkTWtukZLRRtF+EOEzpy/XhKoTZqZ3J23p6AzpGj40Kx5FBnpuTIDJV25sPthDz2IUAumdZNx46wlTCFgksYpd1YQ8T4gPWgaWjAfNCtZPLLiB4YpUO7oTIVIJ2o8xMJ87Ue+p7p9Bn29W9vLP7lNWPsllqJCKIYIeDTRd1YUgzpOBjaEQo4yqEhjCthbqW8zxTjaOJLmxC+P6X/k1oh75zk7etirnw+iyNF9kmWHBKHnJIyuSIVUiWc3JNH8kxerAfryXq13qatC9ZsZo/8gPX+BcpImk4=</latexit>\u00000= 240µeVFigure 2Sa\nAl10 μmδf/f0(×10−4)δf/f0(×10−4)\nFIG. 6. Aluminum hybrid resonators .a.False-colored\nscanning electron micrograph of the Al strip terminating a\nNb resonator otherwise identical to that used in the S/F bi-\nlayer devices, as described at length in the main text. b.\nTemperature-dependent resonance frequency of a device with\na 2.5 µm wide Al strip. The data is fit to Eq. (3), which\nyields a value of ∆ 0= 240 µeV for the superconducting gap.\nc.Temperature-dependent resonance frequency and BCS fit\nfor a device with a 5 µm wide Al strip.\ndirect measurements of the superconducting gap of Al\nthin films, which quantitatively validates our measure-\nment technique and analysis procedure.\nMagnon-photon coupling\nAs described in the main text, by tuning an exter-\nnal field such that the Kittel mode frequency, ωm=\nγq\nµ2\n0H∥(H∥+Ms), coincides with that of the resonator,\nωr, avoided crossings are observed symmetrically around\nzero for the first and third harmonic of the resonator.When the magnons couple to the photons, the hybrid\nmode is highly broadened due to magnon damping. We\nobtain the magnon-photon coupling strength, g, from\nmodeling the two bands, using the following equation for\nthe transmission spectrum [27, 28, 37]\nS21(ω, H∥) =κr,ext\ni(ω−ωr)−κr+g2\ni[ω−ωm(H∥)]−κm/2(4)\nwhere κrandκr,extare the resonator internal and exter-\nnal loss rate respectively and κmis the magnon damping\nrate. From this we determine the total magnon-photon\ncoupling to be g1st= 120 MHz for the first harmonic,\nconsistent with the coupling extracted from anticrossing\nof the third harmonic of the resonator, g3rd= 100 MHz.\nWe note that the extracted saturation magnetization of\nthe first mode, M1st\nsis more than twice the value that\nwe obtain from the third harmonic, µ0M3rd\ns= 1.38 T,\nand what has previously been reported in the literature\n[27, 28]. We speculate that trapped fields could lead to\na seemingly larger saturation magnetization.\nAlignment procedure\nThe measurements described in the main text are per-\nformed in an in-plane magnetic field, although in real-\nity sample misalignment inevitably leads to small out-of-\nplane field components. To eliminate the effects of these\nsmall unwanted out-of-plane fields, we employ the three-\naxis vector magnet in our cryostat to compensate for the\nout-of-plane field and ensure that the magnetic field ex-\nperienced by the sample is entirely in-plane. To do so,\nwe use the resonator frequency as a sensitive measure of\nthe field experienced by the sample. The resonance fre-\nquency of the superconducting resonator decreases with\napplied out-of-plane fields in an approximately parabolic\nfashion for small fields due to field-induced pair break-\ning and the associated decrease in superfluid density (or,\nequivalently, increase in kinetic inductance), as shown in\nFig. 7a. The maximum of this parabola indicates the\n“effective” zero-field where the out-of-plane field experi-\nenced by the sample vanishes.\nTo align the magnetic field, we begin by setting the\nnominal in-plane field to its desired value. At this fixed\nin-plane field, we sweep the out-of-plane field and de-\ntermine the applied out-of-plane field corresponding to\nthe effective zero-field as described above. We then re-\ntrace the out-of-plane field history to avoid any hysteretic\neffects and set the out-of-plane field to it’s effective-\nzero-field value (i.e. the applied field corresponding to\nthe maximum of the parabola). With this field config-\nuration fixed, we then proceed with our temperature-\ndependent scans. Similar techniques for field alignment\nhave been employed in previous studies of superconduct-\ning resonators [49].11\nFigure 3S\n00.10.20.30.40.50.60.70.8T (K)-0.4-0.200.20.40.60.811.21.41.6f/f0 (x 10-5)0 Hy = -3 mT\n00.10.20.30.40.50.60.70.80-0.2-0.40.40.20.80.611.21.41.6δf/f0(×10−5)\nT(K)3.863.873.883.893.9\n-10-50510Hout(mT)\nμ0H∥=40mTμ0H⊥=0\nμ0H∥=40mTμ0H⊥=0−3mTω/2π(GHz)S21(dB)a\nb\nFIG. 7. Field alignment procedure .a.Resonance fre-\nquency of a S/F hybrid resonator as a function of the applied\nout-of-plane magnetic field. The maximum frequency of the\nresonator corresponds to the effective zero-field. b.Example\nof a temperature scan of the resonance frequency when the\nfield alignment procedure is not followed and the temperature-\ndependence exhibits non-systematic behavior.\nIf this alignment procedure is not followed,\ntemperature-dependent resonance frequency traces\noften feature artifacts due to trapped vortices. An\nexample of one such effect, a pronounced “downturn” in\nthe resonance frequency as the temperature is lowered,\nis shown in Fig. 7b. These artifacts are strongly history-\ndependent and non-systematic. In contrast, when the\nalignment procedure described above is followed (as is\nthe case for all data presented throughout the main\ntext), the temperature-dependent traces are free of these\nartifacts and are highly reproducible.\nNiobium resonators in magnetic fields\nIn this section we discuss the phenomenology of bare\nNb resonators (i.e. without S/F bilayers) subject to mag-\nnetic fields, and contrast their behavior to that of the hy-\nbrid S/F resonators studied in the main text. In Fig. 8a\nwe show temperature-dependent traces of the resonantfrequency for a Nb resonator subject to an in-plane field\nofµ0H∥= 20 mT for varying values of the out-of-plane\nmagnetic field. To compare to the results in the main\ntext, we fit the response to a power law δf/f 0=ATn,\nand take the phenomenological approach of considering\nexponents n≈4 equivalent to an activated temperature\ndependence. We also introduce the total frequency shift\nS= [f(55 mK) −f(800 mK)] /f(55 mK) which quantifies\nthe overall size of the frequency shift with temperature\nin each run. We plot the extracted Sandnas a function\nof the out-of-plane field in Fig. 8c,d, where we see that\nthe temperature-scaling exponent is unchanged by the\nout-of-plane field. In contrast, the net frequency shift S\nincreases monotonically with the out-of-plane field, pre-\nsumably due to the reduction of the superfluid density\nand commensurate decrease of f0. To emphasize the in-\nsensitivity of the temperature scaling to magnetic fields,\nin Fig. 8b we normalize the frequency shifts to S, and see\nthat the curves for each out-of-plane field collapse onto\none another. Thus, the sole effect of the out-of-plane\nmagnetic field is to rescale the total size of the frequency\nshift, and does not affect the temperature scaling in any\nway.\nFurther, in Fig. 8e we compare temperature-\ndependent traces in in-plane fields along both orthogonal\ndirections H∥andH⊥studied in the main text. We note\nthat in this particular cooldown, the misalignment in the\nH⊥direction was somewhat large, with Hout≈20 mT\ncorresponding to the effective zero field. Again, indepen-\ndent of the out-of-plane magnetic field, all curves display\nan activated temperature dependence. Normalizing by S\nas before, we find that the temperature-dependent traces\nfor both in-plane field directions collapse onto one an-\nother. That is, the temperature dependent response of\nbare Nb resonators is activated irrespective of applied\nmagnetic fields and isotropic with respect to the orienta-\ntion of in-plane fields.\n∗These authors contributed equally to this work.\nEmail: charlotte.boettcher@yale.edu\nC.G.L.B. present address: Department of Applied\nPhysics, Yale University, New Haven, CT, 06520, USA.\n[1] M. Sato and Y. Ando, Reports on Progress in Physics\n80, 076501 (2017).\n[2] J. Linder and J. W. A. Robinson, Nature Physics 11, 307\n(2015).\n[3] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and\nS. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).\n[4] G. E. Volovik, JETP Letters 90, 398 (2009).\n[5] J. Sauls, Advances in Physics 43, 113 (1994).\n[6] K. E. Avers, W. J. Gannon, S. J. Kuhn, W. P. Halperin,\nJ. A. Sauls, L. DeBeer-Schmitt, C. D. Dewhurst, J. Gav-\nilano, G. Nagy, U. Gasser, and M. R. Eskildsen, Nature\nPhysics 16, 531 (2020).12\n2468101234\n24681011.5200.20.40.60.8T (K)-1-0.8-0.6-0.4-0.200.2(f/f0)/S00.20.40.60.8T (K)-2-1.5-1-0.500.5f/f0 (x 10-5)μ0H∥=20mT    μ0Hout=3mTμ0Hout=4mTμ0Hout=5mTμ0Hout=6mTμ0Hout=9mT\nHoutδf/f0(×10−5)\n(δf/f0)/SS(×10−5)nμ0Hout(mT)μ0Hout(mT)T(K)T(K)ab\ncdFigure 4S\n00.20.40.60.8-4-3-2-10110-5\n0.20.40.60.8-1-0.8-0.6-0.4-0.20T(K)\nT(K)e\nf(δf/f0)/Sδf/f0(×10−5)μ0H∥=20mT,μ0Hout=3mTμ0H⊥=500mT,μ0Hout=30mTμ0H∥=500mT,μ0Hout=5mT\nFIG. 8. Niobium resonators in magnetic fields .a.Temperature-dependent resonance frequency of a Nb resonator subject\nto an in-plane magnetic field µ0H∥= 20mTfor different values of the out-of-plane field. b.Data in panel anormalized to the\nintegrated frequency shift S.c.,d. 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Orso1,∗\n1Universit´ e de Paris, Laboratoire Mat´ eriaux et Ph´ enom` e nes Quantiques, CNRS, F-75013, Paris, France\n2School of Science and Technology, Physics Division,\nUniversit` a di Camerino, 62032 Camerino (MC), Italy\nThe ground-state properties of the Hubbard chain with on-si te repulsion and anisotropic nearest-\nneighbor attraction are investigated by means of density ma trix renormalization group calculations.\nThe non-local attraction acts between fermions of one spin c omponent only, mimicking the effect\nof p-wave Feshbach resonances in cold-atom systems. We anal yze the onset of itinerant ferromag-\nnetism, pinpointing the critical attraction strength wher e partially and fully ferromagnetic states\noccur. In the cold-atom setup, where the two (pseudo) spin po pulations are separately conserved,\nferromagnetism occurs with the nucleation of a fully imbala nced band-insulating domain hosting\nthe attractive component only. The size of this domain grows with the attraction strength, therefore\nincreasing the (opposite) imbalance of the other domain, un til the two spin components are fully\nseparated. In the presence of a harmonic trap, the ferromagn etic state hosts a partially imbalanced\ndomain in the center with an excess of the attractive compone nt and filling lower than one. This\ncentral region is surrounded by fully imbalanced domains, l ocated in the trap tails, hosting only\nfermions belonging to the other component.\nI. INTRODUCTION\nIn various metals, such as nickel, cobalt, and iron,\nitinerant electrons display ferromagnetic behavior. The\nfirst theory introduced to characterize this phenomenon,\nwhich is usually referred to as itinerant ferromagnetism,\nis Stoner’s 1933 continuous-space Hamiltonian [1]. Also\nthe celebrated Hubbard model, which describes elec-\ntrons hopping between the sites of a discrete lattice, was\noriginally introduced to characterize itinerant ferromag-\nnetism [2]. However, it is still unclear if and when the\nconventional Hubbard model – i.e., the one including only\nnearest-neighbor hopping and on-site repulsion – has a\nferromagnetic ground-state [3, 4], beyond the infinite-\ninteraction limit [5, 6]. The important role of additional\nhopping and interaction terms has been stressed in the\ncondensed-matter literature (see, e.g., Ref. [7]).\nIn recent years, cold-atom experiments have emerged\nas the ideal platform to investigate quantum magnetism\nin strongly correlated systems. In particular, deep opti-\ncal lattices have allowed the implementation of Hubbard-\ntype Hamiltonians [8]. Antiferromagnetism has been\nunambiguously observed in deep optical lattices close\nto half-filling [9–12]. Following early attempts [13–\n15], signatures of itinerant ferromagnetism have been\nobserved too [16, 17]. These latter experiments, per-\nformed in a setup without an optical lattice, addressed\nthe metastable upper branch of a resonantly interacting\nFermi gas. The results were consistent with continuous-\nspace quantum Monte Carlo simulations [18–20]. Sev-\neral procedures have been proposed to shift the onset\nof itinerant ferromagnetism to weaker interactions. This\nwould allow experimentalists to avoid the three-body col-\n∗giuliano.orso@univ-paris-diderot.frlisions that plague the strongly-interacting regime and\nprevent the creation of a more stable ferromagnetic\nstate. The list of proposed procedures includes: tun-\ning the interaction strength via narrow Feshbach reso-\nnances [21, 22], adding shallow optical lattices [23, 24],\noptical-flux lattices [25], flat-band optical lattices [26] or\ncorrelated disorder [27], using atomic species with dif-\nferent masses [28, 29], and trapping atoms in confined\nlow-dimensional geometries [30–34]. More recently, it\nhas been proposed to favor itinerant ferromagnetism by\nmeans of an attractive intra-species interaction, tuned via\na p-wave Feshbach resonance [35–37]. This mechanism\nhas been studied only in one-dimensional continuous-\nspace models and its generalization to strongly correlated\nlattice systems, including atoms confined in deep optical\nlattices, remains an open problem. Furthermore, it is not\nclear how phase-separation would occur for bulk systems\nin the standard cold-atom setup, where the (pseudo) spin\npopulations are separately conserved.\nIn this Article, we investigate the ground-state prop-\nerties of a repulsive Hubbard chain augmented with\nan anisotropic nearest-neighbor intra-species attraction.\nOur calculations are based on the numerically-exact\ndensity matrix renormalization group (DMRG) tech-\nnique [38]. Due to the splitting between the Fesh-\nbach resonances in the triplet-states with different spin-\nprojections, p-wave resonant interactions break spin-\nrotation symmetry [39, 40]. To describe this scenario, our\nmodel includes attractive interactions between fermions\nof one spin component only. Due to the presence of\nSU(2) symmetry-breaking interactions, the Lieb-Mattis\ntheorem [41], stating that the ground state of the one\ndimensional Hubbard model is a singlet, does not ap-\nply. Therefore, a ferromagnetic ground-state is in prin-\nciple possible even in one-dimension [42]. To deter-\nmine if and when ferromagnetism occurs, we determine\nthe ground-state energy as a function of the attraction2\nstrength and of the spin-population imbalance. The spin-\nresolved density profiles as well as the double occupancy\nare computed. The critical point where ferromagnetism\noccurs is pinpointed, considering both the typical con-\ndensed matter setup where spin-rotation mechanisms are\npresent, and also the standard cold-atom setup where the\ntwo (pseudo) spin populations are individually conserved,\nthus fixing the global spin polarization. In the latter case,\nferromagnetism manifests as a phase separation. At the\ncritical point, a band-insulating domain appears, hosting\nthe attractive species only. The size of this domain grows\nwith the attraction strength, therefore increasing the (op-\nposite) population imbalance in the other domain, until\nthe two spin components are fully separated. Finally, we\nshow that the presence of an additional harmonic trap\nleads to a different scenario for the formation of the fer-\nromagnetic state. In particular, the band-insulating do-\nmain of the attractive species emerges in the middle of\nthe trap only after fully polarized domains of the opposite\nspin component have formed in the trap tails.\nThe rest of the Article is organized as follows: the\nHamiltonian we study is described in Section II, together\nwith some details on the DMRG technique. The results\nfor the ground-state energy, the spin-resolved density\nprofiles, the double occupancy, as well as the analysis\nof the onset of ferromagnetic behavior, are presented in\nSection III. Section IV reports a summary of our main\nfindings.\nII. MODEL AND COMPUTATIONAL DETAILS\nWe consider a one dimensional spin-1/2 Fermi gas de-\nscribed by the following generalized Hubbard Hamilto-\nnian:\nH=−t/summationdisplay\niσ(c†\niσci+1σ+h.c.) +U/summationdisplay\nin↑in↓i\n+/summationdisplay\niσσ′Vσσ′nσinσ′i+1, (1)\nwherec†\niσ(ciσ) andnσiare the creation (annihilation)\nand number operators of fermions with spin projection\nσ=↑,↓,tis the hopping rate between nearest neighbor-\ning sites, and Uis the strength of the on-site repulsive\ninteraction between fermions with opposite spins. The\nmodel (1) includes additional spin-dependent nearest-\nneighbor interactions of strength Vσσ′, mimicking the\neffect of odd-wave (specifically p-wave) anisotropic inter-\nactions. Since the latter are only relevant for fermion\npairs with a given (total) spin projection, we consider\nVσσ′=Vforσ=σ′=↑andVσσ′= 0 otherwise. Impor-\ntantly, we assume that the nearest neighbor interactions\nare attractive in nature, corresponding to V < 0, so that\nthey might favor ferromagnetism. In this Article, the\nHubbard chain is considered with open boundary con-\nditions, corresponding to the configuration of a flat box\nwith hard wall boundaries. This type of configuration-1 -0.5 0 0.5 1\nP-60-50-40-30 EV=0\n-0.2\n-0.36\n-0.37\n-0.5\n-0.6\n-0.7\n-0.84\n-0.85\n-1.0\nFIG. 1. (Color online) Ground-state energy Eas a function of\nthe spin polarization P= (N↑−N↓)/(N↑+N↓), plotted for\ndifferent values of the interaction strength V, ranging from\nV= 0 (upper curve) to V=−1 (bottom curve). The total\nnumber of fermions is fixed to N=N↑+N↓= 40, the length\nof the chain is L= 60. Here and in all other figures the\non-site Hubbard-repulsion strength is U= 5; furthermore,\nthe connecting lines are a guide to the eye, unless otherwise\nspecified.\ncan be created in cold-atoms experiments using almost\nuniform traps, as in Refs. [43–45]. Below, we will also\nconsider the addition of a harmonic confinement, which\ndescribes the effect of more conventional magneto-optical\ntraps.\nThe ground state properties of the Hamiltonian (1)\nare investigated using the DMRG method, expressed in\nterms of matrix product states [46]. Specifically, we use\nthe open-source code of the ALPS library [47]. To ensure\nproper convergence of the various observables, we allow\nfor bond dimensions up to 4000 and perform a large num-\nber of sweeps (between 60 and 80).\nIII. RESULTS\nHereafter, we fix the energy scale by setting t= 1,\nwhile the strength of the on-site repulsion is fixed to U=\n5, corresponding to the strongly interacting regime.\nOur main aim is to shed light on the effect of the\nnearest neighbor attraction on onset and on the sta-\nbility of itinerant ferromagnetism. In Fig. 1 we show\nthe ground-state energy as a function of the spin po-\nlarization P= (N↑−N↓)/(N↑+N↓), for different val-\nues of the attraction strength V. These data are ob-\ntained by keeping the total number of particles constant\natN=N↑+N↓= 40. The length of the chain is L= 60,\nso that the total density of fermions is N/L = 2/3. For\nfiniteV, the ground state energy is no longer a symmet-\nric function of the spin polarization. Since the nearest\nneighbor attraction affects the spin-up component only,3\n-1-0.95 -0.9-0.85 -0.8-0.75 -0.7\nV-56-54-52-50-48 E40↑\n41↑\n40↑, 1↓\nFIG. 2. (Color online) Ground-state energy Eas a function\nof the nearest neighbor attraction strength V, for three differ-\nent sets of population numbers. The red squares correspond\nto a fully polarized gas of spin-up fermions, with N↑= 40\nandN↓= 0. The other two datasets represent the energy\nof the same system upon addition of, respectively, a further\nspin-up fermion (green circles) or a spin-down fermion (blu e\ntriangles). The dashed vertical segment indicates the posi tion\nof the crossing point, V=−0.849(1). This corresponds to the\ncritical value of the attraction strength for the onset of fu ll\nferrromagnetism. The length of the chain is L= 60.\nits effect on the ground state energy is more sizable for\npositive spin polarizations, where it diminishes the en-\nergy. For V >−0.365, the ground state energy has a\nminimum at P= 0, indicating that the ground state\nis paramagnetic. For stronger attraction, however, the\nminimum shifts to positive values of the spin polariza-\ntion. If spin-rotating processes were present, as in typ-\nical condensed matter systems, the ground state would\nturn (partially) ferromagnetic, with a spin polarization\ncorresponding to the position of the energy minimum.\nBy further increasing the nearest neighbor attraction, so\nthatV/lessorsimilar−0.85, the minimum of the energy occurs at\nP= 1, implying that the ground state is fully ferromag-\nnetic.\nTo precisely determine the critical attraction strength\nwhere the fully ferromagnetic phase occurs, we analyze\nthe energy cost of adding a spin-up or a spin-down\nfermion to a gas of spin-up fermions only. As a refer-\nence, in Fig. 2 we show the ground-state energy of a fully\npolarized gas of N↑= 40 fermions, plotted as a function\nof the interaction strength V(see red squares). The green\nand the blue lines correspond to the ground-state ener-\ngies with an additional spin-up or a spin-down fermion,\nrespectively. The two curves cross at V=−0.849(1).\nFor more negative V, adding a spin-up fermion reduces\nthe energy by a larger amount than adding a spin-down\nfermion, implying that the ground state is fully ferromag-\nnetic. It is worth reminding that the change in energy\ndue to the inclusion of an extra fermion with spin σrep-\nresents the chemical potential µσof the correspondingspin component.\nSo far we have considered how itinerant ferromag-\nnetism occurs assuming that the global spin polarization\ncan vary to minimize the ground-state energy. Hereafter,\nwe discuss the emergence of ferromagnetism under the\nassumption that the numbers of spin-up and spin-down\nfermions are fixed and coincide, N↑=N↓=N/2. There-\nfore, the global spin polarization is always zero. This is\nthe common setup in cold-atom experiments, where the\npopulation of atoms in the two hyperfine states, which\nplay the role of pseudo-spin components, are separately\nconserved. In this setup, ferromagnetic phases corre-\nspond to phase-separated states hosting regions where\nthe local spin densities are finite. To identify such states,\nwe compute the density profiles of the two spin compo-\nnents. In Fig. 3 we show the results for a chain of L= 120\nsites, filled with N= 80 fermions. The four panels cor-\nrespond to as many different values of the strength of\nthe nearest neighbor attraction. For V=−0.5 the den-\nsity profiles of the two spin components essentially co-\nincide, indicating that the system is paramagnetic. The\ntwo profiles substantially differ only close to the walls,\nmainly due to the open boundary conditions and the on-\nsite repulsion between the two spin components. One\nalso observes small out-of-phase oscillations, analogous to\nFriedel oscillations, which are magnified near the system\nboundaries. For V=−1.2 the Friedel-like oscillations\ndisappear; the average densities of the two spin compo-\nnents are only slightly different away from the bound-\naries, implying that the system is still paramagnetic. For\nV=−1.3, the system is no longer homogeneous. A do-\nmain including only spin-up fermions with nearly unit\nlocal filling coexists with a partially imbalanced phase\nhosting a majority of spin-down fermions. The domain\nwith only spin-up fermions migrates towards one edge of\nthe chain to further decrease the kinetic energy of the\nsystem. Finally, for V=−1.4 the same domain has\nenglobed all spin-up fermions, forming a band insulator\noccupying N↑sites. The latter coexists with a fully polar-\nized gas of spin-down fermions occupying the remaining\nsites, with average density N↓/(L−N↑) = 0.5, as dis-\nplayed in the bottom panel of Fig. 3.\nThe emergence of the ferromagnetic phases has a\nclear fingerprint also in the double occupancy d=/summationtext\ni/angbracketleftn↑in↓i/angbracketright/L. This is plotted in Fig. 4 as a function\nof the attraction strength V, for different values of the\nsystem size. As expected, dis an increasing function of\nthe nearest neighbor interaction strength V. For large\nand negative V, where the ground-state is fully ferro-\nmagnetic, the only contribution to the double occupancy\ncomes from the overlapping tails of the two spin-polarized\ndomains. Being a surface effect, this contribution van-\nishes in the thermodynamic limit. To verify this point,\nin the inset of Fig. 4 we plot the double occupancy for\nV=−1.34 as a function of 1 /L. The dashed line cor-\nresponds to a linear fit of the data obtained by retain-\ning only the three largest system sizes considered. The\nintercept is approximately zero, implying that the dou-4\n0.20.30.4\n〈n↑i〉\n〈n↓i〉\n00.20.4\n00.51\n0 30 60 90 120\ni00.51local densities V = -0.5\nV = -1.2\nV = -1.3\nV = -1.4(a)\n(b)\n(c)\n(d)\nFIG. 3. (Color online) Density profiles of spin-up fermions\n(upward triangles) and spin-down fermions (downward trian -\ngles) in a chain with L= 120 sites, labelled by the index i.\nTo improve visibility, only every other point is shown. The\nfour panels (a)-(d) refer to different values of the nearest-\nneighbor attraction strength Vbetween spin-up fermions:\nV=−0.5,−1.2,−1.3,−1.4 (from top to bottom). The spin\npopulations are N↑=N↓= 40. For large and negative V,\na growing fully-imbalanced domain hosting a band insulator\nof spin-up particles (i.e., the attractive component) coex ists\nwith a partially ferromagnetic domain with an excess of spin -\ndown fermions.\nble occupancy vanishes in the thermodynamic limit. In\ncontrast, we see from Fig. 4 that for V=−1.33 the\nsame quantity saturates to a finite value for large system\nsizes. The value V∼=−1.34 can then be identified as\nthe critical attraction strength for the onset of the fully\nferromagnetic state under the constraint of zero global\nspin polarization. For intermediate values of V, where\nthe fully polarized domain of spin-up fermions coexists\nwith the partially ferromagnetic phase, finite-size effects\nare almost negligible for the largest system sizes consid-\nered. At the interaction strength where such domain dis-\nappears, the double occupancy displays a sudden varia-\ntion. For weaker nearest neighbor attractions, where the\nground state is paramagnetic, the density profiles of the\ntwo spin components overlap over the entire chain, lead-\ning to significantly larger values of the double occupancy.\nIn contrast with what observed for the fully ferromag-\nnetic phase, here dincreases with system size. This is\ndue to the fact that the presence of the hard walls and\nthe repulsive on-site interactions cause a depletion of the\ndouble occupancy. Being a surface effect, the depletion\ndiminishes as Lincreases.\nWe also see from Fig. 4 that the position of the sudden\nvariation of the double occupancy shifts towards weakernearest neighbor attractions for larger system sizes, thus\nbroadening the parameter region where the paramagnetic\nphase is unstable against phase separation. We can esti-\nmate the critical value of the interaction strength V=Vc\nas the position of the sudden variation in the thermo-\ndynamic limit, L→+∞. This value is determined by\nperforming a finite-size scaling analysis. For each system\nsizeL, we determine the value V=VLof the interaction\nstrength at which the reduced double occupancy in Fig.\n4 exhibits an inflexion point in the critical region. For\nthe largest system sizes, where the sudden variation is es-\nsentially a vertical jump, we simply identify VLwith the\nposition of the jump. The obtained results are displayed\nin Fig. 5 as a function of the system inverse size 1 /L.\nThe dashed continuous line corresponds to a linear fit of\nthe data, VL=a+b/L, obtained by retaining only the\nthree largest system sizes. This gives a critical interac-\ntion strength Vc=a=−1.2385(3). A closer look to the\ndata reveals that a quadratic fit VL=a+b/L+c/L2\nto the entire data set is also plausible. The result is\nshown in Fig. 5 by the dot-dashed line, from which we\ngetVc=a=−1.233(2). The small difference between\nthe linear and the quadratic extrapolations provides a\nconfidence interval of the estimate of the critical point,\nwhich we finally quote as Vc=−1.235(4). The height of\nthe vertical jump diminishes with the system size. How-\never, our data do not allow us to ascertain if it van-\nishes in the thermodynamic limit, or if it converges to\na small but finite value. For this reason, we cannot un-\nambiguously identify the order of the phase transition. It\nis worth emphasizing that, in the cold-atom setup with\nfixed global spin polarization, the partially and the fully\nferromagnetic states occur at significantly stronger at-\ntraction compared to the case where the system can vary\nthe global spin polarization to minimize its ground-state\nenergy. Furthermore, one notices that the parameter re-\ngion where the ground-state is partially ferromagnetic,\nnamely−1.34/lessorsimilarV/lessorsimilar−1.235, is quite narrow, indicating\nthat the system rapidly transitions from the paramag-\nnetic to the fully ferromagnetic phase.\nNext, we analyze the zero-temperature equation of\nstate. We obtain the ground-state energy per particle\nE/N in the thermodynamic limit via a linear extrapo-\nlation as a function of 1 /L. This is shown in the inset\nof Fig. 6 for V=−1.3. By repeating the same proce-\ndure for the different values of the attraction strength V,\nwe obtain the curve shown in the main panel of Fig. 6.\nThe dashed and the dot-dashed straight lines correspond\nto the asymptotic behavior for strong and weak nearest\nneighbor attraction, respectively. In the regime of large\nnegative V, the band-insulating domain of the attractive\nspin-up fermions only coexists with the fully polarized\nideal gas of spin-down fermions occupying the remaining\nL−N↑lattice sites. Therefore, the ground-state energy\nper particle can be computed as\nE≃V(N↑−1)\nN−2(L−N↑)\nπNsinπN↓\nL−N↑. (2)5\n-1.4 -1.35 -1.3 -1.25 -1.2\nV00.010.020.03 d/L L=60\n72\n90\n102\n1200 0.01 0.02\n1/L00.0020.004\nV=-1.34\nFIG. 4. (Color online) Double occupancy das a function\nof the nearest neighbor attraction strength V. The different\ncurves correspond to calculations for different system size s.\nFor each value of Lwe choose the spin populations so that the\noverall densities are kept constant to N↑/L=N↓/L= 1/3, so\nthat the global spin polarization is P= 0. Notice the sudden\nvariation around V=−1.25, corresponding to the nucleation\nof the band-insulator domain hosting spin-up fermions only ,\nleadingtoanessentiallyverticaldropofthedoubleoccupa ncy.\nThe inset shows the double occupancy as a function of 1 /L\nforV=−1.34, showing that dvanishes for infinite system\nsizes as the ground state becomes fully ferromagnetic.\n0 0.005 0.01 0.015 0.02\n1/L-1.28-1.27-1.26-1.25-1.24-1.23VL\nFIG. 5. Extrapolation to the infinite-size limit of the criti -\ncal strength V=Vcfor the nucleation of the band-insulating\ndomain, corresponding to the phase transition to a partiall y\nferromagnetic state. For each system size L, we extract the\nvalueV=VLat which the double occupancy in Fig. 4 dis-\nplays the sudden variation and plot it as a function of 1 /L.\nThe position of the critical point is inferred by fitting the d ata\nwith a low order polynomial p(x), withx= 1/L, and setting\nVc=p(0). Specifically, a linear fitting function (continuous\nred line), obtained by retaining only the three largest syst em\nsizes, and a quadratic fitting function (dashed green curve)\nare shown, yielding Vc=−1.2385(3) and Vc=−1.233(2),\nrespectively.-2 -1.5 -1 -0.5 0\nV-1.6-1.4-1.2 E / N\n0 0.01 0.02\n1/L-1.29-1.28-1.27E / NV=-1.3\nFIG. 6. Ground-state energy per particle E/N, extrapolated\nto the thermodynamic limit, plotted as a function of the near -\nest neighbor attraction strength V. The asymptotic behavior\nforlarge andnegative V, seeEq.(2), isshownas adashedline.\nThe dot-dashed line is the prediction from first order pertur -\nbation theory, holding for small and negative V. The inset\nshows the dependence of the ground-state energy per particl e\non the inverse system size (data symbol) for V=−1.3; the\nspin densities are fixed at N↑/L=N↓/L= 1/3. The infinite-\nlattice result corresponds to the intercept of the straight line\nfitting the data (dashed curve).\nThe asymptotic behavior (2) is shown in Fig. 6 by the\nblue dashed line (we have neglected the 1 /Nterm which\nvanishes in the thermodynamic limit). One notices that\nthe prediction from Eq. (2) is indeed very close to the\nDMRG result for V/lessorsimilar−1.34, where the ground state of\nthe system is fully ferromagnetic. For small negative V,\nthe effect of the nearest neighbor interactions in Eq. (1)\ncan be taken into account within first order perturba-\ntion theory, yielding E≃E(V= 0) +V/summationtext\ni/angbracketleftn↑in↑i+1/angbracketright,\nwhere the expectation value is computed for the Hub-\nbard model, by assuming V= 0 in Eq. (1). We observe\nthat the perturbative behavior (green dot-dashed line)\nis only recovered for relatively small values of V. It is\nalso worth noting that the two asymptotic lines cross at\nV=−1.2, which is not far from the critical point Vcfor\nthe onset of the partially ferromagnetic phase through\nphase separation.\nFinally, we discuss how ferromagnetism forms when\nthe Hamiltonian includes an additional longitudinal har-\nmonic confinement. Specifically, we include the addi-\ntional term:\nH′=/summationdisplay\niK/parenleftbigg\ni−L\n2/parenrightbigg2\n(ni↑+ni↓), (3)\nwhereKis a constant. This term is designed to de-\nscribe the effect of the most common magneto-optical\ntraps used to confine ultracold atoms. Fig. 7 displays\nthe spin-density profiles for V=−1.35. Notice that in\nthe flat box trap without the harmonic confinement this6\n0 30 60 90 120\nsite i00.20.40.60.81local densitiesni↑\nni↓\nFIG. 7. (Color online) Density profiles of the two spin compo-\nnents in a harmonic trap of intensity K= 0.002. The nearest-\nneighbor attraction strength is V=−1.35, which in the flat\ntrap with open boundaries corresponds to a fully ferromag-\nnetic ground state. The spin populations are N↑=N↓= 40.\nattraction strength is sufficient to induce full phase sep-\naration of the two spin components. In the harmonic\ntrap, the attractive spin-up fermions occupy mostly the\ntrap center. This helps decreasing the interaction energy.\nDifferently from the flat box case, where the spin-up do-\nmain is fully polarized, in the harmonic trap the (cen-\ntral) domain with the majority of spin-up fermions also\nhosts a lower density of spin-down fermions. The trap\ntails host fully polarized domains including spin-down\nfermions only, apart the small regions of interface with\nthe central domain. For more negative V, sayV=−1.4,\na fully polarized spin-up domain emerges in the center of\nthe trap, characterized by a flat density profile with unit\nfilling.\nIV. CONCLUSIONS\nWe have investigated the ferromagnetic proper-\nties of a Hubbard chain with on-site repulsion and\nanisotropic nearest-neighbor attraction between the spin-\nup fermions. The DMRG algorithm allowed us to com-\npute global properties such as the ground-state energy\nand the double occupancy, as well as local properties\nsuch as the spin-resolved density profiles. From energy\ncalculations as a function of the spin-population imbal-\nance we extracted the critical attraction strength for the\ntransition to partially ferromagnetic and to fully fer-romagnetic phases. By inspecting the density profiles\nand the double occupancy we determined how ferromag-\nnetism occurs in the standard cold-atom setup where the\n(pseudo) spin populations are individually conserved. In\nthis case, a significantly stronger attraction is needed\nto induce the separation of domains with non-zero local\nspin-population imbalance. Specifically, in the uniform\nsystem with open boundary conditions ferromagnetism\noccurs with the nucleation of a fully spin-polarized do-\nmain hosting spin-up fermions only. The size of this do-\nmain grows with the attraction strength until all spin-\nup fermions have been absorbed, meaning that the two\nspin components are fully separated. The inclusion of\nan harmonic confinement substantially modifies the sce-\nnario. In this case, a partially imbalanced domain with\nan excess of spin-up fermions is located in the trap cen-\nter, while the trap edges host domains with spin-down\nfermions only. In contrast, the band-insulating domain\nof spin-up fermions only is recovered for stronger nearest\nneighbor attractions.\nWe have presented an application of the DMRG algo-\nrithm to a novel Hamiltonian relevant to describe cold-\natom systems. The computation of local properties al-\nlowed us to investigate the phase separation of different\nspin components and the coexistence of domains with\ndifferent local spin-population imbalances. In particu-\nlar, we have shown that the double occupancy, which is\nexperimentally accessible with cold-atom systems, is a\nkey quantity to investigate phase separation and the for-\nmation of different ferromagnetic phases. Our findings\ncan serve as a guide for future cold-atom experiments fo-\ncussing on itinerant ferromagnetism in one-dimensional\noptical lattices with p-wave resonant interactions, and\nthey complement previous studies on anti-ferromagnetic\ncorrelations in optical lattices [10, 48–50].\nACKNOWLEDGEMENTS\nWe acknowledge fruitful discussions with A. Biella.\nThis work is supported by the ANR project SPIFBOX\nand by the SIRTEQ DIM program of the Region Ile-de-\nFrance through the project 1DFG. M. 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Crowell1\n1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA\n2Department of Electrical and Computer Engineering,\nUniversity of Minnesota, Minneapolis, Minnesota 55455, USA\nWe demonstrate the pumping of phonons by ferromagnetic resonance in a series of\n[Co(0.8 nm)/Pd(1.5 nm)] nmultilayers ( n= 6, 11, 15, and 20) with large magnetostriction and\nperpendicular magnetic anisotropy. The effect is shown using broadband ferromagnetic resonance\nover a range of temperatures (10 to 300 K), where a resonant damping enhancement is observed\nat frequencies corresponding to standing wave phonons across the multilayer. The strength of this\neffect is enhanced by approximately a factor of 4 at 10 K compared to room temperature, which\nis anomalous in the sense that the temperature dependence of the magnetostriction predicts an\nenhancement that is less than a factor of 2. Lastly, we demonstrate that the damping enhancement\nis correlated with a shift in the ferromagnetic resonance field as predicted quantitatively from linear\nresponse theory.\nThe ability to couple the spin degree of freedom with\nother degrees of freedom, such as charge or strain, is cru-\ncial to many spintronic applications. The coupling of spin\nto strain is a phenomenon known as magnetostriction,\nwhich is known to directly influence magnetization dy-\nnamics [1–10]. Some work on dynamical magnon-phonon\ncoupling has focused on the generation of phonons by\nferromagnetic resonance (FMR) in a magnetic thin film\nand subsequent propagation of the phonons into the sub-\nstrate, which is referred to as phonon pumping [2, 11–\n14]. Much of the early work on phonon pumping lacked\nbroadband frequency dependence, which is necessary for\nfully characterizing the effect as well as demonstrating\nthe existence of multiple resonances. Recent experimen-\ntal work on phonon pumping has largely relied on time-\nresolved Kerr measurements [4, 5, 9], which are suscepti-\nble to strain excitation through laser heating rather than\ndue to magnetization dynamics alone. Also, the tempera-\nture dependence of this effect has not been studied, which\nmay provide new insights into the underlying physics.\nIn this Letter, we demonstrate the phonon pumping\neffect by ferromagnetic resonance in a series of [Co/Pd] n\nmultilayers with perpendicular magnetic anisotropy\n(PMA). It is shown that the strength of the effect is\nstrongly temperature dependent (a factor of ∼4 en-\nhancement at 10 K relative to 300 K)—much more than\nwould be expected from the temperature dependence\nof the magnetostriction alone (less than a factor of 2\nenhancement)—which we argue is due to the sensitivity\nof the phonon pumping to the pinning of the dynamic\nmagnetization. We also show that the frequencies of the\nphonon pumping resonances can be tuned by varying n,\nthe number of Co/Pd repetitions. Finally, we show the\ndispersive effect of the phonon pumping through shifts\nin the FMR field, as predicted from the dissipation using\nKramers-Kronig relations.\nCo/Pd multilayers are well-known for their large mag-\nnetostriction and PMA [15] and have been demonstratedfor use in perpendicular magnetic tunnel junctions (p-\nMTJ) [16], including cases where synthetic antiferromag-\nnets (SAF) made from Co/Pd multilayers were used for\nthe reference layers [17, 18]. The PMA is particularly\nsignificant for this application since phonon pumping is\nmore efficient when the magnetization is perpendicular to\nthe film plane [11]. [Co(0.8 nm)/Pd(1.5 nm)] nmultilay-\ners (n= 6, 11, 15, and 20) were grown by dc magnetron\nsputtering at room temperature with a base pressure of\n<5×10−8Torr using Ar gas at a working pressure\nof 2.0 mTorr. The thicknesses of the Co and Pd lay-\ners are 0.8 nm and 1.5 nm, respectively, for all of the\nsamples and will henceforth be omitted. Ferromagnetic\nresonance of the [Co/Pd] nmultilayers [Fig. 1(a)] was\nmeasured using a coplanar waveguide setup with mod-\nulation of the applied magnetic field for lock-in detection\nof the transmitted microwave power, which was rectified\nwith a Schottkey diode detector. The applied magnetic\nfield was swept with the microwave frequency held fixed.\nMagnetometry measurements were performed on all the\nmultilayers using superconducting quantum interference\ndevice (SQUID) magnetometry. The SQUID measure-\nments were performed over a range of temperatures (5 to\n300 K) for both in-plane and out-of-plane applied fields.\nThey confirmed an out-of-plane easy axis in all the sam-\nples, and was also used to measure the saturation mag-\nnetization as a function of temperature in the multilayers\n[19].\nWe first demonstrate the effect of phonon pumping on\nthe FMR linewidths and how it depends on the num-\nber of Co/Pd repetitions in the multilayer stack. Fig.\n1(b) shows a schematic of the phonon pumping process,\nwhere magnetization dynamics are damped by the leak-\nage of magnetoelastically-driven phonons into the sub-\nstrate. Figure 2 shows FMR linewidths measured in a\nperpendicular field as a function of frequency at 150 K for\nfour different [Co/Pd] nmultilayer structures with n= 6,\n11, 15, and 20. The lower frequency limit of the mea-arXiv:2110.01714v1  [cond-mat.mtrl-sci]  4 Oct 20212\n} (a) \n×nM(t) \nu(t) Ta cap \nPd/Ta seed (b) \nSi/SiO 2 substrate phonon pumping m-e coupling \n[Co/Pd] n\nFIG. 1. (a) Stack structure of the [Co/Pd] nmultilayers.\nThicknesses of each layer are given in parentheses and have\nunits of nm. The Co(0.8 nm)/Pd(1.5 nm) bilayer is repeated\na total ofntimes as indicated on the figure. (b) Schematic of\nthe phonon pumping process in the configuration where the\nmagnetization M(t) is normal to the plane of the film. The\nmagnetization depth profile is given by a sine wave (for sim-\nplicity) with pinning at the interfaces. The magnetoelastic\ncoupling (shown by the red arrow) leads to the creation of a\nphonon standing wave with displacement u(t). The phonon\npumping process is shown by the wavy gold arrow represent-\ning the leakage of phonons into the seed layers and substrate.\nsurements is determined by the perpendicular anisotropy\nfield (which sets the zero-field FMR frequency) for the\nn= 6 and 11 samples. For the n= 15 and 20 sam-\nples, the FMR signal disappears at nonzero field, which\nsuggests that the sample becomes unsaturated at fields\nhigher than zero. This observation is corroborated by\nout-of-plane magnetic hysteresis loops, which show the\nnucleation of domains before zero field is reached [19].\nFor all of the samples shown in Fig. 2, there are reso-\nnant linewidth enhancements that appear at specific fre-\nquencies. The linear background is due to the Gilbert\ndamping, for which fits were generated by excluding\npoints within 3 GHz of the center of the peaks. For\nthen= 11 and 20 multilayers, there are two resonant\npeaks in the linewidth. In the n= 11 multilayer, the fre-\nquency of the high frequency peak is double that of the\nlow frequency peak, implying that these represent the\nfirst and second harmonics of a fundamental damping\nresonance. In the n= 20 multilayer, the high frequency\npeak is 3/2 that of the low frequency peak, suggesting\nthat the low and high frequency peaks are the second and\nthird harmonics of a fundamental damping resonance,\nrespectively. We cannot observe the fundamental reso-\nnance, however, since it is expected at a frequency ( /similarequal13\nto 14 GHz) at which the sample is unsaturated. For\nthen= 6 and 15 multilayers, there is one peak in the\nlinewidth. This corresponds to the fundamental damp-\ning resonance in the n= 6 multilayer, and the second\nharmonic in the n= 15 multilayer. The fundamental res-\nonance is undetectable in the n= 15 multilayer because\n/s48/s51/s48/s48/s54/s48/s48/s57/s48/s48/s49/s50/s48/s48/s49/s53/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s51/s48/s48/s54/s48/s48/s57/s48/s48/s72\n/s70/s87/s72/s77/s32/s40/s79/s101/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s72\n/s70/s87/s72/s77/s32/s40/s79/s101/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41[Co/Pd]6\n[Co/Pd]20 [Co/Pd]11 \n[Co/Pd]15 (a)\n(c)(b)\n(d)T = 150 K \n12112\n2\n13FIG. 2. Ferromagnetic resonance linewidths as a function of\nfrequency with applied magnetic field out of plane at T=\n150 K for (a) [Co/Pd] 6(magenta triangles), (b) [Co/Pd] 11\n(red circles), (c) [Co/Pd] 15(blue triangles), and (d) [Co/Pd] 20\n(black squares). The vertical arrows indicate the positions of\nthe phonon pumping resonances, and the green stars indicate\nthe corresponding positions predicted from the positions ob-\nserved in the [Co/Pd] 11multilayer. The numbers labelling the\nstars correspond to the number of half-waves in the thickness\nresonance so that, e.g., “3” means a phonon standing wave\nwith wavelength λ= 3d/2, wheredis the thickness of the\nmagnetic portion of the multilayer.\nit occurs at a frequency ( /similarequal16 GHz) at which the sample\nis unsaturated. We note that the damping resonance in\nthe [Co/Pd] 6multilayer exhibits a twin-peak structure,\nwith the two peaks separated by approximately 4 GHz.\nThis may be due to the existence of standing waves with\nnodes at both the interface between the 2-nm Pd and\n5-nm Ta seed layers and the inteface between the Co\nand 2-nm Pd seed layer [shown in Fig. 1(a)]. Were this\nthe case, one would expect a spacing of about 4 GHz\nas we observe. This hypothesis predicts a peak spacing\nof/lessorsimilar1 GHz for the thicker multilayers, which would ex-\nplain why the twin-peak structure is only observed in the\n[Co/Pd] 6multilayer.\nThe vertical arrows in Fig. 2 indicate the positions of\nthe resonances for each multilayer. Transverse acous-\ntic phonon standing waves are expected at frequencies\nwhered, the thickness of the stack excluding capping\nand seed layers, matches an integer number of phonon\nhalf wavelengths. This condition can be expressed as\nf=ct/(2d/m) (ctis the transverse speed of sound and m\nis a positive integer). Longitudinal phonons are neglected\nbecause they couple to the magnetization at higher order\n[11, 20, 21]. The hypothesis that the multilayer is a half-\nwave resonator is based on the fact that the highly dense\nTa capping and seed layers will lead to pinning of the3\nphonons at these interfaces. The green stars in panels\n(a), (c), and (d) indicate the positions of the resonances\npredicted from the positions observed in the [Co/Pd] 11\nmultilayer in panel (b), where the effect is strongest. The\nnumbers labelling the stars indicate the order of the res-\nonance, so that a resonance of order mcorresponds to a\nphonon standing wave of wavelength λ= 2d/m. From\nthis we note that there is good agreement between the\nobserved and predicted positions of the resonances, which\ndemonstrates that the damping resonances can indeed be\nthought of as “thickness” resonances. The most signifi-\ncant deviation is observed in the [CoPd] 6sample, which\nis the thinnest and therefore most sensitive to changes in\nthe effective thickness at the top and bottom interfaces.\nWe did not observe any thickness resonances for IP mag-\nnetization (shown in the Supplemental Material for the\n[Co/Pd] 11multilayer [19]), which is expected due to the\nfact that the strongest coupling is to phonons propagat-\ning parallel to the static magnetization [11, 20, 21].\nThe temperature dependence of the phonon pumping\ncontribution to the FMR linewidths of the [Co/Pd] 11\nmultilayer is shown in Fig. 3 for temperatures ranging\nfrom 10 to 300 K. The phonon pumping contribution\nis quantified by fitting the full width at half maximum\n(FWHM) FMR linewidths to the form\n∆HFWHM = ∆H0+ 2αω/γ + ∆Hph(ω), (1)\nwhere ∆H0is the frequency-independent inhomogeneous\nbroadening, 2 αω/γ is the contribution from Gilbert\ndamping (αis the Gilbert damping constant and γis\nthe gyromagnetic ratio), and ∆ Hph(ω) is the nonlinear\nfrequency-dependent contribution from phonon pump-\ning. We assume a phenomenological Lorentzian lineshape\nfor the form of ∆ Hph(ω):\n∆Hph(ω) =/summationdisplay\nn2Anδω/2\n(ω−nω0)2+ (δω/2)2,(2)\nwhereδωis the FWHM of the resonance, ω0is the fre-\nquency of the fundamental half-wave resonance, and An\nsets the amplitude (the factor of 2 is needed to convert\nfrom HWHM to FWHM). In the case of the [Co/Pd] 11\nmultilayer, we enforce the constraints that the high fre-\nquency resonance is exactly twice the low frequency res-\nonance and that the amplitudes of both resonances are\nequal. The widths of the resonances are set by the acous-\ntic impedance ratios at the boundaries, so that a strong\nmismatch will yield a sharp resonance [2, 3, 11, 22, 23],\nand should not depend on the order of the resonance.\nThe phonon relaxation rate can also influence the reso-\nnance width [3, 22, 23], but this is probably a secondary\neffect [24].\nIt is clear from Fig. 3 that the intensity of the thick-\nness resonances increases strongly at low temperature\n(whereas the Gilbert damping depends very weakly on\ntemperature, as shown in the Supplemental Material\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s53/s48/s53/s48/s48/s55/s53/s48/s49/s48/s48/s48/s49/s50/s53/s48/s72\n/s70/s87/s72/s77/s32/s40/s79/s101/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s91/s67/s111/s47/s80/s100/s93\n/s49/s49-T \n10 K\n75 K\n150 K\n225 K\n300 KFIG. 3. Evolution of the phonon pumping contribution to the\nFMR linewidths with temperature for the [Co/Pd] 11multi-\nlayer at temperatures of 10 K (blue diamonds), 75 K (green\ntriangles), 150 K (black triangles), 225 K (magenta circles),\nand 300 K (red squares). The vertical dashed lines indi-\ncate the locations of the phonon pumping peaks, 23 GHz and\n46 GHz (at 10 K), which correspond to phonon wavelengths\nofλ= 2dandλ=d, respectively, where dis the thickness\nof the multilayer (excluding capping and seed layers). The\ndata below 300 K are offset vertically so that the individual\ndatasets could be more easily distinguished.\n[19]). The amplitudes of the resonances are about a fac-\ntor of 4 larger at 10 K relative to 300 K. Magnetoelastic\neffects, originating from a magnetic anisotropy energy,\nare expected to increase at low temperature due to a\nreduction of thermal fluctuations of the magnetization\n[10, 25]. A strong temperature dependence of the ampli-\ntude of the resonances is seen for all of the multilayers,\nincreasing at low temperature by a magnitude similar to\nthat seen in Fig. 3 for the [Co/Pd] 11sample. Also note-\nworthy is the small upward shift in the frequency of the\nresonances with decreasing temperature. This is consis-\ntent with the expectation that the elastic moduli should\nincrease at low temperature, causing an increase in the\nspeed of sound (which is proportional to the frequency\nof a given thickness resonance). The frequencies of the\nfirst and second thickness resonances shift from 22 and\n44 GHz to 23 and 46 GHz, respectively.\nThere has been significant theoretical work attempt-\ning to model phonon pumping [11, 13, 21–23, 26–28],\nand so we will not give a comprehensive overview here.\nAll models predict that the phonon pumping amplitude\nshould go as the square of the magnetoelastic coefficient,\nwhich can be understood in terms of Fermi’s golden rule.\nOne of the primary factors influencing the phonon pump-\ning is the nonuniformity of the dynamic magnetization,\nwhich is necessary for exciting acoustic phonons (since\nthey have nonzero wave vector). The model presented by\nStreib et al. [11] assumes uniform magnetization within4\n(a) (b)\n(c) (d)/s48/s55/s53/s49/s53/s48/s50/s50/s53/s51/s48/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s55/s53/s49/s53/s48/s50/s50/s53/s51/s48/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s72\n/s112/s101/s97/s107/s32/s40/s79/s101/s41\n/s91/s67/s111/s47/s80/s100/s93\n/s54/s91/s67/s111/s47/s80/s100/s93\n/s49/s49/s72\n/s112/s101/s97/s107/s32/s40/s79/s101/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s91/s67/s111/s47/s80/s100/s93\n/s49/s53\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s91/s67/s111/s47/s80/s100/s93\n/s50/s48/s48 /s49/s48/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s109/s54 \n/s84/s32/s40/s75/s41/s48 /s49/s48/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s109/s54 \n/s84/s32/s40/s75/s41/s48 /s49/s48/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s109/s54 \n/s84/s32/s40/s75/s41/s48 /s49/s48/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s109/s54 \n/s84/s32/s40/s75/s41\nFIG. 4. Temperature dependence of the peak full width at\nhalf maximum phonon pumping contribution to the FMR\nlinewidth for the (a) [Co/Pd] 6, (b) [Co/Pd] 11, (c) [Co/Pd]15,\nand (d) [Co/Pd] 20multilayers. Insets show m6as a function\nof temperature.\nthe film, with the only nonuniformity coming from the\ndiscontinuity of the magnetization at the interfaces of\nthe film. It is important to note that this model pre-\ndicts the excitation of only odd-integer half-wave reso-\nnances (d=λ/2, 3λ/2, . . . ) due to destructive interfer-\nence at frequencies where the phonons are even-integer\nhalf-waves. Our data show clearly that both even and\nodd resonances are excited, however, which relates to\nthe fact that the dynamic (and static) magnetization in\nthese multilayers is certainly nonuniform. Furthermore,\nthe boundary conditions likely differ at the bounding in-\nterfaces at the top and bottom of the multilayer since\nthe interfaces themselves are different (Ta/Co on top and\nCo/Pd on bottom). The interior of the multilayer also\npromotes nonuniform magnetization due to the nonuni-\nformity inherent in the proximity-induced magnetism in\nthe Pd layers. The differences between our observations\nand the predictions of the model of Streib et al. [11] un-\nderscore the importance of boundary conditions in the\nphonon pumping process, and it is probable that the com-\nplex magnetization depth profile associated with mag-\nnetic multilayers serves to enhance the phonon pumping.\nThe temperature dependence of the phonon pumping\nstrength is expected to be primarily due to the depen-\ndence of magnetostriction on temperature [2, 11]. It\ncan be shown that since the magnetoelastic energy is\nquadratic in the magnetization cosines [20], the magne-\ntoelastic energy should scale with temperature as m3(T)\n[25, 29, 30], where m(T)≡M(T)/M(0) is the reduced\nmagnetization. (It is shown in the Supplemental Mate-\nrial that the interface anistropy energy exhibits m3scal-\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s49/s53/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s40/s98/s41/s72\n/s70/s77/s82/s32/s40/s79/s101/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s84/s32/s61/s32/s49/s48/s32/s75/s91/s67/s111/s47/s80/s100/s93\n/s54\n/s50/s48 /s52/s48\n/s48/s49/s48/s48/s50/s48/s48\n/s72\n/s70 /s87/s72/s77 /s32/s40/s79/s101/s41/s102/s32/s40/s71/s72/s122/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s91/s67/s111/s47/s80/s100/s93\n/s49/s49/s40/s97/s41\n/s50/s48 /s52/s48\n/s48/s49/s48/s48/s50/s48/s48\n/s72\n/s70 /s87/s72/s77 /s32/s40/s79/s101/s41/s102/s32/s40/s71/s72/s122/s41FIG. 5. Shifts in the FMR field as a function of frequency\nat 10 K for the (a) [Co/Pd] 6and (b) [Co/Pd] 11multilayers.\nThe insets in both panels show the corresponding linewidth\nenhancements as a function of frequency at 10 K. The solid\ncurves in the main panels are predictions based on the fits of\nthe linewidths in the insets using Kramers-Kronig relations.\ning as expected [19].) As mentioned earlier, the phonon\npumping amplitude depends on the square of the magne-\ntoelastic energy and would therefore be expected to scale\nwith temperature as m6.\nFigure 4 shows the temperature dependence of the\npeak phonon pumping contribution to the full width\nat half maximum (FWHM) FMR linewidths for all the\nmultilayers. The damping enhancement depends quite\nstrongly on temperature, with the effect being a factor\nof at least 4 greater at low temperature compared to\nroom temperature. This depends on temperature much\nmore strongly than m6(shown in the insets of Fig. 4),\nwhich ranges from /similarequal0.65 to 0.75 at 300 K in the four\nmultilayers. It is unclear what causes the temperature\ndependence of the phonon pumping to be so strong, but\nthere are several factors such as pinning and interlayer\nexchange coupling that also depend on temperature. We\nexpect that the pinning of the dynamic magnetization at\nthe exterior interfaces of the multilayer [Co/Ta on the\ntop and Co/Pd on the bottom, see Fig. 1(a)] becomes\nstronger at low temperature due to an increase in the\ninterfacial anisotropy energy [25]. In addition to the en-\nhanced pinning, the interior Co/Pd interfaces will be af-\nfected by an increase in the interlayer exchange coupling\nat low temperatures that occurs for metallic nonmag-\nnetic spacer layers [31, 32], which may serve to enhance\nthe phonon pumping.\nWe now consider the effect of coupling to phonons on\nthe effective field acting on the dynamic magnetization.\nFigure 5 shows the observed shifts in FMR field as a func-\ntion of frequency for the [Co/Pd] 6and [Co/Pd] 11multi-\nlayers at 10 K. The shifts are quantified by deviations of5\nthe FMR fields from the Kittel dispersion, so that the\nFMR field as a function of frequency is given by\nHFMR =ω/γ−Hk,eff +δHFMR (ω), (3)\nwhereHk,eff is the uniaxial out-of-plane anisotropy\nfield (containing both shape and interface contribu-\ntions, defined here as positive for a PMA material), and\nδHFMR (ω) is the frequency-dependent shift in FMR field\ndue to phonon pumping. The Kramers-Kronig relations\nof linear response theory imply that an absorptive effect,\nhere a resonant enhancement of the FMR linewidths,\nmust be accompanied by a dispersive effect, i.e. a shift\nin the FMR field. Given that the absorptive response\nis a Lorentzian [Eq. (2)], the field shifts caused by the\ndispersive response must be of the form\nδHFMR (ω) =/summationdisplay\nn−Anω−nω0\n(ω−nω0)2+ (δω/2)2,(4)\nwhere the parameters δω,ω0, andAare the same as in\nEq. (2). The solid curves in the main panels of Fig. 5 are\npredictions of the FMR field shifts based on the linewidth\nenhancements (the absorptive response): The parameters\nδω,ω0, andAare determined from fits of the linewidths\nto Eq. (2) [via Eq. (1)], and used to predict the FMR\nfield shifts via Eq. 4 with no free parameters. It can be\nseen from Fig. 5 that there is good agreement between\nthe observed and predicted FMR field shifts. We also\nnote that the twin-peak structure seen in the linewidths\nof the [Co/Pd] 6multilayer [inset of Fig. 5(a)] manifests as\na kink between the two extrema in the FMR field shifts\n[main panel of Fig. 5(a)].\nWe conclude by emphasizing that the Co/Pd multi-\nlayer system is an ideal platform for phonon pumping due\nto the large magnetostriction and PMA, which opens the\npossibility of engineering devices that utilize this effect\nat zero applied field in the advantageous perpendicular\nconfiguration. As we have demonstrated, the frequency\nof the phonon pumping resonance is highly tunable by\nadjusting the number of Co/Pd repetitions—which no-\ntably does not significantly affect the magnitude of the\nPMA. It is therefore feasible to engineer a Co/Pd mul-\ntilayer that experiences a phonon pumping resonance at\nzero external field.\nThis work was supported by SMART, a center funded\nby nCORE, a Semiconductor Research Corporation pro-\ngram sponsored by NIST. Parts of this work were carried\nout in the Characterization Facility, University of Min-\nnesota, which receives partial support from NSF through\nthe MRSEC program, and the Minnesota Nano Cen-\nter, which is supported by NSF through the National\nNano Coordinated Infrastructure Network, Award Num-\nber NNCI - 1542202.[1] H. B¨ ommel and K. Dransfeld, Excitation of Hypersonic\nWaves by Ferromagnetic Resonance, Phys. Rev. Lett. 3,\n83 (1959).\n[2] M. Seavey, Phonon generation by magnetic films, Proc.\nIEEE 53, 1387 (1965).\n[3] R. Weber, Magnon-Phonon Coupling in Metallic Films,\nPhys. Rev. 169, 451 (1968).\n[4] J. V. J¨ ager, A. V. Scherbakov, T. L. Linnik, D. R.\nYakovlev, M. Wang, P. Wadley, V. Holy, S. A. Cavill,\nA. V. Akimov, A. W. Rushforth, and M. Bayer, Picosec-\nond inverse magnetostriction in galfenol thin films, Appl.\nPhys. Lett. 103, 032409 (2013).\n[5] J. V. J¨ ager, A. V. Scherbakov, B. A. Glavin, A. S.\nSalasyuk, R. P. Campion, A. W. Rushforth, D. R.\nYakovlev, A. V. 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Ounadjela, Us-\ning Ferromagnetic Resonance as a Sensitive Method to\nStudy Temperature Dependence of Interlayer Exchange\nCoupling, Phys. Rev. Lett. 73, 336 (1994).Supplemental Material for\n“Anomalous temperature dependence of phonon pumping by ferromagnetic resonance\nin [Co/Pd] nmultilayers with perpendicular anisotropy”\nCONTENTS\nSQUID magnetometry 1\nAbsence of phonon pumping for in-plane magnetization 3\nGilbert damping 3\nInterface anisotropy 3\nReferences 4\nSQUID MAGNETOMETRY\nSuperconducting quantum interference device (SQUID) magnetometry was performed on the [Co/Pd] nmultilayers\nusing a Quantum Design MPMS in the vibrating sample magnetometry (VSM) configuration. Figure S1 shows M-H\nloops for both field in-plane (black points) and perpendicular-to-plane (red points) at temperatures of 10 K (top row)\nand 300 K (bottom row) for the four multilayers. These M-Hloops confirm an out-of-plane easy axis in all of the\nmultilayers.\n/s45/s56 /s48 /s48 /s45/s52 /s48 /s48 /s48 /s52 /s48 /s48 /s56 /s48 /s48 \n/s45/s50 /s48 /s48 /s50 /s48 /s45/s56 /s48 /s48 /s45/s52 /s48 /s48 /s48 /s52 /s48 /s48 /s56 /s48 /s48 \n/s45/s50 /s48 /s48 /s50 /s48 /s45/s50 /s48 /s48 /s50 /s48 /s45/s50 /s48 /s48 /s50 /s48 /s32/s72/s32/s73/s80\n/s32/s72/s32/s80/s80/s77/s32/s40/s101/s109/s117/s47/s99/s99/s41\n/s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41/s77/s32/s40/s101/s109/s117/s47/s99/s99/s41\n/s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41 /s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41 /s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41(a) (b)\n(e)(c) (d)\n(f) (g) (h)T = 10 K \nT = 300 K [Co/Pd]6[Co/Pd]15 [Co/Pd]11 [Co/Pd]20 \n[Co/Pd]6[Co/Pd]11 [Co/Pd]15 [Co/Pd]20 T = 10 K T = 10 K T = 10 K \nT = 300 K T = 300 K T = 300 K \nFIG. S1. (a)–(d) M-Hloops withT= 10 K for field IP (black) and PP (red). (e)–(h) M-Hloops withT= 300 K for field IP\n(black) and PP (red).arXiv:2110.01714v1  [cond-mat.mtrl-sci]  4 Oct 20212\n/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s48 /s55/s53 /s49/s53/s48 /s50/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s48 /s55/s53 /s49/s53/s48 /s50/s50/s53/s32/s109\n/s32/s109/s51\n/s32/s109/s54/s109/s110\n/s32/s109\n/s32/s109/s51\n/s32/s109/s54\n/s32/s109\n/s32/s109/s51\n/s32/s109/s54\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s109/s110\n/s32/s109\n/s32/s109/s51\n/s32/s109/s54\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41(a)\n(c)(b)\n(d)[Co/Pd]6[Co/Pd]11 \n[Co/Pd]15 [Co/Pd]20 \nFIG. S2. Reduced saturation magnetization m≡M(T)/M(0) as a function of temperature raised to powers of 1 (solid line), 3\n(dashed line), and 6 (dotted line) for the (a) [Co/Pd] 6, (b) [Co/Pd] 11, (c) [Co/Pd] 15, and (d) [Co/Pd] 20multilayers\n-T \n10 K\n75 K\n150 K\n225 K\n300 K\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s53/s48/s53/s48/s48/s55/s53/s48/s49/s48/s48/s48/s49/s50/s53/s48/s72\n/s70/s87/s72/s77/s32/s40/s79/s101/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s91/s67/s111/s47/s80/s100/s93\n/s49/s49\nFIG. S3. Ferromagnetic resonance linewidths—with Gilbert damping and inhomogeneous broadening contributions\nsubtracted—as a function of frequency for the [Co/Pd] 11multilayer with magnetization IP at temperatures of 10 K (blue\ndiamonds), 75 K (green triangles), 150 K (black triangles), 225 K (magenta circles), and 300 K (red squares). The solid curves\nare fits to the corresponding FMR linewidths with PP magnetization. The data below 300 K were given a positive vertical\noffset so that the individual datasets could be more easily distinguished.\nSQUID was also used to measure the saturation magnetization as a function of temperature. The reduced saturation\nmagnetization m(T)≡Ms(T)/Ms(0) for the multilayers is shown in Fig. S2, along with the quantities m3andm6.\nThe saturation magnetization at a given temperature was determined by measuring the moment of the sample at four\ndifferent fields (5, 7.5, 10, and 12.5 kOe) at which the sample was saturated for PP applied fields. The moment as a\nfunction of applied field was fit linearly, and the saturation moment was taken as the intercept of that fit since there\nis no diamagnetic or paramagnetic moment at zero applied field. The saturation magnetization was then determined\nby dividing the saturation moment by the volume of the multilayer.3\n/s48/s49/s50/s51/s52\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s50/s51\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s32/s40 /s49/s48/s45/s50\n/s41 /s32/s40 /s49/s48/s45/s50\n/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41 /s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41(a) (b)\n(c) (d)[Co/Pd]6[Co/Pd]11 \n[Co/Pd]15 [Co/Pd]20 \nFIG. S4. Gilbert damping as a function of temperature measured with applied field out of the plane for (a) [Co/Pd] 6, (b)\n[Co/Pd] 11, (c) [Co/Pd] 15, and (d) [Co/Pd] 20multilayers.\nABSENCE OF PHONON PUMPING FOR IN-PLANE MAGNETIZATION\nThe phonon pumping effect is strongly mitigated for in-plane magnetization since the phonons driven by FMR\npropagate parallel to the magnetization (to leading order), and will therefore not propagate into the substrate [S1–S3].\nWe demonstrate this fact in Fig. S3, where the FMR linewidths (with Gilbert damping and inhomogeneous broadening\ncontributions subtracted) of the [Co/Pd] 11multilayer with IP magnetization are shown at different temperatures.\nThe lower frequency limit was set by the field below which the sample was unsaturated. The solid curves are the fits\nobtained from the PP linewidths. The disagreement between the IP FMR linewidths and the solid curves demonstrates\nthe lack of thickness resonances, and therefore phonon pumping, for IP magnetization.\nGILBERT DAMPING\nThe Gilbert damping αdetermines the linear dependence of the FMR linewidths. The FMR FWHM linewidths\nwere used to determine the Gilbert damping through the relation\n∆HFWHM = ∆H0+ 2αω/γ + ∆Hph(ω), (S1)\nwhere ∆H0is the frequency-independent inhomogoneous broadening and ∆ Hph(ω) is the phonon pumping contri-\nbution to the linewidth, the form of which is given in the main text. The Gilbert damping is shown as a function\nof temperature in Fig. S4 for the four multilayers. The Gilbert damping does not show a strong temperature de-\npendence in any of the multilayers, showing a modest increase with temperature in the [Co/Pd] 15multilayer and a\nmodest decrease with temperature in the [Co/Pd] 6, [Co/Pd] 11, and [Co/Pd] 20multilayers.\nINTERFACE ANISOTROPY\nThe interface anisotropy of the multilayers was determined from the relation 2 Ku,int/Ms= 4πMs−Hk,eff, where\nHk,eff is the net perpendicular anisotropy field measured with FMR and Msis the saturation magnetizion determined\nfrom SQUID VSM. The interface anisotropy Ku,int is shown as a function of temperature in Fig. S5 for the four\nmultilayers.\nA uniaxial interface anisotropy is expected to scale with temperature as the cube of the saturation magnetization\n[S4]. The reduced interface anisotropy ku,int≡Ku,int(T)/Ku,int(10 K) is shown as a function of temperature in Fig.\nS6, along with the quantity m3. From Fig. S6 it can be seen that the interface anisotropy shows good agreement with\nthem3scaling law.4\n(a)\n(c)(b)\n(d)[Co/Pd]6[Co/Pd]11 \n[Co/Pd]15 [Co/Pd]20 /s48/s50/s52/s54\n/s48 /s55/s53 /s49/s53/s48 /s50/s50/s53/s48/s50/s52/s54\n/s48 /s55/s53 /s49/s53/s48 /s50/s50/s53/s75\n/s117/s44/s105/s110/s116/s32/s40/s77/s101/s114/s103/s47/s99/s99/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s75\n/s117/s44/s105/s110/s116/s32/s40/s77/s101/s114/s103/s47/s99/s99/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\nFIG. S5. Uniaxial interface anisotropy 2 Ku,int/Ms= 4πMs−Hk,eff as a function of temperature for the (a) [Co/Pd] 6, (b)\n[Co/Pd] 11, (c) [Co/Pd] 15, and (d) [Co/Pd] 20multilayers.\n(a)\n(c)(b)\n(d)[Co/Pd]6[Co/Pd]11 \n[Co/Pd]15 [Co/Pd]20 /s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s48 /s55/s53 /s49/s53/s48 /s50/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s48 /s55/s53 /s49/s53/s48 /s50/s50/s53/s32/s107\n/s117 /s44/s105 /s110 /s116\n/s32/s109/s51 /s107\n/s117/s44/s105 /s110/s116\n/s32/s107\n/s117/s44/s105/s110/s116\n/s32/s109/s51\n/s32/s107\n/s117/s44/s105/s110/s116\n/s32/s109/s51\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s107\n/s117/s44/s105 /s110/s116\n/s32/s107\n/s117/s44/s105/s110/s116\n/s32/s109/s51\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\nFIG. S6. Reduced anisotropy field ku,int≡Ku,int(T)/Ku,int(10 K) (symbols) and m3(dashed lines) as functions of temperature\nfor the (a) [Co/Pd] 6, (b) [Co/Pd] 11, (c) [Co/Pd] 15, and (d) [Co/Pd] 20multilayers.\n[S1] C. Kittel, Interaction of Spin Waves and Ultrasonic Waves in Ferromagnetic Crystals, Phys. Rev. 110, 836 (1958).\n[S2] S. Streib, H. Keshtgar, and G. E. Bauer, Damping of Magnetization Dynamics by Phonon Pumping, Phys. Rev. Lett. 121,\n027202 (2018), arXiv:1804.07080.\n[S3] T. Sato, W. Yu, S. Streib, and G. E. W. Bauer, Dynamic Magnetoelastic Boundary Conditions and the Pumping of\nPhonons, , 1 (2021), arXiv:2104.05992.\n[S4] H. Callen and E. Callen, The present status of the temperature dependence of magnetocrystalline anisotropy, and the\nl(l+ 1)/2 power law, J. Phys. Chem. Solids 27, 1271 (1966)."    },    {        "title": "0807.1555v1.High_temperature_ferromagnetism_in_Co_implanted_TiO2_rutile.pdf",        "content": "High temperature ferromagnetism in Co-implanted\nTiO 2rutile\nNuman Akdogan1;z, Alexei Nefedov2, Hartmut Zabel3,\nKurt Westerholt3, Hans-Werner Becker4, Christoph\nSomsen5, S \u0018afak G ok6, Asif Bashir7, Rustam Khaibullin8;9\nand Lenar Tagirov8;9\n1Department of Physics, Gebze Institute of Technology, 41400 Kocaeli, Turkey\n2Lehrstuhl f ur Physikalische Chemie I, Ruhr-Universit at Bochum, D-44780\nBochum, Germany\n3Institut f ur Experimentalphysik/Festk orperphysik, Ruhr-Universit at Bochum,\nD-44780 Bochum, Germany\n4Institut f ur Physik mit Ionenstrahlen, Ruhr-Universit at Bochum, D-44780\nBochum, Germany\n5Institut f ur Werksto\u000be, Ruhr-Universit at Bochum, D-44780 Bochum, Germany\n6Lehrstuhl f ur Angewandte Festk orperphysik, Ruhr-Universit at Bochum,\nD-44780 Bochum, Germany\n7Lehrstuhl f ur Physikalische Chemie I, Ruhr-Universit at Bochum, D-44780\nBochum, Germany\n8Kazan Physical-Technical Institute of RAS, 420029 Kazan, Russia\n9Kazan State University, 420008 Kazan, Russia\nE-mail: numan.akdogan@ruhr-uni-bochum.de\nAbstract. We report on structural, magnetic and electronic properties of Co-\nimplanted TiO 2rutile single crystals for di\u000berent implantation doses. Strong\nferromagnetism at room temperature and above is observed in TiO 2rutile plates\nafter cobalt ion implantation, with magnetic parameters depending on the cobalt\nimplantation dose. While the structural data indicate the presence of metallic\ncobalt clusters, the multiplet structure of the Co L3edge in the XAS spectra\ngives clear evidence for a substitutional Co2+state. The detailed analysis of the\nstructural and magnetic properties indicates that there are two magnetic phases\nin Co-implanted TiO 2plates. One is a ferromagnetic phase due to the formation\nof long range ferromagnetic ordering between implanted magnetic cobalt ions\nin the rutile phase, and the second one is a superparamagnetic phase originates\nfrom the formation of metallic cobalt clusters in the implanted region. Using x-ray\nresonant magnetic scattering, the element speci\fc magnetization of cobalt, oxygen\nand titanium in Co-implanted TiO 2single crystals are investigated. Magnetic\ndichroism was observed at the Co L2;3edges as well as at the O Kedge.\nThe interaction mechanism, which leads to ferromagnetic ordering of substituted\ncobalt ions in the host matrix, is also discussed.\nPACS numbers: 85.75.-d, 78.70.Dm, 75.50.Pp, 61.72.U-\nSubmitted to: J. Phys. D: Appl. Phys.\nzAuthor to whom correspondence should be addressed.arXiv:0807.1555v1  [cond-mat.mtrl-sci]  9 Jul 2008High temperature ferromagnetism in Co-implanted TiO 2rutile 2\n1. Introduction\nOxide-based diluted magnetic semiconductors (DMSs) have recently attracted\nconsiderable attention because of reports on the room temperature ferromagnetism\n(FM) in several systems and their projected potential for spintronic devices [1, 2].\nSince Matsumoto et al. [3] observed room temperature FM in Co-doped anatase\nTiO 2, much interest has been focused on the titanium dioxide as a host material for\nmagnetic doping. Co-doped TiO 2has been grown by using a wide variety of growth\nmethods, including pulsed laser deposition (PLD) [4{7], laser molecular beam epitaxy\n(LMBE) [8{10], combinatorial LMBE [3, 11], reactive co-sputtering [12, 13], magnetron\nsputtering [14, 15], metal organic chemical-vapor deposition (MOCVD) [16], oxygen\nplasma assisted molecular beam epitaxy (OPA-MBE) [17{19] and as well as the sol-\ngel method [20]. Both the epitaxial TiO 2anatase thin \flm and the single-crystalline\nTiO 2rutile have also been doped by using ion implantation technique [21{27]. In\naddition to di\u000berent growth techniques, di\u000berent substrates such as Al 2O3[28, 29],\nSrTiO 3[4{6, 15, 17{19], LaAlO 3[3, 5, 6, 8, 18], Si [12] and SiO 2/Si [16] have been used\nto synthesize Co-doped TiO 2\flms.\nMany groups have observed room temperature ferromagnetism in Co-doped\nTiO 2for both anatase and rutile phases [2, 3, 6, 12, 13, 22{25, 27, 30, 31]. A Curie\ntemperature of about 650 K [6] and 700 K [22] was reported by di\u000berent groups.\nSubsequent reports have concentrated on the origin of ferromagnetism in this material.\nSpectroscopic studies indicated that cobalt ions in thin TiO 2\flms exist in a +2 formal\noxidation state, consistent with ferromagnetism originating from Co substitution on\nthe Ti site [4]. In other publications it is suggested that the ferromagnetic behavior\nis due to cobalt clustering depending on the growth conditions [3, 8, 18]. Chambers et\nal.[17] reported that the solution of Co in TiO 2is possible at least up to 10% when\nthe TiO 2is deposited on SrTiO 3substrate. However, when the TiO 2\flms grown\non LaAlO 3substrate the solid solution is about 2-7% [3, 6]. Co metal clusters were\nobserved in the as-grown Co-doped TiO 2\flms with a cobalt concentration of 2%.\nPost-annealing of the samples leads to dissolving of clusters in the TiO 2matrix [6].\nFor higher cobalt concentrations, bigger cobalt cluster were reported with a cluster\nsize of about 150 nm [16].\nIf the observed ferromagnetism is actually due to substituted magnetic elements\nin the host matrix, then another important question arises; what is the coupling\nmechanism which leads to ferromagnetism? Recently, we have reported the\nobservation of room temperature FM and in-plane magnetic anisotropy of single-\ncrystalline TiO 2rutile structures after high dose Co implantation [23, 24, 27]. From the\nobservation of the in-plane magnetic anisotropy we concluded that FM in this system\nresults from the incorporation of Co ions in the TiO 2lattice, but a co-existence with\nCo nanoclusters could not be excluded.\nIn order to clarify this situation we studied the structural, magnetic and\nelectronic properties of Co-doped (100)-oriented rutile TiO 2single crystals for di\u000berent\nimplantation doses. The resulting Co:TiO 2samples have been characterized by\nRutherford backscattering spectroscopy (RBS) to obtain the Co depth distribution\npro\fles and by atomic force microscopy (AFM) to check the surface properties after\nimplantation, as well as by x-ray di\u000braction (XRD) and by high resolution transmission\nelectron microscopy (TEM) to reveal the presence of precipitates and metallic Co\nclusters. X-ray absorption spectroscopy (XAS) has also been employed to determine\nwhether the implanted cobalt ions are in the Co2+oxidation state or are in theHigh temperature ferromagnetism in Co-implanted TiO 2rutile 3\nmetallic state. The magnetic properties of TiO 2rutile samples have been investigated\nusing magneto-optical Kerr e\u000bect (MOKE), superconducting quantum interference\ndevice (SQUID) based magnetometry, and x-ray resonant magnetic scattering (XRMS)\ntechniques. In addition, Hall e\u000bect measurements were carried out to verify the\noccurrence of intrinsic ferromagnetism and relate it to the carrier type in the samples.\n2. Sample Preparation\n40 keV Co+ions were implanted into (100)-oriented 15 \u000215\u00021 mm3single-crystalline\nTiO 2rutile substrates (from Moscow Power Engineering Institute in Russia) by using\nthe ILU-3 ion accelerator (Kazan Physical-Technical Institute of the Russian Academy\nof Science) with an ion current density of 8 \u0016A\u0001cm\u00002. The implantation dose varied\nin the range of 0 :25\u00001:50\u00021017ions\u0001cm\u00002. The sample holder was cooled by\n\rowing water during the implantation to prevent the samples from overheating. The\nimplanted plates were cut by a diamond cutter into smaller pieces for structural,\nmagnetic and electronic studies. As a last step, four gold contacts were evaporated on\nthe corners of the samples for Hall e\u000bect measurements. The list of the Co-implanted\nTiO 2samples used in the present study is given in Table 1.\nTable 1. TiO 2samples implanted with 40 keV Co+for di\u000berent Co ion doses.\nSample Dose (\u00021017ion\u0001cm\u00002)\n1 0.25\n2 0.50\n3 0.75\n4 1.00\n5 1.25\n6 1.50\n3. Experimental Results\n3.1. Structural Properties\nIn this section, the structural properties of non-implanted and Co-implanted TiO 2\nrutile plates are presented. The depth distribution of implanted cobalt ions in the\nrutile samples as well as the cobalt concentration for each dose are determined by\nusing the RBS technique. The RBS measurements were carried out at the Dynamic\nTandem Laboratory (DTL) at the Ruhr-Universit at Bochum. Fig. 1 presents the depth\ndependence of the cobalt concentration in Co-implanted TiO 2plates for di\u000berent Co\nion implantation doses. The RBS data show a maximum cobalt concentration of about\n25 at. % for the highest Co dose and it decreases to about 5 at. % for the lowest dose.\nDue to the ion sputtering of the surface during implantation, the maximum slightly\nshifts to the left for higher dose levels. An extended inward tail up to 70 nm due to\ncobalt di\u000busion into the volume of the rutile single crystals is also observed for each\nimplantation dose.\nFig. 2 shows small-angle x-ray re\rectivity taken with synchrotron radiation at\nthe \"Hamburg Synchrotron Radiation Laboratory\" (HASYLAB) with an energy of\nE=8048 eV. The solid line in Fig. 2 is a \ft to the data points for sample 6 (1 :50\u0002\n1017ions\u0001cm\u00002) from Table 2 as is obtained by the commercial software WinGIXA,\nwhich is based on the Parratt formalism [32]. Since the cobalt concentration in theHigh temperature ferromagnetism in Co-implanted TiO 2rutile 4\nFigure 1. The cobalt concentration pro\fle as a function of depth and for di\u000berent\nimplantation doses measured by RBS.\nTiO 2crystals changes with the depth, for \ftting of the re\rectivity data the implanted\narea is sliced into \fve layers. The roughness and electron density values obtained from\nthe \ft are listed in Table 2 for each layer. The model used for \ftting perfectly matches\nthe RBS data and Fig. 3 shows the depth dependence of the cobalt concentration and\nthe normalized electron density ( \u001ae=\u001ae(TiO 2)) obtained from the \ft of re\rectivity. The\nsolid line in Fig. 3 presents the calculated pro\fle using the SRIM ( Stopping and Range\nof Ions in Matter ) software [33], without taking ion sputtering e\u000bects into account.\nTable 2. Fitting parameters of re\rectivity curve for sample 6 (1 :50\u00021017ions\u0001\ncm\u00002).\nLayer Thickness (nm) Roughness ( \u0017A)\u001ae(g\u0001cm\u00003)\n1. layer 11.2 26.87 4.229\n2. layer 11.1 0.50 4.234\n3. layer 12.1 0.71 4.227\n4. layer 14.2 0.09 4.200\n5. layer 20.3 9.49 4.180\nPure TiO 2 | 0.09 4.170\nThe high angle XRD measurements were also carried out at HASYLAB, in order\nto detect possible impurity phases in the samples after implantation. The Bragg scans\nbefore and after implantation with di\u000berent doses are shown in Fig. 4 for (100)-oriented\nTiO 2rutile samples. Increase of the implantation dose up to 1 :50\u00021017ions\u0001cm\u00002\nresults in two additional peaks which correspond to the (10 10) and (0002) re\rections\nof hcp Co. Below the implantation dose of 1 :25\u00021017ions\u0001cm\u00002cobalt nanoclusters\ncannot be detected by x-ray di\u000braction. For every implanted sample a tail, indicated\nby an arrow in Fig. 4, is present around the main peak of the TiO 2(200) re\rection.\nThis tail results from the expansion of the TiO 2lattice upon cobalt implantation andHigh temperature ferromagnetism in Co-implanted TiO 2rutile 5\nFigure 2. Small angle x-ray re\rectivity data and \ft for sample 6 (1 :50\u00021017ions\u0001\ncm\u00002).\nFigure 3. The cobalt concentration (RBS data) and the electron density\ndetermined from the small angel x-ray re\rectivity as a function of depth for sample\n6 (1:50\u00021017ions\u0001cm\u00002). The solid line represents the calculated SRIM pro\fle.\nis not observed before the implantation. In addition, a new peak is present on the low\nangle side which corresponds to the spinel cobalt oxide (Co 3O4) phase reported alreadyHigh temperature ferromagnetism in Co-implanted TiO 2rutile 6\nFigure 4. High-angle Bragg scattering scan for non-implanted (solid line) and\ndi\u000berent dose implanted (100)-TiO 2samples. The presence of Co clusters is\nclearly seen for the highest dose (Sample 6).\nFigure 5. AFM surface topography of (100)-TiO 2rutile after Co ion\nimplantation with a dose of 1 :25\u00021017ions\u0001cm\u00002(Sample 5).\nby Khaibullin et al. in Co-implanted TiO 2[31]. Due to the di\u000berence in etching\nrates of Co and the TiO 2during high dose ion implantation [31], cobalt nanoparticlesHigh temperature ferromagnetism in Co-implanted TiO 2rutile 7\nform on the surface and become oxidized forming antiferromagnetic Co 3O4with a\nNeel temperature of about 40 K. Fig. 5 presents the surface morphology of sample 5\n(1:25\u00021017ions\u0001cm\u00002) probed by AFM (Digital Instruments NanoScope MultiMode\nAFM). The AFM image clearly shows a network of cobalt oxide islands on the surface\nwith a roughness of about 2 :14\u00060:25 nm.\nFigure 6. Cross-sectional TEM images of sample 6 (1 :50\u00021017ions\u0001cm\u00002).\nFor further investigations on the e\u000bects of ion implantation into TiO 2, high\nresolution cross sectional TEM measurements were performed. For the preparation of\nTEM samples, the plates were thinned by focused ion beam (FIB) technique. First, the\nsample surface is covered by a tungsten (W) \flm to prevent charging e\u000bects. Then a\nvery small cross sectional piece of the implanted sample was cut by FIB. Fig. 6 presents\nTEM images of sample 6 (1 :50\u00021017ions\u0001cm\u00002) with an increasing resolution from\n50 nm to 2 nm. In Figs. 6(a) and (b), a general overview of the sample is shown. It\ncan clearly be seen that a surface layer of about 40 nm thickness is strongly damaged\nafter ion bombarding. There are many defects and di\u000berently sized cobalt clusters in\nthis region. However, in Figs. 6(c) and (d) it can be recognized that the structure of\nTiO 2is preserved after implantation. Beneath the surface layer there is another cobalt\nrich layer of about 40 nm thickness. Element speci\fc TEM measurements indicate\nthat the cobalt concentration in this layer is much smaller than in the surface layer in\nagreement with the RBS and x-ray re\rectivity data (Fig. 3).\n3.2. Magnetic Properties\n3.2.1. In-plane magnetic anisotropies and hysteresis measurements In order\nto investigate the in-plane magnetic anisotropy of the implanted samples we used a\nhigh-resolution MOKE setup in the longitudinal con\fguration with s-polarized light\n[34{36]. The MOKE setup allows for a rotation of the sample around its surface\nnormal (by the angle ') in order to apply a magnetic \feld in various in-plane directions\nand thus provide information about the in-plane magnetic anisotropy. The in-plane\nmagnetic anisotropy of the samples doped with di\u000berent doses as determined by the\nMOKE measurements are shown in Fig. 7. For the sample with the highest dose, both,High temperature ferromagnetism in Co-implanted TiO 2rutile 8\nthe remanent Kerr signal normalized to the Kerr signal at saturation ( \u0012rem\nK=\u0012sat\nK) and\ncoercive \feld ( HC), are reduced to almost zero near the hard axis ( '= 0\u000e\u0000180\u000e),\nwhile for the magnetic \feld applied along the easy axis ( '= 90\u000e\u0000270\u000e) they are\nclose to unity. It is evident from Fig. 7 that both \u0012rem\nK=\u0012sat\nKandHCexhibit a strong\ntwo-fold symmetry for the highest dose which decreases with decreasing implantation\ndose.\nFigure 7. Azimuthal dependence of the normalized remanent magnetization\n(left) and the coercive \feld (right) for di\u000berent Co ion doses.\nFigure 8. SQUID hysteresis loops for di\u000berent Co ion doses taken parallel to the\neasy axis at T=300 K.\nHysteresis loops of the Co-implanted TiO 2samples, obtained by using a Quantum\nDesign MPMS XL SQUID magnetometer, are presented in Figs. 8 and 9. Fig. 8\nshows hysteresis curves taken parallel to the easy axis at 300 K. For the highest dose\n(1:50\u00021017ions\u0001cm\u00002) a square-like hysteresis curve is observed with a large coerciveHigh temperature ferromagnetism in Co-implanted TiO 2rutile 9\nFigure 9. SQUID hysteresis loops for Co-implanted TiO 2samples measured at\n5 K along the easy axis.\n\feld ofHC=950 Oe. A rather sharp magnetization reversal takes place for this sample\nwith a small step at 260 Oe. For the samples implanted with intermediate ion doses\n(1:00\u00001:25\u00021017ions\u0001cm\u00002), the recorded M\u0000Hloops also show hysteretic behavior,\nbut the coercive \felds decrease signi\fcantly. The remanent magnetization normalized\nto the saturation magnetization also decreases for the intermediate doses. At 5 K the\ntwo step feature in the hysteresis curve is not only present for the highest dose but\nalso for intermediate doses (Fig. 9). The low dose implanted samples exhibit at low\ntemperatures a typical superparamagnetic behavior with a pronounced paramagnetic\ncontribution to the hysteresis curves .\nIn order to further investigate the e\u000bect of cobalt clusters, we have performed\ntemperature dependent magnetization ( M\u0000T) measurements using a SQUID\nmagnetometer. Fig. 10 presents \feld cooled (FC) and zero \feld cooled (ZFC) plots for\neach sample. For ZFC measurements, the samples are cooled in zero \feld to 5 K and\nthe magnetization is recorded during warming up to 390 K in an applied \feld of 100 Oe\nparallel to the \flm surface. For FC measurements the applied \feld of 100 Oe is kept\nduring cooling to 5 K and the magnetization is recorded during \feld warming with\nthe same \feld value. The FC (black squares) and ZFC (grey circles) curves diverge\nsubstantially for all doses and the peak in the ZFC curve progressively shifts to higher\ntemperatures with increasing cobalt concentration. This behavior is not expected\nfor a ferromagnet and suggests the presence of magnetic cobalt nanoparticles in the\n\flms or a spin-glass like nature of the system [37, 38]. The M\u0000Tcurve of sample 1\n(0:25\u00021017ions\u0001cm\u00002) is rather unusual and may be attributed to the coexistence of\na weak ferromagnetic and a superparamagnetic phase with a transition temperature\nof about 30 K. The M\u0000Tcurves for \flms with higher cobalt concentrations\n(0:50\u00000:75\u00021017ions\u0001cm\u00002) indicate the occurrence of superparamagnetism with a\nblocking temperature of about 100 K and 250 K for sample 2 (0 :50\u00021017ions\u0001cm\u00002)High temperature ferromagnetism in Co-implanted TiO 2rutile 10\nFigure 10. FC (black squares) and ZFC (grey circles) magnetization curves of\nCo-implanted TiO 2rutile samples taken using a SQUID magnetometry.\nand sample 3 (0 :75\u00021017ions\u0001cm\u00002), respectively. The temperature dependence of\nsample 4 is similar to that of sample 2 and sample 3 except the blocking temperature\nis much higher, namely above 390 K. It is also important to note that the FC curve\nof this sample shows a more or less continuous behavior versus temperature which is\ntypical for ferromagnets. The reported room temperature ferromagnetism with a two\nfold in-plane magnetic anisotropy in this sample indicates that for this dose substituted\ncobalt ions start to interact ferromagnetically. The FC and ZFC curves of sample 5\n(1:25\u00021017ions\u0001cm\u00002) and sample 6 (1 :50\u00021017ions\u0001cm\u00002) are much closer to each\nother. This progression indicates that the ferromagnetic phase becomes dominant in\nthese samples. The observation of a two component hysteresis at RT for sample 6\nsupports this argument. Small peaks in the ZFC curves at low temperatures of these\nsamples indicate the existence of superparamagnetic cobalt clusters. These clusters\nare also clearly seen in the TEM images of sample 6 (Fig. 6).\n3.2.2. XRMS and XAS measurements To shed more light on the origin of\nroom temperature ferromagnetism in Co-doped TiO 2, the magnetic properties of Co-\nimplanted TiO 2rutile \flms have also been investigated using the XRMS and XASHigh temperature ferromagnetism in Co-implanted TiO 2rutile 11\ntechniques. Both the XRMS and XAS experiments were carried out at the undulator\nbeam lines UE56/1-PGM and UE52-SGM at BESSY II (Berlin, Germany) using the\nALICE di\u000bractometer [39]. The di\u000bractometer comprises a two circle goniometer and\nworks in horizontal scattering geometry. A maximum magnetic \feld of \u00062700Oe\ncan be applied in the scattering plane along the sample surface either parallel or\nantiparallel to the photon helicity, which corresponds to the longitudinal magneto-\noptical Kerr e\u000bect (L-MOKE) geometry. The magnetic contribution to the scattered\nintensity (XRMS) was always measured by reversing the magnetic \feld direction while\nkeeping the photon helicity \fxed. Thus, by tuning the energy to the Co L3absorption\nedge (780 eV), re\rectivity scans were taken and the magnetic splitting for plus and\nminus \feld was clearly seen (presented in Ref. [25]). As a compromise between high\nscattering intensity and high magnetic sensitivity for the investigation of the magnetic\nproperties via energy scans at the Co Ledges, the scattering angle was \fxed at the\nposition of 2 \u0012= 8:2\u000e(the angle of incidence \u0012= 4:1\u000e) [25]. For measurements at the\nOKedge (E=535 eV) the scattering angle was \fxed at 2 \u0012= 12\u000e, which corresponds\nto the same scattering vector in the reciprocal space.\nFigure 11. Dose dependence of the asymmetry ratio at the Co L2;3edges\nmeasured at saturation \feld.\nFirst, we measured the energy dependence of the scattered intensity (XRMS)\naround the Co L2;3edges. Since the magnetic contribution to the resonant scattering\ncan best be visualized by plotting the asymmetry ratio ( Ar= (I+\u0000I\u0000)=(I++I\u0000)),\nin Fig. 11 we present the dose dependence of the asymmetry ratio at the Co\nLedges measured in saturation at room temperature. Only the lowest dose of\n0:25\u00021017ions\u0001cm\u00002is measured at 30 K. The magnetization of the samples decreases\nby decreasing the Co ion dose in agreement with SQUID hysteresis curves and previous\nMOKE measurements [27]. It is important to note that the \fne structure around theHigh temperature ferromagnetism in Co-implanted TiO 2rutile 12\nFigure 12. X-ray absorption spectra of sample 6 measured at the Co L2;3edges,\ndetermined by total electron yield. \u001b+and\u001b\u0000denote the right and left circular\npolarization of the incident light, respectively.\nCoL3edge, which is clearly seen in the asymmetry ratio in Fig. 11 for sample 6,\nis not typical for metallic cobalt. It is well known that in the case of metallic \flms\nthe absorption spectra around the L3peak of Co consists of a single component [40].\nThis \fne structure of the Co L3peak is similar to that observed before for CoO \flms\n[40], and it is indicative of a Co2+state. To make this situation more clear and to\nrecord comparable results with previous reports [8], we performed XAS experiments.\nThe absorption data were taken by the total electron yield (TEY) method, i.e. by\nmeasuring the sample drain current. Since the external magnetic \feld changes the\nexcited electron trajectories, the XAS spectra were measured with \fxed photon helicity\nin remanence. The angle of incidence was set again to 4 :1\u000ewith respect to the surface.\nThe absorption spectra were normalized to the incoming photon \rux measured from\nthe beam line mirror. The averaged x-ray absorption spectra ( \u001b++\u001b\u0000)/2 at the Co\nL2;3edges is shown in Fig. 12. The XAS spectrum clearly shows a multiplet structure\nat the CoL3edge. This multiplet structure is a strong indication for the Co ions\nbeing in the Co2+state in this sample.\nThe magnetic signal at the Ti L2;3and the O Kedges was also investigated for\nsample 6. Fig. 13 and Fig. 14 show the corresponding asymmetry ratios. Within\nthe sensitivity limit no magnetic signal could be recorded for Ti and O at room\ntemperature. However, at the O Kedge, a small but clearly visible magnetic signal was\nobserved at T=30 K [25]. It should be mentioned that the oxygen polarization has also\nbeen observed for samples with lower dose (dose levels of 1 :00\u00001:25\u00021017ions\u0001cm\u00002).\nFor samples implanted with doses below 1 :00\u00021017ions\u0001cm\u00002, the magnetic signal\nat the O Kedge is below the sensitivity limit of the experimental setup.High temperature ferromagnetism in Co-implanted TiO 2rutile 13\nFigure 13. Asymmetry ratio measured at the Ti L2;3edges for sample 6.\nFigure 14. Asymmetry ratio taken at the O Kabsorption edge for sample 6.\n3.2.3. High temperature magnetization experiments In order to determine\nthe Curie temperatures ( TC) of Co:TiO 2samples, we have carried out thermo-\nmagnetic measurements using the Faraday balance technique [41] by heating the\nsamples from 100 K (ZFC) up to 1000 K with a rate of 100 K/min in air and\nat an applied \feld of 2 kOe. In Fig. 15 we show the magnetization curves forHigh temperature ferromagnetism in Co-implanted TiO 2rutile 14\nTiO 2rutile plates implanted with di\u000berent doses. The sample with a dose of\n0:50\u00021017ions\u0001cm\u00002shows the magnetic/non-magnetic transition temperature\nat about 850 K (curve 1). For the samples 4 (1 :00\u00021017ions\u0001cm\u00002) and 6\n(1:50\u00021017ions\u0001cm\u00002), two magnetic ordering temperatures of TC1\u0018700 K and\nTC2\u0018850 K, were observed (curves 2 and 3). This shows that two ferromagnetic\nphases, a \\low temperature\" and a high-temperature\" phase, coexist in these samples.\nThe contribution to the magnetization from the high-temperature phase decreases\ngradually with increasing the cobalt implantation dose. Finally, for the sample with\nthe highest dose of 1 :50\u00021017ions\u0001cm\u00002, the low-temperature phase dominates, while\nthe high-temperature phase practically disappears (curve 3).\nFigure 15. High temperature magnetization curves measured in a \feld of 2000\nOe for TiO 2rutile samples implanted by cobalt ions with di\u000berent doses.\nIt should be stated here that the high temperature magnetization curves presented\nin Fig. 15 are irreversible, i.e. on cooling down the ferromagnetic signal disappears.\nFrom this we infer that some di\u000busion process or recrystallization may occur, which\nshould not be confused with a real Curie temperature. Furthermore, the magnetization\nversus temperature does not follow the shape of a usual order parameter. On the other\nhand, after vacuum annealing the samples at high temperatures [31], TC1reappears\nbut notTC2, indicating that TC1is more intrinsic than TC2. This point clearly needs\nsome further investigations. For the present purpose, it is important to note that a\nstable ferromagnetic phase exists at room temperature and far beyond, which may be\nvery useful for high temperature applications of DMSs.High temperature ferromagnetism in Co-implanted TiO 2rutile 15\n4. Hall e\u000bect measurements\nThe observation of an anomalous Hall e\u000bect (AHE) is suggested to be one of the\nimportant criteria for DMS materials to be intrinsically ferromagnetic [42, 43]. In\nthe past, several groups reported the AHE in highly reduced TiO 2\flms doped with\neither with Co or Fe, from which they infer the possibility of intrinsic ferromagnetism\nin these samples [7, 9, 44]. However, recently, Shinde et al. [45] reported the co-\noccurrence of superparamagnetism and AHE in highly reduced Co-doped TiO 2rutile\n\flms, raising questions about the usefulness of the AHE as a test of the intrinsic\nnature of ferromagnetism in DMS materials without a detailed characterization of the\nsample.\nIn magnetic materials, in addition to the ordinary Hall e\u000bect (OHE), there is\nan additional voltage proportional to the sample magnetization [46], the so-called\nanomalous Hall e\u000bect. Hence, the Hall voltage can be written as follows [46],\nVH=\u0010R0I\nt\u0011\nHcos\u000b +\u0010RA\u00160I\nt\u0011\nMcos\u0012; (1)\nwheretis the \flm thickness and Iis the current. R0andRAare the ordinary\nand anomalous Hall e\u000bect coe\u000ecients, respectively. \u00160is the permeability of free\nspace.\u000bis the angle between the applied magnetic \feld ( H) and the \flm normal. \u0012\nis the angle between the sample magnetization ( M) and the sample normal. The \frst\nterm in Eq. 1 is the ordinary Hall e\u000bect and arises from the Lorentz force acting on\nconduction electrons. This establishes an electric \feld perpendicular to the applied\nmagnetic \feld and to the current. The second term is the anomalous Hall e\u000bect and\nit is conventionally attributed to spin dependent scattering mechanism involving a\nspin-orbit interaction between the conduction electrons and the magnetic moments\nof the material. At low applied magnetic \felds, the Hall voltage ( VH) is dominated\nby the magnetic \feld dependence of the sample magnetization M. When the applied\nmagnetic \feld is high enough to saturate the sample magnetization, the magnetic \feld\ndependence of the Hall voltage becomes linear due to the ordinary Hall e\u000bect.\nThe Hall e\u000bect measurements were carried out at 4.2 K using a van der Pauw\ncon\fguration presented in Fig. 16 as an inset. In spite of the fact that the structural\nand the magnetization measurements indicate the presence of magnetic nanoparticles\nin the Co-implanted TiO 2\flms, the anomalous Hall e\u000bect is observed for these\nsamples. The Hall e\u000bect data of sample 6 are shown in Fig. 16. As it is explained\nabove, a rapid increase in the Hall voltage at low \felds can be interpreted as an AHE\nwhich is followed by a slow decrease corresponding to the ordinary Hall e\u000bect. It\nis important to note that the negative slope of the high \feld data indicates n-type\ncarriers in Co-implanted TiO 2rutile. The electron density (n), calculated from the\nslope of the curve at higher \felds, is about 3 :75\u00021018cm\u00003.\n5. Discussion\nThe origin of the observed two magnetic phases (ferromagnetism and superparamag-\nnetism) in Co-implanted TiO 2rutile plates is attributed to the formation of two cobalt\nenriched layers with di\u000berent cobalt concentrations and valence states of the cobalt\ndopant. The TEM images (Fig. 6) clearly show that nanosize magnetic particles of\ncobalt metal nucleate in the surface region of the implanted rutile where the cobalt\nconcentration is maximal (see RBS data in Fig. 1). Mostly beneath this layer, in theHigh temperature ferromagnetism in Co-implanted TiO 2rutile 16\nFigure 16. Hall e\u000bect data of sample 6 taken at 4.2 K. Inset shows the\ngeometry of the Hall e\u000bect experiments. His the external magnetic \feld applied\nperpendicular to the \flm surface.\ntail of the depth pro\fle, the implanted cobalt can exist in an ionic state substituting\nthe Ti4+ions in the matrix by Co2+ions. Thus for charge neutrality either two Co2+\nions substitute for one Ti4+ion, or one Co2+ion and one oxygen vacancy are formed.\nAt the lowest dose (0 :25\u00021017ions\u0001cm\u00002) the magnetic contribution from the metallic\ncobalt clusters and the substituted cobalt ions is very small, and hence the MOKE sig-\nnal at room temperature is rather weak. Increasing the cobalt implantation dose leads\nto both, an increase of the Co cluster size, as seen by the increasing blocking temper-\nature, and to more substitutional cobalt in the Co2+state. At certain concentrations\nthe substituted cobalt ions start to interact leading to ferromagnetism at room tem-\nperature in sample 4 (1 :00\u00021017ions\u0001cm\u00002) and sample 5 (1 :25\u00021017ions\u0001cm\u00002). At\ndoses higher than 1 :25\u00021017ions\u0001cm\u00002strong ferromagnetic order is formed due to\nthe ion accumulation and indirect exchange interaction between the Co2+ions. How-\never, TEM images, peaks in the ZFC curves, and two component hysteresis curves for\ndose levels 1 :25\u00001:50\u00021017ions\u0001cm\u00002indicate that the superparamagnetic phase\nis present in these samples and coexists with the ferromagnetic phase.\nRevealing the interaction mechanism of substituted cobalt ions which leads to\nferromagnetism in Co:TiO 2is also an important incentive of this study. Since the\nXAS spectra clearly show the multiplet structure of the Co L3peak (see Fig. 12), it\nis certain that some portion of the implanted cobalt ions in TiO 2rutile are in the\nCo2+oxidation state. When TiO 2is doped with cobalt ions, simultaneously oxygen\nvacancies are also expected to be produced [47]. The observation of the AHE in the\nCo-implanted rutile samples give clear evidence for oxygen vacancies which contribute\nto shallow donor states in TiO 2and increase the carrier density [48]. It was suggestedHigh temperature ferromagnetism in Co-implanted TiO 2rutile 17\nthat these oxygen vacancies strongly promote ferromagnetism in Co-implanted TiO 2\n\flms by an indirect exchange of substituted cobalt ions through electrons trapped by\nneighboring oxygen vacancies [49].\nWe have also noticed a clear polarization of the oxygen p-orbitals in Co-implanted\nTiO 2rutile samples. The shape of the hysteresis curve and the coercive \feld measured\nat the O Kedge is the same as the one recorded at the Co L3edge [25]. This is a clear\nindication that oxygen ions, which are close to Co ions in TiO 2become magnetically\npolarized. Whether the oxygen polarization is essential for supporting ferromagnetic\nexchange is presently not clear.\nAnother important result of this study is the observation of an anomalous Hall\ne\u000bect. The AHE is often taken as an evidence that the charge carriers are polarized\nand that the material is a true DMS. However, after simultaneous observation of\nsuperparamagnetism and AHE in Co-doped TiO 2\flms by Shinde et al. [45] and also\nin this study, the existence of an AHE can be thought of a necessary measurement\ncondition but it is not su\u000ecient by itself to claim the intrinsic nature of ferromagnetism\nin a DMS material.\n6. Summary and conclusions\nIn conclusion, we have studied in detail the structural, magnetic and electronic\nproperties of Co-implanted TiO 2rutile \flms for di\u000berent implantation doses. The\nstructural data clearly show that cobalt clusters are present in the samples after high\ndose cobalt ion implantation. In addition to the cluster formation, substitution of\ncobalt ions into the rutile lattice is also con\frmed by XAS experiments. The origin\nof the observed magnetic behavior in the samples is explained by the coexistence\nof two di\u000berent magnetic phases. Cobalt nanoparticles in the surface layer form a\nsuperparamagnetic phase in the samples implanted with low and intermediate doses.\nIn addition, substitution of Ti4+ions by Co2+ions leads to intrinsic ferromagnetism as\na second magnetic phase. The oxygen vacancies formed by ion implantation provide\ncharge compensation and serve as mediators for the exchange interaction between\nthe Co2+ions in high dose doped samples. The observation of the anomalous Hall\ne\u000bect in Co-implanted TiO 2rutile can also be thought of an important indicator\nfor the observed long range ordered intrinsic ferromagnetism in the rutile phase. At\nthe highest dose, a strong ferromagnetic phase exists with a Curie temperature of\nabove 700 K. This ferromagnetic phase exhibits a perfect uniaxial in-plane magnetic\nanisotropy following exactly the crystal symmetry of the TiO 2rutile. We consider\nthis as the strongest experimental evidence for the intrinsic nature of ferromagnetism\nin the Co-doped TiO 2rutile.\n6.1. Acknowledgments\nWe wish to acknowledge A. Kr oger for preparation of TEM samples, and O. Seeck\nand W. Caliebe (HASYLAB) for their assistance with the beamline operation. This\nwork was partially supported by BMBF through Contracts Nos. 05KS4PCA (ALICE\nChamber) and 05ES3XBA/5 (Travel to BESSY), by DFG through SFB 491, and\nby RFBR through the grant Nos 07-02-00559-a and 04-02-97505-r. N. Akdogan\nacknowledges a fellowship through the IMPRS-SurMat.High temperature ferromagnetism in Co-implanted TiO 2rutile 18\nReferences\n[1] Matsukura F, Ohno H and Dietl T 2002 III-V Ferromagnetic Semiconductors, in Handbook of\nMagnetic Materials, edited by K. H. J. Buschow (Elsevier)\n[2] Janisch R, Gopal P and Spaldin N A 2005 J. Phys.: Condens. 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Rev. 1301711\n[49] Coey J M D, Venkatesan M and Fitzgerald C B 2005 Nature Materials 4173"    },    {        "title": "2402.04715v1.Anatomy_of_localized_edge_modes_in_laterally_coupled_waveguides.pdf",        "content": "Anatomy of localized edge modes in laterally coupled waveguides\nVadym Iurchuk,a)Sven Stienen, J¨ urgen Lindner, and Attila K´ akay\nInstitute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,\nGermany\n(Dated: 8 February 2024)\nWe present a systematic micromagnetic study of standing spin-wave modes in infinitely long Permalloy strips\nwith rectangular cross-section. Using a finite-element dynamic-matrix method, we first calculate the eigen-\nfrequencies and the corresponding eigenvectors (mode profiles), as a function of the in-plane magnetic field\napplied across the strip. The ferromagnetic resonance spectra is computed from the mode profiles, assuming\na homogeneous radio-frequency excitation, equivalently to an experimental ferromagnetic resonance measure-\nment. The investigation of the field-dependent mode profiles enables for the classification of the observed\nresonances, here focusing mostly on the true edge mode localized at the vicinity of strip edges. Furthermore,\nwe study the mode localization in pairs of 50-nm-thick Permalloy strips as a function of the strip width and\ntheir lateral separation. For closely spaced strips, the spatial profile of the quasi-uniform mode is substantially\nmodified due to a significant hybridization with the edge-localized standing spin-wave modes of the neigh-\nbouring strip. We show that a wide-range-tunability of the localized edge-mode resonances can be achieved\nwith a precise control of the magnetostatic coupling between the strips. Extreme sensitivity of the edge mode\nfrequency on the bias field demonstrates a potential of the edge resonances for field sensing. Furthermore,\nfor narrow strips ( ≈100 nm in width), due to the reduced number of the allowed confined modes, a field-\ncontrollable switching between the resonances localized either in the strip center or at the edges of the strips\ncan be achieved.\nSpin-wave conduits based on ferromagnetic microstrips\nconstitute elementary building blocks of the magnonic\narchitectures for spin-based information transport and\nprocessing1. Numerous studies revealed a rich dynam-\nical spin-wave mode spectrum hosted in confined ferro-\nmagnetic microstructures2–6. Arrays of such microstruc-\ntures can act as magnonic crystals with magnetic-field\ncontrollable excitation bandwidths7–11. Among numer-\nous resonances, which can be excited in ferromagnetic\nstrips, edge modes attract a particular attention due to\ntheir localized nature, allowing for spin-wave propaga-\ntion in extremely narrow channels4,6,12, which is benefi-\ncial for spin-wave based logic devices. Recently, a study\nof the quasi-uniform non-propagating modes in arrays of\nferromagnetic strips with various width and separation\nbetween the strips showed a significant impact of the\nmagnetostatic coupling on the spin-wave frequencies11.\nFurthermore, it was also shown that the edge-localized\nresonances (magnetization dynamics) can also be tuned\nin a wide frequency range by changing the distance be-\ntween the neighboring ferromagnetic strips6.\nIn this manuscript, we focus on the behavior of the\nlocalized standing spin waves (SSW) in isolated 50-nm-\nthick Permalloy (Py) strips and strip pairs. We study the\nedge mode localization as a function of a bias magnetic\nfield for different strip widths and separation distances\nbetween the strips. We show that for reduced gaps be-\ntween the strips, the increased magnetostatic coupling\nlargely modifies the resonance fields/frequencies of the\nmodes excited at the inner edges of the strip pair. Fur-\nthermore, we show that the presence of the closely spaced\na)Corresponding author’s e-mail: v.iurchuk@hzdr.deneighboring strips considerably alters the quasi-uniform\nmode profile, manifesting in a mode localization redistri-\nbution and to a significant hybridization with the edge-\nlocalized standing spin-wave modes. Finally, we show the\nmode distribution in the strip pairs with reduced width,\nallowing for a precise confinement of the excited quasi-\nuniform and localized resonances within the cross-section\nof the strip.\nThe study is performed with the TetraX finite-\nelement package13,14, that allows to calculate numerically\nthe eigenvalues (the resonance frequencies) and the cor-\nresponding eigenvectors (the mode profiles) in infinitely\nlong Py strips with rectangular cross section, among\nthe variety of geometries offered by the program library.\nThe strip cross-section is discretized into a regular tri-\nangular mesh with an average cell size of 5 nm [see\nFig. 1(a)]. We used the following magnetic parameters\nfor Py: saturation magnetization Ms= 796 kA/m, ex-\nchange constant Aex= 13 pJ/m, gyromagnetic ratio γ=\n1.76×1011rad/Ts and Gilbert damping constant αG=\n0.008. For all simulations, the static magnetic field Hx\nis applied in-plane along x-direction, i.e. across the strip\nwidth, while the out-of-plane direction is considered to\nbe the Oy[see Fig. 1(a)]. For each Hxvalue, we relax\nthe magnetic state and solve the eigenvalue problem, to\nget the mode profiles and their frequencies. The mode\nprofiles are used to compute the ferromagnetic resonance\n(FMR) absorption spectra assuming a homogeneous out-\nof-plane radio-frequency field hrf(for details see15).\nFig. 1(b) shows the calculated frequency-sweep FMR\nabsorption map of the infinitely long 1 µm wide and\n50 nm thick Py strip as a function of the in-plane mag-\nnetic field Hxapplied across the strip [see Fig. 1(a)]. A\nset of confined spin-wave modes is observed correspond-\ning to the different spin-wave resonances at given mag-arXiv:2402.04715v1  [cond-mat.mes-hall]  7 Feb 20242\n-500 -250 0 250 500-0.45-0.30-0.150.0060 mT\nmz (arb.u.)n=4\nx (nm)100 mT\n150 mT\n192 mT\n250 mT\n297 mT\n400 mT\n600 mT\n-500 -250 0 250 50060 mTn=3\nx (nm)100 mT\n150 mT\n192 mT\n250 mT\n297 mT\n400 mT\n600 mT\n-500 -250 0 250 50060 mTn=2\nx (nm)100 mT\n150 mT\n192 mT\n250 mT\n297 mT\n400 mT\n600 mT\n-500 -250 0 250 500-0.45-0.30-0.150.0060 mT\n600 mT400 mT297 mT250 mT192 mT150 mTmx(arb.u)\nx (nm)n=1\n100 mT\n0100 200 300 400 500 600 700010203040\n0Hx (mT)f (GHz)\n-120-60060120Re(arb. units)hrf\nxy\nz(a)\n50 nm\n1 µm\n 5 nmHx(b)\nFig. 1new (33x19)\n(g) (h)\n-500 -250 0 250 5000.000.250.500.751.00mx \nx (nm) 48 mT\n 60 mT\n 100 mT\n 150 mT\n 192 mT\n 250 mT\n 400 mT\n-500 -250 0 250 500-0.12-0.08-0.040.00mz (arb. u.)\nx (nm)100 mT\n150 mT\n192 mT\n250 mT\n297 mT\n400 mT\n600 mTquasi-uniform(c) (d)\n(f) (e)\nFIG. 1. (a) Schematics of the simulation geometry. The bias in-plane field Hxis applied across the strip. The FMR spectra\nis computed assuming a homogeneous rf field along Oy, the out-of-plane direction. (b) Frequency-field FMR absorption map\nof infinitely long Py strip with 1 µm×50 nm cross section. (c) x-component of the equilibrium magnetization across the strip\nwidth calculated for different Hx. (d) Dynamical profiles of the quasi-uniform Kittel mode for the given values of the bias\nmagnetic field Hx. Here and further, we plot the dynamical mzcomponents taken in the middle of the strip cross section\n(y=0). (e–h) Dynamical mode profiles of the first four standing spin-wave modes ( n⩽4) for different Hx.\nnetic field. The mode with the largest amplitude is the\nquasi-uniform mode or Kittel mode16with a typical hard-\naxis behavior originating from the presence of the shape\nanisotropy in the strip confined along the x-direction.\nThe lowest-frequency (lowest-energy) mode for a given\nHxcorresponds to the so-called true edge mode3,6,17, with\nthe magnetization precession localized at the very edge\nof the strip above saturation [Fig. 1(e)], as opposed to\nthe higher order edge resonances. The frequency band\nbetween the true edge and the quasi-uniform mode is oc-\ncupied by numerous standing spin-wave modes. For these\nmodes, as seen in Figs. 1(f–h), the spatial magnetization\ndynamics shifts from the edges towards the strip center,\ngradually spreading across the strip width with increased\nmode number.\nFigs. 1(e–h) show the mode profiles of the first four\nobserved modes as a function of increased static field.\nIn the absence of a static external field, the waveguide\nis magnetized at equilibrium in the direction of its long\nits axis ( Oz). For already a field of about 48 mT, corre-\nsponding to the bulk saturation field , the magnetization\nin the middle of the strip will rotate into the direction\nof the field [see Fig. 1(c)]. In Fig. 1(b), this is the point\nwhen the lowest mode has its first frequency minimum.\nWhen increasing Hxfrom 48 to 192 mT, still followingFig. 1(b), the resonance frequency of the lowest mode\n(n=1) increases. During this field increase, as seen in\nthe Fig. 1(e), the dynamic regions (or mode amplitudes)\ncontinuously shift towards the opposite strip edges, yet\nthe modes become more and more confined in width. The\nobserved gradual confinement together with the pinning\nof the modes at the edges (where the amplitude is close\nto zero) leads to an increased exchange interaction con-\ntribution to the mode energy. From the other side, in\nthis field region, one can see that the magnetization at\nthe edges continuously rotates in the direction perpen-\ndicular to the edges [see Fig. 1(c)] to reduce the Zeeman\nenergy. However, this results in a continuously increas-\ning demagnetizing field, opposed to the static external\nmagnetic field. Apparently, at about 192 mT, the ex-\nchange contribution to the internal fields (from the edge-\nconfinement) reaches it’s maximum, but the demagne-\ntizing field will further increase with the static magne-\ntization pointing more and more in the direction of the\nexternal field. Moreover, the pinning of the edge modes\nis gradually released, with increased external field, re-\nsulting in a macrospin like dynamics at the edges, from\nthe exchange point of view. Further increasing the ex-\nternal field in the range of 192 mT < H x<297 mT, the\nresonance frequency decreases and shows a well-known3\nFig.2new (33x8)\n0100 200 300 400 500 600 700010203040\n0Hx (mT)f(GHz)w = 1 m; d = 100 nm \n0 200 400 600 800 10000100200300400\n outer edge\n inner edgeedge mode onset (mT)\nd (nm)\n0100 200 300 400 500 600 700010203040\nw = 1 m; d = 20 nm\n0Hx (mT)f(GHz)\n0100 200 300 400 500 600 700010203040\nw = 1 m; d = 5 nm\n0Hx (mT)f(GHz)\n-1200120Re(arb.u.)\n(a) (b) (c) (d)\nFIG. 2. (a–c) Frequency-field FMR absorption maps of the pair of 1- µm-wide and 50-nm-thick Py strips separated by 100 nm\ngap (a), 20 nm gap (b) and 5 nm gap (c). (d) Edge saturation fields, corresponding to the onset of the true edge modes localized\nat the outer (black squares) and inner (red circles) edges of the strip pair as a function of the gap between the strips.\ndipole-dominated non-aligned edge mode behaviour [see\nFig. 1(c)]. The edge saturation eventually occurs at Hx\n= 297 mT, the second minimum of the lowest mode, and\ndue to the zero internal fields the frequency goes down to\nzero, showing a mode softening or representing a so-called\nGoldstone mode. Above saturation, for Hx>297 mT,\nthen=1 resonance is the true edge mode, localized in\na narrow ( ≈15–20 nm) region at the strip edges. The\ntrue edge mode is also a Kittel-like mode, therefore the\nfurther increase of the bias field Hxdoes not impact the\nmode localization and leads only to a frequency increase\n[see Fig. 1(b)] according to the Kittel (approximately lin-\near for large fields) relation16. We note, that for the field\nrange 48 mT < H x<192 mT, the quasi-uniform mode is\nhybridized by the higher-order non-aligned modes, which\nleads to the distinct spatial modulation of its mode pro-\nfile amplitude [see Fig. 1(d)].\nHigher order standing spin-wave modes ( n⩾2) are\ncharacterized by an increased number of nodal points\nand thus the dynamics extends over a larger volume com-\npared to the n=1 mode. They exhibit similar behavior,\ni.e. while nucleated at the strip center as exchange-\ndominated modes, their intensity maxima gradually\npropagate towards the strip edges with increased Hx\n[Figs. 1(d–f)] as a result of the increased dipolar con-\ntribution to the spin-wave energy.\nNow that we discussed the main features of the FMR\nspectra related to the edge modes, we extend our study\nto a pair of Py strips separated laterally by a gap with a\nwidth of d. Figs. 2(a–c) show the frequency-field absorp-\ntion map of a pair of 1 µm wide and 50 nm thick strips\nseparated by 100, 20 and 5 nm gaps. One can see, that\nthe spectral maps are qualitatively different from that of\nthe single strip. A striking difference is the presence of an\nadditional mode in the low frequency band [see Fig. 2(a)].\nAn examination of the mode profiles (not shown here)\nreveals that this resonance is attributed to a mode, clas-\nsified as n=1i mode, localized at the inner edges of the\nstrip pair (see Iurchuk et al.6for more details).\nThen=1i mode is also a dipole dominated true edge\nmode and, therefore, extremely sensitive to the stray\nfield produced by the neighbouring edges. For large gaps(d⩾1µm), the resonance field/frequency of the n=1\nandn=1i modes are equal, as expected for the com-\npletely isolated strips, since the magnetostatic coupling\nis negligible. As the gap between the strips shrinks, the\nmagnetostatic coupling between the inner edges of the\nstrips increases, favouring the parallel alignment of the\ninner edges and, thus leading to the reduction of the in-\nner edge saturation field, as summarized in Fig. 2(d).\nThe reduction of the resonance field is equivalent to an\nincrease of the resonance frequency. The outer edge sat-\nuration field stays unaltered, due to the negligible effect\nof the stray fields on the opposite edges distant by more\nthan 2 µms apart. For extremely small gaps dbelow the\nexchange length lex(lex≈5.7 nm for Permalloy), the\nresonance field of the inner edge mode approaches the\nshape anisotropy field of the strip pair. For example,\natd= 1 nm, the n=1i mode can be excited already at\nµ0Hx= 26 mT, which corresponds to the field needed for\nthe quasi-uniform mode softening. The observed extreme\nsensitivity of the inner edge mode frequency to the stray\nfields allows for an efficient use of the edge resonances for\nfield sensing.\nOur analysis shows that not only edge modes are af-\nfected by the magnetostatic coupling in the strip pairs,\nbut the intensity and the spatial profile of the quasi-\nFig.3new (22x9)\n0 200 400 600 800 10000.000.020.040.060.080.10\nd=1 nmmz (arb.u.)\nx (nm)d=1 m\nd=500 nm\nd=300 nm\nd=200 nm\nd=100 nm\nd=50 nm\nd=20 nm\nd=10 nm\n1 10 100 10000100200300400500600x coordinate of mmax (nm)\nd (nm)\n(a) (b)\nFIG. 3. (a) ∆ mzdynamical component of the quasi-uniform\nmode for µ0Hx=0.6 T as a function of the x-coordinate of\nthe right strip for different d. The plots are vertically offset\nfor better visibility. (b) Coordinate of the mode intensity\nmaximum versus the gap between the strips.4\n-500 -250 0 250 5000.000.150.300.450.60\n........................................................................mz (arb.u.)\nx (nm)quasi-uniform 24.82 GHz\nn=3i 21.76 GHz\nn=3 21.08 GHz\nn=2i 20.67 GHz\nn=2 19.58 GHz\nn=1i 14.02 GHz\nn=1 12.44 GHz\n-250 -125 0 125 2500.000.150.300.450.600.75\n.........................................mz (arb.u.)\nx (nm)n=2i\nn=2\nn=1i\nn=121.18 GHz\n19.31 GHz\n18.81 GHz\n12.91 GHz\n11.84 GHzquasi-uniform\n-150 0 1500.00.30.6\n17.05 GHzmz (arb.u)\nx (nm)n=1n=1i\n10.62 GHz11.41 GHzquasi-\nuniformFig.4new (33x17)\n0100 200 300 400 500 600 700010203040\n0Hx (mT)f (GHz)w = 1 m; d = 100 nm \n0100 200 300 400 500 600 700010203040\nw = 500 nm; d = 100 nm\n0Hx (mT)f (GHz)\n0100 200 300 400 500 600 700010203040\nw = 100 nm; d = 100 nm\n0Hx (mT)f (GHz)\n-1200120Re(arb.u.)\n0100 200 300 400 500 600 700010203040\nw = 200 nm; d = 100 nm\n0Hx (mT)f (GHz)\n(a) (b) (c) (d)\n-1000 -500 0 500 10000.000.150.300.450.60\n22.23 GHz26.18 GHz\n21.27 GHz\n21.09 GHz\n19.75 GHz\n14.43 GHz\n12.60 GHzquasi-uniform\nn=3n=3i\nn=2n=2i\nn=1imz (arb.u.)\nx (nm).........................................................................................\nn=1\n(e) (f) (g) (h)\nFIG. 4. (a–d) Frequency-field FMR absorption maps of the pair of 50 nm thick Py strips with different widths wseparated by\na 100 nm gap: (a) w=1µm, (b) w=500 nm, (c) w=200 nm, and (d) w=100 nm. (e–h) Corresponding mode profiles (at µ0H=\n0.6 T) of the first six localized modes and a quasi-uniform mode for w=1µm (e) and w=500 nm (f); first four localized modes\nand a quasi-uniform mode for w=200 nm (g); first two localized modes and a quasi-uniform mode for w=100 nm (h).\nuniform mode is substantially modified too. Fig. 3(a)\nshows the quasi-uniform mode profiles versus the strip\nwidth for µ0Hx=0.6 T and for different gaps dbetween\nthe strips, ranging from 1 nm to 1 µm. Note, that here\nwe show the spatial mode intensity distribution in the\nright strip only, whereas the left one shows a completely\nsymmetric picture. We observe two effects of the gap size\non the quasi-uniform mode profiles. Firstly, the gradual\nshift of the intensity maximum in the direction towards\nthe neighbouring strip, plotted on Fig. 3(b). This behav-\nior is in agreement with the results of Rych ly-Gruszecka\net al.11, where a similar spatial intensity redistribution\nfor the reduced distance between the ferromagnetic strips\nwas reported. It can be explained in terms of the reduced\nSW pinning at the inner edges of the strips for the in-\ncreased magnetostatic coupling.\nSecondly, we observe a significant modulation of the\nquasi-uniform mode intensity along the strip width, with\na modulation amplitude enhancement for increased mag-\nnetostatic coupling [Fig. 3(a)]. As the gap ddecreases,\nthe frequency of the edge modes localized at the in-\nner edges at given Hxincreases, gradually approaches\nthe resonance frequency of the quasi-uniform mode [see\nFigs. 2(a–c)]. In fact, the frequency of the inner edges will\nbe always higher compared to the frequency of the outer\nedges for a fixed external field value. Thus the higher or-\nder edge modes will be pushed towards the main mode.\nThe accumulation of the higher order modes at the quasi-\nuniform mode as well as the strong dipole-dipole interac-tion lead to the hybridization of the quasi-uniform mode\nwith the inner edge-localized modes, which results in the\nmodulation of the mode intensity across the strip width.\nNote, that the quasi-uniform mode hybridization, due to\nthe presence of the neighbouring strip, is observed even\nat large bias fields Hx(well above the saturation field).\nTherefore, this effect is qualitatively different from the\nhybridization with the higher order modes in the single\nstrip at low bias fields [see Fig. 1(d)].\nFinally, we investigate the mode localization in the\nstrip pairs with a fixed gap das a function of the strip\nwidth w. The primary purpose of this investigation is\nto explore the effect of the width on the localization of\nthe true edge modes and the quasi-uniform mode. On\none hand, it is expected that the true edge modes will\nnot anymore localize to the very edge of the sample be-\nlow a given strip width. On the other hand, the ”mode\nsoftening” of the quasi-uniform mode is expected to dis-\nappear for narrow strips, and the magnetization across\nthe strip width should behave more homogeneously due\nto the increased proportion of the exchange interaction\nto the total energy.\nFigs. 4(a–d) show the frequency-field absorption maps\nfor the pairs of 50 nm-thick Py strips with w=1µm,\n500 nm, 200 nm and 100 nm for d= 100 nm. The corre-\nsponding mode profiles of the true edge modes, standing\nspin-wave modes and the quasi-uniform modes versus the\nstrip width are shown in Figs. 4(e–h). For completeness,\nwe start with the absorption map for the Py strip pair5\nwith w=1µm and d= 100 nm [as also shown in Fig. 2(a)].\nThe corresponding mode profiles across the width of the\nwaveguides (the strip pair) for a static external field of\n0.6 T are extracted and summarized in Fig. 4(e) for the\nfirst six localized modes. The lowest mode in frequency\nis the true outer edge mode n=1, the second lowest is\nthe true inner edge mode n=1i, followed by the higher\norder modes localized close to the edge and, finally, the\nquasi-static mode. As the strip width is reduced, we ob-\nserve a gradual decrease of the frequency gap between\nthen=1 and n=1i modes for a given field, due to the de-\ncreased effective coupling between the inner edges, since\nthe inner-outer edge interaction becomes more signifi-\ncant. Furthermore, when looking on the mode profiles\nsummarized in Fig. 4(e–h), one can easily deduce that the\nedge mode volume increases with decreasing strip width.\nAtw=100 nm, the edge mode significantly extends over\nthe strip cross section, as expected, due to the increased\nexchange interaction.\nAn intricate plethora of the large- nstanding spin-wave\nmodes with closely spaced resonant fields/frequencies is\nobserved in relatively wide strips (e.g. w= 500 nm\nand above) for fields around and above the edge satu-\nration field. For reduced strip width, the number of the\nmodes the strip pair can host decreases, and for wbelow\n∼200 nm, only few standing spin waves are present [see\nFig. 4(c,d)]. Note, for the pair of 100-nm-wide strips only\nthree modes can be efficiently excited with spatially ho-\nmogeneous rf-field at high static fields [see Fig. 4(d,h)],\ni.e. the two true edge modes ( n=1 and n=1i) localized\nat the outer and inner edges (with the closely spaced fre-\nquencies) and the quasi-uniform mode localized at the\nstrip center. Furthermore, for such narrow strips only\nthe edge mode softening remains, showing that the mag-\nnetization of the strip across the width rotates almost\nhomogeneously into the direction of the static field. At\nabout 360 mT the frequency drops to a minimum, close\nto zero. This field now is related to the shape anisotropy\nof the strip pair.\nIn conclusion, we studied the spatial distribution of\nthe standing spin-wave modes in infinitely long Permal-\nloy strips with rectangular cross-section. We define the\nresonance conditions (fields and frequencies) for the exci-\ntation of the true edge modes localized at both inner and\nouter edges of the neighbouring strips. Furthermore, we\nstudy the mode localization in pairs of Permalloy strips\nas a function of the strip width and the lateral sepa-\nration between the strips. We show that a wide-range-\ntunability of the localized edge mode resonances can be\nachieved with a precise control of the magnetostatic cou-\npling between the strips. The observed mode confine-\nment allows for an efficient control of the SW localiza-\ntion in the pair of sub-100-nm waveguides, with the field-\ncontrollable switching between the propagation channels\nlocalized either in the strip center or at the edges of the\nstrips [see Fig. 4(h)]. This study provides a micromag-netic background for the understanding of the interaction\nbetween the standing (as well as propagating) spin waves\nin closely packed magnonic waveguides.\nACKNOWLEDGMENTS\nFinancial support by the Deutsche Forschungsgemein-\nschaft (DFG) within the programs IU 5/2-1 (project\nnumber 501377640) and KA 5069/3-1 (project number\n444929866) is gratefully acknowledged.\n1A. V. Chumak et al. , IEEE Transactions on Magnetics 58, 1\n(2022), conference Name: IEEE Transactions on Magnetics.\n2C. Bayer, J. Jorzick, B. Hillebrands, S. O. Demokritov, R. Kouba,\nR. Bozinoski, A. N. Slavin, K. Y. Guslienko, D. V. Berkov, N. L.\nGorn, and M. P. Kostylev, Physical Review B 72, 064427 (2005),\npublisher: American Physical Society.\n3K. Lenz, T. Schneider, G. Hlawacek, R. Narkowicz, S. Stienen,\nA. K´ akay, M. Lenz, J. Fassbender, and J. Lindner, Trimming of\npermalloy stripes to enhance the localized edge mode spectrum\nprobed by ferromagnetic resonance, Magnonics 2019, 28.07.-\n01.08.2019, Carovigno, Italy (2019).\n4Z. Zhang, M. Vogel, M. B. Jungfleisch, A. Hoffmann, Y. Nie, and\nV. Novosad, Physical Review B 100, 174434 (2019), publisher:\nAmerican Physical Society.\n5S. Pile, S. Stienen, K. Lenz, R. Narkowicz, S. Wintz, J. F¨ orster,\nS. Mayr, M. Buchner, M. Weigand, V. Ney, J. Lindner, and\nA. Ney, Physical Review B 105, 094415 (2022), publisher: Amer-\nican Physical Society.\n6V. Iurchuk, J. Pablo-Navarro, T. Hula, R. Narkowicz,\nG. Hlawacek, L. K¨ orber, A. K´ akay, H. Schultheiss, J. Fassben-\nder, K. Lenz, and J. Lindner, Scientific Reports 13, 764 (2023),\nnumber: 1 Publisher: Nature Publishing Group.\n7M. Krawczyk and D. Grundler, Journal of Physics: Condensed\nMatter 26, 123202 (2014), publisher: IOP Publishing.\n8R. A. Gallardo, T. Schneider, A. Rold´ an-Molina, M. Langer,\nA. S. N´ u˜ nez, K. Lenz, J. Lindner, and P. Landeros, Physical\nReview B 97, 174404 (2018), publisher: American Physical So-\nciety.\n9R. A. Gallardo, T. Schneider, A. Rold´ an-Molina, M. Langer,\nJ. Fassbender, K. Lenz, J. Lindner, and P. Landeros, Physical\nReview B 97, 144405 (2018), publisher: American Physical So-\nciety.\n10M. Langer, R. A. Gallardo, T. Schneider, S. Stienen, A. Rold´ an-\nMolina, Y. Yuan, K. Lenz, J. Lindner, P. Landeros, and J. Fass-\nbender, Physical Review B 99, 024426 (2019), publisher: Amer-\nican Physical Society.\n11J. Rych ly-Gruszecka, J. Walowski, C. Denker, T. Tubandt,\nM. M¨ unzenberg, and J. W. K los, Scientific Reports 12, 20678\n(2022), number: 1 Publisher: Nature Publishing Group.\n12A. Lara, J. Robledo Moreno, K. Y. Guslienko, and F. G. Aliev,\nScientific Reports 7, 5597 (2017), number: 1 Publisher: Nature\nPublishing Group.\n13L. K¨ orber, G. Quasebarth, A. Otto, and A. K´ akay, AIP Advances\n11, 095006 (2021).\n14L. K¨ orber, G. Quasebarth, A. Hempel, F. Zahn, O. Andreas,\nE. Westphal, R. Hertel, and A. K´ akay, TetraX: Finite-Element\nMicromagnetic-Modeling Package (2022).\n15L. K¨ orber, M. Zimmermann, S. Wintz, S. Finizio, M. Kronseder,\nD. Bougard, F. Dirnberger, M. Weigand, J. Raabe, J. A. Ot´ alora,\nH. Schultheiss, E. Josten, J. Lindner, I. K´ ezsm´ arki, C. H. Back,\nand A. K´ akay, Physical Review B 104, 184429 (2021).\n16C. Kittel, Introduction to Solid State Physics , 8th ed. (John Wi-\nley & Sons, 2005).\n17R. D. McMichael and B. B. Maranville, Physical Review B 74,\n024424 (2006), publisher: American Physical Society."    },    {        "title": "1705.02261v2.Light_induced_anisotropic_skyrmion_and_stripe_phases_in_a_Rashba_ferromagnet.pdf",        "content": "Light-induced anisotropic skyrmion and stripe phases in a Rashba ferromagnet\nDmitry Yudin,1Dmitry R. Gulevich,1and Mikhail Titov2, 1\n1ITMO University, Saint Petersburg 197101, Russia\n2Radboud University Nijmegen, Institute for Molecules and Materials, NL-6525 AJ Nijmegen, The Netherlands\n(Dated: November 13, 2018)\nAn external o\u000b-resonant pumping is proposed as a tool to control the Dzyaloshinskii-Moriya\ninteraction (DMI) in ferromagnetic layers with strong spin-orbit coupling. Combining theoretical\nanalysis with numerical simulations for an s-d{like model we demonstrate that linearly polarized o\u000b-\nresonant light may help stabilize novel non-collinear magnetic phases by inducing a strong anisotropy\nof the DMI. We also investigate how with the application of electromagnetic pumping one can\ncontrol the stability, shape, and size of individual Skyrmions to make them suitable for potential\napplications.\nPACS numbers: 12.39.Dc, 75.70.Tj, 78.20.Ls, 75.70.-i\nLow-dimensional magnetic structures provide an excit-\ning playground for condensed matter physics and tech-\nnology applications. Some of them, e. g., helical mag-\nnets [1{4], are known to support topologically nontriv-\nial magnetic textures [5{7]. Such noncollinear states\nemerge as a result of the competition between Heisen-\nberg exchange, antisymmetric Dzyaloshinskii-Moriya ex-\nchange, and magnetocrystalline anisotropy yielding mag-\nnetic ground states that are far more intricate than those\nin homogeneous ferromagnets [8{12]. Recent progress in\nthe fabrication of magnetic materials motivated an inter-\nest in particlelike domains, such as magnetic Skyrmions\n[13{25], that are typical for nonhomogeneous ferromag-\nnets. Skyrmions and other topologically protected mag-\nnetic textures have been proposed as building blocks\nfor logical operations and information storage in the\nrapidly advancing \felds of magnon spintronics [26] and\nSkyrmionics [27, 28].\nSpintronics, as a branch of applied science, is traced\nback to the pioneering work on giant magnetoresistance\nby Gr unberg [29, 30] and Fert [31]. The subsequent\ndiscovery of spin-transfer torque [32, 33] provoked the\nidea to exploit the spin degree of freedom rather than\nthe charge for processing and transferring information.\nThis idea is currently being extended to magnetic ma-\nterials supporting localized magnetic excitations such as\nSkyrmions.\nWays to create and control Skyrmions and other non-\ncollinear magnetic textures are essential for the practical\nimplementation of this emerging technology. In this Let-\nter, we investigate microscopically how the noncollinear\nmagnetic domains in thin ferromagnet layers with strong\nspin-orbit coupling may be controlled by linearly polar-\nized light. The e\u000bects predicted may be observed in thin\n\flms such as Co/Pt heterostructures subject to short\nlight pulses [34].\nThe exchange interaction alone may lead only to a\ncollinear orientation of magnetic moments in a cubic crys-\ntal. Spatially inhomogeneous magnets are usually as-\nsociated with a lack of lattice inversion symmetry. Inhis seminal work on noncentrosymmetric magnets [35],\nDzyaloshinskii identi\fed the one-dimensional magnetic\nspiral states stabilized by the Dzyaloshinskii-Moriya in-\nteraction (DMI) [36] that favors the noncollinear orien-\ntation of neighboring spins. A nontrivial ground state\narises in helical magnets as a consequence of the com-\npetition between the Heisenberg exchange and the DMI.\nIn two-dimensional structures, this competition leads to\na helical spin-spiral ground state con\fguration that be-\ncomes unstable in the presence of a magnetic \feld with\nthe tendency towards the formation of Skyrmions [37].\nQuite generally, Skyrmions correspond to the solitonlike\nsolutions of the \feld equations of Dzyaloshinskii's theory\nthat destroy the homogeneity of magnetic order [38]. The\nexistence of such localized states and the mechanism of\ntheir nucleation as mesoscopic objects are rather common\nfor continuum systems described by the free energy func-\ntional with Lifshitz invariants [39]. The strength of DMI\ncan be rigorously approached by the microscopic theory,\nas an indirect exchange interaction between two neigh-\nboring spins facilitated by itinerant (conduction) elec-\ntrons [40{42], as well as adopted from the \frst-principles\nsimulations [43{45]. In a thin \flm, the DMI can be in-\nduced by the Rashba spin-orbit coupling.\nAn external electromagnetic \feld may strongly modify\nthe properties of an electronic system providing an im-\nportant tool for manipulating materials in a controllable\nfashion [46{48]. It has been recently shown that the ef-\nfect of o\u000b-resonant electromagnetic radiation (with the\nfrequency exceeding the bandwidth of the system) may\nbe described by e\u000bective time-independent models with\nstrongly renormalized parameters [49{58].\nIn this Letter, we derive the e\u000bective s-dexchange\nmodel of a Rashba ferromagnet in the regime of strong\ncoupling to external radiation. We \fnd that the DMI\nstrength can be e\u000bectively controlled by the application\nof o\u000b-resonant pumping that opens up exciting oppor-\ntunities for controlling the stability, size, and shape of\nindividual metastable Skyrmions. Also, we show that\nthe application of linearly polarized radiation inducesarXiv:1705.02261v2  [cond-mat.mtrl-sci]  7 Oct 20172\nanisotropy of the DMI that not only provides a \fner\ncontrol over the individual DMI strengths in two orthog-\nonal directions but also leads to the appearance of novel\nanisotropic phases.\nFrom the theory point of view, the e\u000bect of time-\nperiodic \felds may be described, with some reservations,\nby the so-called Floquet theory [59, 60]. Periodicity of\nthe driving \feld enables one to map the original time-\ndependent problem to the eigenvalue problem of Floquet\nstates. O\u000b-resonant pumping takes place if the frequency\nof the driving \feld is so high that electrons are not able\nto follow \feld oscillations. In this case, real absorption\nor emission of light quanta cannot happen due to re-\nstrictions imposed by energy conservation for radiation\nfrequency exceeding the electron bandwidth. Still, such\no\u000b-resonant radiation a\u000bects the system via virtual pro-\ncesses leading to a signi\fcant renormalization of the pa-\nrameters of the initial Hamiltonian of an electron sub-\nsystem. Below, we restrict our attention to the e\u000bects of\nlinearly polarized light, since it has a greater impact on\nthe noncollinear magnetic textures. For the sake of mi-\ncroscopic treatment, we rely upon the Floquet-Magnus\nexpansion and its generalizations that have been devel-\noped in Refs. [61{64].\nFor microscopic analysis, we consider a weak two-\ndimensional ferromagnet that yields an s-d-like Rashba\nmodel for conduction electrons:\nH=p2=2m+\u000b\u001b\u0001(p\u0002^z) + \u0001m\u0001\u001b; (1)\nwheremis the unit (jmj= 1) local magnetization vector\ndue to, e. g., localized delectrons, \u0001 is the s-d-like ex-\nchange energy, pstands for the momentum operator for\nconduction electrons with an e\u000bective mass m,\u000bis the\nRashba spin-orbit interaction strength, ^zis the unit vec-\ntor in the direction perpendicular to the two-dimensional\nelectron gas, and \u001b= (\u001bx;\u001by;\u001bz) denotes the vector of\nPauli matrices. Models of the type of Eq. (1) were orig-\ninally proposed to explain the physics of ferromagnetic\nmetals beyond the Heisenberg exchange picture [65{68].\nThis approach relies upon a formal distinction between a\nlocalized (classical) magnetic subsystem (e. g., dorfelec-\ntrons, that are described by an m\feld which is governed\nby a classical Heisenberg model) an itinerant subsystem\n[e. g.,selectrons described by Eq. (1)] that are coupled\nto each other by means of exchange interaction.\nAn external acelectromagnetic \feld of frequency !\n[69] is introduced in the model of Eq. (1) by means of\nthe Peierls substitution p!p+eA0cos!t, whereA0=\nE0=!andE0is the electric \feld component of the \feld.\nIn what follows, we restrict our analysis to the case of\nlinearly polarized light by choosing E0=E0^y, where ^y\nis the in-plane unit vector (in the ydirection).\nThe Hamiltonian of Eq. (1) is conveniently rotated as\nFIG. 1: Elliptic Skyrmion arising in the presence of the DMI\nanisotropy induced by the linearly polarized light. The \fgure\nrepresents the results of our numerically simulations for the\nmodel of Eq. (11) at Hext= 0:01Jand\r= 1:5 for the DMI\nstrength \fxed at Dx=Dy= 0:18Jfor\r= 0. Arrows indicate\nthe in-plane components of the average magnetization in a\nblock of 3 \u00023 spins.\nH!Uy\ntHUtwith the time-dependent unitary operator\nUt= 2\u00001=2(\u001bx+\u001bz)e\u0000it[ut+\u001bze\u000bA 0sin(!t)=!t]=~;(2a)\nut=e2A2\n0\n4m\u0012\n1 +sin(2!t)\n2!t\u0013\n+epyA0\nmsin(!t)\n!t: (2b)\nThe transformed Hamiltonian yields the matrix Floquet\nmodel of the form\nH=p2\n2m+ (\u000bpy+ \u0001mx)\u001bz+1X\nn=\u00001hnein!t;(3)\nwith the coe\u000ecients hnde\fned by\nhn= [\u0001mz\u001bx+ (\u000bpx\u0000\u0001my)\u001by]Jn(\r); (4)\nwhere the parameter \r= 2e\u000bE 0=~!2describes the e\u000bec-\ntive light-matter coupling and Jn(\r) stands for the nth\norder Bessel function of the \frst kind.\nThe high-frequency expansion in the form of the\nBrillouin-Wigner perturbation theory recently developed\nfor this class of problems [70] maps Eq. (3) onto an ef-\nfective time-independent Hamiltonian:\nHe\u000b=p2=2m+\u000b[py\u001bx\u0000px\u001byJ0(\r)] +V; (5a)\nV= \u0001fmx\u001bx+ [my\u001by+mz\u001bz]J0(\r)g; (5b)\nthat is valid away from resonance frequencies. The ef-\nfective model fails only in a tiny vicinity \u000e\rof the ze-\nros of the Bessel function, \u000e\r\u001910\u00005\u00012=(~!)2, which\nis well beyond our numerical resolution [71]. The model\nis equivalent to that of Eq. (1) with anisotropic renor-\nmalization of coupling constants: Rashba spin-orbit in-\nteraction strength and s-dexchange coupling. In what3\nfollows, we assume a weak ferromagnet and treat the ex-\nchange interaction term Vperturbatively.\nBased on the symmetry analysis, Dzyaloshinskii dis-\ncovered that the e\u000bective Ginzburg-Landau free energy\nfunctional may allow for terms linear in magnetization\ngradients provided the absence of lattice inversion sym-\nmetry [36]. Later, Moriya argued, on the basis of An-\nderson's theory of superexchange, that the microscopic\nmechanism of spin-orbit coupling is responsible for such\nan interaction [72, 73]. The latter can also be thought as\na coupling between an excited state of a magnetic ion and\nthe ground state of the neighboring ion. Such a coupling\ncan be derived microscopically from the correction to the\nbare actionS0[m] (that collects all terms corresponding\nto magnetic subsystem) computed to the second order\nwith respect to the perturbation V[71].\nTo construct the perturbation theory, we take advan-\ntage of the bare Matsubara Green's function for the\nHamiltonian of Eq. (5a):\nG0(i!;k) =1\n2X\ns=\u0006\u0003s(\u0012k)\ni~!\u0000\"s\nk; (6a)\n\u0003\u0006(\u0012) = 1\u0006[\u001bxsin\u0012\u0000\u001byJ0(\r) cos\u0012]=g(\u0012): (6b)\nwhereg(\u0012) =\u0002\nsin2\u0012+J2\n0(\r) cos2\u0012\u00031=2and the spec-\ntrum\"\u0006\nk=~2k2=2m\u0006\u000b~kg(\u0012k) acquires the depen-\ndence on the direction of the wave vector k=p=~=\nk(cos\u0012k;sin\u0012k) due to the linear polarization of light.\nThe second-order contribution to the e\u000bective action\nin the imaginary time representation is given by\n\u000eS[m] =1\n\fX\ni!;kX\nij\u0005ij(i!;k)mi(k)mj(\u0000k);(7)\nwhere\fstands for the inverse temperature while the in-\ndexesiandjdenote the Cartesian vector components.\nThe polarization operator is expressed as\n\u0005ij(i!;k) =1\n4\u0001i\u0001jX\nqX\ns;s0=\u0006f(\"s\nk+q)\u0000f(\"s0\nq)\n\"s\nk+q\u0000\"s0\nq\u0000i~!\n\u0002Tr [\u0003s(\u0012k+q)\u001bi\u0003s0(\u0012q)\u001bj];(8)\nwheref(\") is the Fermi-Dirac distribution function and\n\u0001x= \u0001, \u0001y= \u0001z=J0(\r)\u0001.\nThe straightforward expansion of \u0005 ij(!= 0;k) around\nk= 0 up to the terms linear in kyields a fully antisym-\nmetric contribution to the e\u000bective action [71]:\n\u000eS[m] =Z\nd2r\u0010\nDxL(x)\nxz+DyL(y)\nyz\u0011\n; (9)\nwhere we introduce the DMI couplings\nDx=\u00012\n\u0019\u000b~jJ0(\r)j\n1 +jJ0(\r)j; Dy=\u00012\n\u0019\u000b~J2\n0(\r)\n1 +jJ0(\r)j;(10)\nand Lifshitz invariants L(l)\nij=mi@lmj\u0000mj@lmi.In the absence of an electromagnetic \feld, i. e., for\n\r= 0, we obtain an isotropic DMI with Dx=Dy=\n\u00012=(2\u0019\u000b~). In the presence of linearly polarized light,\nthe DMI coupling becomes essentially anisotropic as\ngiven by Eq. (10). We stress that the employed high-\nfrequency expansion is legitimate only as far as there are\nno resonant transitions and the parameter \ris away from\nzeros of the Bessel function J0(\r) [71].\nTo illustrate our results, we consider a classical two-\ndimensional Heisenberg exchange model on a square lat-\ntice, that is given by the total energy\nE=\u0000JX\nrSr\u0001[Sr+^x+Sr+^y]\u0000HextX\nr(Sr)z\n\u0000DxX\nr(Sr\u0002Sr+^x)y+DyX\nr(Sr\u0002Sr+^y)x;(11)\nwhereSris the spin on a lattice site r,Hextis an external\nmagnetic \feld (in energy units) perpendicular to the two-\ndimensional plane, ^xand^ystand for the unit vectors in x\nandydirection, correspondingly, and the lattice constant\nis set to unity.\nWith the help of the numerical approach described\nin Ref. [71], we analyze the in\ruence of the anisotropic\nDMI on the Skyrmion pro\fle. The pro\fle is obtained\nby relaxing a trial Skyrmion ansatz using the dynami-\ncal Landau-Lifshitz-Gilbert equation until the stationary\nstate is reached. To avoid nonuniversal e\u000bects of bound-\nary conditions, the numerical simulation is performed in\na box of a large size that exceeds characteristic size of a\nSkyrmion by a large factor (only the central part of the\nbox is shown in Fig. 1).\nWe \fnd that the anisotropic renormalization of the\nDMI strength of Eq. (10) results in the anisotropic\nsqueezing of a Skyrmion. The Skyrmion becomes elon-\ngated along the light polarization direction and develops\nan elliptic pro\fle as shown in Fig. 1. In the model with\n\r.1 and positive DxandDy, the in-plane spin projec-\ntions are directed towards the Skyrmion center. Such a\ncon\fguration may be referred to as the inverted hedge-\nhog Skyrmion that is distinguished from the hedgehog\nSkyrmion in which the in-plane spin projections point\noutwards. The Skyrmion type is, therefore, de\fned by\nthe overall sign of the DMI coupling.\nIt is also worth stressing that the model (11) supports\nonly Ne\u0013 el-type Skyrmions which are the Skyrmions with\na radial orientation of spins. Such Ne\u0013 el-type Skyrmions\nwere observed recently in GaV 4S8[74]. Those should be\ncontrasted with Bloch-type Skyrmions that are charac-\nterized by spin orientations perpendicular to the radial\ndirection. The Bloch-type Skyrmions are thought to be\ncharacteristic for materials like FeGe [75] and MnSi [76].\nIt can be shown using the methodology of Ref. [71]\nthat individual Skyrmions are metastable only in certain\nareas of the parameter space as illustrated in Fig. 2(a).\nThe metastability regions of individual Skyrmions in the4\nFIG. 2: (a) Region of metastability of an isolated Skyrmion in the model of Eq. (11) with the anisotropic DMI strength given\nby Eq. (10). (b) Phase diagram as a function of the parameter \rand external magnetic \feld Hext. Both \fgures represent the\nresults of numerical simulations of the model of Eq. (11) with the DMI strength given by Dx=Dy= 0:18Jat\r= 0.\nmodel of Eq. (11) must be distinguished from the phases\nthat characterize the absolute minimum of the energy\nfunctional. By extending our numerical analysis to search\nfor a ground state [71], we obtain the phase diagram de-\npicted in Fig. 2(b). The diagram consists of three phases:\n(i) the homogeneous ferromagnetic order phase denoted\nby points, (ii) the Skyrmion crystal phase (a crystal of\nelliptic Skyrmons) denoted by circles, and (iii) the stripe\ncrystal phase (periodic stripes in the direction of light\npolarization) denoted by vertical lines.\nAnisotropy induced by pumping distorts the symmetry\nof the Skyrmion crystal from the equilateral triangular at\n\r= 0 to an isosceles triangular at nonzero \r[71]. The\nstripe crystal phase is analogous to the helical phase dis-\ncussed, e. g., in Ref. [77], although, in contrast to the\nconventional helical phases, the stripe phase arising for\nthe model (11) is the N\u0013 eel type, with spins rotating in the\nradial direction, as opposed to a helix. The orientation of\nthe stripe phase depends on the direction of the induced\nanisotropy of the DMI. This provides a control over the\norientation of the stripe phase by changing the polariza-\ntion of the applied radiation | a property which may be\nemployed in future light-controlled magnetic logic gates.\nInterestingly, the range of metastability of individual\nSkyrmions does not generally coincide with the phase\nboundaries. However, we \fnd that metastable Skyrmions\ngenerally do not exist in a stripe crystal phase that is\ndominant at low values of the magnetic \feld. In this re-\ngion, individual Skyrmions quickly become unstable with\nrespect to stretching in the ydirection to form a stripe.\nFor intense light with the parameter \rexceeding the \frst\nzero ofJ0(\r), i. e., for\r&2:4, the phase diagram is dom-\ninated by the stripe phase at small magnetic \felds. The\nSkyrmion crystal phase is limited to a moderate lightintensity as shown in Fig. 2(b).\nThe numerical studies of Skyrmion dynamics in the\nabsence of the \feld \r= 0 were performed in Refs. [77{\n81]. In the absence of light, i. e., for \r= 0, the phase\ndiagram of Fig. 2(b) reproduces these known results. In-\ndeed, the obtained values of the critical \felds for the tran-\nsition between the stripe and Skyrmion-crystal phases\n(Hc1= 0:0072J) and between the Skyrmion-crystal and\nferromagnetic phases ( Hc2= 0:026J) are very close to\nthose given in Ref. [78]. To simplify the comparison with\nRef. [78], we have used the same parameter values of the\nDMI strength Dx=Dy= 0:18Jat\r= 0.\nIn conclusion, the \feld of magnetic Skyrmions has at-\ntracted considerable attention due to the potential ap-\nplications of Skyrmions in information processing. The\nmajor advantage of such noncollinear spin con\fgurations\nas compared to domain walls is the possibility to make\nthe Skyrmion size as small as a few nanometers with-\nout losing its stability. In this Letter, we employ the\ns-d-like exchange model for a weak two-dimensional fer-\nromagnet with strong spin-orbit coupling to show that\nthe o\u000b-resonant linearly polarized light can be used to\ntune the strength of the DMI and induce a large DMI\nanisotropy in the two orthogonal directions. This e\u000bect\nleads to the appearance of novel anisotropic phases |\nan elliptic Skyrmion crystal phase and a stripe phase |\nand can provide a new tool to control the stability, size,\nand shape of individual Skyrmions, as well as a control\nover the stripe phases by changing the light polarization\ndirection.\nThe predicted e\u000bects may be observed in thin \flms\nsuch as Co/Pt using femtosecond laser pulses [34]. The\nlight pulses must be su\u000eciently long to drive the struc-\nture into a nonequilibrium state that can be considered5\nquasistationary. The typical experimental facility with\nthe pump \ruence 2-3 mJ/cm2should be su\u000ecient to test\nthe theoretical results. We also expect that qualitatively\nthe same physics persists at room temperature, helping\nto create a controlled set of Skyrmions that can be used\nto make the concept of Skyrmion racetrack memory vi-\nable [82].\nWe thank Alexey Kimel for helpful discussions. 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Phys. 54,\n053001 (2015).\n[81] R. Zhu and Y.-Y. Zhang, Eur. Phys. J. B 89, 262 (2016).\n[82] T. L. Monchesky, Nature Nanotech. 10, 1008 (2015).s1\nONLINE SUPPLEMENTAL MATERIAL\nLight-induced anisotropic skyrmion and stripe phases in a Rashba ferromagnet\nDmitry Yudin, Dmitry R. Gulevich, and Mikhail Titov\nIn this Supplemental Material we provide technical details for analytical calculations and numerical routine\ndelivered in the main text.\nI. DERIVATION OF THE EFFECTIVE HAMILTONIAN\nThe model of Eq. (1) of the main text is transformed by the unitary transformation (2a) to the model (3) which\ncan also be written as\nH=1X\nn=\u00001hnein!t;withhn=\u0014p2\n2m+ (\u000bpy+ \u0001mx)\u001bz\u0015\n\u000en;0+ [\u0001mz\u001bx+ (\u000bpx\u0000\u0001my)\u001by]Jn(\r): (s1)\nwhereJn(\r) is then-th order Bessel function of the \frst kind and \r= 2e\u000bA 0=~!.\nIf the frequency !of a driving \feld is the largest energy scale in the system (the frequency !is much larger than the\nbandwidth so that optical excitations of electrons are forbidden) one may replace the time-dependent problem with\nthe time-independent Hamiltonian of an e\u000bective stationary model. The corresponding formalism of high-frequency\nexpansion in the form of Brillouin-Wigner perturbation theory [s70] has recently been developed for this class of\nproblems. This formalism applied to the model of Eq. (s1) gives the following e\u000bective model\nHe\u000b=H(0)+H(1)+H(2)+:::=h0+1\n~!X\nm6=0hmh\u0000m\nm+1\n(~!)2\u0012X\nm;n6=0hmhn\u0000mh\u0000n\nmn\u0000X\nm6=0hmh\u0000mh0\nm2\u0013\n;(s2)\nwhere we kept terms up to the second order in 1 =!. Substituting hnfrom Eq. (s1) into the general expression of\nEq. (s2) we obtain the e\u000bective Hamiltonian\nHe\u000b=p2\n2m+\u0012\n1\u00002\fX\nn6=0Jn(\r)J\u0000n(\r)\nn2\u0013\n(\u000bpy+ \u0001mx)\u001bz\n+\u0012\nJ0(\r) +\fX\nm6=n\nm;n6=0Jm(\r)Jn\u0000m(\r)J\u0000n(\r)\nmn\u0013\n(\u0001mz\u001bx+ [\u000bpx\u0000\u0001my]\u001by); (s3)\nwhere\f= (\u00012m2\nz+ [\u000b~ky\u0000\u0001my]2)=(~!)2\u001c1 under o\u000b-resonant radiation conditions. It is also clear that we can\nneglect the terms proportional to the sum of the Bessel functions in Eq. (s3). Moreover, if the parameter \rdoes not\nlay in the immediate vicinity of zeros of the Bessel function J0(\r) (the second line in Eq. (s3)) the system in question\ncan be e\u000bectively described (in the original basis) by the Hamiltonian\nHe\u000b=p2\n2m+ (\u000bypy+ \u0001xmx)\u001bx\u0000(\u000bxpx\u0000\u0001ymy)\u001by+ \u0001zmz\u001bz; (s4)\nwhich is the e\u000bective model of Eq. (5a) considered in the main text. Similarly, one can show that all high order terms\narising in 1 =!expansion are negligible away from zeros of J0(\r).\nIn a close vicinity \u000e\rof the zeros of J0(\r) the high order terms are formally important. However, it can be shown\nthat the actual value of \u000e\ris negligibly small. To provide a quantitative estimate of \u000e\rwe evaluate the quantity\nC=\f\f\fX\nm6=n\nm;n6=0Jm(\r0)Jn\u0000m(\r0)J\u0000n(\r0)\nmn\f\f\f= 6\u000110\u00006; (s5)\nat the \frst zero \r0= 2:4048 of the function J0(\r). Thus, the e\u000bective model of Eq. (5) formally breaks down only in\na tiny region around \r0that is determined by \u000e\r=j\r\u0000\r0j\u001810\u00005\f, which is well beyond our numerical resolution.s2\nII. DERIVATION OF THE DMI STRENGTH\nIn this section we present the derivation of antisymmetric exchange interaction which is linear in magnetization\ngradients and can be identi\fed as the DMI. To analyze the polarization operator (8) we compute the quantity\nNi(k) =1\n2Zd2q\n(2\u0019)2X\ns;s0=\u0006f(\"s\nk+q)\u0000f(\"s0\nq)\n\"s\nk+q\u0000\"s0\nq\u0012\ns^ei\u0001nk+q\ng(\u0012k+q)\u0000s0^ei\u0001nq\ng(\u0012q)\u0013\n; (s6)\nwhere\"\u0006\nq=~2q2=(2m)\u0006\u000b~qg(\u0012q),g(\u0012) = [sin2\u0012+J2\n0(\r) cos2\u0012]1=2,f(\") is the Fermi-Dirac distribution, ^ex=^x,\n^ey=^y, andnq=q=jqj. Furthermore, one can show that\n\u0005xz(k) =\u0000\u0005zx(k) =J0(\r)\u0001x\u0001zNx(k);and \u0005yz(k) =\u0000\u0005zy(k) = \u0001y\u0001zNy(k); (s7)\nwhere we kept the notations of the main text. Taking the integral in Eq. (s6) we conclude that the quantities N(1)\ni(k)\nin the linear order with respect to the momentum kare given by\nN(1)\nx(k) =\u00001\n\u0019\u000b~2\u0019Z\n0d\u0012\n2\u0019kxcos2\u0012+kysin\u0012cos\u0012\nsin2\u0012+J2\n0(\r) cos2\u0012=\u0000kx\n\u0019\u000b~1\njJ0(\r)j1\n1 +jJ0(\r)j; (s8)\nand\nN(1)\ny(k) =\u00001\n\u0019\u000b~2\u0019Z\n0d\u0012\n2\u0019kysin2\u0012+kxsin\u0012cos\u0012\nsin2\u0012+J2\n0(\r) cos2\u0012=\u0000ky\n\u0019\u000b~1\n1 +jJ0(\r)j: (s9)\nThus, the second order correction \u000eS[m] to the bare action S0[m] reads\nS[m] =S0[m] +\u000eS[m] =S0[m] +Z\nd2r\u0010\nDx\u0003(x)\nxz+Dy\u0003(y)\nyz\u0011\n; (s10)\nwhich coincides with Eq. (9) of the main text with renormalized DMI strength Dx,Dyand Lifshitz invariants \u0003(l)\nij\nde\fned by Eqs. (10) of the main text.\nIII. DETAILS OF THE NUMERICAL SIMULATIONS\nIn our numerical calculations we use the lattice model given by the Eq. (11). To \fnd the stationary states minimizing\nthe total energy (11) we evolve the overdamped Landau-Lifshitz-Gilbert (LLG) equation\n(1 +\u000b2)dSr\ndt=\u0000\rSr\u0002Heff\u0000\u000b\rSr\u0002Sr\u0002Heff; (s11)\nuntil the stationary state is reached. Here, the e\u000bective \feld Heffis de\fned by the functional derivative of the\nenergy (11) over the local magnetic moment,\nHeff=\u00001\n~\r\u000eE\n\u000eSr=Hext^z+JX\nd=\u0006^x;^ySr+d+Dx(Sr+^x\u0000Sr\u0000^x)\u0002^y\u0000Dy(Sr+^y\u0000Sr\u0000^y)\u0002^x: (s12)\nThe stationary state corresponds to the minimum of the total energy (11). To ensure that a particular state is a\nglobal minimum rather than the local minimum we compare the energies of essentially di\u000berent solutions obtained\nfor each known nontrivial phase (skyrmion crystal, stripe phase or a ferromagnetic phase). To solve numerically the\nLLG equation (s11) we use an explicit \fnite element method implemented in C and parallelized with OpenMP. Below\nwe discuss how each of the two \fgures presented in Fig. 2 of the main text was obtained.\nThe Fig. 2a representing stability of a skyrmion is obtained as follows. We start from a seed approximate solution\nin the form of a skyrmion and evolve the LLG equation (s11) for su\u000eciently long time to obtain the numerically exact\nstationary solution for an isolated skyrmion. On the second stage, the stationary skyrmion solution is distorted by\nadding a random noise and by varying the parameters \randHext. The solution has been accepted as a stable if,s3\nFIG. s1: Deformation of the skyrmion lattice with the parameter \rat a \fxed magnetic \feld Hext= 0:015J. The plots are\nobtained by numerical analysis of the skyrmion lattice con\fgurations which minimize the total energy (11). With increasing \r\nthe skyrmion lattice deforms from the equilateral triangular lattice (at \r= 0:0) to a isosceles triangular lattice (at \r= 0:5 and\n\r= 1:0). Square parts of a skyrmion lattice with 210 \u0002210 spins are shown. The arrows represent the in-plane components of\nthe average magnetization in a block of 5 \u00025 spins.\nupon addition of the noise and variation of the parameters it settles down and does not decay during su\u000eciently long\nevolution time of about \u00183000 ~=J.\nTo calculate the phase diagram in Fig. 2b we proceed as follows. Con\fgurations minimizing (11) in a periodic\nrectangular domain of Nx\u0002Nycites are found as stationary solutions of the Eq. (s11) obtained by evolving (s11) from\nan initial seed solution for all known nontrivial phases (skyrmion crystal or stripe phase). In case of the modulated\nphases (skyrmion and stripe phases) the periodicity of the system for an in\fnite system is not known beforehand and\nshould be found by minimizing the lattice parameters. To perform this task numerically we analyze a rectangular\nregion of the skyrmion lattice made of Nx\u0002Nyspins, which contains two skyrmions. The energy con\fguration\nminimizing the energy density E=NxNyis then found by the coordinate descent method in the con\fguration space\n(Nx,Ny). The minimal energy con\fgurations obtained for di\u000berent values of the parameter \rand for a \fxed magnetic\n\feldHext= 0:015Jare shown in Fig. s1 where a part of the skyrmion lattice with 210 \u0002210 spins is shown. As\nseen from the Figure, the increase of \rdeforms the skyrmion con\fguration from the equilateral triangular lattice (at\n\r= 0:0) to a isosceles triangular lattice (at \r= 0:5 and\r= 1:0). Similar procedure is implemented for the stripe\ncrystal. In this case the problem is simpli\fed since the optimization is required only along one direction (due to the\ntendency of stripes to align along the \\easy\" direction de\fned by the polarization of the applied radiation). The\nenergies of the stationary solutions obtained are, then, compared to the minimal energy identi\fed as a ground state."    },    {        "title": "1402.5844v2.Coplanar_waveguide_based_ferromagnetic_resonance_in_ultrathin_film_magnetic_nanostructures__impact_of_conducting_layers.pdf",        "content": "arXiv:1402.5844v2  [cond-mat.mes-hall]  9 May 2014Coplanar waveguide based ferromagnetic resonance in ultra thin film\nmagnetic nanostructures: impact of conducting layers\nH. G/suppress lowi´ nski,1M. Schmidt,1I. Go´ scia´ nska,2J. Dubowik,1and J-Ph. Ansermet3\n1)Institute of Molecular Physics, Polish Academy of Sciences , PL-60179 Pozna´ n,\nPoland\n2)Faculty of Physics, A.Mickiewicz University, Umultowska 8 5, PL-61614 Pozna´ n,\nPoland\n3)Institute of Condensed Matter Physics, Station 3, Ecole Pol ytechnique F´ ed´ erale de Lausanne EPFL,\nCH-1015 Lausanne, Switzerland\n(Dated: 12 May 2014)\nWe report broadband ferromagnetic resonance (FMR) measurem ents based on a coplanar waveguide (CPW)\nof ultrathin magnetic film structures that comprise in-plane/out-o f-plane decoupled layers deposited on non-\nmagnetic buffer layers of various thickness or other buffer struct ures with a diverse sheet resistance. We show\nthat the excitation of the fundamental mode can be substantially ( up to 10 times) enhanced in the structures\ndeposited on buffer layers with a low sheet resistance in comparison t o the structures deposited on thin or\nweakly conducting buffer layers. The results are analyzed in terms o f shielding of the electromagnetic field of\nCPW by the conducting buffer layers. The effect of enhancement of FMR absorption can be attractive for\napplications in spintronic devices that utilize magnetization dynamics o f ultrathin ferromagnetic layers.\nPACS numbers: 76.50.+g, 75.40.Gb, 75.40.Mg\nI. INTRODUCTION\nFerromagneticresonance(FMR)basedonavectornet-\nwork analyzer (VNA) and a coplanar waveguide (CPW)\nhas become a common experimental tool for studying\nmagnetic films and nanostructures.1–4A ferromagnetic\nfilm is placed close to the surface of the CPW so that a\nsubstrate is the furthest outer medium from the CPW.\nA microwave magnetic field ˜hproportional to the rf cur-\nrent in the central the CPW line excites the precession of\nthe magnetization ˜ m, which in turn induces a microwave\nvoltage in CPW. The FMR response is commonly ex-\ntractedfromthereflection( S11) ortransmission( S21)co-\nefficients ofscatteringparametersusingVNAand, hence,\nthe technique is referred to as the VNA-FMR3or the\nbroadbandFMR.5Inpractice,onlythechangesin S21(or\nS11) due to the FMR absorption are of interest and they\nare detected in a frequency-swept mode2or in a field-\nswept mode.4It has been proved that a change ∆ S21(or\n∆S11) due to microwave absorption of a single ferromag-\nnetic film is proportional to the complex susceptibility\nχ(ω) orχ(H).1,6,7The imaginary part of ∆ S21vs.H\nreflects the Lorentzian curve characteristic of the FMR\nabsorption. TheFMRabsorptionismeasuredatdifferent\nfrequencies to determine the effective saturation magne-\ntization 4 πMeff, the damping constant αas well as the\ninhomogeneous contribution to the linewidth ∆ H0.8In\nthe presentpaper weratherfocuson alessrecognizedpo-\ntential of the FMR technique: evaluation of the intensity\nof the FMR absorption defined as the integrated FMR\nabsorption, which if properly employed, can be used to\ndetermine the total magnetic moment.9\nInterpretation of the VNA-FMR experimental results\nhas been well established.3However, for metallic multi-\nlayers or magnetic films5,10in contact with conductingnonmagnetic layers11analysis of the experimental data\nis more complicated. In opposite to the standard FMR\nexperiments based on microwave cavities with a homoge-\nneous microwave field, in the CPW the microwave field\nis asymmetric relative a magnetic thin film and inhomo-\ngeneous due to the shielding of microwaves by the eddy\ncurrents.5,10In particular, image currents generated in\na floating ground conductor increased a pulsed induc-\ntive microwave magnetometer sensitivity, as well as the\nfield strength, resulting in a fourfold increase in over-\nall signal-to-noise ratio.12Recently, Bailleul11has shown\nwiththe aidoffinite-elementelectromagneticsimulations\nthat the propagation of microwave fields along the CPW\nis strongly modified when a nonmagnetic film is brought\nclose to it. This effect has been attributed to the shield-\ning of the electric ˜ eand/or magnetic field ˜hof the CPW\ndepending on the thickness of the metallic film. The\nshielding is expected to have important consequences for\nthe CPW based VNA-FMR experiments.11For example,\nit has been reported that the CPW efficiently excites\nhigher order standing spin wave modes across the film\nwith thickness of 30 - 90 nm and the amplitude of the\nmodes depends on ordering of FM layers with respect to\nthe CPW.5,13–15\nThis paper aims at broadening the above experiments\nto ultrathin ferromagnetic layers for which macrospin\nmodel is regarded to be fulfilled.8For such thin layers\n(a few nm in thickness) ∆ S21can sometimes be hardly\ndistinguished from the noise. Therefore, any enhance-\nment of the FMR response (∆ S21) is of importance for\nthe VNA-FMR measurements. The purpose of the pa-\nper is to investigate the effect of a nonmagnetic buffer\nlayer on the FMR response of systems that include a\nstack of ultrathin (buried) exchange decoupled ferromag-\nnetic layers with distinct effective magnetization (mag-2\n0.0 0.5 1.0 1.5 2.0 2.5 3.010-510-310-1Normalized Intensity\nθ (deg)0 10 20 30 40\nAFScattering Length\nDensity\nDepth (nm) BUFFER\nCAPIrMn\nSiTi 4Au 10Co 0.5\nPy 3Cu 3Co 3IrMn 10\nCu 4Au 5\nFreeAnalyz\nBuffer\nFIG. 1. X-ray reflectivity of the multilayer sample SC that\ncomprises an analyzer and a free layer on a Au buffer layer\n(lower inset). For simulation (blue curve) the same parame-\nters were used as that presented in Tab. I and the interface\nroughness of 0.5 - 0.7 nm. The composition profile of the\nwhole structure is shown in the upper inset.\nnetic anisotropy)sothatthe FMR responsesofeachlayer\nare well separatedin the field scale. In particular, we will\nexamine how the FMR absorption intensity of each mag-\nnetic layer depends on the thickness of the conducting\nbuffer layer and on their arrangement with respect to\nthe buffer.\nII. EXPERIMENTAL DETAILS\nThe multilayer thin films investigated in the present\npaper by using the VNA-FMR are intended for spin-\ntransfer oscillators16that comprise a [Au/Co] ×4 perpen-\ndicular polarizer, an in-plane magnetized [Py/Co] free\nlayerwithPermalloy(Py),andanin-planeCoanalyzerin\ncontactwithaIrMnantiferromagneticlayer. Thecompo-\nsition of the multilayers with the thickness of individual\nlayers in nanometers is shown in Tab. I. The multilayer\nfilms were deposited in a Prevac sputtering system onto\nthe high resistivity ( ρ >2 kΩ cm) Si/SiO 2substrates\nthat include [Ti/Au] buffer layers of various thickness.\nSince the Ti 4 nm films are used only for an improve-\nment in adhesive strength of our multilayer structures,\nthe [Ti/Au] layers will be referred to as the Au buffers.\nThe base pressure was less than 1 ×10−6Pa and the Ar\npressure was approximately10−2Pa. All structures were\ncovered with a 5 nm Au cap layer. For the VNA-FMR\ninvestigations, the films on the substrates of 19 ×15 mm\nwere cut to approximately 10 ×15 mm samples. The to-\ntal thickness of the structures investigated (30 - 100 nm),\nincluding the conducting buffer layers, is well below the\nskin depth at the microwave frequencies of 20 - 30 GHz.\nTwo reference samples, which comprise a 2.5 nm Co on\nthe Au (10 - 40 nm) and Au (30 - 60 nm) wedge buffers,\nwere deposited in the same conditions.Crystalline structure was determined using X-ray\ndiffraction. Diffraction profiles were measured in the\nBragg-Brentano geometry and analysis of the diffraction\nprofiles indicates that the Au buffers show a strong (111)\ntexture since only the (111) and (222) peaks were vis-\nible. The width of diffraction lines are 0.7 deg, 1 deg,\nfor the free Py/Co layer and IrMn, respectively. For the\n10 and 40 nm thick Au buffer, the width is 1.1 deg and\n0.4 deg, respectively. Using the Scherrer formula we es-\ntimate crystallite size as 9, 25 nm for the 10 and 40 nm\nAu buffer, respectively. On the basis of a fitting pro-\ncedure with the use of SimulReflec17software for x-ray\nreflectivity data the composition profiles of a few cho-\nsen multilayers were determined as it is shown in Fig. 1,\nas a typical example. Thickness of the individual lay-\ners (Fig. 1, the upper inset) is in agreement with that\nassumed from technological parameters. The roughness\nestimated from the fitting is of 0.5 - 0.7 nm.\nFIG. 2. A sketch of a piece of coplanar waveguide (CPW)\nwith typical thin film structure that comprises a magnetic\nmultilayer and a conducting buffer layer. A substrate above\nthe buffer is not shown for clarity.\nThemagnetizationreversalsofthe multilayerswereex-\namined using a standard vibrating sample magnetometer\nat room temperature. The measurements confirmed that\nthe multilayers comprise the perpendicular magnetized\npolarizer with the effective magnetization 4 πMeff≈ −3\nkG, the in-plane magnetized free layer with 4 πMeff≈8\nkG, and the analyzer with 4 πMeff≈12 kG and the\nexchange-bias field of 200 Oe. The individual P, F, and\nAmagneticlayershaveapproximatelythesamemagnetic\nmoments.\nIII. CPW VNA-FMR MEASUREMENT TECHNIQUE\nA broadband FMR spectrometer based on the VNA-\nFMR technique was used to measure the FMR spectra\nof multilayers in the in-plane geometry with an external\nmagnetic field applied in the sample plane. An in-plane\nmicrowave field with a frequency of 20 - 30 GHz was\napplied to the sample using a grounded CPW with a\n0.45 mm wide central strip. Our CPW design is simi-\nlar to that of Southwest Microwave Inc.18The samples\n(10×15 mm) were placed face down on the CPW so\nthat the buffer layer was always the furthest layer in the\ninvestigated multilayer structures as it is schematically3\nTABLE I. Composition of multilayer structures which compri se the polarizer (P), the analyzer (A), and the free layer (F)\nseparated in between by the Cu spacers. All samples are cover ed with the Au 5 nm cap layers. The polarizer consists of the\nAu/Co bilayers repeated four times. The reference samples c omprise only the buffer layer, the free Co layer, and the cap la yer.\nsample buffer sequence of the layers in the stack cap layer\nP F A\nSAaTi4/Au40/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n(Au1/Co0.7)4Cu4/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nPy3/Co0.5 Cu3/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nCo3/IrMn15 Au5\nP F A\nSA 1a(Ti2/Au2) 5b/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n(Au1/Co0.7)4Cu4/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nPy3/Co0.5 Cu3/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nCo3/IrMn15 Au5\nP F A\nSA 2a(Ti2/Au2) 10/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n(Au1/Co0.7)4Cu4/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nPy3/Co0.5 Cu3/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nCo3/IrMn15 Au5\nA F P\nSB Ti4/Au40/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nIrMn10/Co3 Cu3/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nCo0.5/Py3 Cu4/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n(Au1/Co0.7)4Au5\nF A\nSC Ti4/Au10 - Cu3/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nCo0.5/Py3 Cu4/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nCo3/IrMn10 Au5\nA F P\nSD Ti4/Au10/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nIrMn10/Co3 Cu3/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nCo0.5/Py3 Cu4/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n(Au1/Co0.7)4Au5\nF\nRef-1 Ti4/Au wedge(10-40) -/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nCo2.5 - Au5\nRef-2 Ti4/Au wedge(30-60) - Co2.5 - Au5\naSamples SA, SA1 and SA2 have the same structure except buffer l ayers.\nbThe subscripts denote the number of repetition and the other numbers denote thickness in nanometers.\nshown in Fig. 2. The complex transmission parameter\nS21was measured with the VNA at a fixed frequency -\ntypically 20 GHz - while the external magnetic field was\nswept between +10, 0, and -10 kOe. Since the FMR\nsignals were measured in two quadrants, the number of\nFMR peaks is doubled as it is shown in Fig. 3. We find\nsuch a measurement procedure helpful in estimating er-\nrors in calculations of the area under absorption curve.\nFigure3(a)showstypicalrealpartRe S21andimaginary\npart ImS21of a complex scattering factor S 21×exp(iφ)\nwith a phase φcorrection.19Two main features charac-\nterize Im S21vs.H: a quasi-parabolic background Im\nS0\n21which is related to a nonresonant background of the\nwhole microwave track and six characteristic absorption\npeaks (three for positive and three for negative H, re-\nspectively) plus one central peak. The three peaks P+,\nF+, and A+(or, P−, F−, and A−, for the negative field\ndirection) are related to the FMR absorption of the ex-\nchange decoupled polarizer, the free layer, and the an-\nalyzer, respectively. The central peak at H= 0 Oe is\nrelated to a additional absorption due to magnetization\nreversal of the F+A structure. The central peak will not\nbe discussed further in this paper.\nFigure 3 (a) clearly shows that the experimental data\nof ImS21can be broken into magnetic and nonmagnetic\ncontributions assuming that a reflection of microwave\npower is weak in our VNA-FMR set-up. Therefore, fol-\nlowing a similar analysis discussed in Refs. 6 and 7, thecomplex S21scattering term may be expressed as\nS21(H,t)≈S0\n21(H,t)+χ(H)\nχ0, (1)\nwhereχ(H) is complex microwave susceptibility, χ0is\ncomplex function of the experimental parameters , such\nas frequency and film thickness7. Furthermore, time t\ntakes into account some drift of S0\n21during measure-\nments. Assuming that S0\n21(H,t) depends on Hin a non-\nresonant way and also depends on t, we can reasonably\napproximate S0\n21(H,t)≈AH+BH2so that we arrive at\na simple relation\nχ(H)≈χ0∆S21(H), (2)\nwhere ∆ S21=S21−S0\n21. Figure 3 (b) shows the mea-\nsured Im ∆ S21(H)) after subtraction of the nonresonant\nbackground Im S0\n21(H). It can be easily shown that\nthe experimental spectrum can be deconvoluted using a\nset of Lorentzians. Slight differences in the height and\nlinewidth ∆ H(FWHM) for P+, P−and F+,F−peaks\nservehereasaroughestimateofuncertaintiesindetermi-\nnation of the ∆ S21absorptions in our VNA-FMR set-up.\nOntheotherhand, thesubstantialdifferencebetweenA+\nand A−is due to the unidirectional anisotropy of the an-\nalyzer. Keeping in mind that χ′′≈χ0Im∆S21(H), we\ncan further express the area under the FMR absorption\npeakIas\nI∝/integraldisplay\nχ′′dH. (3)4\n-0.165-0.164\n Im S21 (V/V)\n-10 -5 0 5 100.0000.001\nA+F+\nP+ A-F-Im ∆S21 (V/V)\nH (kOe)P--0.043-0.042\n(b)\nRe S21 (V/V)Re\nIm(a)\nFIG. 3. (a) Typical in-plane VNA-FMR spectrum of the SA\nstructure with the real Re S21and imaginary Im S21parts\nof the complex transmission parameter S21. (b) The same\nspectrum with the values of Im ∆ S21adjusted by removing a\nbackground (red dashed curve). The spectrum was measured\nwith the magnetic field sweep from + 10 kOe to -10 kOe\nso that the number FMR absorptions is doubled. The central\npeak atH= 0 is related to an absorption due to magnetiza-\ntion reversal. Blue lines (in colors - online) show the fits of\nthe spectra with the Lorentzians.\nIn theory, the intensity of the FMR absorption Imea-\nsuredinamicrowavecavityisproportionalthetotalmag-\nnetic moment.20However, in this case the FMR intensity\nstudies require a microwave system which can provide\nreproducible results with a special emphasis on a mi-\ncrowave cavity coupling and a cavity quality factor.9In\ncontrast to the discussion in Ref. 4, we have found the\nmagnitude of the FMR absorption ( ∝Im ∆S21) quite\nstable for the structures of the same size and the same\ncomposition. It suggests that we can compare the in-\ntensities of FMR absorption of various samples provided\nthat the measurement conditions in the CPW set-up are\nthe same.\nIV. VNA-FMR RESULTS\nUsing our CPW-VNA set-up, we have measured FMR\nofthree SA, SA1, and SA2 structuresthat compriseiden-0 2 4 6 8 100.00000.00050.00100.0015\n10.68 10.70 10.720.00000.00010.0002(b)Im ∆S21 (V/V)\nH (kOe) (Ti 2/Au 2)x5\n (Ti 4/Au 40)\n (Ti 2/Au 2)x10\n20 GHz(a)\n8 10Im ∆S21 (V/V)\nH (kOe)Im ∆S21 (V/V)\nH (kOe) DPPH on Ti(4)/Au(40) buffer\n DPPH on silicon \n30 GHz\nFIG. 4. (a) Comparison of the FMR absorptions in three\nmagnetic structure (SA, SA1, and SA2) in contact with buffer\nlayersofvariousstructures(sheetresistance). (b)Compa rison\nof EPR signal from a DPPH film on a 40 nm Au buffer and\non a bare Si substrate.\ntical P, F, and A but different buffer layers (see Tab. I,\nfor details). As it is shown in Fig. 4 (a), the same posi-\ntions of the resonance field of the P, F, and A layers in\nthe three structures prove that the magnetic layers have\nthe same magnetic properties (e.a., the same values of ef-\nfective magnetization and the same exchange bias of the\nanalyzer) In contrast, the signal amplitude of Im ∆ S21\nfor SA with the thickest Au buffer is nearly 6 - 7 times\nhigher than those of SA1 and SA2. The effect of signal\nenhancement is even more pronounced for the polarizer\nP with the perpendicular anisotropy(see the inset in Fig.\n4 (a)). While the FMR absorptions of the polarizer are\nbarely seen from the noise for the samples SA1 and SA2,\ntheFMR absorptionofPin SAissubstantialandcompa-\nrablewiththoseofAandF.Asitwasshowninourrecent\npaper,21the enhancement in this case is presumably ad-\nditionally influenced by a better texture and crystallite\nsize of the polarizer, which was grown on the thick Au\n40 nm buffer layer. Therefore, Fig. 4 (a) can be regarded\nas an experimental evidence of shielding of the electro-\nmagnetic field in the CPW by a conducting film with a\nlow sheet resistance. It is shown that a highly conduct-\ning buffer layer that is the outer conducting layer from5\n4\n5\n60.00000.00050.0010\n102030405060\n0.02 0.04 0.060.20.40.60.8(b)Im ∆ S21 (V/V)\nd Au (nm)H (kOe)20 GHz (a)ln (I/I0)\n1/d Au (nm-1)\nFIG. 5. (a) 3-D plot of FMR absorption of a 2.5nm Co film\ndeposited onto Ti 4/Au (10 - 40) (red curves - sample Ref-1)\nand Ti 4/Au (30 - 60) wedge (blue curves - sample Ref-2),\nrespectively. (b) Log-lin dependence of the normalized FMR\nintensity I/I0on the inverse thickness of Au buffer layer.\nCPW can beneficiary affect the excitation of the funda-\nmental mode(s) in our ultrathin film structure. We have\nchecked with a four-point probe that the sheet resistance\nof SA, SA2, and SA1 buffer layers is of 0.5, 15 and 30 Ω,\nrespectively.\nSuch a change in the sheet resistance has recently been\nshown to strongly affect shielding of either ˜hand/or ˜e\nfields.11To check if the field ˜his really enhanced due\nto a conducting buffer alone, we performed a similar\nexperiment using DPPH - the common EPR standard\ncompound - dissolved in a nonconducting glue and then\ndeposited on a bare Si substrate and on a Si substrate\ncovered with a Au 40 nm buffer layer, respectively. As it\nis shown in Fig. 4 (b), there is no substantial signal en-\nhancement due to the thick Au buffer. Since the DPPH\n”layers”are insulating and very thick (of 100 µm), we at-\ntribute the FMR signal enhancement for the conducting\nSA, SA1, and SA2 structures to their close proximity of\nconducting nonmagnetic layers.\nThe VNA-FMR measurements of a single ultrathin Co\nfilm in contact with a wedged Au buffer give reference\ndata for quantitative analysis. For this purpose, we mea-\nsured Im ∆ S21for the 2.5 nm Co layer deposited on the\n10 - 40 nm and Au 30 - 60 nm Au wedges (see Tab. I).\nTheresultsoftheFMRmeasurementsareshowninFig.5\n(a) as a 3-D plot. It is clearly seen that the amplitudeTABLE II. Ratio of magnetic moments and FMR intensity\nratios of the P, F, and A layers in the SA, SB, and SD struc-\ntures.\nSample mP:mF:mA\nSA,SB,SD 1 0.8 1.07\nIP:IF:IA\nSA - inverse structure 1 0.84 0.97\nSB - simple structure 1 0.86 0.58\nSD - simple structure 1 0.64 0.40\nof the FMR absorption of the Co 2.5 nm layer increases\nwith the thickness of Au buffer layer and saturates at its\nthickness of 40-60 nm. It appears that a dependence of\nthe intensity Ivs. 1/dAu(see Fig. 5 (b)) can be approx-\nimated by the following expression:\nI=I0exp (−do\ndAu) =I0exp (−Rsr\nRosr),(4)\nwheredois of 38 nm and I0≈1.7 are the fitting param-\neters. If we define the sheet resistance of Au buffer layer\nasRsr=ρ/dAuEq. (4) can be alternatively expressed in\nterms of the sheet resistance. By assuming the resistiv-\nity of bulk gold as 3 µΩcm,Rsrof the Au buffer varies\nfrom 2 Ω to 0.33 Ω for the Au thickness of 10 and 60\nnm, respectively. Fitting to the experimental data using\nEq. (4) gives Ro\nsr≈0.8 Ω.\nFig.6showsthe effectoftheFMR absorptionenhance-\nment observed in more complex structures SA, SB, SC,\nand SD that include the P, F, and A magnetic layers in\nvarious arrangements with respect to the buffer layers\n(Tab. I). For some purposes, which are out of scope of\nthe present paper, the sample SC has no polarizer. Com-\nparing Figs. 6 (SA), (SB) with (SC), and (SD), one can\nsee that the FMR amplitudes for the samples SA and SB\ndeposited on the Au 40 nm buffers are about ten times\nhigher than those of the samples SC and SD deposited\nonto the Au 10 nm buffers. Besides, a clear decrease in\nthe signal-to-noise ratio is seen in Fig. 6 for SA and SB\nin comparison with the SC and SD structures.\nAs can be seen in Tab. I, the multilayer samples differ\nin sequences ofthe magnetic layers. The P-F-Aand A-F-\nP structures are referred to as the simple and the inverse\nstructures,respectively. Letusexaminetheimpactofthe\narrangements of magnetic layers on their FMR intensity\nratios. According to Eq. (3), the FMR intensity ratio\nofIPtoIFtoIAshould be the same as the magnetic\nmoments ratio mP:mF:mA(Tab. II) provided the\ndynamic field ˜his homogeneous. However, Tab. II shows\nthatIP:IF:IAratios depend on an arrangement of\nmagneticlayers. ForthesimplestructuresSBandSDthe\nintensity ratios are distinctly different from that of mP:\nmF:mA. Moreover, the intensity of FMR absorption is\nstrongly diminished for the analyzer A with a IrMn layer\nin close contact with the Au buffer. On the other hand,\nfor the inversestructure SA the FMR intensity ratio does\nnot differ much from the ratio of the magnetic moments.6\n0.05.0x10-41.0x10-31.5x10-3\nSD SBSCAbsorption (arb.u.)SA\n0.01.0x10-42.0x10-4Absorption (arb.u.)\n-10 0 100.05.0x10-41.0x10-31.5x10-3Absorption (arb.u.)\nMagnetic field (kOe)-10 0 100.01.0x10-42.0x10-4Absorption (arb.u.)\nMagnetic field (kOe)F-\nP-A-\nA+F+\nP+P-A-F-\nF+\nA+\nP-A-F-\nF+\nA+P+P-\nA-F-\nF+\nA+ P+A-F-F+\nA+\nFIG. 6. FMR absorption vs. Hof a series of SA - SD multilayer thin film structures prepared for spin-transfer-torque oscillators\nwith a perpendicular polarizer P, a free layer F, and an in-pl ane analyzer A pinned to IrMn layer. Blue lines (in colors - on line)\nshow the fits of the spectra with the Lorentzians. The ”centra l”peak is subtracted for clarity.\nV. DISCUSSION\nDiscussion of our experimental data is based on the\nessential results of Ref. 14. ( i) In contrast to the com-\nmon cavity FMR measurements, a conducting thin film\nsample in the CPW - FMR is illuminated by microwaves\nasymmetrically from the front surface of the film as it\nis shown in Fig. 7. ( ii) In such a geometry, the thin\nfilm sample with a thickness dless than the skin depth\nthe microwave magnetic field decays more strongly than\nexponentially. ( iii) In a highly conducting film the mi-\ncrowave magnetic field is strongly inhomogeneous.\nWe can write the scattering parameter S21in terms\nof the complex reflection coefficient Γ and the complex\npropagation factor γ0γf.3\nS21\nS0\n21=Γ2−1\nΓ2exp(−γ0γfδ)−exp(γ0γfδ).(5)\nγ0=iω/υphis the complex propagation factor of the\nunloaded CPW, where ωis the angular frequency of mi-\ncrowave field and υphis the phase velocity of microwaves\ninthetheCPW. γfisthepropagationindexoftheloaded\nCPW.δis the film width and S0\n21is the scattering pa-\nrameter of the empty CPW. Keeping only linear term in\nthe expansion of Eq. (5) and assuming that |Γ| ≪1,5we\nobtain\nS21\nS0\n21= exp(−γ0γfδ). (6)The propagation index γfcan be further approximated\nin termsofthe characteristicimpedance γf=/radicalBig\nZ0−Zr\nZ0≈\n1 +Zr\n2Z0, (Eq. (13) in Ref. 14), where Z0is the charac-\nteristic impedance of the unloaded CPW and Zris the\nsurface impedance of a thin film placed on the CPW.\nHence,\nS21\nS0\n21= exp/parenleftbigg\n−γ0Zr\n2Z0δ/parenrightbigg\n. (7)\nAccording to Ref. 11\nZr=Rsrδ\nw=ρ\ndδ\nw, (8)\nwhereRsris the sheet resistance and wis a width of the\ncentral line of CPW (see. Fig. 2 in Ref. 11). Eventually,\nin agreement with Ref. 14, we can express the measured\nscattering coefficient S21in terms of geometrical param-\neters of CPW and the film placed on it\nS21\nS0\n21∝exp/parenleftBigg\n−γ0ρδ2\n2Z0w\nd/parenrightBigg\n∝exp/parenleftbigg\n−d0\nd/parenrightbigg\n,(9)\nwhich has the same form as the fitting formula Eq. (4)\nto the experimental data shown in Fig. 5. Let us esti-\nmated0. ForZ0= 25−50 Ω,|γ0|=ω/υph= 7 cm−1,\nδ= 0.4 cm,υph= 1.8×1010cm/s, and ω/2π= 20 GHz\nthe estimated range of d0is between 8 and 15 nm if we\nassume the resistivity ρ= 3µΩcm of the gold buffer the7\n1 2 3\ndy\nFIG. 7. Sketchshowing astructurethatcomprise an ultrathi n\nCo film and a Au wedge 10 < d <50 nm. Microwaves are\npartially reflected and transmitted through the wedge.\nsame as for bulk. In practice, the resistivity of several\nnanometers thick gold films is several times higher22so\nthatdo= 38 nm estimated from fitting of the experi-\nmental data (Fig. 5) according to Eq. (4) is in agreement\nwith the above model.\nLet us consider an ultrathin Co 2.5 nm film deposited\non a gold wedged buffer layer of thickness 10 < d <50\nnm as it shown in Fig. 7. The Co film plays the role of a\ntag useful for monitoring of the dynamic magnetic field\n˜h. Since the Co film is very thin in comparison to the\nAu buffer, we assume that the entire structure has the\nconductivity of gold wedge (region 2). A transverse wave\nwith a wavenumber k1and with the amplitude equal to\nunity incidents perpendicular to the film surface from re-\ngion 1 with the permittivity ǫ1the same asfor region3.14\nThe permittivity of region 2 ǫ2is complex. Taking con-\ntinuity boundary conditions at the boundaries of regions\n1, 2, and 3 for ˜hxand ˜ezwe have\n˜hx1= exp(−ik1y)+B1exp(ik1y), (10)\n˜hx2=A2exp(−ik1y)+B2exp(ik1y),(11)\n˜hx3=A3exp(−ik1y) (12)\nwith\nB1=(r−1)(exp(2ik2d)−1)\nD, (13)\nA2=2(1+√r)exp(2ik2d)\nD, (14)\nB2=2(√r−1)\nD, (15)\nA3=4√rexp(ik2d(1+√r))\nD, (16)\nwherer=ǫ1/ǫ2andD= exp(2ik2d)(1+√r)2−(1−√r)2.\nSinceǫ1= 1 and ǫ2of gold is imaginary and very large,5\n|r|≪1 so that after expanding exponential functions\nandkeepingonlythelineartermsofexpansionsweobtain\nthat˜hvaries linearly with yas\n˜hx2≈2d−y\nd, (17)where 2 on the right-hand side denotes that the ampli-\ntude at the front of the structure is doubled due to the\npositive interference.\nIn order to extend the model of inhomogeneous dy-\nnamic magnetic field within more complicated structure\nthat comprise P+F+A layers and the Au buffer layer,\nlet us compare the FMR intensity of the CPW-FMR re-\nsponses shown in Fig. 6. In contrast to the single ultra-\nthin Co film on the Au buffer layer, the P+F+A ferro-\nmagnetic structure is more extended (of ∼20 nm) and\nconsists of the exchange decoupled (the Cu spacers are 3\n- 4 nm thick) Co and Permalloy layers with diverse effec-\ntive anisotropies. This makes possible observation of the\nwell separated the FMR absorptions of each layer. Ex-\nact calculations of the electromagnetic field distribution\nin such structures would require a set of many boundary\nconditions15with several material parameters. However,\nwe make use of Eq. (17) taking into account that ˜hin\nthe multilayer is a linear combination of ˜hin individual\nlayers and its slope scales with a sheet resistance ( ρ/d)\nof the individual layers.5Hence, the lower Rrsthe higher\nis the slope of ˜hwithin a layer. Rrsvalues of the indi-\nvidual layers in the entire stack are quite diverse - Rrs\nis the highest for the IrMn layer23and the lowest for the\nAu buffers. Possible distributions of the dynamic field ˜h\n(black lines) are shown in Fig. 8 (a), (b), and (c) for SA,\nSB, and SD structures, respectively. In other words, we\n0306090\n h\nIrMn A+F+P Au(a)\n0306090\n~\n~~\n \n h (arb. units)Rsr (Ohm)h\nIrMn P+F+A Au(b)~\n0 10 20 30 40 50 60 700306090\ny(nm)h\nIrMn P+F+A Au(c)\nFIG. 8. Distribution of the dynamic magnetic field ˜hinside\nSA (a), SB (b), and SD (b) structures with diverse sheet\nresistance Rrsof individual layers.8\nassume in accordance with Eq. (17) that ˜h(0) = 2 and\n˜h(d) = 0 what is generally not true for very thin stacks\n(see Fig. 6 in Ref. 14) so that the microwavescan be par-\ntially transmitted out of the stack as it is shown in Fig. 8\n(c). Nonetheless, the distributions of ˜hjust depict graph-\nically that its slope in the P+F+A stacks is the highest\nfor the SD sample and the lowest for the SA sample so\nthat the magnetic field inside the SA sample is the most\nhomogeneous. It is also seen that the position of antifer-\nromagnetic IrMn pinning layer plays an important role\nin the distribution of ˜hbecause its resistivity23is about\n100 times higher than that of Au.22Hence, in accordance\nwith the sketches shown in Fig. 8, the dynamic magnetic\nfield is the most inhomogeneous for the SD sample with\nthe simple P+F+A structure. In contrast, for the SA\nsample with the inverse A+F+P structure the FMR in-\ntensity ratio is in a fairly good agreement with the ratio\nof magnetic moments estimated from geometry of P, F,\nand A layers.\nVI. CONCLUSIONS\nWe have expanded former1–7,10,13,14coplanar waveg-\nuide based VNA-FMR studies of thin magnetic films to\nultrathinmagneticstructuresdepositedonthe bufferlay-\nerswith diversesheetresistance Rsr. We showedthatthe\nintensity of the FMR absorption of the single ultrathin\nCo layer depends on the thickness dAuof the conduct-\ning Au buffer ∝exp(−d0/dAu) or, equivalently, on the\nbuffer sheet resistance ∝exp(−Rsr/R0\nsr). We showed\nthat the measured FMR absorption intensities of struc-\ntures composed of several exchange decoupled ultrathin\nmagnetic layers do not scale in proportion to their mag-\nnetic moments as would be expected. On the contrary,\nthe ratios of FMR absorption intensity of the individual\nP, F, and A layers depend on their arrangement with\nrespect of the buffer layer. The above mentioned find-\nings are interpreted in terms of the microwave shielding\neffect by the conducting nonmagnetic buffers and the in-\nhomogeneous dynamic field ˜h. The coplanar waveguides\n(micro-antennas) are widely used in numbers of spin-\ntronic devices24and the enhancement of FMR response\nhas potential to be applied in spintronic devices.ACKNOWLEDGMENT\nThis research has been conducted in framework of\nProject NANOSPIN PSPB-045/2010 supported by a\ngrant from Switzerland through Swiss contribution to\nthe enlarged European Union. The authors thanks Dr\nB. Szyma´ nski for assistancewith x-raymeasurements, A.\nKrysztofik for assistance with FMR measurements, and\nDr P. Balaˆ z for his help with some calculations.\n1G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shigeto,\nand Y. Otani, J. Appl. Phys. 95(2004).\n2I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gub-\nbiotti, and C. Back, J. Magn. Magn. Mater. 307, 148 (2006).\n3C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, a nd\nP. Freitas, J. Appl. Phys. 101, 074505 (2007).\n4I. Harward, T. O’Keevan, A. Hutchison, V. Zagorodnii, and\nZ. Celinski, Rev. Sci. 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S. Maksymov and M. Kostylev, J. Appl. Phys. 113, 043927\n(2013).\n16D. Houssameddine, U. Ebels, B. Dela¨ et, B. Rodmacq, I. Fi-\nrastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel ,\nL. Prejbeanu-Buda, et al. , Nature materials 6, 447 (2007).\n17SimulReflec, “Lab. Leon Brillouin CEA/CNRS UMR12,” (2007),\nhttp://www-llb.cea.fr/prism/programs/simulreflec/sim ulreflec.\n18B. Rosas, “The Design and Test of Broadband Launches\nup to 50 GHz on Thin and Thick Substrates,” (2011),\nhttp://mpd.southwestmicrowave.com/resources/.\n19S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schne i-\nder, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys.\n99, 093909 (2006).\n20A. Gurevich and G. Melkov, Magnetization Oscillations and\nWaves (CRC Press, 1996).\n21M. Matczak, B. Szyma´ nski, M. Urbaniak, M. Nowicki,\nH. G/suppress lowi´ nski, P. Ku´ swik, M. Schmidt, J. Aleksiejew, J. Du bowik,\nand F. Stobiecki, J. Appl. Phys. 114, 093911 (2013).\n22J. R. Sambles, K. C. Elsom, and D. J. Jarvis, Phil. Trans. of th e\nRoyal Soc. of London A 304, 365 (1982).\n23R. Y. Umetsu, K. Fukamichi, and A. Sakuma, Mater. Trans. 47,\n2 (2006).\n24V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Appl.\nPhys.115, 043917 (2014)."    },    {        "title": "1909.08004v1.Microwave_induced_tunable_subharmonic_steps_in_superconductor_ferromagnet_superconductor_Josephson_junction.pdf",        "content": "arXiv:1909.08004v1  [cond-mat.supr-con]  17 Sep 2019Microwave induced tunable subharmonic steps in\nsuperconductor-ferromagnet-superconductor Josephson j unction\nM. Nashaat,1,2,∗Yu. M. Shukrinov,2,3,†A. Irie,4A.Y. Ellithi,1and Th. M. El Sherbini1\n1Department of Physics, Cairo University, Cairo, 12613, Egy pt\n2BLTP, JINR, Dubna, Moscow Region, 141980, Russian Federati on\n3Dubna State University, Dubna, 141982, Russian Federation\n4Department of Electrical and Electronic Systems Engineeri ng, Utsunomiya University, Utsunomiya, Japan.\nWe investigate the coupling between ferromagnet and superc onducting phase dynamics in\nsuperconductor-ferromagnet-superconductor Josephson j unction. The current-voltage character-\nistics of the junction demonstrate a pattern of subharmonic current steps which forms a devil’s\nstaircase structure. We show that a width of the steps become s maximal at ferromagnetic reso-\nnance. Moreover, we demonstrate that the structure of the st eps and their widths can be tuned\nby changing the frequency of the external magnetic field, rat io of Josephson to magnetic energy,\nGilbert damping and the junction size.\nThis paper is submitted to LTP Journal.\nI. INTRODUCTION\nJosephson junction with ferromagnet layer (F) is\nwidely considered to be the place where spintronics and\nsuperconductivity fields interact1. In these junctions\nthe supercurrent induces magnetization dynamics due\nto the coupling between the Josephson and magnetic\nsubsystems. The possibility of achieving electric con-\ntrol over the magnetic properties of the magnet via\nJosephson current and its counterpart, i.e., achieving\nmagnetic control over Josephson current, recently at-\ntracted a lot of attention1–7. The current-phase rela-\ntion in the superconductor-ferromagnet-superconductor\njunction (SFS) junctions is very sensitive to the mutual\norientation of the magnetizations in the F-layer8,9. In\nRef.[10] the authors demonstrate a unique magnetization\ndynamics with a series of specific phase trajectories. The\norigin of these trajectories is related to a direct coupling\nbetween the magnetic moment and the Josephson oscil-\nlations in these junctions.\nExternal electromagnetic field can also provide a cou-\npling between spin wave and Josephson phase in SFS\njunctions11–17. Spin waves are elementary spin excita-\ntions which considered to be as both spatial and time\ndependent variations in the magnetization18,19. The fer-\nromagnetic resonance(FMR) correspondsto the uniform\nprecession of the magnetization around an external ap-\nplied magnetic field18. This mode can be resonantly ex-\ncited by ac magnetic field that couples directly to the\nmagnetization dynamics as described by the Landau-\nLifshitz-Gilbert (LLG) equation18,19.\nIn Ref.[18] the authors show that spin wave resonance\nat frequency ωrin SFS implies a dissipation that is mani-\nfested as adepressionin the IV-characteristicofthe junc-\ntion when /planckover2pi1ωr= 2eV, where/planckover2pi1is the Planck’s constant,\ne is the electron charge and Vis the voltage across the\njunction. The ac Josephson current produces an oscil-\nlating magnetic field and when the Josephson frequencymatches the spin wave frequency, this resonantly excites\nthe magnetization dynamics M(t)18. Due to the non-\nlinearity of the Josephson effect, there is a rectification\nof current across the junction, resulting in a dip in the\naverage dc component of the suppercurrent18.\nIn Ref.[13] the authors neglect the effective field due\nto Josephson energy in LLG equation and the results re-\nveal that even steps appear in the IV-characteristic of\nSFS junction under external magnetic field. The ori-\ngin of these steps is due to the interaction of Cooper\npairs with even number of magnons. Inside the ferro-\nmagnet, if the Cooper pairs scattered by odd number of\nmagnons, no Josephson current flows due to the forma-\ntion of spin triplet state13. However, if the Cooper pairs\ninteract with even number of magnons, the Josephson\ncoupling between the s-wave superconductor is achieved\nand the spin singlet state is formed, resulting in flows of\nJosephsoncurrent13. In Ref.[20]weshowthat takinginto\naccount the effective field due to Josepshon energy and\nat FMR, additional subharmonic current steps appear in\nthe IV-characteristic for overdamped SFS junction with\nspin wave excitations (magnons). It is found that the po-\nsition of the current steps in the IV-characteristics form\ndevil’s staircase structure which follows continued frac-\ntion formula20. The positions of those fractional steps\nare given by\nV=\nN±1\nn±1\nm±1\np±..\nΩ, (1)\nwhere Ω = ω/ωc,ωis the frequency of the external ra-\ndiation, ωcis the is the characteristic frequency of the\nJosephson junction and N,n,m,pare positive integers.\nIn this paper, we present a detailed analysis for the\nIV-characteristics of SFS junction under external mag-\nnetic field, and show how we can control the position\nof the subharmonic steps and alter their widths. The\ncoupling between spin wave and Josephson phase in SFS\njunction is achieved through the Josephson energy and\ngauge invariant phase difference between the S-layers. In\nthe framework of our approach, the dynamics of the SFS2\njunction isfully describedbytheresistivelyshuntedjunc-\ntion (RSJ) model and LLG equation. These equations\nare solved numerically by the 4thorder Runge-Kutta\nmethod. The appearance and position of the observed\ncurrent steps depend directly on the magnetic field and\njunction parameters.\nII. MODEL AND METHODS\nF\nss\nHacxyz\nH0I\nI\nFIG. 1. SFS Josephson junction. The bias current is applied\nin x-direction, an external magnetic field with amplitude Hac\nand frequency ωis applied in xy-plane and an uniaxial con-\nstant magnetic field H0is applied in z-direction.\nIn Fig 1 we consider a current biased SFS junction\nwhere the two superconductors are separated by ferro-\nmagnet layer with thickness d. The area of the junction\nisLyLz. An uniaxial constant magnetic field H0is ap-\nplied in z-direction, while the magnetic field is applied in\nxy-plane Hac= (Haccosωt,Hacsinωt,0)withamplitude\nHacand frequency ω. The magnetic field is induced in\nthe F-layer through B(t) = 4πM(t), and the magnetic\nfluxes in z- and y-direction are Φ z(t) = 4πdLyMz(t),\nΦy(t) = 4πdLzMy(t), respectively. The gauge-invariant\nphase difference in the junction is given by21:\n∇y,zθ(y,z,t) =−2πd\nΦ0B(t)×n, (2)\nwhereθis the phase difference between superconducting\nelectrodes, and Φ 0=h/2eis the magnetic flux quantum\nandnis a unit vector normal to yz-plane. The gauge-\ninvariantphasedifference in terms ofmagnetizationcom-\nponents reads as\nθ(y,z,t) =θ(t)−8π2dMz(t)\nΦ0y+8π2dMy(t)\nΦ0z,(3)\nwhere Φ 0=h/(2e) is the magnetic flux quantum.\nAccordingtoRSJ model, the currentthroughthe junc-\ntion is given by13:\nI\nI0c= sinθ(y,z,t)+Φ0\n2πI0cRdθ(y,z,t)\ndt,(4)\nwhereI0\ncis the critical current, and R is the resistance\nin the Josephson junction. After taking into account thegaugeinvarianceincludingthemagnetizationoftheferro-\nmagnetandintegratingoverthejunction areatheelectric\ncurrent reads13:\nI\nI0c=Φ2\nosin(θ(t))sin/parenleftBig\n4π2dMz(t)Ly\nΦo/parenrightBig\nsin/parenleftBig\n4π2dMy(t)Lz\nΦo/parenrightBig\n16π4d2LzLyMz(t)My(t)\n+Φ0\n2πRI0cdθ(y,z,t)\ndt. (5)\nThe applied magnetic field in the xy-plane causes pre-\ncessionalmotionofthemagnetizationinthe F-layer. The\ndynamics of magnetization Min the F-layer is described\nby LLG equation\n(1+α2)dM\ndt=−γM×Heff−γ α\n|M|[M×(M×Heff)](6)\nThe total energy of junction in the proposed model is\ngivenby E=Es+EM+EacwhereEsistheenergystored\nin Josephson junction, EMis the energy of uniaxial dc\nmagnetic field (Zeeman energy) and Eacis the energy of\nac magnetic field:\nEs=−Φ0\n2πθ(y,z,t)I+EJ[1−cos(y,z,t)],\nEM=−VFH0Mz(t),\nEac=−VFMx(t)Haccos(ωt)−VFMy(t)Hacsin(ωt)(7)\nHere,EJ= Φ0I0\nc/2πis the the Josephson energy, H0=\nω0/γ,ω0is the FMR frequency, and VFis the volume of\nthe ferromagnet. We neglect the anisotropy energy due\nto demagnetizing effect for simplicity. The effective field\nin LLG equation is calculated by\nHeff=−1\nVF∇ME (8)\nThus, the effective field Hmdue to microwave radiation\nHacand uniaxial magnetic field H0is given by\nHm=Haccos(ωt)ˆex+Hacsin(ωt)ˆey+H0ˆez.(9)\nwhile the effective field ( Hs) due to superconducting part\nis found from\nHs=−EJ\nVFsin(θ(y,z,t))∇Mθ(y,z,t).(10)\nOne should take the integration of LLG on coordinates,\nhowever, the superconducting part is the only part which\ndepends on the coordinate so, we can integrate the ef-\nfective field due to the Josephson energy and insert the\nresult into LLG equation. Then, the y- and z-component\nare given by\nHsy=EJcos(θ(t))sin(πΦz(t)/Φ0)\nVFπMy(t)Φz(t)/bracketleftbigg\nΦ0cos(πΦy(t)/Φ0)\n−Φ2\n0sin(πΦy(t)/Φ0)\nπΦy(t)/bracketrightbigg\nˆey, (11)\nHsz=EJcos(θ(t))sin(πΦy(t)/Φ0)\nVFπMz(t)Φy(t)/bracketleftbigg\nΦ0cos(πΦz(t)/Φ0)\n−Φ2\n0sin(πΦz(t)/Φ0)\nπΦz(t)/bracketrightbigg\nˆez. (12)3\nAs a result, the total effective field is Heff=Hm+\nHs. In the dimensionless form we use t→tωc,ωc=\n2πI0\ncR/Φ0is the characteristic frequency, m=M/M0,\nM0=∝ba∇dblM∝ba∇dbl,heff=Heff/H0,ǫJ=EJ/VFM0H0,hac=\nHac/H0, Ω =ω/ωc, Ω0=ω0/ωc,φsy=4π2LydM0/Φo,\nφsz=4π2lzdM0/Φo. Finally, the voltage V(t) =dθ/dtis\nnormalized to /planckover2pi1ωc/(2e). The LLG and the effective field\nequations take the form\ndm\ndt=−Ω0\n(1+α2)/parenleftbigg\nm×heff+α[m×(m×heff)]/parenrightbigg\n(13)\nwith\nheff=haccos(Ωt)ˆex+(hacsin(Ωt)+ΓijǫJcosθ)ˆey\n+ (1+Γ jiǫJcosθ)ˆez, (14)\nΓij=sin(φsimj)\nmi(φsimj)/bracketleftbigg\ncos(φsjmi)−sin(φsjmi)\n(φsjmi)/bracketrightbigg\n,(15)\nwherei=y,j=z. The RSJ in the dimensionless form is\ngiven by\nI/I0\nc=sin(φsymz)sin(φszmy)\n(φsymz)(φszmy)sinθ+dθ\ndt.(16)\nThe magnetization and phase dynamics of the SFS\njunction can be described by solving Eq.(16) together\nwith Eq.(13). To solve this system of equations, we em-\nploy the fourth-order Runge-Kutta scheme. At each cur-\nrent step, we find the temporal dependence of the volt-\nageV(t), phase θ(t), andmi(i=x,y,z) in the (0 ,Tmax)\ninterval. Then the time-average voltage Vis given by\nV=1\nTf−Ti/integraltext\nV(t)dt, whereTiandTfdetermine the in-\nterval for the temporal averaging. The current value is\nincreased or decreased by a small amount of δI (the bias\ncurrent step) to calculate the voltage at the next point\nof the IV-characteristics. The phase, voltage and mag-\nnetization components achieved at the previous current\nstep are used as the initial conditions for the next cur-\nrent step. The one-loop IV-characteristic is obtained by\nsweeping the bias current from I= 0 toI= 3 and back\ndown to I= 0. The initial conditions for the magnetiza-\ntion components are assumed to be mx= 0,my= 0.01\nandmz=/radicalBig\n1−m2x−m2y, while for the voltage and\nphase we have Vini= 0,θini=0. The numerical param-\neters (if not mentioned) are taken as α= 0.1,hac= 1,\nφsy=φsz= 4,ǫJ= 0.2 and Ω 0= 0.5.\nIII. RESULTS AND DISCUSSIONS\nItiswell-knownthatJosephsonoscillationscanbesyn-\nchronized by external microwave radiation which leads\nto Shapiro steps in the IV-characteristic22. The position\nof the Shapiro step is determined by relation V=n\nmΩ,\nwheren,mare integers. The steps at m= 1 are calledharmonics, otherwise we deal with synchronized subhar-\nmonic (fractional) steps. We show below the appearance\nof subharmonics in our case.\nFirst we present the simulated IV-characteristics at\ndifferent frequencies of the magnetic field. The IV-\ncharacteristics at three different values of Ω are shown\nin Fig 2(a).\nFIG. 2. (a) IV-characteristic at three different values of Ω.\nFor clarity, the IV-characteristics for Ω = 0 .5 and Ω = 0 .7\nhave been shifted to the right, by ∆ I= 0.5 and ∆ I= 1,\nrespectively with respect to Ω = 0 .2; (b) An enlarged part\nof the IV-characteristic with Ω = 0 .7. To get step voltage\nmultiply the corresponding fraction with Ω = 0 .7.\nAs we see, the second harmonic has the largest step\nwidth at the ferromagnetic resonance frequency Ω = Ω 0,\ni.e., the FMR is manifested itself by the step’s width.\nThere are also many subharmonic current steps in the\nIV-characteristic. We have analyzed the steps position\nbetween V= 0 and V= 0.7 for Ω = 0 .7 and found dif-\nferent level continued fractions, which follow the formula\ngiven by Eq.(1) and demonstrated in Fig.2(b). We see4\nthe reflection of the second level continued fractions 1 /n\nand 1−1/nwithN= 1. In addition to this, steps with\nthird level continued fractions 1 /(n−1/m) withN= 1\nis manifested. In the inset we demonstrate part of the\nfourth level continued fraction 1 −1/(n+ 1/(m+1/p))\nwithn= 2 and m= 2.\nIn case of external electromagnetic field which leads to\nthe additional electric current Iac=AsinΩt, the width\nof the Shapiro step is proportional to ∝Jn(A/Ω), where\nJnis the Bessel function of first kind. The preliminary\nresults (not presented here) show that the width of the\nShapiro-like steps under external magnetic field has a\nmore complex frequency dependence20. This question\nwill be discussed in detail somewhere else.\nThe coupling between Josephson phase and magneti-\nzation manifests itself in the appearance of the Shapiro\nsteps in the IV-characteristics at fractional and odd mul-\ntiplies of Ω20. In Fig.3 we demonstrate the effect of the\nratio of the Josephson to magnetic energy ǫJon appear-\nance of the steps and their width for Ω = 0 .5 where the\nenlarged parts of the IV-characteristics at three differ-\nent values of ǫJare shown. As it is demonstrated in\nthe figures, at ǫJ= 0.05 only two subharmonic steps\nappear between V= 1 and V= 1.5 (see hollow ar-\nrows). An enhanced staircase structure appears by in-\ncreasing the value of ǫJ, which can be see at ǫJ= 0.3\nand 0.5. Moreover, an intense subharmonic steps appear\nbetween V= 1.75 andV= 2 forǫJ= 0.5. The posi-\ntions for these steps reflect third level continued fraction\n(N−1)+1/(n+1/m)withN=4 andn=1 [see Fig.3(b)].\nLet us now demonstrate the effect of Gilbert damping\non the devil’s staircase structure. The Gilbert damping\nαis introduced into LLG equation23?to describe the\nrelaxation of magnetization dynamics. To reflect effect\nof Gilbert damping, we show an enlarged part of the IV-\ncharacteristic at three different values of αin Fig.4.\nThewidthofcurrentstepat V= 2Ωisalmostthesame\nat different values of α(e.g., see upward inset V= 2Ω).\nThe subharmonic current step width for V= (n/m)Ω (n\nis odd,mis integer) is decreasing with increasing α. In\naddition a horizontal shift for the current steps occurs.\nWe see the intense current steps in the IV-characteristic\nfor small value of α= 0.03 (see black solid arrows). With\nincrease in Gilbert damping (see α= 0.1, 0.16 and 0 .3)\nthe higher level subharmonic steps disappear. It is well-\nknownthatatlargevalueof αtheFMRlinewidthbecome\nmore broadening and the resonance frequency is shifted\nfrom Ω 0. Accordingly, the subharmonic steps disappear\nat large value of α. Furthermore, using the formula pre-\nsented in Ref.[20] the width at Ω = Ω 0for the fractional\nand odd current steps is proportional to (4 α2+α4)−q/2\n×(12+3α2)−k/2, whereqandkare integers.\nFinally, we demonstrate the effect of the junction size\non the devil’s staircase in the IV-characteristic under ex-\nternalmagneticfield. Thejunction sizechangesthe value\nofφsyandφsz. In Fig.5(a) we demonstrate the effect of\nthe junction thickness by changing φsz(φsyis qualita-\nFIG. 3. (a) An enlarged part of the IV-characteristic at\ndifferent values of ǫJin the interval between V= 1 and V=\n1.5; (b)Thesameintheintervalbetween V= 1.75andV= 2.\nFor clarity, the IV-characteristics for ǫJ= 0.3, and 0 .5 have\nbeen shifted to right, by ∆ I= 0.07, and 0 .14, respectively\nwith respect to the case with ǫJ= 0.05.\ntively the same).\nWe observe an enhanced subharmonic structure with\nincrease of junction size or the thickness of the ferro-\nmagnet. In Ref.[13] the authors demonstrated that the\ncritical current and the width of the step at V= 2Ω as a\nfunction of Lz/Lyfollow Bessel function of first kind. In\nFig.5(b), we can see the parts of continued fraction se-\nquences for subharmonic steps between V= 1 andV= 2\natφsz=φsy= 6. Current steps between V= 1 and\nV= 1.5 reflect the two second level continued fractions\n(N−1)+ 1/nandN−1/nwithN= 3 in both cases,\nwhile for the steps between V= 1.5 andV= 2 follow\nthe second level continued fraction ( N−1) + 1/nwith\nN= 4.\nFinally, wediscussthepossibilityofexperimentallyob-5\nFIG. 4. An enlarged part of IV-characteristic for four differ -\nent values of Gilbert damping for Ω = 0 .5. The inset shows an\nenlargedpartofcurrentstepwithconstantvoltage at V= 2Ω.\nserving the effects presented in this paper. For junction\nsized= 5nm, Ly=Lz= 80nm, critical current I0\nc≈\n200µA, saturation magnetization M0≈5×105A/m,\nH0≈40mT and gyromagnetic ratio γ= 3πMHz/T,\nwe find the value of φsy(z)=4π2Ly(z)dM0/Φ0= 4.8 and\nǫJ= 0.1. With the same junction parameters one can\ncontrol the appearance of the subharmonic steps by tun-\ning the strength of the constant magnetic field H0. Esti-\nmations showthat for H0= 10mT, the value of ǫJ= 0.4,\nand the fractional subharmonic steps are enhanced. In\ngeneral, the subharmonic steps are sensitive to junction\nparameters, Gilbert damping and the frequency of the\nexternal magnetic field.\nIV. CONCLUSIONS\nIn this work, we have studied the IV-characteristics\nof superconductor-ferromagnet-superconductor Joseph-\nson junction under external magnetic field. We used a\nmodified RSJ model which hosts magnetization dynam-\nics in F-layer. Due to the external magnetic field, the\ncouplingbetweenmagneticmomentandJosephsonphase\nis achieved through the effective field taking into account\nthe Josephson energy and gauge invariant phase differ-\nence between the superconducting electrodes. We have\nsolvedasystemofequationswhichdescribethe dynamics\nof the Josephson phase by the RSJ equation and magne-\ntization dynamics by Landau-Lifshitz-Gilbert equation.\nThe IV-characteristic demonstrates subharmonic current\nsteps. The pattern of the subharmonic steps can be con-\ntrolled by tuning the frequency of the ac magnetic field.\nWe show that by increasing the ratio of the Josephson to\nmagneticenergyanenhancedstaircasestructureappears.\nFinally, we demonstrate that Gilbert damping and junc-\nFIG. 5. (a) IV-characteristic at three different values of\nφsz= 0.7,3,6 andφsy=φsz. (b) An enlarged part of the IV-\ncharacteristic at φsz=φsy=6. The hollow arrows represent\nthe starting point of the sequences. To get step voltage we\nmultiply the corresponding fraction by Ω = 0 .5.\ntion parameters can change the subharmonic step struc-\nture. The observed features might find an application in\nsuperconducting spintronics.\nV. ACKNOWLEDGMENT\nWe thank Dr. D. V. Kamanin and Egypt JINR col-\nlaboration for support this work. The reported study\nwas partially funded by the RFBR research Projects No.\n18-02-00318 and No. 18-52-45011-IND. Numerical cal-\nculations have been made in the framework of the RSF\nProject No. 18-71-10095.6\nREFERENCES\n∗majed@sci.cu.edu.eg\n†shukrinv@theor.jinr.ru\n1J. Linder and K. Halterman, Phys. Rev. B 90, 104502\n(2014).\n2Yu. M. Shukrinov, A. Mazanik, I. Rahmonov, A. Botha,\nand A. Buzdin, EPL122, 37001 (2018).\n3Yu. M. Shukrinov, I. Rahmonov, K. Sengupta, and A.\nBuzdin, Appl. Phys. Lett. 110, 182407 (2017).\n4A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008).\n5A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n6F. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys.77, 1321 (2005).\n7A. A. Golubov, M. Y. Kupriyanov, and E. IlIchev, Rev.\nMod. Phys. 76, 411 (2004).\n8M. A. Silaev, I. V. Tokatly, and F. S. Bergeret, Phys. Rev.\nB95, 184508 (2017).\n9I. Bobkova, A. Bobkov, and M. Silaev, Phys. Rev. B 96,\n094506 (2017).\n10Y. M. Shukrinov, I. Rahmonov, and K. Sengupta, Phys.\nRev. B99, 224513 (2019).\n11M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D.\nKoelle, R. Kleiner, and E. Goldobin, Phys. Rev. Lett. 97,\n247001 (2006).\n12J. Pfeier, M. Kemmler, D. Koelle, R. Kleiner, E. Goldobin,\nM. Weides, A. Feofanov, J. Lisenfeld, and A. Ustinov,\nPhys. Rev. B 77, 214506 (2008).\n13S. Hikino, M. Mori, S. Takahashi, and S. Maekawa, Super-\ncond. Sci. Technol. 24, 024008 (2011).14G. Wild, C. Probst, A. Marx, and R. Gross, Eur. Phys. J.\nB78, 509523 (2010).\n15M. Kemmler, M. Weides, M. Weiler, M. Opel, S. Goennen-\nwein, A. Vasenko, A. A. Golubov, H. Kohlstedt, D. Koelle,\nR. Kleiner, et al., Phys. Rev. B 81, 054522 (2010).\n16A. Volkov and K. Efetov, Phys. Rev. Lett. 103, 037003\n(2009).\n17S. Mai, E. Kandelaki, A.Volkov, andK. Efetov, Phys. Rev.\nB84, 144519 (2011).\n18I. Petkovic, M. Aprili, S. Barnes, F. Beuneu, and S.\nMaekawa, Phys. Rev. B 80, 220502 (2009).\n19B. Hillebrands and K. Ounadjela, Spin dynamics in con-\nfined magnetic structures II , Springer-Verlag Berlin Hei-\ndelberg (2003) Vol. 83.\n20M. Nashaat, A. Botha, and Y. M. Shukrinov, Phys. Rev.\nB 97, 224514 (2018).\n21K. K. Likharev, Dynamics of Josephson junctions and cir-\ncuits, Gordon and Breach science publishers -Switzerland\n(1986).\n22S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).\n23T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443-\n3449 (2004).\n24M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102,\n137601(2009)."    },    {        "title": "1405.5242v6.Interaction_energy_and_itinerant_ferromagnetism_in_a_strongly_interacting_Fermi_gas_in_the_absence_of_molecule_formation.pdf",        "content": "arXiv:1405.5242v6  [cond-mat.quant-gas]  26 Nov 2014Interaction energy and itinerant ferromagnetism ina stron gly interacting Fermi gas\ninthe absence ofmoleculeformation\nLianyi He\nTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: May 7, 2021)\nWe investigate the interactionenergy and the possibility o f itinerant ferromagnetism ina strongly interacting\nFermi gas at zero temperature in the absence of molecule form ation. The interaction energy is obtained by\nsumming the perturbative contributions of Galitskii-Feyn man type to all orders in the gas parameter. It can\nbe expressed by a simple phase space integral of an in-medium scattering phase shift. In both three and two\ndimensions (3D and 2D), the interaction energy shows a maxim um before reaching the resonance from the\nBose-Einsteincondensate side, whichprovides a possible e xplanation of the experimental measurements of the\ninteraction energy. This phenomenon can be theoretically e xplained by the qualitative change of the nature of\nthe binary interaction in the medium. The appearance of an en ergy maximum has significant e ffects on the\nitinerantferromagnetism. In3D, theferromagnetic transi tionisreentrant anditinerant ferromagnetism existsin\na narrow window around the energy maximum. In 2D, the present theoretical approach suggests that itinerant\nferromagnetism does not exist, which reflects the fact that t he energy maximum becomes much lower than the\nenergyof the fullypolarized state.\nPACS numbers: 03.75.Ss,05.30.Fk, 64.60.De,67.85.–d\nI. INTRODUCTION\nA repulsively interacting Fermi gas can be realized by\nrapidly quenching the atoms to a metastable state in the\nabsence of bound-state (molecule) formation at the Bose-\nEinstein condensate (BEC) side of a Feshbach resonance [1–\n4]. Animportantgoalistostudytheitinerantferromagneti sm\nin repulsive Fermi systems [5–8], which is a long-standing\nproblem in many-body physics. The interaction energy has\nbeenmeasuredbystudyingtheexpansionproperties[2]orby\nusing rf spectroscopy [3, 4]. In an expansion experiment on\na6Li Fermi gas [2] around its broad Feshbach resonance at\nmagnetic field B≃834 G and at temperature T≃0.6TF,\nwhereTFistheFermitemperature,itwasfoundthattheinter-\nactionenergyoftherepulsivebranchsuddenlyjumpstonega -\ntivevaluesatmagneticfield B≃720G,whichliesattheBEC\nsideoftheresonance. Thesamefeaturewasalsoindicatedby\nrfspectroscopymeasurementinatwo-dimensionalFermigas\n[4, 9].\nA repulsive Fermi gas was previously suggested to exist\nin the upper branch of a strongly interacting Fermi gas [10].\nHowever, the “upper branch” is well defined only for two-\nbodysystems. Exact solution of the energylevels of three at -\ntractively interacting fermions in a harmonic trap [11] sho ws\nthattherearemanyavoidedcrossingsbetweenthelowesttwo\nbranchesas one approachesthe resonance,makingit di fficult\ntoidentifyarepulsiveFermisystem. Sofarthereisnopreci se\nformulation of it for many-body systems. In this paper, we\nstudy a metastable many-body state of a strongly interactin g\nFermi gas in the absence of molecule formation, or contain-\ningonlyscatteringstates[12]. Inthehigh-temperatureli mitit\ncan beformulatedbyusingthe virial expansionto the second\norderin thefugacitybecausethetwo-bodycontributiondom -\ninates [13]. Moreover, in the weak-coupling limit (both the\nBCS and BEC limits), the equation of state of such a system\ncanbedescribedperturbatively[14,15].\nThe suddenjumpof the interactionenergyat the BEC sideof the resonance can be qualitatively explained by the stron g\natom loss around B=720 G, where the system may be re-\ngarded as a mixture of atoms and weakly bound molecules\n[2]. Shenoy and Ho [12] rather suggested that the interac-\ntionenergyofthe repulsivebranchwasfoundto increaseand\nthendecreaseas one approachesthe resonancefromthe BEC\nside, showinga maximumbefore reachingthe resonance. By\nusing a generalized Nozi` eres-Schmitt-Rink (NSR) approac h\nwhere the molecular contribution is subtracted, they found\nthat an energymaximumalreadyappearsat high temperature\nT∼3TF[12]. However, the NSR approach to the repul-\nsive branch is limited to the high-temperature region where\nthe chemical potential becomes negative and the fugacity is\nsmall. It becomes less accurate and predicts artificial disc on-\ntinuitiesand instability at low temperature[12]. Moreove r,at\nlow temperature, since the compressibility becomes negati ve\nin a large forbidden area, the number equation has no solu-\ntion and hencethe generalizedNSR approachcannot provide\nquantitativepredictions.\nInthispaper,wefollowShenoyandHo’sexplanationofthe\nbehavior of the interaction energy but employ an alternativ e\nnonperturbativeapproachatzerotemperaturetoovercomet he\ndifficulty of a negative compressibility. The basic idea is to\nsum some certain type of perturbative contributionsto all o r-\ndersinthegasparameter[16–18]. Theinteractionenergy Eint\ncanbeformallyexpressedas\nEint(g)=∞/summationdisplay\nn=1cngn, (1)\nwheregis the gas parameter. Obviously, the result becomes\nperturbativeat weak coupling |g|≪1. The basic requirement\nfor this resummation is that the interaction energy converg es\nin the strong-coupling limit g→ ∞. According to Bishop\n[19], thereexist two di fferentschemesto calculatethe pertur-\nbativeequationofstate[14,15],theBethe-Goldstonesche me\nand the Galitskii-Feynman (GF) scheme. If the perturbative\ncontributions can be computed and summed precisely to all2\norders in g, they should agree with each other. However, this\nis impossible since the problemis not exactly soluble. In th is\nworkweemploytheGFscheme. Inthisscheme,boththepar-\nticle andhole partsof the single-particlepropagatorare u sed.\nBy summing the perturbative contributions of the GF type,\nthe contributionsfrom the particle-particle ladders, hol e-hole\nladders, and mixed particle-particleand hole-holeladder sare\nresummedself-consistentlytoall ordersin g[18].\nThe paper is organized as follows. In Sec. II, we briefly\nintroduce the description of the two-bodyscattering by usi ng\na contact interaction. In Sec. III, we study the binary scat-\ntering at finite density, i.e., in the presence of Fermi surfa ces.\nThe interactionenergyis calculated in Sec. IV. We apply the\ntheory to study the itinerant ferromagnetism in Sec. V. We\nsummarizeinSec. VI.\nII. BASICS:TWO-BODYSCATTERING\nTwo-componentatomicFermigasesacrossa broad s-wave\nFeshbachresonancecanbedescribedbytheHamiltonian\nH=/summationdisplay\nσ=↑,↓/integraldisplay\ndrψ†\nσ(r)/parenleftBigg\n−/planckover2pi12∇2\n2M/parenrightBigg\nψσ(r)+Hint (2)\nwitha contactinteraction[20]\nHint=U/integraldisplay\ndrψ†\n↑(r)ψ†\n↓(r)ψ↓(r)ψ↑(r). (3)\nHereψσare the fermionfields with σ=↑,↓denotingthe two\ncomponents, Mis the fermion mass, and Uis a contact cou-\nplingwhich representsthe short-rangedattractive intera ction.\nThefreefermionpropagatorin vacuumisgivenby\nG0(p0,p)=1\np0−εp+iǫ, (4)\nwherep0andpdenote the energy and momentum of a\nfermion,ǫ=0+, andεp=p2/(2M). For convenience, we\nusethe units/planckover2pi1=M=1throughout.\nThe advantage of using the contact interaction is that the\nLippmann-Schwingerequationforthetwo-body s-wave scat-\nteringTmatrix becomes a simple algebraic equation. In the\ndiagrammatic representation, it is equivalent to resummat ion\nof particle-particle ladder diagrams to all orders in U. The\noff-shellTmatrixcanbeexpressedas\nT2B(P0,P)=U\n1−UΠ0(P0,P), (5)\nwhereP0andParethetotalenergyandmomentumofthetwo\nscattering fermions. The two-bodybubblediagram Π0(P0,P)\nisgivenby\nΠ0(P0,P)=i/integraldisplay∞\n−∞dq0\n2π/summationdisplay\nqG0(q+,q+)G0(q−,q−)\n=/summationdisplay\nq1\nP0+iǫ−P2\n4−2εq. (6)Here we have defined the notations q±=P0/2±q0and\nq±=P/2±q. We notice that the two-body bubble function\nΠ(P0,P) and hence T2B(P0,P) depend only on the combina-\ntionP0−P2/4 because of the Galilean invarianceof the two-\nbody system. The scattering amplitude f(k) can be obtained\nby imposing the on-shell condition P0=P2/4+E, where\nE=k2isthescatteringenergyinthecenter-of-massframe.\nThe cost of using the contact interaction is that the inte-\ngral over qbecomes divergent. This divergence can be re-\nmovedthroughthe renormalizationofthecontactcoupling U\nintermsofthephysicalscatteringlength. Tothisend,wefir st\nregularizethedivergencebyintroducingacuto ffΛfor|q|. We\nobtain\nΠ0(P0,P)=−1\n2π2Λ+1\n4π/radicalbigg\n−P0−iǫ+P2\n4(7)\nforthreedimensions(3D)and\nΠ0(P0,P)=−1\n2πlnΛ+1\n4πln/parenleftBigg\n−P0−iǫ+P2\n4/parenrightBigg\n(8)\nfor two dimensions(2D). To renormalize the contact interac -\ntion,wematchthe TmatrixT2B(P0,P)onthescatteringmass\nshellP0=P2/4+k2to the known scattering amplitude f(k)\n[20]. In general, we find that only the coupling constant U\nneedsrenormalization. In3D,wehave\nf(k)=4π\na−1+ik, (9)\nwhereais the 3D scattering length. A boundstate with bind-\ning energy εB=1/a2exists only for a>0. The coupling\nconstantisgivenby\nU(Λ)=−4π\n2Λ/π−a−1. (10)\nIn 2D, a two-body bound state exists for arbitrarily weak at-\ntraction. Thescatteringamplitudereads[21]\nf(k)=4π\n−ln(E/εB)+iπ, (11)\nwhereεBis thebindingenergyoftheboundstate. Forconve-\nnience, we define a 2D scattering length a2. There exist two\npopular definitions of a2in the literature. In this paper, we\nemploythedefinition εB=1/a2\n2inaccordancewithearlythe-\noreticalstudies[22,23]andrecentexperimentalstudies[ 4,9].\nNotice that a2is always positive. From this definition of a2,\nthecouplingconstantisgivenby\nU(Λ)=−2π\nln(Λa2). (12)\nAnotherpopulardefinitionofthe2Dscatteringlengthisgiv en\nbyεB=4/(a2\n2e2γ), whereγ≃0.577isEuler’sconstant. Con-\nvertingthe theoretical results from one definition to the ot her\nisrathersimple.3\nIII. BINARY SCATTERINGIN MEDIUM\nAtfinitedensity,thepropagatorsofnoninteractingfermio ns\naregivenby\nGσ(p0,p)=1−nσ(p)\np0−εp+iǫ+nσ(p)\np0−εp−iǫ,(13)\nwherenσ(p)≡Θ(kσ\nF−|p|). Herek↑,↓\nFare the Fermi momenta\nof the two spin components. For convenience, we express\nthemaskσ\nF=kFησ,wheretheaverageFermimomentum kFis\ndefinedbythetotaldensity n=n↑+n↓andthedimensionless\nquantitiesησdependonthepolarization x=(n↑−n↓)/(n↑+n↓).\nIn 3D we have n=k3\nF/(3π) andη↑,↓=(1±x)1/3. In 2D,\nn=k2\nF/(2π)andη↑,↓=(1±x)1/2. Thegasparameterisdefined\nasg=kFain 3D and g=−1/ln(kFa2) in 2D. It is convenient\nto use an alternativeformofthe propagator. It is givenbyth e\nvacuum-mediumdecomposition\nGσ(p0,p)=G0(p0,p)+Gσ\nm(p0,p), (14)\nwhere\nGσ\nm(p0,p)=2πiδ(p0−εp)nσ(p) (15)\niscalleda “mediuminsertion”(MI)[18].\nTo sum certain types of perturbativecontributions, we em-\nploytheGFscheme[15,19,24],whichtakesintoaccountthe\npropagationsof both particles and holes and is exact to orde r\nO(g2). Themany-body Tmatrixisgivenbysummationofthe\nGF ladderdiagramsto allordersin U. We have\nTm(P0,P)=U\n1−UΠ(P0,P), (16)\nwherethebubblediagram Π(P0,P)isnowgivenby\nΠ(P0,P)=i/integraldisplay∞\n−∞dq0\n2π/summationdisplay\nqG↑(q+,q+)G↓(q−,q−).(17)\nAccording to the vacuum-medium decomposition, it can be\ndecomposedintothreeparts,\nΠ(P0,P)=Π0(P0,P)+Π1(P0,P)+Π2(P0,P),(18)\nwhereΠl(l=0,1,2) stands for the bubble diagram with l\nMIs. Thevacuumcontribution Π0(P0,P)naturallycancelsthe\ncutoffdependenceof U. The medium contributionsare finite\nandcanbeevaluatedas\nΠ1(P0,P)=−/summationdisplay\nqn↑(q+)+n↓(q−)\nP0+iǫ−P2\n4−2εq(19)\nand\nΠ2(P0,P)=−2πi/summationdisplay\nqn↑(q+)n↓(q−)δ/parenleftBigg\nP0−P2\n4−2εq/parenrightBigg\n.(20)\nWe notice that in the presence of the medium, the two-body\nbubble functionΠ(P0,P) and hence the Tmatrix depend notonlyonthecombination P0−P2/4butalsoonthemomentum\nPthroughthe distributionfunctions n↑(q+)andn↓(q−).\nTheTmatrixTm(P0,P) characterizes the energy spectrum\nof the system in the GF approach. We note that the imagi-\nnary part ofΠ(P0,P) vanishesfor P0−P2/4<0. The bound\nstates ormoleculestatescorrespondto the polesofthe Tma-\ntrix in the region P0−P2/4<0. Since we consider only the\nscatteringpartofthemany-bodyenergyspectrum,whichcor -\nresponds to the two-particle continuum P0−P2/4>0, we\nimpose the on-shell condition P0=P2/4+k2. For conve-\nnience,we definetwo dimensionlessvariables\ns=|P|\n2kF,t=|k|\nkF. (21)\nIn analogy to the vacuum case, the in-mediumscattering am-\nplitudecanbeexpressedas\nfm(s,t)=4π\nf1(s,t)+if2(s,t)(22)\nwheref1(s,t) andf2(s,t) are the real and imaginary parts of\nthedenominator,respectively. Theycanbeexpressedas\nf1(s,t)=4π/bracketleftBigg\nU−1−ReΠ/parenleftBigg\nP0=P2\n4+k2,P/parenrightBigg/bracketrightBigg\n,\nf2(s,t)=−4πImΠ/parenleftBigg\nP0=P2\n4+k2,P/parenrightBigg\n. (23)\nWe note that fm(s,t) recovers the two-body scattering ampli-\ntudeatvanishingdensity( kσ\nF→0).\nIn Sec. III we will show that the interaction energy of the\nmany-body scattering state can be expressed in terms of the\nin-mediumscatteringphaseshift\nEint=−4π/summationdisplay\nP/summationdisplay\nkn↑(k+)n↓(k−)φm(s,t)\nf2(s,t),(24)\nwherethe in-mediumscatteringphaseshiftis definedas\ne−2iφm(s,t)=f1(s,t)+if2(s,t)\nf1(s,t)−if2(s,t). (25)\nAtvanishingdensity,itrecoversthetwo-bodyscatteringp hase\nshiftφ2B(k), whereφ2B>0 andφ2B<0 correspondto attrac-\ntion and repulsion, respectively. In 3D, we have φ2B(k)=\n−arctan(ka). Notethat it isdifferentfromthe usualdefinition\nofthescatteringphaseshift φm=−Imln(−f1−if2). Fromthis\ndefinition, we have φ2B(0)=πfora>0 andφm(s,t)→πin\nthe BEC limit, which clearly shows the existence of a bound\nstate. Sinceweareconsideringasystemcontainingonlysca t-\nteringstates, we shouldexcludetheinfluenceofthe molecul e\nboundstate andusethedefinition(25).\nIn the followingwe analyze the behaviorof the phase shift\nφminthephasespaceSdefinedas\nS=/braceleftBig\n(k,P)/vextendsingle/vextendsingle/vextendsingle|P/2+k|<k↑\nF,|P/2−k|<k↓\nF/bracerightBig\n.(26)\nItiscrucialforustounderstandthebehavioroftheinterac tion\nenergyacrossa Feshbachresonance.4\n0 0.2 0.4 0.6 0.8 1−0.500.5φm / π\n0 0.2 0.4 0.6 0.8 1−0.500.5\nk/kFφm / π\n(b) 2D(a) 3D\n1/g=2\n1/g=20\n1−11 0\n−1\nFIG. 1: (Color online) The in-medium scattering phase shift φmat\nzero center-of-mass momentum Pfor various values of the inverse\ngas parameters gin3D (g=kFa)and 2D [ g=−1/ln(kFa2)].\nA. Threedimensions\nIn 3D and in the phase space S, the functions f1(s,t) and\nf2(s,t)canbe expressedas\nf1(s,t)=a−1−kF[R↑(s,t)+R↓(s,t)],\nf2(s,t)=kFI(s,t), (27)\nwhereRσ(s,t) andI(s,t) aregivenby\nRσ(s,t)=ησ\nπ+η2\nσ−(s+t)2\n4πsln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleησ+s+t\nησ−s−t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+η2\nσ−(s−t)2\n4πsln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleησ+s−t\nησ−s+t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nI(s,t)= Θ(η2\n↑+η2\n↓−2s2−2t2)/productdisplay\nσ=↑,↓Θ(ησ−|s−t|)\n×t+/summationdisplay\nσ=↑,↓η2\nσ−(s+t)2\n4sΘ(s+t−ησ).(28)\nLet us focus on the balanced case x=0. The functions f1\nandf2atP=0 (s=0)canbesimplifiedas\nf1\nkF=1\ng−4\nπ/parenleftBigg\n1−t\n2ln1+t\n1−t/parenrightBigg\n,f2\nkF=t(29)\nfor 0<t<1. Here g=kFais the gas parameter in\n3D. In the weak coupling limit where g→0, we haveφm≃−arctan(ka),whichcoincideswiththetwo-bodyscatter-\ningphaseshift φ2B(k)intheabsenceofboundstate. However,\nwhen approachingthe resonance, the nature of the binary in-\nteractionisqualitativelychangedbythemediume ffect. Atthe\nBECside( g>0)oftheresonance,asimplemathematicalex-\nercise shows that the function f1has a zero t=t0∈(0,1)for\ng> π/4. At the BCS side ( g<0) of the resonance, this zero\nalwaysexists. IntheBCS limit,thezerocanbeexpressedas\nt0=/radicalbigg\n1−εc\n2EF, (30)\nwhereεc≃8EFexp(π\n2g−2) is the Cooper-pair binding en-\nergy. Here EF=k2\nF/2 is the Fermi energyof the noninteract-\ningsystem.\nThe numerical results for φmatP=0 are shown in Fig.\n1(a). Once f1has a zero t=t0, it is easy to show that f1<0\nfor0<t<t0andf1>0fort0<t<1. Accordingly,wehave\nφm>0 for 0<t<t0andφm<0 fort0<t<1. A jump of\nπatt=t0appears. Therefore,the binaryinteractionbecomes\nmixed in the region −∞<1/g<4/π. It is attractive at low\nenergy(0<t<t0) and repulsive at high energy ( t0<t<1).\nEven though we have shown the results only for P=0, the\nqualitative behavior for P/nequal0 is similar. From the results\nshown in Fig. 1(a), it is intuitive that when approaching the\nresonancefromthe BEC side, the attractiveregionwith φm>\n0becomeslargerandlarger.\nB. Two dimensions\nIn2D,thefunctions f1(s,t)andf2(s,t) becomedimension-\nless. Inthephasespace S, theycanbeexpressedas\nf1(s,t)=−2ln(ka2)−[R↑(s,t)+R↓(s,t)],\nf2(s,t)=I(s,t), (31)\nwhereRσ(s,t)andI(s,t) aregivenby\nRσ(s,t)=/integraldisplayπ\n0dθ\nπΘ(ησ−ssinθ)ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(u+\nσ)2Θ(u+\nσ)−t2\n(u−σ)2Θ(u−σ)−t2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nI(s,t)= Θ(1−s2−t2)/productdisplay\nσ=↑,↓Θ(ησ−|s−t|)\n×π−/summationdisplay\nσ=↑,↓arccosη2\nσ−s2−t2\n2stΘ(s+t−ησ).(32)\nHereu±\nσ=scosθ±(η2\nσ−s2sin2θ)1/2. Note that we have\n−2ln(ka2)=2/g−2lntby using the gas parameter g=\n−1/ln(kFa2)in 2D.\nForthebalancedcase x=0,we have\nf1=2\ng+2lnt\n1−t2,f2=π (33)\nin the phase space 0 <t<1 forP=0. In contrast to the\n3D case, the function f1has a zero t=t0∈(0,1) for arbi-\ntrary values of a2>0. We have φm>0 for 0<t<t0and5\nφm<0 fort0<t<1. In the BCS limit a2→+∞, we have\nt0=√1−εc/(2EF) whereεc≃√2εBEFis the Cooper-pair\nbinding energy in 2D [21, 25, 26]. Therefore, there exists a\nqualitative difference between 2D and 3D: the in-mediumbi-\nnary interaction shows attraction at low energy for arbitra ry\nvalue ofa2in 2D. The results for φmare shown in Fig. 1(b).\nFor arbitrary value of the gas parameter g, we find that the\nphaseshiftφmispositivein theregion0 <t<t0.\nIV. INTERACTIONENERGY\nThe interaction energy Eintcan be obtained by summing\nthe perturbative contributions of the GF type to all orders i n\ng, which has been done by Kaiser [18] in 3D. Consider the\nopenladderdiagramwith ncontactinteractions. It is roughly\ngiven by the ( n−1)th power ofΠtimesUn. When closing\nthe two open fermion lines, we replace them by two medium\ninsertions. Thisintroducesanintegration\n−/integraldisplay∞\n−∞dP0\n2π/integraldisplay∞\n−∞dk0\n2π/summationdisplay\nP/summationdisplay\nkG↑\nm(k+,k+)G↓\nm(k−,k−)···\n=/summationdisplay\nP/summationdisplay\nkn↑(k+)n↓(k−)/integraldisplay∞\n−∞dP0δ/parenleftBigg\nP0−P2\n4−k2/parenrightBigg\n···.(34)\nThedeltafunction δ(P0−P2/4−k2)clearlyshowsthatthein-\nteraction energycontainsonly the contributionfrom the sc at-\nteringstates.\nOn the scattering mass shell P0=P2/4+k2, we have\nΠ(s,t)=U−1−(f1+if2)/(4π). We note that only the\nclosedladdersthathaveatleast onepairofadjacentMIscon -\ntributetotheinteractionenergy. Byusingthespecialprop erty\nΠ−Π2=Π∗we can take into account the symmetry factors\nwhich will also correct for the overcounting of certain dia-\ngrams. Afteracarefulcombinatorialanalysis,wefindthatt he\ninteractionenergydensityisgivenby[27]\nEint=/summationdisplay\nP/summationdisplay\nkn↑(k+)n↓(k−)∞/summationdisplay\nn=1Cn(s,t),(35)\nwherethe nth-ordercontributionis\nCn(s,t)=−[UΠ(s,t)]n−[UΠ∗(s,t)]n\n2in4π\nf2(s,t).(36)\nCompletingthesummationover n,weobtain\n∞/summationdisplay\nn=1Cn(s,t)=4π\nf2(s,t)1\n2ilnf1(s,t)+if2(s,t)\nf1(s,t)−if2(s,t)\n=−4πφm(s,t)\nf2(s,t). (37)\nThe above result shows that the interaction energy is an inte -\ngrationofthephaseshift φmoverthephasespace S.\nFor the 3D case, the perturbative expansion of the interac-\ntionenergycanbeexpressedas\nEint=∞/summationdisplay\nn=1cn(kFa)n, (38)wherethe expansioncoe fficients\ncn=4π/summationdisplay\nP/summationdisplay\nkn↑(k+)n↓(k−)Hn(I,R).(39)\nHereR(s,t)=R↑(s,t)+R↓(s,t)and\nHn(I,R)=(R+iI)n−(R−iI)n\n2inI. (40)\nUptothe sixthorder,we have\nH1=1,H2=R,H3=R2−1\n3I2,\nH4=R3−RI2,H5=R4−2R2I2+1\n5I4,\nH6=R5−10\n3R3I2+RI4. (41)\nBy using a detailed diagrammatic analysis, Kaiser has care-\nfullycheckedupto thesixth orderthatthe aboveperturbati ve\nexpansion includes precisely the perturbative contributi ons\nfrom particle-particle ladders, hole-hole ladders, and mi xed\nparticle-particleandhole-holeladders[18].\nA. Threedimensions\nIn3D,theinteractionenergydensitycanbe expressedas\nEint\nE0=−80\nπ/integraldisplay /integraldisplay\nSs2tφm(s,t)dsdt, (42)\nwhereE0=3\n5nEFis the energy density of a noninteracting\nFermigas. Forsmall g,we have\n−φm=gI+g2I(R↑+R↓)+O(g3). (43)\nUsing this expansion, we can recover precisely the second-\norder perturbation theory [14, 28, 29]. For the balanced cas e\nx=0,we obtain\nEint\nE0=10\n9πg+4(11−ln2)\n21π2g2+O(g3). (44)\nFortheimbalancedcase x/nequal0,we have\nEint\nE0=10(1−x2)\n9πg+ξ(η↑,η↓)\n21π2g2+O(g3),(45)\nwherethe second-ordercoe fficientξ(η↑,η↓) reads\nξ=22η3\n↑η3\n↓(η↑+η↓)−4η7\n↑lnη↑+η↓\nη↑−4η7\n↓lnη↑+η↓\nη↓\n+1\n2(η↑−η↓)2η↑η↓(η↑+η↓)[15(η2\n↑+η2\n↓)+11η↑η↓]\n+7\n4(η↑−η↓)4(η↑+η↓)(η2\n↑+η2\n↓+3η↑η↓)ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleη↑−η↓\nη↑+η↓/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(46)\nThe perturbative result for x/nequal0 agrees with the result first\nevaluatedbyKanno[29].6\n−5 0 5 −4−3−2−1 1234−0.4−0.200.20.40.6\n−1/(kFa)Eint / E0\n−5 0 5 −4−3−2−1 1234−0.6−0.4−0.200.20.4\nln(kFa2)Eint / E0\n(b) 2D(a) 3D\nFIG. 2: (Color online) The interaction energy for the balanc ed case\nx=0 as a function of −1/(kFa) in 3D (a) and of ln( kFa2) in 2D.\nThe blue dashed lines are the results from second-order pert urbation\ntheory. The red dash-dotted line in (a) corresponds to the en ergy of\nthe fullypolarized statein3D, Efp=22/3E0.\nThe interaction energy for the balanced case is shown in\nFig. 2(a). It reachesa maximum0 .62E0atg=0.88and then\ndecreases. This can be clearly understood by the qualitativ e\nchange of the binary interaction in medium: the phase shift\nφmexperiences more and more attraction when approaching\nthe resonancefromthe repulsiveside a>0. We notice that a\nrecent quantum Monte Carlo study of the 3D dilute Hubbard\nmodel also found that the interaction energy shows a max-\nimum at some interaction strength [30]. The energy maxi-\nmumisonly0 .034E0largerthantheenergyofthefullypolar-\nized state. At unitarity the Bertsch parameter (for the norm al\nphase) reads ξ=0.507, which agrees with the experimental\nresultξ=0.51(2)[31] and the Monte Carlo results: ξ≃0.54\n[32],ξ=0.56 [33], and ξ=0.52 [34]. Notice that our ap-\nproach at T=0 does not predict any discontinuity of the en-\nergyanditsslope,incontrasttotheresultsfromagenerali zed\nNSR approach[12]. In particular, we have checked the com-\npressibility κwhichis definedas 1 /(n2κ)=∂2E/∂n2. We find\nthatit isalwayspositivein ourapproach.\nIt is also intuitively easy to understand the behavior of the\ninteractionenergybyusingan“e ffective”scatteringlength aeff\nin the medium. At vanishing center-of-massmomentum P=\n0andatsmallscatteringenergy E=k2≪EF,thein-mediumscatteringamplitudecanbeexpressedas\nfm(k)=4π\n1\na−4kF\nπ+4\nπkFk2+ik. (47)\nIn analogy to the vacuum case, the e ffective scattering length\naeffin the medium can be defined as aeff=1/(a−1−4kF/π).\nIt is also interesting that the medium e ffect generates an ef-\nfective range. The e ffective scattering length aeffdiverges at\ng=π/4≃0.79. The location of the energymaximum corre-\nspondsto 1/(kFaeff)=−0.14, which lies at the “BCS side” in\ntermsofaeff.\nB. Two dimensions\nIn2D,theinteractionenergydensityis givenby\nEint\nE0=−32\nπ/integraldisplay /integraldisplay\nSstφm(s,t)dsdt, (48)\nwhereE0=1\n2nEFistheenergydensityofanoninteracting2D\nFermigas. Forsmall g,we have\n−φm=1\n2gI+1\n4g2I(2lnt+R↑+R↓)+O(g3).(49)\nForthebalancedcase x=0,we obtain\nEint\nE0=g+3−4ln2\n4g2+O(g3). (50)\nThe coefficient of the second-order term, (3 −4ln2)/4≃\n0.057, agrees with the result by Engelbrecht, Randeria, and\nZhang [22] but disagrees with Bloom’s numerical result 0 .28\n[23]. The 2D scattering length is also defined as εB=\n4/(a2\n2e2γ) in some papers, where γ≃0.577 is Euler’s con-\nstant. Forthisdefinition,we have\nEint\nE0=g+/parenleftBigg\nγ+3\n4−2ln2/parenrightBigg\ng2+O(g3). (51)\nwhere the second-order coe fficientγ+3\n4−2ln2≃−0.059\nbecomes negative. For the imbalanced case, we do not have\nananalyticalresult fortheperturbativeexpansion.\nThe interaction energy for the balanced case is shown in\nFig. 2(b). It reaches a maximum 0 .47E0atg=0.71 or\nln(kFa2)=−1.4. The energy curve around the maximum be-\ncomesmuchflatterthaninthe3Dcase. Asaresult,theenergy\nmaximum becomes much lower than the energy of the fully\npolarized stateEfp=2E0. These results can be understood\nintuitivelythroughthebehaviorof φm: In2D,thebinaryinter-\nactionisqualitativelychangedevenintheBEClimit a2→0+.\nV. ITINERANTFERROMAGNETISM\nFinallywestudythepossibilityofitinerantferromagneti sm\nin the many-body scattering state. It is intuitively clear t hat\nexistenceofanenergymaximumat the BECside of thereso-\nnancehasasignificante ffectontheitinerantferromagnetism.\nTo study the itinerant ferromagnetism, we study the system\nwith a finite polarization x=(n↑−n↓)/(n↑+n↓) and analyze\nthelandscapeoftheenergydensity E(x).7\nA. Threedimensions\nLet us first assume that the many-bodyscattering state can\nbe prepared in equilibrium. By analyzing the energy curve\nE(x),we find that the systemundergoesa second-orderphase\ntransition to the ferromagnetic phase at g=0.79 where the\nspinsusceptibility χdivergesandthenafirst-orderorderphase\ntransition to the paramagnetic phase at g=0.96. This reen-\ntrant phenomenon can be clearly understood from the exis-\ntence of an energymaximum at g=0.88. The spin suscepti-\nbilityχcanbeobtainedbymakinguseofasmall-polarization\nexpansionoftheenergydensity,\nE(x)=E(0)+αx2+···. (52)\nWe haveχ0/χ∝α. Hereχ0is the spin susceptibility of\nnoninteractingFermigases. Thenormalizedinversespin su s-\nceptibilityχ0/χis shown in Fig. 3(a). In a narrow region\n0.79<g<0.82 whereχ0/χ <0, the system phase separates\ninto partially polarized domains. We notice that the second -\norderferromagnetictransition at g=0.79 is veryclose to the\ngasparameter g=π/4wherethein-mediumscatteringlength\naeffdiverges.\nThemaximumcriticaltemperatureofferromagnetism Tmax\nc\nbecomes constrained by the energy maximum. Since χ0/χ >\n0 near the energy maximum, Tmax\nccan be roughly estimated\nby using second-order perturbation theory. By equating the\nenergy of the second-order perturbation theory to the energ y\nmaximumEmax=1.62E0, we estimate Tmax\nc≃0.2TF. Above\nthis temperature,the ferromagneticphase disappearsand o ne\ncan never observe a diverging spin susceptibility. We note\nthat the lowest temperature realized in the first experiment of\nKetterle and co-workers [1] is about T=0.12TFand a later\nexperiment[35] at T=0.23TFdid not observeany diverging\nbehaviorofthespinfluctuation.\nOntheotherhand,themany-bodyscatteringstateisnotsta-\nbleandsuffersfromvariousdecayprocesses. InthedeepBEC\nregionwhere gissmall,ithasbeenshownthatthethree-body\nrecombinationrate is proportionalto ¯ ε(na3)2[36], where ¯ εis\ntheaveragekineticenergyofafermion. InadegenerateFerm i\ngas at zero temperature, ¯ εis given by 3 EF/5. This indicates\nthat the decay rate of the repulsive branch is quite small for\na small positive gas parameter g. Recent experiments on the\nrepulsive branch [35, 37] found that equilibrium study of th e\nrepulsiveFermigasispossibleonlyfor g<0.25fortempera-\nturearound0 .3TF. At largegasparameter g,fast decayofthe\ngaspreventstheobservationoftheequilibriumprofiles.\nFor large gas parameter g, it seems impossible to present\nan accurate theoretical study of the decay rate. However,\nthe present many-body approach allows us to study the pair\n(molecule) formation rate or pairing decay rate from an in-\nmedium two-body picture [38, 39]. It has been shown that\nthis pairing decay picture can qualitatively explain the ex per-\nimental observations of the fast decay at large g[38]. The\npair formation rate is characterized by the imaginary part o f\nthe pole of the in-medium TmatrixTm(P0,P). For a fixed\npair momentum P, we make an analytical continuationof the\nvariableP0to the complex plane. The pole can be expressed\nasP0=ΩP+i∆P, where the imaginarypart ∆Pcharacterizesthe pair formation rate [38, 39]. The strongest decay occurs\natP=0 for balanced populations. The result of the pair-\ning decay rate∆≡∆P=0at zero temperatureis shown in Fig.\n3(a). It arises at g=0.93 and rapidly reaches a maximum\natg=1.8. In the BCS limit, the pairing decay rate coin-\ncideswiththesuperfluidgap, ∆≃8EFexp(π\n2g−2). Thesharp\nonset atg=0.93 is expected to be smoothed by three-body\nprocesses, since the three-body decay rate is proportional to\n¯ε(na3)2forsmallpositive g[36].\n−4 −2 0 2 4 −3 −1 1 3300.20.40.60.81\n−1/(kFa)\n−4 −2 0 2 4 −3 −1 1 300.511.52\nln(kFa2)∆/EFχ0/χ\nχ0/χ∆/EF\n(b) 2D(a) 3D\nFIG. 3: (Color online) The normalized inverse spin suscepti bility\nχ0/χ(blue solid lines) and the pairing decay rate ∆divided by EF\n(black solid lines) as functions of the gas parameters in 3D ( a) and\n2D (b). The green dotted lines show schematically the behavi or of\nthe decay ratewhen three-body processes are taken intoacco unt.\nThe study of the equilibrium properties of the repulsive\nFermi gas is therefore limited within the time scale of pair\nformation. From the result of the pairing rate ∆shown in\nFig. 3(a), we find that pair formation occurs in a time scale\n2/planckover2pi1/EFfor a wide range of the gas parameter around the fer-\nromagnetic phase. For typical atom density nrealized in ex-\nperiments,we estimate that this time scale is of order0.1ms ,\nwhichiseveryshortforexperimentalobservationoftheequ i-\nlibriumprofiles. Actually,arecentexperimenthasobserve da\nrapiddecayintoboundpairsovertimesontheorderof10 /planckover2pi1/EF\nfor a wide range of the interaction strength [35]. Future ex-\nperimental studies of repulsive Fermi gases should overcom e\nthefastdecayrate. Theoreticalstudieshavesuggestedsev eral\nways to suppressthe decay rate: narrowresonance [40], high\ntemperature[38], low dimensionality[41], populationimb al-\nance[39],massimbalance[36,42],andlatticeandbandstru c-\nture [43]. It will be interesting to extend the present nonpe r-\nturbativeapproachto studythe abovee ffects.8\nB. Two dimensions\nThemean-fieldtheoryin2Dpredictsaferromagneticphase\ntransition at g=1 or ln(kFa2)=−1 since the energy density\nisgivenby\nEmf(x)\nE0=1+x2+(1−x2)g. (53)\nHowever,thepresentnonperturbativeanalysisrathersugg ests\nthat there exists no itinerant ferromagnetism in a 2D Fermi\ngasatzerotemperature. We havecarefullystudiedtheenerg y\ncurveE(x) and found that the minimum is always located at\nx=0. The normalized inverse spin susceptibility χ0/χis\nshown in Fig. 3(b). It never reaches zero, which indicates no\nferromagnetic transition. This can be intuitively underst ood\nby the fact that the energy maximum is much lower than the\nenergy of the fully polarized state. We notice that a recent\nquantum Monte Carlo study of a two-component Fermi gas\nwith hard-coreinteractionsalso suggestedan absence of it in-\nerant ferromagnetism in 2D [44]. The pairing decay rate in\n2Dcanbeanalyticallyevaluatedas\n∆=Θ(8EF−εB)/radicalbigg\n2εBEF−1\n4ε2\nB, (54)\nwhichshowsa maximumat ln( kFa2)=−0.35.\nVI. SUMMARY\nIn this work, we have studied the behavior of the interac-\ntion energy and the possibility of itinerant ferromagnetis minastronglyinteractingFermigasatzerotemperatureinthea b-\nsence of molecule formation. The interaction energy of the\nsystemisobtainedbysummingtheperturbativecontributio ns\noftheGFtypetoallordersinthegasparameter. Weshowthat\nin both 3D and 2D, the interaction energy arrives at a maxi-\nmum before reaching the resonance from the BEC side. This\nphenomenoncan be understood qualitatively through the na-\nture of the binary interaction in the medium: the in-medium\nscatteringphaseshiftshowsattractionatlowenergyandhe nce\nreducesthe interaction energybefore reachingthe resonan ce.\nTheappearanceofanenergymaximumhassignificante ffects\non the possibility of itinerant ferromagnetism in the syste m\nwe study. In 3D, the ferromagnetictransitionis reentranta nd\nitinerantferromagnetismexistsinanarrowrangeoftheint er-\naction strength. 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Exchange-free description of magnetization d ynamics in the films allowed us to \nobtain simple analytical expressions. They enable q uick and efficient numerical \nsimulations of the dynamics. With this model we stu died the contribution of radiation \nlosses to the ferromagnetic resonance linewidth, as  measured with the stripline FMR. \nWe found that for films with large conductivity of metals the radiation losses are \nsignificantly smaller than for magneto-insulating f ilms. Excitation of microwave eddy \ncurrents in these materials contributes to the tota l microwave impedance of the system. \nThis leads to impedance mismatch with the film envi ronment resulting in decoupling \nof the film from the environment and, ultimately, t o smaller radiation losses. We also \nshow that the radiation losses drop with an increas e in the stripline width and when \nthe sample is lifted up from the stripline surface.  Hence, in order to eliminate this \nmeasurement artefact one needs to use wide striplin es and introduce a spacer between \nthe film and the sample surface. The radiation loss es contribution is larger for thicker \nfilms. \n \n \n \n1. Introduction \nThe microwave conductivity contribution to the stri pline broadband ferromagnetic \nresonance (FMR) response of highly-conducting (meta llic) magnetic multilayers and \nnanostructures of sub-skin-depth thicknesses has at tracted significant attention in \nrecent years [1-17]. It has been shown that these e ffects are important when the \nmicrowave magnetic field is incident on only one of  the two surfaces of a planar \nmetallic material (see e.g. [17]).  \nThe geometry of a stripline ferromagnetic resonance  experiment [17-21] is \ncharacterized by such single-surface incidence of t he microwave magnetic field on the \nsample. This experiment usually employs a macroscop ic coplanar (CPW) or \nmicrostrip (MSL) stripline through which a microwav e current flows (Fig. 1). A \nsample - a film or a nanostructure - sits on top of  this line, often separated by an \ninsulating spacer. The stripline with the sample is  placed in a static magnetic field \napplied along the stripline. The microwave current in the stripline drives \nmagnetization precession in the ferromagnetic mater ial. The complex transmission \ncoefficient S21 of the stripline is measured either  as a function of microwave \nfrequency f for a given applied magnetic field H=e zH (“frequency resolved FMR”), or \nas a function of H for given f (“field-resolved FMR”) to produce FMR traces. The \nFMR absorption by the material is seen as a deep in  the Re(S21) vs. H or f trace. \n                                                 \n*mikhail.kostylev@uwa.edu.au  2 In our previous work [16] we constructed a quasi-an alytical theory of stripline \nbroadband FMR of single-layer metallic ferromagneti c films. A drawback of the \nconstructed theory is that it is not self-consisten t. It uses the same approach as \npreviously employed to calculate the impedance of m icrostip transducers of travelling \nspin waves in thick magneto-insulating films [22,23 ]. The central point of this \napproach is assumption of some realistic distributi on for the microwave current \ndensity across the width of a microstrip (“Given Cu rrent Density” (GCD) method). \nThe next step of the solution is calculation of the  amplitude of the dynamic \nmagnetization in the film driven by the Oersted fie ld of the assumed microwave \ncurrent density. Finally, the microwave electric fi eld induced in the stripline by the \nfound dynamic magnetization in the film is determin ed. This 3-step analysis allows \none to obtain the value of the complex impedance in serted into the microwave path \ndue to loading of a section of the microstrip line by the ferromagnetic film.  \nLater on a self-consistent approach to calculation of the inserted impedance was \nsuggested [24,25]. In the framework of the self-con sistent approach the distribution of \nthe microwave current density is obtained by solvin g an integral equation. Then the \nfound distribution is used to calculate the impedan ce with one of the same GCD \ntheories [22,23]. \nIn this work we use a similar approach of an integr al equation to obtain a self-\nconsistent solution for the broadband stripline FMR  of highly conducting \nferromagnetic films with nanometre-range thicknesse s. To simplify the problem, \ncontrary to [16], we neglect the exchange interacti on. In this way we are able to treat \nthe fundamental (dipole) mode of FMR response of th in magnetic films only; \nresponses of the higher-order standing spin wave mo des across the film thickness \ncannot be obtained with this theory. Given the impo rtance of the fundamental mode \nfor various applications of FMR [19,26], this does not represent a major drawback. \nFurthermore, simple analytical description in the F ourier space which follows from \nthe exchange-free approximation results in an effic ient and quick numerical algorithm \nfor solution of the integral equation for the micro wave current density in the stripline.  \nIn our discussion we will focus on the effect of co upling of the magnetization \ndynamics in the film to the microwave current in th e stripline. Inclusion of the \nconductivity effect will allow us to judge whether the conductivity may influence the \nstrength of this coupling. Experimentally, the prob lem of coupling of the probing \nstripline to the FMR in a film has been addressed i n a recent paper [27]. It has been \nshown that strong coupling leads to additional reso nance linewidth broadening called \n“radiation damping”. This damping mechanism is rela ted to radiation of the \nmagnetization precession energy back into the probi ng system – the stripline, because \nof non-negligible coupling between the two. Basical ly, one deals with the fact that the \nexternal and unloaded Q-factors of a resonator are different [28] if coupling of the \nresonator to environment is not vanishing. The theo ry developed in the present work \nincludes naturally the radiation damping, as well a s damping due to eddy currents and \nexcitation of travelling spin waves.  \n \n \n2. Numerical model \n \n To solve the problem we make use of the idea first  proposed in [9]. We extend it to \nthe case of electromagnetic boundary conditions app ropriate for excitation of \nmagnetization dynamics in a ferromagnetic film by a  stripline [16]. These boundary  3 conditions include microwave shielding effect by th e eddy currents in conducting \nfilms [10,11].  \n We consider a model in which the y-axis is perpendicular to the surfaces of a \nconducting magnetic film (Fig. 1(a)). For small ele vations s of the film from the \nstripline surface ( s< <w) the microwave field of a microstrip line is local ized within \nthe area of the width on the order 2 w. This allows us to consider the sample size in the  \ndirection x as infinite. \n \n \nFig.1. (a) Sketch of the modelled geometry. 1: grou nd plane of the microstrip line. 2: \nsubstrate of the microstrip line of thickness d. 3: infinitely thin strip of width w \ncarrying a microwave current. 4: Spacer of thicknes s s. 5: ferromagnetic film of \nthickness  L . \n \n  \n  Thus, the film of thickness L is assumed to be continuous in the x- and z-directions. \nThe external magnetic field zH=H u is applied in the positive direction of the z-axis. \nAll the dynamic variables depend harmonically on th e time - exp( ) i t ω, where ω is \nthe microwave frequency. In order to include the ma gnetization dynamics in the \nferromagnetic layer into the model, we employ the l inearized Landau-Lifshitz \nequation \n0 | |( ) z eff i t M ω γ =− × + × m m H u h . (1) \nIn (1) the dynamic magnetization vector m has only two non-vanishing components \n(mx , m y) that are perpendicular to the static magnetizatio n 0zMu, where 0M is the \nsaturation magnetization for the ferromagnetic film , and zu is a unit-vector in the \ndirection z. The dynamic effective field heff  has only one component - the dynamic \nmagnetic field h which includes the magnetostatic field of the dyna mic magnetization \nand the Oersted field of the eddy currents which ma y circulate in the film because of \nnon-vanishing film conductivity σ.  \n In this approximation, the Linearized Landau-Lifsh its equation reduces to the \nPolder Microwave Susceptibility Tensor ˆχ: \n \nˆχ=m h  (2) \n \nwith  \n  4 ˆa\nai\niχ χ χ\nχ χ =\n−, \nwhere  \n \n2 2 H M \nHω ω χω ω =−, 2 2 M\na\nHωω χω ω =−, \n \n0 MMω γ =, H G H i ω γ α ω = + , γ is the gyromagnetic ratio, Gα is the Gilbert magnetic \ndamping constant, and i is the imaginary unit. Note that the second term i n the last \nexpression accounts for magnetic losses in the syst em. The losses are included by \nallowing the applied field to take complex values ( /GH i α ω γ + ) [29].  \n The dynamic magnetic field h is sought as solution of Maxwell Equations \n \n σ∇× = h e ,  (3) \n0( ) iωµ ∇× =− + e h m ,  (4)  \n∇ =−∇ h m ,  (5) \n \nwhere 0µ is the magnetic permittivity of vacuum and h = (hx, h y). Because the \nmicrowave magnetic field of MSL is quasi-static and  also because the ferromagnetic \nfilm is metallic, we neglected the term involving t he electric permittivity on the right-\nhand side of Eq.(3) and took into account only the term involving the film \nconductivity σ.  \n We also assume that all dynamical variables do not  depend on z (the quasi-static \napproach to description of microwave transmission l ines), therefore our problem is \ntwo-dimensional. The standard real-space methods fa ce serious difficulties [1,30] \neven in two dimensions because of incompatibility o f the length scales in Fig. 1(a) – \ncompare L with w d, and  s . To get around this problem and ultimately to acce lerate \nthe numerical solution we take advantage of the tra nslational symmetry of the sample \nin the direction x. To this end we Fourier-transform Eqs.(3-5) and (1 ,2) with respect to \nx.  \n To implement the Fourier transformation we assume that  \n, , exp( ) k k ikx dx ∞\n−∞ = − ∫m h m h ,  \nwhere \n, 1/(2 ) , exp( ) k k ikx dx π∞\n−∞ =∫m h m h . \n This procedure results in a system of equations:  \n/x y z h y ikh e σ ∂ ∂ + =−  , (6)  \n0 / ( ) z x x e y i h m ωµ ∂ ∂ =− + ,  (7)  \n0( ) z y y ke h m ωµ =− + ,   (8) \n  / / x y y x ikh h y m y ikm − +∂ ∂ =−∂ ∂ + .   (9) \n \nHere and at many places below we drop the subscript  “ k” to simplify notations.  \n We differentiate (6) and substitute (7) into the r esulting differentiated equation and \nmake use of (9). This gives:   5  \n2 2 2 2 ( / ) / 0 x x y y K h K m ik m y ∂ ∂ − − − ∂ ∂ = ,  (10) \n \nwhere 2 2 \n0 K k i σωµ = + .  \n We seek solutions of the system of equations (2,9, 10) for the area inside the film \nwith the spatial variation exp(± Qy ), as suggested in [9]. After a brief calculation o ne \nfinds that  \n \n2\n0V Q k i µ µ ωσ = −  , (11) \n \nwhere \n \n2 2 ( 1) /( 1) V a µ χ χ χ   = + − +   . (12) \n \nAccordingly, the complete solution reads  \n \nexp( ) exp( ) \nexp( ) exp( ) x\nyh A Qy B Qy \nh iAC Qy iBC Qy + − = + − \n= + − , (13) \n \nwhere  \n \n( )( ) ( 1) / ( 1) a a C k Q k Q χ χ χ χ ±= + ± ± + . (14) \n \nThis result is the same as in Ref.[9].  \n We now need electromagnetic boundary conditions wh ich will relate the \nelectromagnetic fields inside and outside the film.  The microwave magnetic field \noutside the film is given by the same Eq.(10), but for 0σ= = m. Let us first consider \nthe area above the film y>L. From (10) and the condition of vanishing of the \nmicrowave magnetic field at y=+ ∞ one easily finds that for y>L  \n | |\ny x kh i h k=− ,  (15) \nand at the film surface ( y=L)  \n \n| |( ) 0 yi yi xi kh m ih k+ + = . (16) \nIn this expression the subscript “ i” indicates that these field components are taken a t \nthe film surface from inside  the film. Eq.(16) represents the electromagnetic b oundary \ncondition at y=L which excludes the area y>L from consideration. \n Substitution of Eqs.(13,14) into Eq.(16) allows on e to eliminate the unknown \nintegration constant B \n \n0( ) B AB k = , (17) \n \nwhere  6 0(1 ) ( ) ( ) exp(2 ) (1 ) ( ) a\naC k i k k B k QL C k i k k χ χ \nχ χ +\n−+ + − =+ + − . (18) \n \n A similar boundary condition can be obtained for t he area in front of the film ( y<0). \nThis area contains the strip and the ground plane of  the MSL. We model the strip as \nan infinitely thin sheet of a microwave current zIu. The linear current density is ( ) j x  \n(Fig. 1). The width of the sheet along the x-axis is w; hence /2 \n/2 ( ) w\nwI j x dx \n−=∫. The \nsheet is infinite in the z direction to ensure continuity of the current. It is located at a \ndistance s from the film surface y=0 (Fig. 1). An electromagnetic boundary condition \nat the strip reads \n \n( 0) ( 0) xk xk k h y s h y s j =− + = =− − − .  (19) \n \n At a distance s+d from the strip the MSL ground plane is located. Th e ground \nplane is modelled as a surface of a metal with infi nite conductivity (“ideal metal”) \nlocated at y=−s−d. At the ideal-metal surface hyk =0. By solving (10) for −d−s<y<0 \ntaking into account (8) together with this boundary  condition and the condition (19), \none obtains a boundary condition at y=0, as follows: \n \n | | sinh(| | ) | |( )coth(| |( )) sinh(| |( )) yi yi xi k k k d k h m k d s i h i j k k d s k + + − = +. (20) \n \nSubstitution of (13) and (14) into (20) taking into  account (17) results in an expression \nfor the integration constant A (see Eq.(13)) \n \n0( ) k A A k j = . (21)  \n \n This concludes solution of the system of equations  (2), (6-9). We do no not show \nthe expression for 0( ) A k  here because it is extremely cumbersome (but easy to derive \nwith some standard software for analytical calculat ions). We note that for σ=0, \nequating the denominator of 0( ) A k  to zero produces the Damon-Eshbach dispersion \nrelation for the surface magnetostatic wave [31]. T his denominator form is typical for \nsolutions of problems of wave and oscillations exci tation – zeros of the denominator \nof an excitation-problem solution correspond to eig en-frequencies and eigen wave \nnumbers of the respective waves or oscillations.  \n Once the expression for A has been obtained, it is a short exercise to deriv e an \nexpression for the microwave electric field induced  at the strip surface by the \ndynamical processes in the magnetic film. This expr ession reads \n \n( ) zk k k e y s G j =− = . (22) \n \nwhere \n \n( )0\n0 0 0 sinh(| | ) cosh(| | ) ( ) ( ) ( ) | | cosh(| |( )) k k k d G k s A k A k B k j k k d s ωµ =− + + +. (23) \n  7 (Here we included the subscript “ k” for the electric field, in order to recall that a ll the \ndynamic quantities above are actually spatial Fouri er components of the respective \nfields.) \n The electric field in the real space is obtained b y Fourier-transforming Eq.(22) \nnumerically. This is the only numerical step in the  solution, provided the Fourier \nimage of the distribution of the microwave current density kj is assumed to be known \n(GCD approach). In our previous work [16] we employ ed the GCD approximation. \nThe main reason for using that method was a very sl ow numerical code resulting from \nEqs.(1,9,10) when the effective exchange field is i ncluded in the model.  \n If the exchange-free limit, the availability of th e analytical solution (22) makes the \nnumerical solution very fast, since numerics is nee ded just to carry out the inverse \nFourier transformation of Eq.(22). This allows us t o go beyond the approximation of \nthe given current density and, ultimately, to solve  the problem of calculation of the \nelectric field self-consistently in the present wor k. In order to obtain the self-\nconsistent solution, we use the fact that the norma l component of the microwave \nmagnetic field hy should vanish at the surface of the ideal metal of  the strip. Then \nfrom Eq.(4) it follows that the z-component of the electric field ez induced in the strip \nby the dynamic magnetisation in the film should be uniformly distributed across the \nstrip width ( ez(x,y =−s)=const( x) for −w/2< x<w/2). Without any loss of generality we \nmay set  \n \n( , ) 1, /2 /2 ze x y s w x w =− = − ≤ ≤ . (24) \n \nThis results in an integral equation  \n \n/2 \n/2 1 ( ') ( ') ', /2 /2 w\ne\nwG x x j x dx w x w \n−= − − ≤ ≤ ∫,   (25) \n \nwhere  \n \n( ) exp( ) e k G p G ikp dk ∞\n−∞ = − ∫.  (26) \n \nIn this work the Green’s function of the electric f ield eG is obtained by carrying out \nthe integration in Eq.(26) numerically. We calculate  ( ) eG p values for 2 N equidistant \npoints ip in the range iw p w − ≤ ≤ . This transforms Eq.(25) into a vector-matrix \nequation \n \n, ' '\n' 1 1N\ni i i \nij x \n== Γ Δ ∑ , (27) \n  \nwhere , ' '( ) i i E i i G x x Γ = − , ( ) i i j j x = is the unknown distribution of the microwave \ncurrent density for N equidistant mesh points ix (/2 /2, 1,2,... iw x w i N − ≤ ≤ = ) and \nxΔ is the mesh step. Solving Eq.(27) for ij using numerical methods of linear \nalgebra results in self-consistent determination of  the complex impedance of MSL  8 loaded by the film Zr. The latter quantity is a measure of the microwave  magnetic \nabsorption by the film [6]. It may be defined as fo llows: \n \n rUZI=− , (28) \n \nwhere the linear voltage U (measured in V/m) is the mean value of the total e lectric \nfield induced at the surface of the strip of MSL:  \n \n/2 \n/2 1( , ) w\nz\nwU e x y s dx w−= =− ∫.  (29) \n \nGiven (24), (27) and (29), Eq.(28) reduces to \n \n11\nr N\ni\niNZwj\n==\n∑, (30) \n \nwhere the vector ij is the numerical solution of the vector-matrix equ ation (27). \n Once Zr has been computed, it is a straightforward procedu re to calculate the \ntransmission coefficient S21 of the stripline loade d by the ferromagnetic film [10]. \nThe formalism has been explained in detail in [16].  We briefly repeat it here, since it \nis important for understanding of the mechanism of radiation losses. \n We now assume that the film has a finite length sl along MSL (i.e. in the y-\ndirection). The presence of the film on top of the MSL divides MSL into 3 sections – \na section covered by the film (“loaded MSL section”  or “loaded microstrip” for \nbrevity), a section between the input port of the m icrostrip fixture and the front edge \nof the film (“unloaded” microstrip section), and an other unloaded section – between \nthe far edge of the film and the output port of the  MSL fixture. The unloaded sections \nof MSL have the same characteristic impedance cZ. The characteristic impedance fZ \nof the loaded section is different because of the f ilm presence.  \n The complex transmission coefficient of the loaded  section of MSL reads \n2\n2121 exp( ) exp( ) f s f s Sl l γ γ Γ − =Γ − −  , (31) \n \nwhere Γ is the complex reflection coefficient from the fro nt edge of the loaded MSL \nsection \n \nf c \nf c Z Z \nZ Z −Γ= +,  (32) \n \n0( ( ) f r c Z Y Y γ= +    (33) \n \n is the complex propagation constant of the loaded microstrip, \n  9 0/( ) f r c Z Z Y Y = + , (34) \n \n0Y is the intrinsic (i.e. in the absence of the sampl e) parallel capacitive conductance of \nMSL, and cY is the parallel capacitive conductance due to the electric shielding effect \n[1]. As shown in [16], cY is negligible for microstrip lines, therefore we w ill neglect it \nbelow. \n \n 3. Discussion \n A. Self-consistent approach vs. GCD approach \nThis algorithm has been implemented as a MathCAD wo rksheet. The numerical \ninverse Fourier transform (Eq.(26)) has been carrie d out using tools for numerical \nintegration built-in MathCAD. The range of integrat ion was 100 / 100 / w k w π π − ≤ ≤ . \nTo produce the numerical solution, a mesh containin g 100 points equidistantly \ndistributed over the strip width was utilized ( N=100). The linear system of equations \n(27) was solved using a MathCAD built-in function e mploying the LU-decomposition \nmethod. Since only two steps of the calculation req uire numerical approaches, the \ncomputation is short – a result for 150 values of t he applied field H is obtained within \n30 min.  \n Fig. 2 shows an example of the self-consistent cal culation of the microwave \ncurrent density distribution across the strip cross -section for two different film \nthicknesses and two different values of conductivit y of the ferromagnetic film. In that \ngraph it is compared with the analytical solution f or an unloaded stripline [32]  \n \n2( ) 1/ 1 (2 / ) j x x w = − . \n \nThe spatial Fourier transform of this distribution reads \n02 2 kw kw j J π  =    ,  (35) \n \nwhere 0( ) J p  is the Bessel function of first kind of zeroth ord er.  \n From this figure one sees that the current distrib utions are qualitatively similar, the \nmain difference being a much stronger increase in t he current density towards the \nstrip edges for the analytical model [32]. One also  sees that for non-conducting films \nthe current value needed to obtain 1 Volt/m of micr owave electric field at the strip \nsurface is about two times smaller than for ones wi th conductivity of Permalloy. For \nthe conducting films this value grows with an incre ase in the film thickness, but for \nthe magneto-insulating ones it remains almost the s ame (compare the traces for L=40 \nnm and L=100nm in Fig. 2).  \n The similarity of the current density distribution s in Fig. 2 translates into similarity \nof the applied field dependences of the stripline l inear impedance (Fig. 3). One clearly \nsees that the results of the self-consistent calcul ation and of the GCD model \nemploying Eq.(35) to calculate Zr are quite close. This justifies the use of the GCD  \napproach in our previous work [16]. \n It is worth noticing excellent agreement of the re sult in Fig.(3) obtained in the \nframework of the GCD model with a respective calcul ation with the dipole-exchange \nnumerical code [16] (not shown). The amplitude of t he resonance peaks in Fig. 3 and \nthe off-resonance level of Zr are the same in both cases, the only difference bet ween  10  the two calculations being the presence of a higher -order standing-spin-wave peak in \nthe dipole-exchange model (see e.g. Fig. (2) in [16 ]). The peak appears for large \nsample conductivity values and is due to the eddy-c urrent contribution to excitation of \nmagnetization dynamics by stripline transducers [10 ].  \n Also, like in [16], the obtained value of Im( Zr) off resonance is in agreement with a \nresult of calculation with the known analytical for mula for the linear inductance for \nmicrostrip lines (see [16] fore more detail of this  property of Zr for non-conducting \nfilms). For instance, in Fig. 3(b) the off-resonanc e Im( Zr)=327 Ohm/cm (as measured \nfor H=0), and the independently calculated linear induct ance for this microstrip \ngeometry is 332 Ohm/cm. \n(a) Current density (A/cm) \n01234\n(b) \nCo-ordinate across strip width ( µm) -40 -20 0 20 40 Current density (A/cm) \n01234(c) \n(d) \nCo-ordinate across strip width ( µm) -40 -20 0 20 40 \n \n \nFig. 2. Distributions of microwave current density across the width of the strip. Solid \nlines: results of the self-consistent calculation. Dashed lines: calculated with the \nanalytical formula (35). For all panels, film thick ness L=40nm. (a) and (b): strip width \nw=100 micron. (c) and (d): w=1500 micron. (a) and (c): film conductivity σ=4.5x10 6 \nS/m. (b) and (d) σ=0.  \nMicrowave frequency is 9.5 GHz, film saturation mag netization (4 pM 0) is 10 kG, \ngyromagnetic ratio is 2.8 MHz/Oe, Gilbert magnetic loss parameter is 0.008.  \n \n B. Radiation losses \n Let us now discuss the important aspect of couplin g of magnetization dynamics in \nthe film to the electromagnetic field of the stripl ine transducer. As mentioned in the \nintroduction, non-negligible coupling results in ad ditional contribution to the \nresonance linewidth due to “radiation losses” [27].  Below we will carry out \ncalculations with the constructed model in order to  elucidate importance of the \nradiation losses effect and its potential correlati on with the eddy-current shielding  11  effect [10]. Above we have established that the GCD  model and the self-consistent \none deliver practically equivalent results. Therefo re we will use numerical results \nobtained with the given-current density model for t his discussion. A big advantage of \nthe GCD method is that software based on it is very  fast. Given that  \n \n(a) Re( Zr) (Ohm/cm) \n110 115 120 125 130 135 \n165 170 175 180 185 190 \n(b) Re( Zr) (Ohm/cm) \n0510 15 20 \n315 320 325 330 335 340 \n(c) Re( Zr) (Ohm/cm) \n010 20 30 40 \n80 90 100 110 120 130 \n(d) \nApplied field (Oe) 0 500 1000 1500 Re( Zr) (Ohm/cm) \n80 100 120 140 \n300 320 340 360 (g) \n10 12 14 16 18 \nIm( Zr) (Ohm/cm) \n46810 \n(h) \nApplied field (Oe) 0 500 1000 1500 0.0 0.5 1.0 1.5 \nIm( Zr) (Ohm/cm) \n64 66 68 70 (f) \n0.0 0.2 0.4 0.6 \nIm( Zr) (Ohm/cm) \n64 65 66 67 68 69 (e) \n22 23 24 25 26 \nIm( Zr) (Ohm/cm) \n14 15 16 17 18 \n \nFig. 3. Comparison of results of the self-consisten t calculation (thick lines) with \nresults obtained with the Given Current Density mod el (thin lines). Solid lines: real \nparts of impedance Z r. (left-hand vertical axes). Dashed lines: imaginar y parts of the \nimpedance (right-hand vertical axes.) Left-hand col umn: strip width w=100 micron. \nRight-hand column: w=1500 micron. (a),(e), (c), and (g): film conductiv ity σ=4.5x10 6 \nS/m. (b), (f), (d), and (h): σ=0. (a),(b), (e) and (f): film thickness L=40 nm. (c), (d), (g) \nand (h): L=100 nm. All other parameters of calculation are th e same as for Fig. 2. \n \n( , ) ( , )( ) z zk e x y s e x y s ikx dk ∞\n−∞ =− = =− − ���, (36) \n \nin the framework of the GCD approach Eq.(29) reduce s to   12   \n0sin( /2) ( , ) 2 /2 2 zk w kw kw U e x y s J dk kw π∞\n−∞   = =−     ∫. (37) \n \nTaking the integral in (37) is the only numerical s tep of this calculation. Therefore, \nthe corresponding MathCAD worksheet is very quick –  it takes about 10 seconds to \ncomplete a program run for 150 values of the applie d field. \n \n Fig. 4 compares results of calculation of Re(S21) and Re( Zr) [33]. One sees a \nnoticeable difference in linewidths for the two pea ks.  We fit complex S21 and Zr with \na complex Lorentzian \n \n1\n0\n2( ) DF H D H D = + −, (38) \n \nwhere D0, D1 and D2 are allowed to take complex values. The fits revea l that the \nresonance linewidths (given by Im( D2)) are different for S21 and Zr – 45 Oe and 31 \nOe respectively.  \n Fig. 5(a) demonstrates the extracted resonance lin ewidth dependence on the strip \nwidth. Clearly, the difference in the linewidths fo r the two quantities decreases with \nan increase in w. From this figure one also sees that an increase i n the thickness s of \nthe spacer between the stripline and the film also decreases the difference. \nApplied field (Oe) 0 500 1000 1500 Re(S21) (unitless) 0.000 0.001 0.002 0.003 \nReal part of \nimpedance (ohm/cm) 120 130 140 150 160 \n \nFig. 4. Comparison of Re(Z r) (dashed line, right-hand axis) and Re(S21) (solid l ine. \nleft-hand axis) traces.  Film thickness is 60 nm, s trip width is 100 micron, \nconductivity σ=4.5x10 6 S/m. Film length along the strip is 7 mm. All othe r \nparameters of calculation are the same as for Fig. 2. \n \n Let us now discuss the origin of this behaviour. T he relation between S12 and Zr is \ngiven by Eqs.(31-34).  From these formulas it becom es clear that the linewidth of the \npeak in Zr(H) represents a kind of “internal” linewidth for the  ferromagnetic \nresonance in the film if we extend the notion of th e “external quality factor” [28] onto \nthe linewidth parameter. Similarly, the linewidth o f the resonance peak in the S21( H) \ndependence is then the “external linewidth”, since it includes the strength of the \ncoupling of the sample to the input and the output ports of the probing stripline fixture. \nActually, Eqs.(31-34) take into account how strongl y the section of the microstrip line \ncovered by the sample (“loaded section”) couples to  the sections of the stripline in \nfront and behind the sample (“unloaded sections”). Important is the mismatch in the \ncharacteristic impedances of the loaded section fZ and the unloaded ones cZ (see  13  Eq.(32)). As follows from Fig. 2, for magneto-insul ating films, off resonance, \nf c Z Z =. Thus, a perfect impedance match takes place. (In real experiments it will be \nsome extra mismatch due to the dielectric constants  of the film and the film substrates. \nThese constants are not taken into account in the p resent theory.) The perfect \nimpedance match means strong coupling and, conseque ntly, significant broadening of \nthe external resonance line. On the contrary, for t he films with large conductivity of \nmetals, due to large contribution of microwave eddy  currents in the films to rZ off \nresonance, there is an impedance mismatch for any v alue of the applied field. \nConsequently, the internal and the external resonan ce linewidths are closer to each \nother.  \n Physically, an impedance mismatch between the load ed and the unloaded sections \nof the stripline implies that a wave of electric fi eld ze (Eq.(36)) induced in the strip by \nthe magnetization dynamics in the film gets partly trapped within the loaded section. \nThis happens because the mismatch leads to a non-va nishing reflection from the edges \nof this section (i.e. from the stripline cross-sect ions along the y-z plane corresponding \nto the edges of the film sample) and, consequently,  to internal reflection of the wave \nincident on the section edges from inside the loade d section. The energy of the \nmicrowave electric field is a part of the total ene rgy of the ferromagnetic resonance in \nthe sample. The larger the mismatch, the more energ y of the electric field of dynamic \nmagnetization is trapped below the film, the less r esonance energy escapes into the \ntwo unloaded sections of the stripline. Consequentl y, with an increase in the mismatch, \nthe loaded quality factor of the ferromagnetic reso nance increases which ultimately \nleads to a smaller external resonance linewidth. \n One more important observation from Fig.(5a) is th at the linewidth of the \nresonance peak in Zr  (thick dashed line) varies noticeably as a functio n of w for s=0. \nOne also sees that this trace converges with the re spective trace for s=33 µm (thin \ndashed line) for large values of w. The two facts suggest that the radiation losses d o \nnot depend solely on coupling between the loaded an d unloaded stripline sections. \nThere exists one more contribution to the total rad iation losses and this contribution \nreduces with an increase in s. This contribution is coupling of the microwave fi eld of \nthe loaded section of the stripline to the magnetiz ation dynamics in the film. The \nelectric field zeof dynamic magnetization represents an evanescent w ave outside the \nfilm – it decays exponentially with the distance fr om the film surface. Therefore, the \nfarther the stripline is located from the film surf ace, the smaller the microwave \nvoltage induced by the dynamic magnetization across  the length of the loaded section \nof the stripline is. This implies that increasing s removes one more contribution to the \nradiation losses – the one related to coupling of t he film to the stripline beneath it. \n Furthermore, for the large separation s=33 micron the external linewidth (as \nextracted from the S21 traces) becomes very close t o the one extracted from the Zr \ndata (compare the two thin lines in Fig. 5(a)). Thi s suggests that the coupling of the \nferromagnetic resonance to the ports of the stripli ne fixture is a two-step process. The \nfirst step is coupling of the electric field of the  dynamic magnetization to the strip \nlocated below the film. The second step of the proc ess is coupling of the loaded \nsection of MSL to the two unloaded ones via the mic rowave voltage induced across \nthe loaded section.  14  (a) \nstrip width ( µm) 0 500 1000 1500 2000 Resonance \nlinewidth (Oe) 25 30 35 40 45 50 55 \n(c) \nFilm conductivity (S/m) 10 010 110 210 310 410 510 610 710 8Resonance \nlinewidth (Oe) 20 30 40 50 60 70 80 (b) \nFilm thickness (nm) 0 50 100 150 Resonance \nlinewidth (Oe) 20 30 40 50 60 70 80 90 100 \n(d) \nFilm length (mm) 0 2 4 6 8 10 12 Resonance \nlinewidth (Oe) 20 40 60 80 100 120 \n \nFig. 5. (a) Resonance linewidth as a function of th e strip width. Thick lines: no spacer \nbetween the film and the stripline (but no electric  contact between the two). Thin \nlines: spacer thickness s=33 micron. Solid lines lines: S21; dashed lines: Zr. (b) \nResonance linewidth vs. film thickness. Thick lines : σ=4.5x10 6 S/m; thin lines: σ=0. \nSolid lines: S21; dashed lines: Zr. Strip width is 100 micron. s=0 for all plots in this \npanel. (c) Resonance linewidth as a function of fil m conductivity. Solid lines: S21; \ndashed lines: Zr. Thin lines: s=33 micron; thick lines: s=0. Strip width is 100 micron. \n(d) Linewidth of the peak in S21 traces as a functi on of the sample size ls along the \nmicrostrip. Open circles: σ=0; L=150nm; filled circles: the same, but σ=4.5x10 6 S/m. \nOpen triangles down: σ=0; L=40nm; filled triangles down: the same, but σ=4.5x10 6 \nS/m. Filled triangles up: the same as filled triang les down, but the strip thickness is \nnow finite and is equal to 16 micron. For all graph s in (d) w=100 micron and s=0. All \nother parameters of the calculation are the same as  for Fig. 2. ls for (a)-(c) is 7 mm.  \n  15   If the first step is inefficient, the overall coup ling strength will be small, \nindependently from the efficiency of the second ste p. This explains why the \ndifference in the linewidths for the peaks in S21 a nd Zr decreases with an increase in s. \nThe increase in s leads to a decrease in the efficiency of the first  step of the coupling \nprocess and hence to a decrease in the total coupli ng strength.  \n The analysis above suggests that in order to measu re the unloaded (or intrinsic) \nresonance linewidth of a sample very precisely, it is important to keep the strength of \ncoupling for the first step of the coupling process  weak. The latter is achieved by \nlifting the sample from the film surface. Also, the  coupling is weaker for wider \nstriplines. This is related to smaller current dens ity in a wider stripline for the same Zc \nand the same microwave voltage applied to the input  port of the line. As ez scales as j \n(see Eq.(22)), increasing w decreases ez  and hence decreases the efficiency of the first \nstep of the coupling process. For conducting sample s, the eddy-current contribution to \nthe FMR dynamics helps to further decrease the coup ling, since this contribution \nincreases with an increase in w [16]. Stronger microwave shielding by the eddy \ncurrents in the film for larger w values leads to a stronger decrease in rZ off \nresonance and hence to stronger impedance mismatch leading to a decrease in the \nstrength of the second stage of coupling.  \n Unfortunately, both measures – lifting the sample and increasing the stripline \nwidth - will also result in a decrease in the stren gth of the FMR response – the height \nof the peak in S21. On the other hand, the peak hei ght grows with an increase in the \nfilm thickness L (see Fig. 3). Fig. (5b) demonstrates the effect of  the film thickness on \nthe resonance linewidth. One sees that the resonanc e linewidth grows with an increase \nin L. The effect is much stronger for S21 than for Zr. Hence, the second step of the \ncoupling process contributes more to this effect. I nterestingly, for S21 the linewidth \nbroadening is larger for magneto-insulating films ( thin solid line) than for ones with \nconductivity of metals (thick solid line). For Zr it is other way around – the linewidth \nbroadening is stronger for conducting films (compar e the two dashed lines).  \n This is due to contribution of eddy current losses  to the intrinsic linewidth for \nthicker films. For thinner films the two dashed lin es overlap with graphic accuracy. \nThe latter confirms the known fact that contributio n of eddy-current losses to the \nintrinsic FMR linewidth  for metallic films with small thicknesses is negli gible. It also \nsuggests that on resonance the contribution of eddy  currents to ez is the same as off-\nresonance. Hence, the field which is responsible fo r the first step of the coupling \nprocess is the electric field of dynamic magnetizat ion. In other words, the contribution \nto the first step of the coupling process by the el ectric field induced by the Oersted \nfield of the eddy current in the film through Farad ay induction is negligible.  \n The fact that the linewidth broadening for the pea k in S21 is larger for insulating \nfilms confirms our conclusion above that shielding by eddy currents reduces the \nstrength of coupling for the second step of the cou pling process. A calculated \ndependence of the resonance linewidth on film condu ctivity is shown in Fig. 5 (c ). \nThe main observations from this figure are that in the 10-GHz frequency range, only \nthe large conductivity of metals matters. Conductiv ity values below 10 4 S/m do not \naffect the broadband ferromagnetic resonance linewi dth. Noteworthy is a significant \ndrop in the peak width seen in the S21 traces betwe en 10 4 and 10 5 S/m. This is due to \nthe onset of the above-discussed eddy-current induc ed decoupling of the \nferromagnetic resonance in the film from the enviro nment. Also, noteworthy is \nsimultaneous growth of all four curves in Fig. 5(c ) for σ>10 6. As this effect is also \npresent for a strongly decoupled resonance ( s=33 µm, thin lines), it is not due to the  16  radiation losses, but to an increase in the eddy-cu rrent contribution to intrinsic FMR \nlosses for larger film conductivities. \n Fig. 5(d) demonstrates the effect of the sample le ngth along the stripline ls on the \nwidth of the external resonance line. This paramete r enters Eq.(31) only, hence it \naffects the second step of the coupling process onl y. One sees that the impacts of ls on \nmagneto-insulating and conducting films are quite d ifferent. For the insulating films \nthe model shows strong periodic variation of the li newidth, but for the films with \nlarge conductivity the dependence is smooth and the  change in the linewidth across \nthe displayed range of film lengths is much smaller . The quasi-periodic character may \nbe explained taking into account that ls enters the arguments of the exponential \nfunctions in Eq.(31). Given that cY is negligible and 0Y is an imaginary quantity, the \npropagation constant fγ is imaginary for σ=0 and off resonance. A small real part is \nadded to it on resonance (compare the scales of the  right-hand and left-hand vertical \naxes in Fig. 3(b)). However, because on resonance t he imaginary part of f s lγ still \ndominates, exp( ) f s lγ remains a (quasi)periodical function. In Fig. 5(d)  this character \nis seen as a significant scatter of data points for  σ=0. For the films with large \nconductivity of metals, fγ is essentially complex both on and off resonance ( see e.g. \nFigs. 3(a) or (e)). As a result, the S21 dependence  on f s lγ does not have a pronounced \nquasi-periodic character.  \n So far we have treated the stripline as infinitely  thin in the y-direction. The GCD \nmodel allows one to consider the effect of the stri p thickness, provided the \ndistribution of the microwave current density j(x,y) is known. For simplicity, let us \nassume that it is uniform across the strip thicknes s ts, Then, Zr may be estimated as  \n \n1( , ', , ') ( ', ') ' '\n( , ) e\nS\nr\nSG y y x x j y x dy dx SZ\nj y x ds =− ∫\n∫, (39) \n \nwhere sS wt = is the strip cross-section area. The Green’s funct ion ( , ', , ') eG y y x x  is \nobtained from (17,23) by considering the electric f ield of dynamic magnetisation at \nthe point ( y,x) of the strip cross-section, provided that the dyn amic magnetisation is \ndriven by an infinitely thin wire of current at the  point ( y’, x’), also belonging to the \nsame cross-section. \n An example of numerical calculation by using (39) is shown in the same Fig. 5(d). \nOne sees that inclusion of the final strip thicknes s drastically reduces the external \nlinewidth broadening. The effect is very similar to  lifting an infinitely thin strip by \nsome distance s on the order of ts/2 from the film surface. Thus, the finite thicknes s of \nreal striplines is a very important factor that nat urally reduces coupling of the film to \nthe environment in real experiments.   \n \n4. Conclusion \n In this work we constructed a quasi-analytical sel f-consistent model of strip-line \nbased broadband ferromagnetic resonance experiment.  With this model we studied the \ncontribution of radiation losses to the ferromagnet ic resonance linewidth. We found \nthat for films with large conductivity of metals th e radiation losses contribution is  17  significantly smaller. This is because of impedance  mismatch due to excitation of \nmicrowave eddy currents in these materials. We also  show that the radiation losses \ndrop with an increase in the stripline width and wh en the sample sits at some \nelevation from the stripline surface. Furthermore, the radiation losses contribution is \nlarger for thicker films.  \n Two consecutive steps of coupling of the ferromagn etic resonance to the \nenvironment leading to the radiation losses have be en identified. The first one is \ncoupling of the dynamic magnetization to the stripl ine section on top of which the \nfilm sits (“loaded” section). This coupling proceed s via the microwave electric field \nassociated with magnetization precession. The secon d step of the process is coupling \nof the microwave electric voltage induced in the lo aded section to unloaded sections \nof the stripline which join the loaded section to t he input and the output ports of the \nstripline fixture.  \n The impedance mismatch affects the second step of the coupling process. The \nstripline width and its distance from the film surf ace are important for the first step. \nBy minimizing the coupling strength for the first s tep of the process it is possible to \nsignificantly reduce total radiation losses.  \n Thus, in order to eliminate the measurement artefa ct of radiation losses in real \nbroadband stripline ferromagnetic resonance experim ents one needs to employ wide \nstriplines and introduce a spacer between the film and the sample surface.  \n \n \nAcknowledgment \n Financial support by the Australian Research Counc il, the University of Western \nAustralia (UWA) and the UWA’s Faculty of Science is  acknowledged.  \n \nReferences \n1. 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H.A. Wheeler, IEEE Trans. Microwave Theory Tech . 13 , 172 (1965). \n33. In order to allow the comparison of the two com plex functions, the phase of S21 \nwas rotated by 130 degree before taking the real pa rt of this complex quantity. "    },    {        "title": "1504.01820v2.Selective_interlayer_ferromagnetic_coupling_between_the_Cu_spins_in_YBa__2__Cu__3__O___7_x___grown_on_top_of_La___0_7___Ca___0_3___MnO__3_.pdf",        "content": "Selective interlayer ferromagnetic coupling between \nthe Cu spins in 𝐘𝐁𝐚𝟐𝐂𝐮𝟑𝐎𝟕−𝐱grown on top of \n𝐋𝐚𝟎.𝟕𝐂𝐚𝟎.𝟑𝐌𝐧𝐎𝟑  \n \n𝐒.𝐖.𝐇𝐮𝐚𝐧𝐠𝟏,𝟐,𝟑,𝐋.𝐀𝐧𝐝𝐫𝐞𝐰  𝐖𝐫𝐚𝐲𝟏,𝟒,𝟓, 𝐇𝐨𝐫𝐧𝐠−𝐓𝐚𝐲 𝐉𝐞𝐧𝐠𝟔,𝟕, 𝐕.𝐓.𝐓𝐫𝐚𝟖, \n𝐉.𝐌.𝐋𝐞𝐞𝟗, 𝐌.𝐂.  𝐋𝐚𝐧𝐠𝐧𝐞𝐫𝟐, 𝐉.𝐌.𝐂𝐡𝐞𝐧𝟗, 𝐒.𝐑𝐨𝐲𝟏, 𝐘.𝐇.𝐂𝐡𝐮𝟏𝟎, 𝐑.𝐖.𝐒𝐜𝐡𝐨𝐞𝐧𝐥𝐞𝐢𝐧𝟐, \n𝐘.−𝐃.𝐂𝐡𝐮𝐚𝐧𝐠𝟏,∗, and 𝐉.−𝐘  𝐋𝐢𝐧𝟖,𝟏,+    \n \n1Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  \n2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  \n3MAX IV Laboratory, Lund University, P . O. Box 118, 22100  Lund, Sweden  \n4Department of Physics, New York University, New York, New York 10003, USA  \n5Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, \nUSA  \n6Department of Physics, National Tsing Hua Un iversity, Hsinchu 30013, Taiwan  \n7Institute of Physics, Academia Sinica, Taipei 11529, Taiwan  \n8Institute of Physics, National Chiao Tung University, Hsinchu 30010, Taiwan  \n9National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan  \n10Departmen t of Materials Science and Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan  \n*ychuang@lbl.gov  \n+ago@cc.nctu.edu.tw  \n \nABSTRACT  \n \nStudies to date on ferromagnet/d -wave superconductor heterostructures  focus mainly on the effects at or near the \ninterfaces while the response of bulk properties to heterostructuring is overlooked. Here we use resonant soft x -ray \nscattering spectroscopy to reveal a novel c -axis ferromagnetic coupling between the in -plane Cu  spins in \nYBa2Cu3O7−x (YBCO) superconductor when it is grown on top of ferromagnetic La0.7Ca0.3MnO3 (LCMO) manganite \nlayer. This coupling, present in both normal and superconducting states of YBCO, is sensitive to the interfacial \ntermination such that it i s only observed in bilayers with MnO2but not with La0.7Ca0.3O interfacial termination. Such \ncontrasting behaviors, we propose, are due to distinct energetic of CuO chain and CuO2plane at the La0.7Ca0.3O and \nMnO2 terminated interfaces respectively, therefor e influencing the transfer of spin -polarized electrons from \nmanganite to cuprate differently. Our findings suggest that the superconducting/ferromagnetic bilayers with proper \ninterfacial engineering can be good candidates for searching the theorized Fulde -Ferrel -Larkin -Ovchinnikov (FFLO) \nstate in cuprates and studying the competing quantum orders in highly correlated electron systems.  \n \nIntroduction  \nFerromagnetism and d -wave superconductivity are often viewed as antagonistic orders as the spin exchange fie ld from \nferromagnetism can introduce energy difference between electrons in the spin -singlet Cooper pair. Although the \nCooper pair can also be formed with electrons from Zeeman splitted Fermi surfaces, an approach that gives finite \ncenter of mass momentum to the Cooper pair and leads to a spatially modulated superconducting order parameter, \nsuch state (Fulde -FerrelLarkin -Ovchinnikov, or FFLO, state1–3) remains to be identified in the high temperature \nsuperconducting cuprates. The coexistence of ferromagneti sm and superconductivity has been reported in some \nuranium -based superconductors,4, 5 but these superconductors have p -wave pairing symmetry and will not be the right \ncandidates for studying the competitive interactions between ferromagnetism and d -wave su perconductivity.  \nWith advanced thin -film growth technology,6–10 heterostructures made out of superconducting cuprates and \nferromagnetic manganites can serve as an ideal platform for such studies. Measurements on \nYBa2Cu3O7−x/La0.7Ca0.3MnO3 (YBCO/LCMO) hete rostructures have revealed an induced ferromagnetic Cu spin moment in the interfacial CuO2 plane that couples antiferromagnetically to the underlying Mn spin moment.6 \nInteresting phenomena such as ferromagnetism/superconductivity proximity11, 12 and invers e proximity effect,13 \ntransfer of spin -polarized electrons from manganite to cuprate,14 and the electronic orbital reconstruction at the \ninterface7 were also observed and proposed to account for the suppression of both ferromagnetism and d -wave \nsuperconduc tivity upon forming such heterostructures (Fig. 1(a)). However, studies to date on these heterostructures \nfocus mainly on the effects at or near the interfaces while the response of bulk properties to heterostructuring is \noverlooked. Here we use bulk -sensi tive resonant soft x -ray scattering spectroscopy (RSXS) and first principles \ncalculations to show that beyond the interface, a novel ferromagnetic order can be established within the YBCO layer \npossibly even in the superconducting state. This c -axis ferrom agnetic coupling between the in -plane Cu spins is subtle \nand can be effectively controlled by the interfacial termination.  \n \nResults  \nX-ray reflectivity of the bilayer  \nDue to very small mismatch between the in -plane lattice constants of YBCO and LCMO, epi taxial growth of \nYBCO/LCMO bilayers with atomically smooth interfaces can be achieved. As illustrated in Fig. 1(b), the crystallinity \nrequires that CuO2 plane and CuO chain to be at the cuprate/manganite interface in MnO2 (left panel) \nandLa0.7Ca0.3O (right  panel) terminated bilayers, respectively. After growth, the film quality was checked separately \nwith synchrotron reflectivity and the result from MnO2 terminated bilayer at 80 K is shown in Fig. 1(c). The \nmeasurement photon energy was 1240 eV, far away from Mn and Cu resonances. Clear intensity oscillations known as \nKiessig fringes from the constructive and destructive interferences between the reflecte d x-rays off different interfaces \nconfirm the high film quality. By fitting these Kiessig fringes, the thickness of each layer is determined be 15 nm (YBCO) \nand 7.5 nm (LCMO), in agreement with the growth conditions (see section V in Supplementary Informat ion).  \nOne immediately notices that the intensity of YBCO (001) structural Bragg peak, marked by the arrow in the figure, \nis only slightly stronger than the Kiessig fringes at this photon energy. This peak is even weaker than the (002) Bragg \npeak measured at larger 2θ angle. The weak (001) Bragg peak is due to the unique YBCO form factor such that the \nscattered x -rays from charges in two CuO2 planes and one CuO chain in the unit cell interfere destructively, and is the \nkey to allow us to identify the even w eaker magnetic contribution.  \n \nComparison of XAS spectra and RSXS resonance profiles  \nThe electronic structure of YBCO is changed when it is grown on top of the LCMO layer. These changes can even be \nseen at 300 K where the LCMO layer remains paramagnetic. Thin solid lines in Fig. 2 represent the Cu L -edge x -ray \nabsorption (XAS) spectra from a pure YBCO film (green, top panel) and the bilayers with MnO2 (red, middle panel) and \nLa0.7Ca0.3O (blue, bottom panel) interfacial terminations recorded in the total el ectron yield mode. Although the \nmain peak at 925.75 eV, ascribed to in -plane Cu2+ with a single 3 dx2−y2 hole,15 is very similar between these samples, \nthe shoulder near 927.5 eV and the high energy feature around 928.75 eV exhibit small differences. The sh oulder \nstructure, relevant to the ligand holes in CuO2 planes (Cu2+L, see inset in Fig. 2), is suppressed in both bilayers. The \nsuppression is consistent with their lower superconducting transition temperatures from effectively lower hole doping \nlevels.16 In addition, the high energy feature from Cu+ shows an enhancement, implying a noticeable charge -transfer \neffect taking place in CuO chains.  \nEven though these changes in XAS spectra upon heterostructuring are subtle, they can be clearly seen in the \nresona nce profiles. In RSXS spectroscopy, the resonance profile  |F(q⃑ ,E)|2 of an electronic order is sensitive to the \nelectronic states that coherently scatter the x -rays into the specific ordering wave vector q⃑  (see Methods and the \nschematic of experimental setup in Fig. 3(e)).17, 18 F(q⃑ ,E) depends on the form factor of scattering channel(s) of ith \natom  fi(ε̂i,ε̂0) (E), which can be charge, spin and orbital degrees of freedom, and the spatial arrangement of these \nscatterers eiq⃑⃑ ∙r⃑ iein the following way: F(q⃑ ,E)=∑fi(ε̂i,ε̂0)(E)eiq⃑⃑ ∙r⃑ ii . Herer i is the position vector of ith atom, E is the \nexcitation photon energy and the summation is carried out over the superlattice. fi(ε̂i,ε̂0)  depends on the incident ( ε̂i) \nand scattered ( ε̂0) photon polarizations, and is the sum of r eal (f′\ni(E)) and imaginary ( f′′i(E)) parts that are related to \neach other through Kramers -Kronig relations. Because of such dependence, RSXS resonance profile often exhibits a \nhigher degree of sensitivity to changes in local electronic structure than XAS. In the curr ent study, we choose YBCO \n(001) Bragg peak ( q⃑ = (001) in (HKL) unit) since the ferromagnetic coupling between magnetic unit cells along c -axis \nwould have the same wave vector as the charge unit cell.  \nThe resonance profiles of these samples are shown as thi ck solid lines in Fig. 2. For bilayers, irrespective to which \ninterfacial termination, the resonance profiles are very different from that of the pure YBCO film. Their resonance \nprofiles show two prominent peaks instead of one at both Cu L 3 and L 2 edges (f eatures labelled A and B at L 3 edge). \nThis double -peak structure is intrinsic and is neither caused by the self -absorption effect in the fluorescence yield measurements, nor by the presence of two types of Cu2+ in CuO2 planes. Further discussions can be fo und in section \nIII in Supplementary Information. Comparison to the maximum in XAS spectra shows that feature A is shifted t owards \nlower photon energy by 〜0.5 eV. Since XAS and RSXS spectra in Fig. 2 were recorded simultaneously, this energy shift \nis not an  experimental artifact. In fact, similar energy shift has been reported in some transition metal oxides that \nexhibit electronic ordering phenomena and in the case of cuprates, it was previously attributed to the subtle spatial \nvariation in the local energe tic of Cu 3d and O 2p states.19–21 The detailed calculation of resonance profiles remains an \nactively developing field and will not be discussed in this paper. For example, one may attempt to first use the x -ray \nmagnetic reflectivity (XRMR) to fit the Kies sig fringes to obtain the complex index of refraction n(E)=1−δ(E)+\niβ(E), which contains the information of charge and magnetic density profile over the thickness of layers, and then use \nthe optical theorem to link δ(E) and β(E) to f′(E) and f’’(E) respectively .22 However, this approach in only valid in the \nforward scattering geometry where Kiessig fringes are the dominant features in the spectra. In addition, XRMR cannot \nreveal the local spin moment. For rigorous treatment of resonant x -ray scattering using the  Kramers -Heisenberg \nformalism, see ref.23, 24 Here we set the measurement photon energy at 925.25 eV to focus on feature A to study the \nmagnetic interactions in CuO2 planes. At this photon energy, the 2θ angle is around 70 °. Inset in Fig. 1(c) shows the \nexemplary q -scans at 80 K (blue) and 300 K (red) from MnO2 terminated bilayer at this photon energy. As one can see, \nthe Kiessig fringes that overwhelm the (001) Bragg peak in Fig. 1(c) become negligible. The correlation length, \ndetermined from the inverse o f half -width -half-maximum of the peak, is >12 nm and is in agreement with the YBCO \nlayer thickness. This again confirms the bulk nature of recorded RSXS signal.  \n \nAdditional magnetic component in the YBCO (001) Bragg peak  \nFig. 3 shows the main experimental findings of this paper: the temperature dependence of normalized YBCO (001) \nBragg peak intensity (red markers, left axis; the intensity is normalized to 1.0 at 300 K) overlaid with the magnetization \ncurves (blue lines, ri ght axis) from these samples. The data points shown here are from the Lorentzian fitting of q -scan \nspectra (see Methods). For pure YBCO film, the temperature -independent Bragg peak intensity implies that the \nchanges in Cu charge scattering form factor, as well as their spatial arrangement, are negligible between .   80 K and \n300 K as expected (Fig. 3(a)). On the other hand, the Bragg peak intensity from MnO2 terminated bilayer shows \nintriguing temperature dependence where two step -like increases can be corre lated with characteristic temperatures \nin the magnetization curve: the . 190 K is the Curie temperature of LCMO layer and the . 105 K is related to the \nstructural phase transition of STO substrate (Fig. 3(b)).25 Since we do not expect to see changes in Cu charge scattering \nas temperature is lowered, the increases can be attributed to an additional order induced by the ferromagnetism in \nLCMO layer.  \nThe nature of this additional order can be further investigated by looking at the Bragg peak intensity as a fu nction \nof sample orientation relative to the photon polarization (see Fig. 3(e) for experimental geometry and the Methods). \nUnder the resonance condition, the scattering intensity |F(E)|2 can come from following terms that involve (unit \nvector) ε̂i, ε̂0 and sp in moment ŝ: ε̂i∙ε̂0,(ε̂i×ε̂0)∙ŝ and (ε̂i∙ŝ)(ε̂0∙ŝ) .17, 18 These three terms contribute to the \nmonopole charge scattering, circular and linear dichroism respectively. Since the incident photon energy is tuned close \nto Cu2+ L3 absorption edge, we only consider t he dipole (E1) transition. Following the treatment by Hill & McMorrow,17 \nthe scattering tensor can be expressed with components that depend on the incident and scattered photon \npolarizations (here, for example, σi and π0 refer to incident σ and scattered π  polarization respectively):  \n𝑓(𝜀̂𝑖,𝜀̂0)→(𝑓𝜎𝑖→𝜎0    𝑓𝜋𝑖→𝜎0\n𝑓𝜎𝑖→𝜋0    𝑓𝜋𝑖→𝜋0)\n=𝐹(0)(E)(1            0      \n0    cos(2𝜃))⏟        \n𝜀̂𝑖∙𝜀̂0+i𝐹(1)(𝐸)(                     0                           𝑧1𝑐𝑜𝑠(𝜃)+𝑧3𝑠𝑖𝑛(𝜃)\n𝑧3𝑠𝑖𝑛(𝜃)−𝑧1𝑐𝑜𝑠(𝜃)            −𝑧2𝑠𝑖𝑛(2𝜃) )\n⏟                                \n(𝜀̂𝑖×𝜀̂0)∙𝑠̂\n+𝐹(2)(𝐸)(                𝑧22                                  −𝑧2(𝑧1sin(𝜃)−𝑧3cos(𝜃)\n𝑧2(𝑧1sin(𝜃)+𝑧3cos(𝜃))         −𝑐𝑜𝑠2(𝜃)(𝑧12𝑡𝑎𝑛2(𝜃)+𝑧32))\n⏟                                        \n(𝜀̂𝑖∙𝑠̂)(𝜀̂0∙𝑠̂)                               (1) \n \nF(0)(E),F(1)(E) and F(2)(E)are defined in ref.17zi are the spin moments projected along three Cartesian axes and θ is \nthe grazing incidence angle ( 〜35° in the current study).  \nUsually, the F(i)(E) terms do not mix with each other except that the 0th harmonic component of F(2)(E) can \nmix with F(0)(E). But for ferro magnetic coupling where the magnetic unit cell coincides with the charge unit cell, the \nKronecker δ conserving the wave vectors in each term becomes 1 and these three terms need to be considered all together as shown above.  \nIn calculating the scattering i ntensity by taking the square of this f(ε̂i,ε̂0) matrix in equation (1), the cross terms \n[F(1,2)∗F(0)] will enhance the weak magnetic signal. Keeping the leading terms, the scattering intensity will vary with \nsample azimuthal angle (φ) as following: a+b∙sin(φ)+c∙cos2(φ) . The parameters b and c depend on the spin  \nprojection angle α and are proportional to |F(1)\nF(0)| and |F(2)\nF(0)| (see Section IV in Supplementary Information).  \nIn Fig. 3(f), the (001) Bragg peak intensity at 80 K shows the strong φ dependence, which can be fitted with the \naforementioned functional form (blue curve in Fig. 3(f)). One should note that such φ  dependence cannot come \nfrom Cu2+ charge scattering or the spin component normal to the CuO2 plane because these two contributio ns do not \ndepend on φ. On the other hand, it can come from Cu2+ spin component in the CuO2 plane. This finding implies the \nmagnetic origin for the additional order seen in the YBCO (001) Bragg peak. The (001) wave vector further tells us that \nit is caused b y the inter -unit-cell (c -axis) ferromagnetic coupling between the in -plane Cu spins. With the p -scattering \ngeometry, the maxima around 0 ° and 180° allow us to determine that the projected in -plane component is along the \nCu-O bond direction (Fig. 3(e) shows  the geometry at φ=0°). This spin alignment is 45 ° away from the easy axis of Mott \nantiferromagnetism, which is along the Cu -Cu or (110) direction, but follows the easy axis of ferromagnetism in the \nLCMO layer and is likely the outcome of strong couplings b etween manganite and cuprate.6, 29, 30  \nAlthough the induced ferromagnetic spin moments in the interfacial CuO2 plane below the Curie temperature of \nLCMO layer have been previously reported,6, 26 –28 our RSXS data suggests that they may have coupled ferroma gnetically \nalong c -axis throughout the YBCO layer as the diffraction peak width is limited by the layer thickness. This c -axis \nferromagnetic spin coupling is established at 80 K, just above the 〜 70 K superconducting transition temperature of \nYBCO under st udy. But the continued increase in RSXS intensity in Fig. 3(b) points to the scenario that this coupling can \npersist down to 30 K for its absence would put the normalized intensity value back to 〜1.0. But this coupling is subtle \nand can be greatly influence d by the interfacial termination. We have performed the same RSXS measurements on \nLa0.7Ca0.3O terminated bilayer and the results are shown in Fig. 3(c). Despite showing very similar characteristic \ntemperatures in the magnetization curve as those in the MnO2 terminated bilayer, the (001) Bragg peak intensity \nremains nearly temperature independent within our measurement resolution.  \n \nDFT calculations  \nTo investigate these contrasting behaviors, we have carried out the DFT calculations (for details, see Method s). The \nschematic in Fig. 4 shows the stacking order of CuO chains (black filled circles with vertical bars) and  CuO2 planes (red \nopen circles with horizontal bars) along c -axis in the calculations. The labeling of Cu sites is guide for readers to \nassociat e the calculated spin moments with their spatial arrangement. The positive and negative moments refer to the \nCu spins that are parallel and antiparallel to the Mn spins, respectively. The DFT calculations show that the magnitude \nof Cu spin moment in the MnO2 terminated bilayer (Fig. 4(a)) is on the order of 〜0.02𝜇𝐵/Cu in CuO2 planes (red \nopen circles), consistent with the XMCD measurements. However, the moment in CuO chains is negligible (black filled \ncircles). The antiferromagnetic coupling between Mn an d Cu spin moments near the interface is correctly reproduced \nin our DFT calculations (Cu site #1 in Fig. 4(a)). This coupling remains antiferromagnetic between the first two CuO2 \nplanes, a phenomenon that was previously predicted by the model calculations and was attributed to an anomalous \nscreening effect.31 Besides the first unit cell, our DFT calculations predict the ferromagnetically coupled spin moments \nthrough out the rest of YBCO layer. In contrast, for La0.7Ca0.3O terminated bilayer, only the chain Cu right next to the \ninterface exhibits a finite spin moment that couples ferromagnetically to the Mn spin moment (Fig. 4(b)). In that regard, \nthe La0.7Ca0.3O terminated bilayer is not expected to have ferromagnetic order inside the YBCO layer. Our DFT \ncalculations have given the results that are in agreement with the RSXS data in Fig. 3(b) and 3(c).  \n \nDiscussion  \nWe have identified a novel inter -unit-cell ferromagnetic coupling between the in -plane Cu spins in YBCO based on the \nclear φ and temperature depe ndence of YBCO (001) Bragg peak intensity recorded at Cu2+ resonance energy. This \nferromagnetic coupling is sensitive to the interfacial termination such that we could not detect its presence in the \nLa0.7Ca0.3O terminated bilayer under the same experimenta l condition. Observation of this coupling can also be \ncompared with previous model calculations.31 Here, we propose that the distinct energetic of CuO2 plane and CuO \nchain at the manganite/cuprate interface in the MnO2 and La0.7Ca0.3O terminated bilayers r espectively is \nresponsible for the contrasting behaviors.  \nFor MnO2 terminated bilayer, we propose that the double -exchange interaction in the itinerant eg bands of \nYBCO,  which is absent in the pristine YBCO and was omitted in the previous model calculations,31 emerges due to the induced \nCu spin moments and the influence from the ferromagnetism in poorly screened LCMO underlayer.11 For La0.7Ca0.3O \nterminated bilayer, the situation is rather different. Although the chain Cu has higher affinity to a ttract the \nspin-polarized electrons from LCMO layer, as evident from its lower superconducting transition temperature, the larger  \ninduced moment ( 〜0.04μB/Cu) and the ferromagnetic coupling to the Mn spin moment, the lower eg electron  \nitineracy in quasi -1D chains plus the strong electron affinity can localize the transferred electrons to the interface and \nweaken the double -exchange interaction in the remaining YBCO layer (Fig. 4(b)). Furthermore, different occupancy on \nthe respective orbitals of chain and plane Cu can also disrupt the double -exchange mediated ferromagnetic coupling.  \n \nSince previous XMCD measurements adopted specific geometries to suppress the ferromagnetic signal from the bulk \nlayers so as to enhance the contrast from the interface region,  these XMCD results are not able to substantiate or \ndispute our findings.6, 26, 27, 32 Therefore, observing this c -axis ferromagnetic coupling calls for further investigation on \nthe MnO2 terminated bilayers using XMCD with geometries that emphasize the sen sitivity to bulk ferromagnetic \nsignal from YBCO layer. In addition, our DFT calculations suggest that the induced Cu spin moment can couple \nferromagnetically to the Mn spin moment in the La0.7Ca0.3O terminated bilayer. Thus performing the XMCD \nmeasurements  on this type of bilayer can serve as an independent check to the theories. Our results also highlight the \nfact that the bulk properties of constituting layers will respond to heterostructuring, an aspect that is largely \noverlooked, and it has recently bee n shown that interesting phenomena can be manifested beyond the interfaces by \napplying the perturbations in the heterostructures.33  \n \nIn conjunction with previous XMCD measurements,6, 26, 27, 32 our results suggest that the induced ferromagnetic Cu spin \nmoments in CuO2 planes may couple ferromagnetically along the c -axis, forming the bulk ferromagnetism in YBCO. \nThis ferromagnetism can even exist in the superconducting state. It is known that the correlated transition metal \noxides can exhibit strong tendenc y towards electronic phase separation, which leads to other intriguing phenomena \nlike colossal magnetoresistance,37 but the possibility of coexistence of ferromagnetism and d -wave superconductivity \n(ferromagnetic d -wave superconductor) makes this type of b ilayer an exciting platform for investigating novel phases \nassociated with high temperature superconductivity. Introducing ferromagnetism to relax the asymptotic confinement \nthat limits the carrier mobility in the underdoped regime not only serves as an al ternative to induce the emergent \nquantum states besides the conventional hole doping approach, but can also reveal a new dimension in cuprate phase \ndiagram. Subjecting d -wave superconductivity to strong exchange field weakens its strength and enhances the \ncompetition with other ground states such as charge checkerboard and/or stripes,34, 35 and examining the extent of \nsuch competitions can be facilitated by using the heterostructures. Moreover, realizing the prerequisites for FFLO state \nimplies that cuprate /manganite heterostructures may exhibit other exotic properties like non -centrosymmetric Cooper \npairs and stripe -like inhomogeneity.36 Exploring their non -trivial interplay with inherent charge inhomogeneity will \nexpand the boundaries of d -wave superconduc tivity theories. To rule out the ubiquitous phase separation \nphenomenon38 and unambiguously identify this ferromagnetic d -wave superconductor, spectromicroscopy with \nsufficient energy and spatial resolutions such as spin -resolved nano -ARPES can be the deci sive tool for this task.39  \n \nMethods  \nMaterials  \nYBa2Cu3O7−x/La0.7Ca0.3MnO3 bilayers were fabricated on top of the (100) oriented SrTiO3 (STO) single crystal \nsubstrate using pulsed laser deposition (PLD) method. A KrF  (λ = 248 nm) excimer laser, with 10 Hz repetition rate and \n250 mJ power, was used to evaporate the targets. Before growing the bilayers, the substrate was first treated with \nHF-NH4F buffer solution to produce a uniform TiO2 termination at the surface. In -situ reflection high energy electron \ndiffraction (RHEED) was used to monitor the layer growth. The LCMO and YBCO layers were deposited at 700 °C and \n750°C, and 80 mTorr and 150 mTorr oxygen pressures respectively. After growth, these films were annealed in a 700 \nTorr oxygen atmosphere at 550 °C for one hour followed by slow cooling down to the room temperature to achieve \nfull oxygenation for the YBCO layer. To produce the MnO2 termination at the manganite/cuprate interface, we started \nwith TiO2 terminated ST O substrate and deposited the LCMO layer directly on top of it, then followed by the \ndeposition of YBCO layer. To switch to the La0.7Ca0.3O termination at the interface, a SrRuO 3 buffer layer (SRO, 1.5 \nu.c.) was deposited between the STO and LCMO layer. Th e schematic crystal structures shown in Fig. 1(b) were \nconfirmed by the high angle annular dark -field scanning transmission electron microscopy (HDDAF -STEM) (see Section \nI in Supplementary Information).  \n \n Transport measurements  \nThe resistivity of bilayer  samples was measured using the standard four -point method. The superconducting transition \ntemperatures were determined to be 〜70 and 55 K for MnO2 and La0.7Ca0.3O terminated bilayers respectively. The \nmagnetization measurements were carried out using the superconducting quantum interference device \nmagnetometer (SQUID, Quantum Design MPMS). 1000 Oe magnetic field was applied perpendicular to the surface of \nbilayers during the measurements and we only show the zero -field cooling data in the paper. The Curie temperature of \nLCMO layer is estimated by intersecting the linear extrapolation of high temperature leading edge of d M(T)/dT curve \nto zero, and is 〜190 K (see Section II in Supplementary Information).  \n \nX-ray spectroscopy  \nX-ray absorption (XAS) and resona nt soft x -ray scattering (RSXS) were performed at Beamline 8.0.1 of Advanced Light \nSource (ALS) using the RSXS endstation.40 During the measurements, the photon energy resolution was better than 0.3 \neV at Cu L -edges and the beam spot on sample was around 4 0 μm (v) by 500 μm (h). In XAS, the spectra shown in Fig. \n2 were recorded in the total electron yield mode (photo -current from sample to the ground). The spectra were \nnormalized by the photon flux determined from the photo -current of an upstream Au mesh. The atomic contributions \nresponsible for step -like intensity increases around Cu L 3 and L 2 edges were further removed.  \nIn RSXS, both incident photon polarization and scattering plane were horizontal ( π-scattering geometry). A \nphotodiode with front Al wind ow to block out ambient light (primarily the visible light around the chamber) was used \nto record the scattered x -rays from the bilayers. This detector, which does not have the selectivity on the polarization \nor energy of scattered photons, records signal from specular reflection, fluorescence and the YBCO (001) Bragg peak \nsimultaneously. Two types of scan, h ν and q -scans, were used to produce the spectra in Fig. 2 and Fig. 3 respectively. In \nthe momentum -space scan (q -scan), the sample and detector were pl aced at the specular geometry such that their \nangles relative to the incident photon beam followed the θ −2θ relationship. Since the signals from specular \nreflection and fluorescence have monotonic angular dependence, they can be separated from the Bragg p eak by fitting \nq-scan spectra with a Lorentzian function on top of a quadratic background. The Kiessig fringes at this photon energy \nare negligible and do not affect the fitting results. The Bragg peak intensity presented in Fig. 3(a) -(c) and 3(f) is the \npeak area. The resonance profile (h ν-scan) was measured with sample and detector angles  \nfollowing the θ −2θ relationship and tracking the incident photon energy so that the photon momentum transfer 𝑞  \nstayed at the (001) Bragg peak wave vector.  \nIn the az imuthal dependence measurements, the x -ray beam was defocused to a size of 〜2 mm square to \nmitigate the sample spatial inhomogeneity issue. The zero of azimuthal angle is defined with the crystallographic a /b \naxis in the horizontal scattering plane.   \n \nDFT calculations  \nThe first principles calculations were performed using the accurate full -potential augmented wave method, as \nimplemented in the VASP package within the GGA+U schemes. The calculations were performed over an 8 ×  8 ×  \n1 Monkhorst -Pack k point  mesh in the irreducible Brillouin zone. Both bilayers contained 4 u.c. of YBCO and 6 u.c. \nof LCMO. The structure of bilayer was optimized with the residual atomic forces less than 0.05 eV/ A° . U values \nused for La  4f and Mn  3d orbitals are 5.0 eV and 4.0 eV respectively.  \n \nReferences  \n \n1. Bianchi, A., Movshovich, R., Capan, C., Pagliuso, P . G. and Sarrao, J. L., Possible Fulde -Ferrell -Larkin -Ovchinnikov \nsuperconducting state in CeCoIn 5, Phys. Rev. Lett. 91, 187004 (2003).  \n2. Fulde, P . &  Ferrell, R. A., Superconductivity in a strong spin -exchange field, Phys. 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Dagotto, E., Hotta, T . and Moreo, A., Colossal magnetoresistant materials: the key ro le of phase separation, \nPhysics Reports 344, 1 (2001).  38. Bert, Julie A. et al., Direct imaging of the coexistence of ferromagnetism and superconductivity at the \nLaAlO3/SrTiO3 interfaces, Nature Phys. 7, 767 -771 (2011).  \n39. Bostwick, A., Rotenberg, E., Avila, J. & Asensio, M.C., Zooming in on electronic structure: nanoARPES at SOLEIL \nand ALS, Synchrotron Radiation News 25, issue 5, 19 (2012)  \n40. Doering, D. et al., Development of a compact fast CCD camera and resonant soft x -ray scattering endstation fo r \ntime -resolved pump -probe experiments, Rev. Sci. Instrum. 82, 073303 (2011).  \n \nAcknowledgements  \nThe Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the \nU.S. Department of Energy under Contract No. DE -AC02 -05CH11231. S.W.H. would like to thank Ruimin Qiao and \nWanli Yang for supporting the beam line operation.  This work was also  supported by MOST of Taiwan, R.O.C. \nunder Grants 103 -2112 -M-009 -007 -MY3 and the MOE ATU  program.  \n \n  Author contributions statement  \nS.W.H. and J. -Y .L. conceived the idea. V.T.T., J.M.L., J.M.C., Y .H.C. and J. -Y .L. provided bilayer samples and transport, \nmagnetization characterization. S.W.H., M.C.L., Y . -D.C. and J. -Y .L. carried out th e experiments. H.T.J. performed the DFT \ncalculations. S.W.H. and S.R. performed fitting to the x -ray reflectivity measurements. S.W.H., L.A.W., Y .H.C., R.W.S., \nY.-D.C. and J. -Y .L. interpreted the data. S.W.H., Y . -D.C and J. -Y .L. wrote up the manuscript.  \n \nAdditional information  \nCompeting financial interests  The authors declare that there is no competing financial interests.  \n \nFigure 1.  (color online) (a) Schematics illustrating various effects at the interface of a heterostructure . (b) Crystal \nstructure near the interface of YBCO/LCMO bilayers with MnO2 (left panel) and La0.7Ca0.3O (right panel) interfacial \nterminations. The arrows indicate the orientation and magnitude (not in proportion) of Mn and induced Cu spin \nmoments. (c) Syn chrotron reflectivity measurement on the MnO2 terminated bilayer at 80 K and 1240 eV. The YBCO \n(001) Bragg peak is indicated by the arrow. Inset shows the q -scans at 80 K (blue) and 300 K (red) from the MnO2 \nterminated bilayer at 925.25 eV. The photodiode intensity was normalized by the incident photon flux only and no \nbackground subtraction was applied. The markers are data points and lines are Lorentzian fitting results.  \n \n  \n \nFigure 2.  (color online) Thin and thick solid lines are the x -ray absorption sp ectra (XAS) and the resonance profiles of \nYBCO (001) Bragg peak at 300 K from the pure YBCO film (green, top panel), MnO2 (red, middle panel) and \nLa0.7Ca0.3O terminated (blue, bottom panel) bilayers. Inset shows the Cu L 3 edge XAS of the pure YBCO film wit h ticks \ndenoting the three states of Cu.  \n  \n \nFigure 3.  (color online) Red markers (left axis) and blue lines (right axis) represent the normalized (001) Bragg peak \nintensity and magnetization for (a) the pure YBCO film, (b) MnO2 and (c) La0.7Ca0.3O termin ated bilayers. The \nintensity of (001) Bragg peak is normalized to 1.0 at 300 K. The superconducting transition, STO structural phase \ntransition and Curie temperatures are marked by green, black and open arrows respectively. (d) Resistivity of the \nbilayers with MnO2 (red) and La0.7Ca0.3O (blue) interfacial terminations. (e) Schematic illustration of experimental \ngeometry with  φ angle at 0 °. (f) Azimuthal angle φ dependence of the normalized (001) peak intensity (red markers) \noverlaid with a sinusoidal functional form (blue line, see text). The data were taken at 80 K.  \n \n  \n \nFigure 4.  The calculated spin moment on the Cu sites in the CuO  chains (black filled circles) and CuO2 planes (red open \ncircles) for (a) MnO2 and (b) La0.7Ca0.3O terminated bilayers. The positive (negative) spin moment is defined as the \nCu spin parallel (antiparallel) to the Mn spin. The schematic next to these two fi gures shows the tacking order of the \nCuO chains (black filled circles with vertical bars) and CuO2 planes (red open circles with horizontal bars) along the \nc-axis in the calculations. The labelling of Cu sites is for the readers to associate the calculated  spin moments with their \nspatial arrangement.  \n \n  \nSupplementary Information for ”Selective interlayer  \nferromagnetic coupling between the Cu spins in  \n𝐘𝐁𝐚𝟐𝐂𝐮𝟑𝐎𝟕−𝐱 grown on top of 𝐋𝐚𝟎.𝟕𝐂𝐚𝟎.𝟑𝐌𝐧𝐎𝟑 ”  \n \nS. W . Huang1,2,3, L. Andrew Wray1,4,5, Horng -Tay Jeng6,7, V. T . Tra8, J. M. Lee9, M. C.  \nLangner2, J. M. Chen9, S. Roy1, Y . H. Chu10, R. W . Schoenlein2, Y .-D. Chuang1,*, and J. -Y  \nLin8,1,+  \n \n1Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  \n2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  \n3MAX IV Laboratory, Lund University, P . O. Box 118, 22100 Lund, Sweden  \n4Department of Physics, New York University, New York, New York 10003, USA  \n5Stanford Insti tute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, \nUSA  \n6Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan  \n7Institute of Physics, Academia Sinica, Taipei 11529, Taiwan  \n8Institute of Physics, National Chiao Tung University, Hsinchu 30010, Taiwan  \n9National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan  \n10Department of Materials Science and Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan  \n*ychuang@lb l.gov  \n+ago@cc.nctu.edu.tw  \n \nABSTRACT  \n \nI. Structure of YBCO/LCMO bilayers grown by pulse laser deposition (PLD)  \nThe ability to control the growth process down to atomic level is the key to the success of this study. In the PLD system, \na 10 Hz 250 mJ  pulsed laser beam from KrF (λ = 248 nm) excimer laser was used to evaporate the targets. The (100) \noriented SrTiO3 (STO) single crystal was used as the substrate. Before growing the bilayers, the substrate was first \ntreated with HF -NH4F buffer solution to  produce a uniform TiO2 termination. Depositing La0.7Ca0.3MnO3 on top of \nthis TiO2 terminated STO substrate produces the MnO2 termination at the YBa2Cu3O7−x -La0.7Ca0.3MnO3 \n(YBCO -LCMO) interface. To produce the La0.7Ca0.3O termination at the manganite/cupr ate interface, a 1.5 unit cell \n(u.c.) SrRuO3 buffer layer was deposited on top of the STO substrate before growing the LCMO layer. The LCMO (YBCO) \nlayer was deposited at 700 °C (750 °C), 80 mTorr (150 mTorr) oxygen pressures. After growth, the bilayer samp les \nwere further annealed in a 550 °C, 700 Torr oxygen atmosphere for one hour followed by a slow cool down to the \nroom temperature to achieve the full oxygenation for the YBCO layer. We used the in -situ reflection high -energy \nelectron diffraction (RHEED) to monitor the growth process. Clear RHEED oscillations confirmed the layer -by-layer \ngrowth mode (for examples, see Fig. S1). To determine the quality of YBCO -LCMO interface, high angle annular \ndark -field scanning transmission electron microscopy (HDDAF -STEM) was used. The HDDAF -STEM images confirmed \nthe structures illustrated in Fig. 1(b) in the manuscript.1  \n \nII. Transport properties of YBCO/LCMO bilayers  \nThe resistivity of bilayer samples was measured using the standard four -probe method. The results ar e shown in Fig. \nS2(a) as well as in Fig. 3(d) in the manuscript. The superconducting transition temperatures are determined to be 〜70 \nand 55 K for MnO2 and La0.7Ca0.3O terminated bilayers respectively. The Curie temperature, which is the onset \ntemperature of ferromagnetism in the LCMO layer, is estimated by intersecting the linear extrapolation of the high \ntemperature leading edge of dM(T )/dT curve to zero. The extrapola tion is shown as the dashed line in Fig. S2(b) and \nthe Curie temperature is estimated to be 190 K, with an uncertainly on the order of 10 K.  \nIn Fig. 3(b) in the manuscript, the onset of RSXS intensity (red markers) is slightly higher t han the Curie temper ature     \nFigure 1.  (color online) Examples of the growth of (a) the STO/LCMO 10nm/YBCO d structure with MnO2-terminated \ninterface and (b) the STO/SRO 1nm/LCMO 10nm/YBCO d structure with La0.7Ca0.3O terminated interface. Top panels show \nthe layer -by-layer growt h mode monitored by the RHEED.  \n \n(open arrow) and this can be understood as the following. For magnetization, the recorded signal is the sum of \nmagnetic moments from different ferromagnetic domains. When these magnetic domains are not properly aligned, \nsay pointing along the easy axes (Mn -O bond direction) but antiparallel to each other, cancellation can l ead to a \nsmaller reading in M(T ). On the other hand, the RSXS intensity measures primarily the sum of the square of these \nmoments. Thus even in this scena rio, as long as the c -axis ferromagnetic coupling is established, the RSXS intensity \nfrom these anti -aligned domains will add up. Thus it is plausible that in the temperature regime between 200 K and \n150 K, the discrepancy between M(T ) and RSXS data is cau sed by the intricate re -alignment of microscopic \nferromagnetic domains. Moreover, a short range ordering preceding the establishment of bulk ferromagnetism would \nalso lead to a higher onset temperature for the RSXS intensity than the Curie temperature by a  similar mechanism as \nmentioned above.  \n \nIII. Two -peak structures in the resonance profile of YBCO (001) Bragg peak  \nIn Fig. 2 of the manuscript, we show the resonance profiles (thick solid lines) of YBCO (001) Bragg peak from MnO2 \nand La0.7Ca0.3O terminated bilayers. Unlike pure YBCO film which shows just one feature at Cu L 3 and L 2 edges, \nbilayer samples exhibit two peaks at these edges (the additional peak at L 3 edge is labelled B).  \nThe two -peak structure is intrinsic to the bilayers. Although one might speculate that it could come from two \ntypes of Cu2+ with different binding energies, this scenario can be ruled out as the similarity between the XAS spectra \nfrom these three samples does not support two distinct Cu2+ states with such large energ y difference ( 〜 2.5 eV apart). \nFurthermore, we also try to simulate the RSXS spectra of bilayers by using the ones from pure YBCO film with a 2.5 eV \nrelative energy shift. Although the simulated spectrum (black curve, Fig. S3) seems to capture the gross sp ectral \nlineshape, the differences can still be clearly seen at selected photon energies (see arrows in Fig. S3).  \nThe self -absorption effect, where x -rays emitted from deep inside the sample are re -absorbed when they come out \nof the sample, tends to suppre ss the high intensity features in XAS spectra recorded in the fluorescence yield mode. \nThis effect becomes appreciable around the elemental absorption edges at which the x -ray penetration depth is \nsignificant reduced. Self -absorption correction is often pe rformed when the x -ray penetration depth is comparable to \nor shorter than the thickness of sample. In our case where the thickness of YBCO film is around 15 nm (at 35 ° grazing \nincidence angle, the effective thickness is around 26 nm) and the minimum x -ray penetration depth at Cu L 3 edge is \naround 140 nm (attenuation length determined from the CXRO website; http://henke.lbl.gov/optical constants/), the \nself-absorption effect is not expected to significantly alter the intensity ratio between features A and B.  Thus it cannot \nbe used to account for the observed two -peak structure in RSXS data.  \nThe intensity of XAS is proportional to the imaginary part of atomic scattering form factor f′′(E), whereas the RSXS \nintensity is related to its square ( f’(E))2+(𝑓′′(𝐸))2 modulated by a phase factor from the spatial arrangement of \nthese scatterers  \n \nFigure 2.  (color online) (a) The in -plane resistivity of MnO2 (red curve) and La0.7Ca0.3O (blue curve) terminated \nbilayers showing the respective superconducting transition tem peratures T c 〜70 K and 55 K determined by the \nmid-point of the transition. (b) Magnetization (blue curve, right axis) and its first derivative (red curve, left axis) from \nMnO2 terminated bilayer. The dashed line indicates the linear extrapolation of the leading edge of the red curve.  \n \n(see description in the manuscript). f’(E) and f ’’(E)are related to each other through Kramers -Kronig  relations. We \nnotice that the intensity of YBCO (001) Bragg peak is much weaker than other (00L) Bragg peaks and this is due to an \neffective destructive interference between Cu charge scatterings from the CuO chain and two CuO2 planes within the \nunit cell . Such interference can be disrupted by a slight shift in the resonance energies or variations in the spatial \narrangement of the scatterers. The former one will affect the energy denominator in f’(E) and f’’ (E), whereas the later \none will affect the phase factor. Simulating the RSXS lineshape will require the full knowledge of the spatial \narrangement of Cu charges within the unit cell and their energetic upon heterogeneity, but to lowest order, these two \nfactors can explain the relative intensity change bet ween feature A and B in Fig. 2 in the manuscript. Irrespective to \nwhich origin, the distinct RSXS resonance profiles seen in the bilayer samples implies that the local energetic of \nelectronic states is altered upon heterostructuring.  \n \nIV. Azimuthal angle dependence of the RSXS intensity  \nUnlike the magnetization, RSXS has the unique elemental, chemical and bonding specificity to differentiate the origins \nof the magnetic moments. Complementary to the x -ray magnetic circular dichroism (XMCD), RSXS intensity \ndependence on the tensorial nature of scattering channels can be helpful in identifying the magnetic couplings \nbetween the CuO2 planes when the ordering vector overlaps with the structural Bragg peak whose intensity is \ndominated by the charge scattering.  \nIn the current study, the scattering plane is horizontal and the incident photon polarization is in this scattering \nplane (p -polarization). We have used the single channel detector (photodiode) without polarization analyzer to record \nthe scattering signal.  The recorded signal will contain both s -and p -polarization components. Although this may \ncomplicate the analysis of spin states, we will show that it still can offer some useful insight.  \nWe follow the formalism outlined in Hill & McMorrow2 and Lovesey & Collins.3 Since the incident photon energy is \ntuned close to Cu2+ L3 absorption edge, we only consider the dipole (E1) ransition and neglect the much weaker \nquadrupole (E2) transition. Equation (15) in ref2 is reproduced here:  \n𝑓(𝜀̂𝑖,𝜀̂0)→(𝑓𝜎𝑖→𝜎0    𝑓𝜋𝑖→𝜎0\n𝑓𝜎𝑖→𝜋0    𝑓𝜋𝑖→𝜋0)\n=𝐹(0)(E)(1            0      \n0    cos(2𝜃))⏟        \n𝜀̂𝑖∙𝜀̂0+i𝐹(1)(𝐸)(                     0                           𝑧1𝑐𝑜𝑠(𝜃)+𝑧3𝑠𝑖𝑛(𝜃)\n𝑧3𝑠𝑖𝑛(𝜃)−𝑧1𝑐𝑜𝑠(𝜃)            −𝑧2𝑠𝑖𝑛(2𝜃) )\n⏟                                \n(𝜀̂𝑖×𝜀̂0)∙𝑠̂\n+𝐹(2)(𝐸)(                𝑧22                                  −𝑧2(𝑧1sin(𝜃)−𝑧3cos(𝜃)\n𝑧2(𝑧1sin(𝜃)+𝑧3cos(𝜃))         −𝑐𝑜𝑠2(𝜃)(𝑧12𝑡𝑎𝑛2(𝜃)+𝑧32))\n⏟                                        \n(𝜀̂𝑖∙𝑠̂)(𝜀̂0∙𝑠̂)                      (1) \nwith F(0),F(1),and F(2) defined in ref.2 We need to include these three terms because the Kronecker δ that co nserves the  \n \nFigure 3.  (color online) Resonance profiles of YBCO (001) Bragg peak and simulated spectrum. The black curve is \nproduced by the sum of two spectra from pure YBCO (green curve) with a 2.5 eV relative energy shift. The \ndiscrepancies between th e simulated spectrum and the ones from bilayers (red and blue curves for MnO2 and \nLa0.7Ca0.3O terminated bilayers) are highlighted by arrows.  \n \nwave vectors becomes 1 in this case. We only need to consider the second column in the matrix because these \nelem ents are relevant to the signal in the 𝜋𝑖→𝜎0 and 𝜋𝑖→𝜋0 channels (here, 𝜋𝑖 and 𝜎0 refer to the incident π \nand scattered σ polarizations). θ is the YBCO (001) Bragg angle and is 〜34.79o in the current study. z𝑖 are the \ncomponents of spin unit vector projected onto three crystalline axes. They are:  \n \nz 1=sin(α)cos(φ)  \nz 2=sin(α)sin(φ)                                                (2) \nz 31=cos(α)  \n \nHere α is the angle between the unit vector and c -axis, and φ is the sample azimuthal angle. Firstly, it is clear that if the \nmoment is completely along the c -axis (α = 0o), there will be no azimuthal angle dependence in the RSXS intensity. To \nsimplify the discussion, we consider the extreme case where α = 90o. Putting these terms together, we have:  \n \nf〜(𝑓𝜎𝑖→𝜎0    −𝑖(𝑧1cos(𝜃))𝐹(1)−𝑧2(𝑧1sin (𝜃))𝐹(2))\n𝑓𝜎𝑖→𝜋0    𝐹(0)cos(2𝜃)+𝑖𝑧2𝑧1sin(2𝜃)𝐹(1)−𝑧12(𝑠𝑖𝑛2(𝜃))𝐹(2))                          (3) \n \nThe RSXS intensity is proportional to sum (over the superlattice ) of the square of f  \n \n|𝑓|2=|𝐹(0)|2{𝑐𝑜𝑠2(2𝜃)−2ℑ(𝐹(0)∗𝐹(1)\n|𝐹(0)|2cos(2𝜃)sin(2𝜃)sin(𝜑)−2ℜ(𝐹(0)∗𝐹(2))\n|𝐹(0)|2cos(2𝜃)𝑠𝑖𝑛2(𝜃)𝑐𝑜𝑠2(𝜑)\n+2ℑ(𝐹(2)∗𝐹(1)\n|𝐹(0)|2(2𝑠𝑖𝑛2(𝜃)−1)sin(𝜃)cos(𝜃)𝑐𝑜𝑠2(𝜑)sin(𝜑)\n+|𝐹(1)|2\n|𝐹(0)|2(𝑐𝑜𝑠2(𝜃)𝑐𝑜𝑠2(𝜑)+𝑠𝑖𝑛2(2𝜃)𝑠𝑖𝑛2(𝜑))\n+|𝐹(2)|2\n|𝐹(0)|2(𝑠𝑖𝑛4(𝜃)𝑐𝑜𝑠4(𝜑)+𝑠𝑖𝑛2(𝜃)𝑠𝑖𝑛2(𝜑)𝑐𝑜𝑠2(𝜑))}                                                        (4) \n \n \n \nFigure 4. (color online) Synchrotron reflectivity measurement.  The agreement between data (blue markers) and fit (red \nline) implies that the choice of 15 nm YBCO and 7.5 nm LCMO layer thickness with 0.6 nm roughness in fitting is \nreasonable. The discrepancy around 50o is due to the presence of YBCO (001) Bragg peak.   \n \nUsually, F(0) is much larger than F(2) and F(1) so that the ferromagnetic signal would be very weak \ncompared to the charge signal in the Bragg peak. However, the destructive interference between \ncharge scatterings leads to a much weaker YBCO (001) Bragg peak (see previous discussion). This \nmakes the ratio ∑ [F(1,2)/F(0)] superlattice  not so negligible. But even so, we do not expect the ratio \ncan be on the order of 1. Thus we argue that the high order terms (last three terms in the equation) \ncan be dropped out  in the following discussion. By doing so, the scattering intensity will have the \nazimuthal angle dependence of a+b∗sin(𝜑)+𝑐∗𝑐𝑜𝑠2(𝜑) where the coefficients b and c are  \nrelated to 𝐹(1)\n𝐹(0) and 𝐹(2)\n𝐹(0). \n \nFrom Fig. 3(f) of the manuscript, the Bragg peak intensity changes from 〜1.5at0o to〜1.1 at \n90o above the charge background of 〜1.0. Having the spin moment along the c -axis would \nincrease the constan t base line and reduce the [F(1,2)/F(0)] ratio. The strong sinusoidal oscillation \nimplies that the in -plane spin component is larger than the out -of-plane component. It also \nsuggests that F(1) is much smaller than F(2), as expected from the extremely weak Cu XMCD versus \nXLD signal.  \nAlthough YB CO has CuO chains that naturally break the four -fold symmetry, the bilayer samples \nunder study are twinned. It is possible that the twinned domains with two distinct CuO chain \norientations have unequal volume fractions that give the observed two -fold symme try, we also \nnotice that the 80 K measurement temperature is below the structural distortion of underlying STO \nsubstrate around 105 K.4, 5 This distortion naturally breaks the four -fold symmetry and further \naligns the ferromagnetism in LCMO layer as shown in the magnetization.  \n \nFigure 5. (color online) q -scans at 80 K (blue) and 300 K (red) from the MnO2 terminated bilayer at 925.25 eV. The \nphotodiode intensity was normalized by the incident photon flux only (photocurrent from upstream Au mesh), and no \nbackground subtraction was applied.  \nV. Determining the layer thickness using synchrotron reflectivity  \nSynchrotron reflectivity is used to determine the YBCO and LCMO layer thickness in the heterostructures. The \nmeasurement temperature was set to 80 K and t he incident photon energy was tuned to 1240 eV, well above the Mn \nand Cu resonances. The blue markers in Fig. S4 are the data while the red curve is the fitting with YBCO / LCMO layer \nthickness of 15 nm / 7.5 nm respectively with roughness around 0.6 nm. T he agreement between data and fit justifies \nthe fitting parameters of layer thickness and roughness. Note that the discrepancy around 2θ 〜50o is caused by the \nYBCO (001) Bragg peak.  \nThe Kiessig fringes that overwhelm the YBCO (001) Bragg peak become negli gible when the incident photon energy \nis tuned to Cu L 3 resonance. As can be seen in Fig. S5, the spectra recorded in q -scan can be nicely fitted by a \nLorentzian function on top of a monotonic background.  \n \nReferences  \n1. Tra, V.T. et al., Termination control of charge transfer in YBa2Cu3O7−x/La0.7Ca0.3MnO3 \n  heterostructures, Submitted to Adv. Mater.  \n2. Hill, J. & McMorrow, D. F., X -ray resonant exchange scattering: polarization dependence and \ncorrelation functions, Acta Cry st. A 52, 236 -244 (1996).  \n3. Lovesey, S. W. and Collins, S. P ., X -ray scattering and absorption by magnetic materials, Oxford \nseries on synchrotron radiation, Clarendon Press, Oxford 1996.  \n4. Bell, R. O. and Ruppercht, G., Elastic constants of Strontium Titanate, Phys. Rev. 129, 90 -94 \n(1963).  \n5. Loetzsch, R. et al., The cubic to tetragonal phase transition in SrTiO3 single crystals near its \nsurface under internal and external strains, Appl. Phys. Lett. 96, 071901 (2010).  \n "    },    {        "title": "1610.01072v2.Magnetomechanical_coupling_and_ferromagnetic_resonance_in_magnetic_nanoparticles.pdf",        "content": "Magnetomechanical coupling and ferromagnetic resonance in magnetic nanoparticles\nHedyeh Keshtgar,1Simon Streib,2Akashdeep Kamra,3Yaroslav M. Blanter,2and Gerrit E. W. Bauer2, 4\n1Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran\n2Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n3Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany\n4Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai 980-8577, Japan\n(Dated: March 20, 2017)\nWe address the theory of the coupled lattice and magnetization dynamics of freely suspended\nsingle-domain nanoparticles. Magnetic anisotropy generates low-frequency satellite peaks in the\nmicrowave absorption spectrum and a blueshift of the ferromagnetic resonance (FMR) frequency.\nThe low-frequency resonances are very sharp with maxima exceeding that of the FMR, because\ntheir magnetic and mechanical precessions are locked, thereby suppressing the effective Gilbert\ndamping. Magnetic nanoparticles can operate as nearly ideal motors that convert electromagnetic\ninto mechanical energy. The Barnett damping term is essential for obtaining physically meaningful\nresults.\nPACS numbers: 75.10.Hk, 75.80.+q , 75.75.Jn , 76.50.+g\nI. INTRODUCTION\nMagnetic nanoparticles (nanomagnets) are of funda-\nmental interest in physics by forming a link between the\natomic and macroscopic world. Their practical impor-\ntance stems from the tunability of their magnetic prop-\nerties [1], which is employed in patterned media for high\ndensity magnetic data storage applications [2] as well as\nin biomedicine and biotechnology [3–6]. Superparamag-\nnetic particles are used for diagnostics, stirring of liq-\nuids, and magnetic tweezers [7]. The heat generated by\nthe magnetization dynamics under resonance conditions\nis employed for hyperthermia cancer treatment [8–10].\nMolecular based magnets can cross the border from the\nclassical into the quantum regime [11, 12]. The magnetic\nproperties of individual atomic clusters can be studied by\nmolecular beam techniques [13–15].\nEinstein, de Haas, and Barnett [16, 17] established the\nequivalence of magnetic and mechanical angular momen-\ntum of electrons by demonstrating the coupling between\nmagnetization and global rotations. Spin and lattice are\nalso coupled by magnetic anisotropy, induced either by\ndipolar forces or crystalline fields. A quite different in-\nteraction channel is the magnetoelastic coupling between\nlattice waves (phonons) and spin waves (magnons) with\nfinite wave vectors. This magnetoelastic coupling be-\ntween the magnetic order and the underlying crystalline\nlattice has been explored half a century ago by Kittel [18]\nand Comstock [19, 20]. The coupling between spin and\nlattice causes spin relaxation including Gilbert damping\nof the magnetization dynamics [21, 22].\n“Spin mechanics” of thin films and nanostructures en-\ncompasses many phenomena such as the actuation of the\nmagnetization dynamics by ultrasound [23–25], the dy-\nnamics of ferromagnetic cantilevers [26–28], spin current-\ninduced mechanical torques [22, 29], and rotating mag-\nnetic nanostructures [30]. The Barnett effect by rotation\nhas been observed experimentally by nuclear magnetic\nresonance [31]. The coupled dynamics of small magneticspheres has been studied theoretically by Usov and Li-\nubimov [32] and Rusconi and Romero-Isart [33] in clas-\nsical and quantum mechanical regimes, respectively. A\nprecessing single-domain ferromagnetic needle is a sen-\nsitive magnetometer [34], while a diamagnetically levi-\ntated nanomagnet can serve as a sensitive force and in-\nertial sensor [35]. A stabilization of the quantum spin of\nmolecular magnets by coupling to a cantilever has been\npredicted [36, 37] and observed recently [38].\nHere we formulate the dynamics of rigid and single-\ndomain magnetic nanoparticles with emphasis on the\neffects of magnetic anisotropy and shape. We derive\nthe equations of motion of the macrospin and macro-\nlattice vectors that are coupled by magnetic anisotropy\nand Gilbert damping. We obtain the normal modes and\nmicrowave absorption spectra in terms of the linear re-\nsponse to ac magnetic fields. We demonstrate remark-\nable changes in the normal modes of motion that can be\nexcited by microwaves. We predict microwave-activated\nnearly undamped mechanical precession. Anisotropic\nmagnetic nanoparticles are therefore suitable for stud-\nies of non-linearities, chaos, and macroscopic quantum\neffects.\nIn Sec. II we introduce the model of the nanomag-\nnet and give an expression for its energy. In Sec. III we\ndiscuss Hamilton’s equation of motion for the magneti-\nzation of a freely rotating particle, which is identical to\nthe Landau-Lifshitz equation. We then derive the cou-\npled equations of motion of magnetization and lattice in\nSec. IV. Our results for the easy-axis and easy-plane con-\nfigurations are presented in Secs. V and VI. We discuss\nand summarize our results in Secs. VII and VIII. In the\nAppendices A to D we present additional technical de-\ntails and derivations.arXiv:1610.01072v2  [cond-mat.mes-hall]  28 Apr 20172\nz\ny\nxnyb\nxbzb\nθ(a)\nn\nm\nm\nn(b)\n(c)\nFigure 1. (a) Laboratory frame ( x,y,z) and (moving) body\nframe (xb,yb,zb) of a nanomagnet with principal axis nalong\nthezb-axis. The directions of nand magnetization mare\nshown for (b) oblate and (c) prolate spheroids with dipolar\nmagnetic anisotropy.\nII. MACROSPIN MODEL\nWe consider a small isolated nanomagnet that justi-\nfies the macrospin and macrolattice approximations, in\nwhich all internal motion is adiabatically decoupled from\nthe macroscopic degrees of freedom, rendering the mag-\nnetoelastic coupling irrelevant.\nWe focus on non-spherical nanoparticles with mass\ndensity\u001a(r)and tensor of inertia\nI=Z\nd3r\u001a(r)\u0002\n(r\u0001r)^1\u0000r\nr\u0003\n;(2.1)\nwhere ^1is the 3x3 unit matrix. The mechanical proper-\nties of an arbitrarily shaped rigid particle is identical to\nthat of an ellipsoid with a surface that in a coordinate\nsystem defined along the symmetry axes (in which Iis\ndiagonal) reads\n\u0010x\nc\u00112\n+\u0010y\nb\u00112\n+\u0010z\na\u00112\n= 1; (2.2)\nwherea;b;care the shape parameters (principal radii).\nThe volume is V= 4\u0019abc= 3, total mass Q=\u001aV,\nand principal moments of inertia I1=Q\u0000\na2+b2\u0001\n=5;\nI2=Q\u0000\na2+c2\u0001\n=5;I3=Q\u0000\nb2+c2\u0001\n=5. We focus in the\nfollowing on prolate (a > b =c)and oblate (a < b =c)\nspheroids, because this allows analytic solutions of the\ndynamics close to the minimum energy state.\nWe assume that the particle is smaller than the crit-\nical sizedcr\u001836pAKA=(\u00160M2\ns)for magnetic domain\nformation [39], where Ais the exchange constant, KA\nthe anisotropy constant, Msthe saturation magnetiza-\ntion, and\u00160= 4\u0019\u000210\u00007N A\u00002the vacuum permeability.\nFor strong ferromagnets these parameters are typically in\nthe rangeA2[5;30] pJ m\u00001,KA2[10;20000] kJ m\u00003,\nMs2[0:4;1:7] MA m\u00001, leading to dcr2[1;500] nm\n[39]. For a spherical particle of radius Rwith sound\nvelocityv, the lowest phonon mode frequency is approx-imately [40]\n!ph\n2\u0019\u0019v\n4R= 0:25\u0012v=(103m\ns)\nR=nm\u0013\nTHz;(2.3)\nwhile the lowest magnon mode (for bulk dispersion rela-\ntion ~!mag=Dk2)\n!mag\n2\u0019\u0019\u0019D\n8~R2= 0:6\u0012D=(meV nm2)\nR2=nm2\u0013\nTHz;(2.4)\nwhere the spin wave stiffness D= 2g\u0016BA=Msis typically\nof the order meV nm2[39], e.g.,D= 2:81 meV nm2for\niron [41]. We may disregard spin and lattice waves and\nthe effects of their thermal fluctuations when the first\nexcited modes are at sufficiently higher frequencies than\nthat of the total motion (the latter is typically in the\nGHz range) and therefore adiabatically decoupled [33,\n40], i.e. the macrospin and macrolattice model is valid.\nThermal fluctuations of the magnetization with respect\nto the lattice do not play an important role below the\nblocking temperature, TB\u0018KAV=(25kB)[42], where\nkBis the Boltzmann constant. For kBT\u001cVMs\u00160H0,\nthermal fluctuations of the magnetization with respect to\nthe static external magnetic field H0are suppressed.\nUnder the conditions stipulated above the classical dy-\nnamics (disregarding translations of the center of mass)\nis described in terms of the magnetization vector M=\nMsm(withjmj= 1) and the three Euler angles ( \u0012;\u001e; )\nof the crystal orientation direction in terms of the axis\nn(\u0012;\u001e)and a rotation angle  around it (see Appendix A\nfor details). The total energy can be split up into several\ncontributions,\nE=ET+EZ+ED+EK: (2.5)\nET=1\n2\nTI\nis the kinetic energy of the rotational mo-\ntion of the nanomagnet in terms of the angular frequency\nvector \n.EZ=\u0000\u00160VM\u0001Hextis the Zeeman energy\nin a magnetic field Hext.ED=1\n2\u00160VMTDMis the\nmagnetostatic self-energy with particle shape-dependent\ndemagnetization tensor D.EK=K1V(m\u0002n)2is the\n(uniaxial) magnetocrystalline anisotropy energy, assum-\ning that the easy axis is along n, andK1is the material-\ndependent anisotropy constant.\nWe consider an inertial lab frame with origin at the\ncenter of mass and a moving frame with axes fixed in the\nbody. The lab frame is spanned by basis vectors ex,ey,\nez, and the body frame by basis vectors exb,eyb,ezb(see\nFig. 1). The body axes are taken to be the principal axes\nthat diagonalize the tensor of inertia. For spheroids with\nb=cthe inertia and demagnetizing tensors in the body\nframe have the form\nIb=0\n@I?0 0\n0I?0\n0 0I31\nA;Db=0\n@D?0 0\n0D?0\n0 0D31\nA;(2.6)\nwithI?=Q\u0000\na2+b2\u0001\n=5andI3= 2Qb2=5; the elements\nD?andD3for magnetic spheroids are given in [43]. The3\nparticle shape enters the equations of motion via I?,I3,\nand the difference D3\u0000D?, the latter reduces to \u00001=2\nfor a thin needle and 1for a thin disk. When\nE?\u0000Ek=KAV=K1V\u00001\n2\u00160VM2\ns(D3\u0000D?)(2.7)\nis larger than zero, the configuration mknis sta-\nble (“easy axis”); otherwise m?n(“easy plane”).\nThe anisotropy constant KAincludes both magnetocrys-\ntalline and shape anisotropy.\nIII. LANDAU-LIFSHITZ EQUATION\nFor reference we rederive here the classical equation of\nmotion of the magnetization. The magnetization of the\nparticle at rest is related to the angular momentum S=\n\u0000VMsm=\r, where\r= 1:76\u00021011s\u00001T\u00001is (minus) the\ngyromagnetic ratio of the electron. The Poisson bracket\nrelations for angular momentum are\nfS\u000b;S\fg=\u000f\u000b\f\rS\r: (3.1)\nHamilton’s equation of motion reads\nd\ndtS=fS;Hg; (3.2)\nwhereH\u0011Eis the Hamiltonian. We consider a general\nmodel Hamiltonian of a single macrospin coupled to the\nmacrolattice,\nH=X\ni;j;k2N0aijk(n;L)Si\nxSj\nySk\nz; (3.3)\nwhere the coefficients aijk(n;L)may depend on the ori-\nentation nof the lattice and its mechanical angular\nmomentum L=I\n. Since lattice and magnetization\nare different degrees of freedom, the Poisson brackets\nfn;Sg=fL;Sg= 0and thereforefaijk(n;L);Sg= 0.\nWe derive in Appendix B\nfS;Hg=X\ni;j;k2N0aijk(n;L)0\n@iSi\u00001\nxSj\nySk\nz\njSi\nxSj\u00001\nySk\nz\nkSi\nxSj\nySk\u00001\nz1\nA\u0002S;(3.4)\nwhich is the Landau-Lifshitz equation [44],\nd\ndtS=rSHjn;L=const:\u0002S: (3.5)\nIn accordance with Eq. (3.4), the gradient in Eq. (3.5)\nhas to be evaluated for constant nandL.\nThe rotational kinetic energy ET=1\n2\nTI\ndoes\nnot contribute to this equation of motion directly since\nfS;ETg= 0. However, ETis crucial when considering\nthe energy of the nanomagnet under the constraint of\nconserved total angular momentum J=L+S. Minimiz-\ning the energy of the nanomagnet under the constraint\nof constant Jis equivalent to\n~He\u000b=\u00001\n\u00160VMsrmE\f\f\f\f\nJ=const:= 0;(3.6)where the rotational kinetic energy ETcontributes the\nBarnett field\nHB=\u00001\n\u00160VMsrmET\f\f\f\f\nJ=const:=\u0000\n\r\u00160;(3.7)\nwhich gives rise to the Barnett effect (magnetization by\nrotation)[17]. AlthoughtheBarnettfieldappearsherein\nthe effective field ~He\u000bwhen minimizing the energy, it is\nnot part of the effective field He\u000bof the Landau-Lifshitz\nequation,\nHe\u000b=\u00001\n\u00160VMsrmE\f\f\f\f\nn;L=const:;(3.8)\nwhere Lis kept constant instead of J. In the Landau-\nLifshitz-Gilbert equation in the laboratory frame the\nBarnett effect operates by modifying the Gilbert damp-\ning torque as shown below.\nIV. EQUATIONS OF MOTION\nWe now derive the coupled equations of motion of the\nmagnetization mand the Euler angles ( \u001e;\u0012; ). The\nmagnetization dynamics is described by the Landau-\nLifshitz-Gilbert equation [21, 44]\n_m=\u0000\r\u00160m\u0002He\u000b+\u001c(\u000b)\nm; (4.1)\nwhere the effective magnetic field Eq. (3.8) follows from\nthe energy Eq. (2.5),\nHe\u000b=Hext+HD+HK; (4.2)\nand\u001c(\u000b)\nmis the (Gilbert) damping torque. The external\nmagnetic field Hextis the only source of angular mo-\nmentum; all other torques acting on the total angular\nmomentum J=L\u0000VMsm=\rcancel. From\n_J=\u00160VMsm\u0002Hext; (4.3)\nwe obtain the mechanical torque as time-derivative of the\nmechanical angular momentum, which leads to Newton’s\nLaw\n_L=VMs\n\r_m+\u00160VMsm\u0002Hext:(4.4)\nThe dissipation parameterized by the Gilbert constant\n[21] damps the relative motion of magnetization and lat-\ntice. In the body frame of the lattice [30]\n\u001c(\u000b)\nm;b=\u000bmb\u0002_mb; (4.5)\nwherethesubscript bindicatesvectorsinthebodyframe.\nTransformed into the lab frame (see Appendix A)\n\u001c(\u000b)\nm=\u000b[m\u0002_m+m\u0002(m\u0002\n)]:(4.6)\nThis torque is an angular momentum current that flows\nfrom the magnet into lattice [22]. Angular momentum4\n2200 2300 2400 2500\nω/(2π) [MHz]01234567−ωImχxx[1013s−1]\nQf= 3900\nQf= 2900\n40 45 50 55\nω/(2π) [GHz]0.00.51.01.52.0−ωImχxx[1013s−1]\nQf= 50\nFigure 2. Low- and high-frequency resonances in the FMR\nspectrum of an Fe nanosphere of 2 nmdiameter in a static\nmagnetic field of 0:65 Twith Gilbert damping constant \u000b=\n0:01; quality factor Qf=!=(2\u0011).\nis conserved, but the generated heat is assumed to ulti-\nmately be radiated away. In vacuum there is no direct\ndissipation of the rigid mechanical dynamics.\nThe Barnett field \u00160HB=\u0000\n=\renters in the lab\nframe only in the damping term \u001c(\u000b)\nm. To leading order\nin\u000b\n_m\u0019\u0000\r\u00160m\u0002He\u000b\u0000\u000b\r\u0016 0m\u0002[m\u0002(He\u000b+HB)]+O(\u000b2):\n(4.7)\nThe contribution of HBin the damping term causes the\nBarnett effect [17]. We find that this Barnett damping is\nvery significant for the coupled dynamics even though no\nfast lattice rotation is enforced: without Barnett damp-\ning the FMR absorption of the low-frequency modes de-\nscribed below would become negative.\nV. EASY-AXIS CONFIGURATION\nWe first consider an easy-axis configuration ( mknk\nez) in the presence of an external magnetic field with\na large dc component H0along ezand a small trans-\nverse ac component, Hext=\u0000hx(t); hy(t); H 0\u0001T, with\nhx(t)/hy(t)/ei!t:Linearizing the equations of mo-\ntion in terms of small transverse amplitudes, we can solve\n(4.1) and (4.4) analytically to obtain the linear response\ntoh(see Appendix C for the derivation), i.e. the trans-\nverse magnetic susceptibility. Since we find _\nz= 0, we\ndisregard an initial net rotation by setting \nz= 0. Forsmall damping \u000b\u001c1, the normal modes are given by\nthe positive solutions of the equations\n!3\u0007!2!0\u0000!!c!A\u0006!c!A!H= 0;(5.1)\nwhere!H=\r\u00160H0,!A= 2\rKA=Ms,!0=!H+!A, and\n!c=MsV=(\rI?)is the natural mechanical frequency\ngoverned by the spin angular momentum. Note that the\nequivalent negative solutions of Eq. (5.1) have the same\nabsolute values as the positive solutions. We find that\nthe FMR mode !0is blueshifted to !k=!0+\u000e!kwith\n\u000e!k\u0019!2\nA!c\n!2\n0>0; (5.2)\nwhich is significant for small nanomagnets with large sat-\nuration magnetization and low mass density. It is a coun-\nterclockwise precession of mwithnnearly at rest.\nTwo additional low-frequency modes emerge. For !\u001c\n!0;!Awe may disregard the cubic terms in Eq. (5.1) and\nfind\n!l1;2\u0019s\u0012!c!A\n2!0\u00132\n+!H!c!A\n!0\u0006!c!A\n2!0:(5.3)\nAt low frequencies, the magnetization can follow the lat-\ntice nearly adiabatically, so these modes correspond to\nclockwise and counterclockwise precessions of nearly par-\nallel vectors mandn, but with a phase lag that gener-\nates the splitting. The frequency of the clockwise mode\n!l1> !l2(see Fig. 3). Since magnetization and mass\nprecess in unison, the effective Gilbert damping is ex-\npected to be strongly suppressed as observable in FMR\nabsorption spectra as shown below.\nThe absorbed FMR power is (see Appendix D)\nP=\u0000\u00160V\n2!Im\u0000\nh\u0003T\n?\u001fh?\u0001\n; (5.4)\nwhere h?is the ac field normal to the static magnetic\nfieldH0ezand\n\u001f\u000b\f=M\u000b\nh\f\f\f\f\f\nh?=0(5.5)\nis the transverse magnetic susceptibility tensor ( \u000b;\f =\nx;y). The diagonal ( \u001fxx=\u001fyy) and the off-diagonal\ncomponents( \u001fxy=\u0000\u001fyx)bothcontributetotheabsorp-\ntion spectrum near the resonance frequencies, jIm\u001fxxj\u0019\njRe\u001fxyj. For\u000b\u001c1, we find that the sum rule\nZ1\n0d!(\u0000!Im\u001fxx(!))\u0019\u0019\n2!0!M;(5.6)\nwhere!M=\r\u00160Ms, does not depend on !c, meaning\nthat the coupling does not generate oscillator strengths,\nonly redistributes it. Close to a resonance\n\u0000!Im\u001fxx(!)\u0018F\u00112\n(!\u0000!i)2+\u00112;(5.7)5\n10−310−210−1100101\nωH/ωA0.00.51.01.52.02.5angular frequency [1010s−1]\nωl1: clockwise mode\nωl2: countercl. mode\nFigure 3. Low-frequency magnetomechanical modes !l1and\n!l2of an Fe nanosphere of 2 nmdiameter.\nwith integral \u0019\u0011F. For the low-frequency modes the\nmaximumF\u00181\n2!M!2\nA=(\u000b!2\nH)with broadening \u0011\u0018\n1\n2\u000b!c!2\nH=(!A+!H)2; for the FMR mode F\u00181\n2!M=\u000b\nwith\u0011\u0018\u000b!0.\nLetusconsideranironspherewith 2 nmdiameter(a=\nb= 1 nm) under\u00160H0= 0:65 Tor!H=(2\u0019) = 18:2 GHz.\nIts magnetization !M=(2\u0019) = 60:33 GHz, crystalline\nanisotropy !A=(2\u0019) = 29:74 GHz [45], and the mag-\nnetomechanical coupling !c=(2\u0019) = 0:5(nm=a)2GHz.\nThe blocking temperature is TB\u001811(a=nm)3Kand\njEZj=(kBTB)\u001930, while the critical size for domain\nformationdcr\u001820 nm[46, 47]. We adopt a typi-\ncal Gilbert damping constant \u000b= 0:01. The calcu-\nlated FMR spectra close to the three resonances are\nshown in Fig. 2. Both low-frequency resonances are very\nsharp with a peak value up to 3.5 times larger than\nthat of the high-frequency resonance, although the in-\ntegrated intensity ratio is only 0.2 %. Long relaxation\ntimes of low-frequency modes that imply narrow reso-\nnances have been predicted for spherical nanomagnets\n[32]. The blueshift of the high-frequency resonance is\n\u000e!k=(2\u0019)\u00190:2(nm=a)2GHz. In Fig. 3 we plot the low-\nfrequency modes !l1and!l2as a function of !H=!A. For\n!H=!A!0,!l1\u0019!cand!l2!0. The low-frequency\nmodes become degenerate in the limit !H=!A!1.\nIn\"-Fe2O3[48] magnetization is reduced, resulting in\n!M=(2\u0019) = 2:73 GHz and!c=(2\u0019) = 35(nm=a)2MHz.\nFor the single-molecule magnet TbPc 2[38], we esti-\nmate!A=(2\u0019)\u00185 THz[49],!M=(2\u0019)\u001810 GHz,\n!c=(2\u0019)\u0018100 MHz [50], giving access to the strong-\nanisotropy regime with ultra-low effective damping.\nVI. EASY-PLANE CONFIGURATION\nAn easy-plane anisotropy aligns the equilibrium mag-\nnetization normal to the principal axis ( m?n), which is\ntypically caused by the shape anisotropy of pancake-like\noblate spheroids corresponding to !A<0. We choose an\nexternal magnetic field with a static component in the\nplaneH0eyand an ac field along xandz, while the equi-\n284.40 284.45 284.50 284.55 284.60\nω/(2π) [MHz]024681012141618FMR spectrum [1013s−1]\n−ωImχxx\n−ωImχzz\n−ωReχxz\n+ωReχzx\n11.011.512.012.513.013.514.0\nω/(2π) [GHz]0.00.51.01.52.02.53.0FMR spectrum [1013s−1]\n−ωImχxx\n−ωImχzz\n−ωReχxz\n+ωReχzxFigure 4. FMR spectrum of an Fe disk with 15 nmdiameter\nand2 nmthickness in a static magnetic field of 0:25 Twith\nGilbert damping constant \u000b= 0:01.\nlibrium npoints along ez(see Fig. 1(b)). For \u0012\u001c1,\nmy\u00191,nz\u00191, we again obtain analytic solutions for\nmandn(see Appendix C). We find two singularities in\nthe magnetic susceptibility tensor with frequencies (for\n\u000b\u001c1)\n!?\u0019!Hr\n1\u0000!A\n!H\u0000!c!A\n!2\nH; (6.1)\n!l\u0019s\n!2\nH!c!A\n!A!H\u0000!2\nH+!c!A: (6.2)\nSincenxdoes not depend on time there is only one\nlow-frequency mode !l, viz. an oscillation about the x-\naxis of the nanomagnet. Linearization results in _Ly\u0019\nVMs_my=\r\u00190and implies _Ly\u0019I?nx\u00190. The high-\nfrequency resonance !?is blueshifted by \u000e!?\u0018!c. As\nbefore, the lattice hardly moves in the high-frequency\nmode, while at low frequencies the magnetization is\nlocked to the lattice.\nIn Fig. 4 we plot the FMR spectrum of an Fe nan-\nodisk with shape parameters a= 1 nmandb= 7:5 nm\nunder\u00160H0= 0:25 Tor!H=(2\u0019) = 7 GHz . The\ncharacteristic frequencies are !c=(2\u0019) = 17:2 MHzand\n!A=(2\u0019) =\u000014:4 GHz. The blocking temperature with\njEZj=(kBTB)\u001924is now about 300 K:Again, the low-\nfrequency resonance is very sharp and relatively weak.\nThe contribution of Im\u001fxxto the low-frequency reso-\nnance is by a factor of 600 smaller than the dominant\nIm\u001fzzand therefore not visible in the plot.6\nVII. DISCUSSION\nThe examples discussed above safely fulfill all condi-\ntions for the validity of the theory either at reduced tem-\nperatures (T < 11 K, Fe sphere with 2 nmdiameter) or\neven up to room temperature ( 2 nm\u000215 nmFe disk).\nThe levitation of the particle can be achieved in cluster\nbeams [13, 15, 51], in aerosols [52], or by confinement to\na magnetic trap [33, 35, 53]. FMR experiments should\npreferably be carried out in a microwave cavity, e.g., a\ncoplanar wave guide that can also serve as a trap [54].\nMetal oxide nanoparticles, such as \"-Fe2O3[48], have\ncrystal anisotropies of the same order as that of pure\niron but smaller magnetization, which reduces the mag-\nnetomechanical coupling strength, leading to similar re-\nsults for somewhat smaller particles. The strongest\nanisotropies and couplings can be found in single-\nmolecule magnets, e.g., TbPc 2[49], but FMR experi-\nmentshavetobecarriedoutatlowtemperaturesinorder\nto suppress thermal fluctuations.\nOur theory holds for isolated particles at suffi-\nciently low temperatures and disregards quantum ef-\nfects. According to the fluctuation-dissipation theorem\na Gilbert damping is at finite temperatures associated\nwith stochastic fields [55]. A full statistical treatment of\nthe dynamics of magnetic nanoparticles at elevated tem-\nperatures, subject to microwaves, and weakly coupled to\ntheenvironmentisbeyondthescopeofthepresentpaper.\nWhen not suspended in vacuum but in, e.g., a liquid, the\nmechanical motion encounters viscous damping and ad-\nditional random torques acting on the lattice. Vice versa,\nthe liquid in proximity of the particle will be stirred by\nits motion. These effects can be included in principle\nby an additional torque term in Eq. (4.4). The external\ntorque will cause fluctuations in \nzand a temperature\ndependent broadening of the low-frequency resonances.\nMicrowave cavities loaded with thin films or spheres of\nthe high-quality ferrimagnet yttrium iron garnet have re-\nceived recent attention because of the relative ease with\nwhich the (ultra) strong coupling between magnons and\nphotons can be achieved (for references and evidence\nfor coherent magnon-phonon interaction, see [56]). The\nsharp low-frequency modes of free magnetic nanoparti-\nclescoupledtorfcavitymodesat10-100MHzcorrespond\nto co-operativities that are limited only by the quality\nfactor of the cavity. This appears to be a promising\nroute to access non-linear, chaotic, or quantum dynami-\ncal regimes. This technique would work also for magnets\nwith large damping and could break the monopoly of\nyttrium iron garnet for quantum cavity magnonics. Ma-\nterials with a large anisotropy are most attractive by the\nenhanced magnetization-lattice coupling.\nVIII. SUMMARY\nIn conclusion, we discussed the effect of the mag-\nnetomechanical coupling on the dynamics of levitatedsingle-domain spheroidal magnetic nanoparticles, e.g., in\nmolecular cluster beams and aerosols. We predict a blue\nshift of the high-frequency resonance and additional low-\nfrequency satellites in FMR spectra that reflect parti-\ncle shape and material parameters. In the low-frequency\nmodes the nanomagnet precesses together with the mag-\nnetization with strongly reduced effective damping and\nthereby spectral broadening.\nACKNOWLEDGMENTS\nThis work is part of the research program of the Sticht-\ning voor Fundamenteel Onderzoek der Materie (FOM),\nwhich is financially supported by the Nederlandse Or-\nganisatie voor Wetenschappelijk Onderzoek (NWO) as\nwellasJSPSKAKENHIGrantNos. 25247056,25220910,\n26103006. A. K. acknowledges financial support from the\nAlexander v. Humboldt foundation. H. K. would like to\nexpress her gratitude toward her late supervisor Malek\nZareyan for the opportunity to collaborate with the TU\nDelft researchers. S. S. is grateful to Alejandro O. León\nfor insightful discussions.\nAppendix A: Coordinate systems and\ntransformations\nWe derive the coordinate transformation from the lab\nwith basis vectors ex,ey,ezto the body frame exb,eyb,\nezb. The position of the particle is specified by the three\nEuler angles ( \u001e;\u0012; ). These three angles are defined by\nthetransformationmatrixfromthelabtothebodyframe\n(rb=Ar),\nA=0\n@cos sin 0\n\u0000sin cos 0\n0 0 11\nA0\n@1 0 0\n0 cos\u0012sin\u0012\n0\u0000sin\u0012cos\u00121\nA\n\u00020\n@cos\u001esin\u001e0\n\u0000sin\u001ecos\u001e0\n0 0 11\nA: (A1)\nThe main axis nof the particle is given by the local zb-\naxis in the body frame and can be directly obtained via\nthe inverse transformation AT,\nn=0\n@sin\u0012sin\u001e\n\u0000sin\u0012cos\u001e\ncos\u00121\nA: (A2)\nThe angular velocity vector of the rotating particle reads\nin the lab frame\n\n=_ AT0\n@0\n0\n11\nA+_\u00120\n@cos\u001e\u0000sin\u001e0\nsin\u001ecos\u001e0\n0 0 11\nA0\n@1\n0\n01\nA+_\u001e0\n@0\n0\n11\nA\n=0\n@_\u0012cos\u001e+_ sin\u0012sin\u001e\n_\u0012sin\u001e\u0000_ sin\u0012cos\u001e\n_\u001e+_ cos\u00121\nA; (A3)7\nand in the body frame,\n\nb=A\n=0\n@_\u001esin\u0012sin +_\u0012cos \n_\u001esin\u0012cos \u0000_\u0012sin \n_\u001ecos\u0012+_ 1\nA:(A4)\nThe mechanical angular momentum Land the principal\naxisnof the nanomagnet can be related by considering\nthe mechanical angular momentum in the body frame\nLb=Ib\nb: (A5)\nTransforming (A5) to the lab frame and expanding for\nsmall angles \u0012,\nLx\u0019I?d\ndt(\u0012cos\u001e)\u0019\u0000I?_ny;(A6a)\nLy\u0019I?d\ndt(\u0012sin\u001e)\u0019I?_nx; (A6b)\nLz\u0019I3(_\u001e+_ )\u0019I3\nz; (A6c)\nwhich is a valid approximation when \nz=O(\u0012):Fur-\nthermore,nz\u00191and _nz\u00190is consistent with \u0012\u001c1.\nThe Gilbert damping is defined for the relative motion\nof the magnetization with respect to the lattice, i.e. in\nthe rotating frame. The damping in the lab frame is\nobtained by the coordinate transformation\n\u001c(\u000b)\nm=AT\u001c(\u000b)\nm;b=AT(\u000bmb\u0002_mb);(A7)\nwhere mb=Am. Expanding the time derivative\n\u001c(\u000b)\nm=\u000bm\u0002_m+\u000bm\u0002\u0010\nAT_Am\u0011\n:(A8)\nThe angular frequency vector \nis defined by\n_r=\n\u0002r; (A9)\nwhere ris a point in the rotating body, i.e. _rb= 0, and\n_r=_ATrb=_ATAr: (A10)\nUsingd\ndt(ATA) =AT_A+_ATA= 0and comparing\nEqs. (A9) and (A10),\nAT_Ar=r\u0002\n; (A11)\nand therefore\n\u001c(\u000b)\nm=\u000bm\u0002_m+\u000bm\u0002(m\u0002\n):(A12)\nAppendix B: Poisson bracket in Hamilton’s equation\nIn the following, we show how to derive Hamilton’s\nequation of motion (3.4). Using the linearity of the Pois-\nson bracket together with the product rule\nfAB;Cg=AfB;Cg+fA;CgB; (B1)andfaijk(n;L);Sg= 0, we get\nfS;Hg=X\ni;j;k2N0aijk(n;L)\b\nS;Si\nxSj\nySk\nz\t\n:(B2)\nWe only consider the x-component, as the other compo-\nnents can be derived similarly. Using the product rule\n(B1), we may write\n\b\nSx;Si\nxSj\nySk\nz\t\n=Si\nx\b\nSx;Sj\nySk\nz\t\n=Si\nxSj\ny\b\nSx;Sk\nz\t\n+Si\nxSk\nz\b\nSx;Sj\ny\t\n:\n(B3)\nNext, we prove by induction that\n\b\nSx;Sk\nz\t\n=\u0000kSySk\u00001\nz; (B4)\nwhere the base case ( k= 0)\n\b\nSx;S0\nz\t\n= 0 (B5)\nand the inductive step ( k!k+ 1)\n\b\nSx;Sk+1\nz\t\n=Sz\b\nSx;Sk\nz\t\n+Sk\nzfSx;Szg\n=\u0000(k+ 1)SySk\nz (B6)\ncomplete the proof. Similarly, it follows\n\b\nSx;Sj\ny\t\n=jSj\u00001\nySz: (B7)\nSummarizing\n\b\nSx;Si\nxSj\nySk\nz\t\n=jSi\nxSj\u00001\nySk+1\nz\n\u0000kSi\nxSj+1\nySk\u00001\nz; (B8)\nwhich gives with Eq. (B2) the x-component of Eq. (3.4).\nAppendix C: Linearized equations of motion\n1. Easy-axis configuration\nIn the easy-axis case ( mknkez), the linearized equa-\ntions of motion of the magnetization mand mechanical\nangular momentum Lread\n_mx=\u0000!Hmy+!Mhy\nMs\u0000!A(my\u0000ny)\u0000\u000b( _my\u0000_ny);\n(C1a)\n_my=!Hmx\u0000!Mhx\nMs+!A(mx\u0000nx) +\u000b( _mx\u0000_nx);\n(C1b)\n_mz= 0; (C1c)\n_Lx=\u0000I?ny; (C2a)\n_Ly=I?nx; (C2b)\n_Lz=I3_\nz= 0; (C2c)8\n0.0 0.5 1.0 1.5 2.0\nωH/ωA050100150200\nReχxx(ωl1)\nReχxx(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−50000−40000−30000−20000−100000\nImχxx(ωl1)\nImχxx(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−20000−15000−10000−500005000100001500020000\nReχxy(ωl1)\nReχxy(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−200−150−100−50050100150200\nImχxy(ωl1)\nImχxy(ωl2)\nFigure 5. Real and imaginary parts of the magnetic susceptibility tensor \u001f(!)of the low-frequency modes !l1and!l2for an\nFe nanosphere of 2 nmdiameter with Gilbert damping \u000b= 0:01.\nwith\nnx=!2\nN(mx\u0000nx) +\u000b!c( _mx\u0000_nx);(C3a)\nny=!2\nN(my\u0000ny) +\u000b!c( _my\u0000_ny);(C3b)\nnz= 0; (C3c)\nwhere!2\nN=!c!A. Since _\nz= 0and with initial condi-\ntion\nz= 0, there is no net rotation \nz. Introducing the\nchiral modes,\nm\u0006=mx\u0006imy; n\u0006=nx\u0006iny; h\u0006=hx\u0006ihy;(C4)\nwecanwritetheequationsofmotioninthecompactform\n_m\u0006=\u0006i\u0012\n!0m\u0006\u0000!Mh\u0006\nMs\u0000!An\u0006\u0013\n\u0006i\u000b\u0000\n_m\u0006\u0000_n\u0006\u0001\n;\n(C5)\nn\u0006=!2\nN\u0000\nm\u0006\u0000n\u0006\u0001\n+\u000b!c\u0000\n_m\u0006\u0000_n\u0006\u0001\n: (C6)\nFor ac magnetic fields\nh\u0006(t) =h\u0006\n0ei!t; (C7)\nwe solve the equations of motion by the ansatz\nm\u0006(t) =m\u0006\n0ei!t; n\u0006(t) =n\u0006\n0ei!t:(C8)\nThe observables correspond to the real part of the com-\nplexm,n;andh. The susceptibilities are defined\nm\u0006=\u001f\u0006h\u0006=Ms; n\u0006=\u001f\u0006\nnm\u0006;(C9)and read\n\u001f\u0006\nn(!) =!2\nN+i\u000b!!c\n\u0000!2+!2\nN+i\u000b!!c;(C10)\n\u001f\u0006(!) =\u0007!M(\u0000!2+!2\nN+i\u000b!!c)\n\u0002\u0002\n(!\u0007!0\u0007i\u000b!)(\u0000!2+!2\nN+i\u000b!!c)\n\u0006!c(!A+i\u000b!)2\u0003\u00001: (C11)\nClose to a resonance of \u001f\u0006at!ithe absorbed microwave\npower is determined by the contributions\n\u0000!\n2Im\u001f\u0006(!)\u0018F\u0006 (\u0011\u0006)2\n(!\u0000!i)2+ (\u0011\u0006)2;(C12)\nwith\n\u0011\u0006=\u0006\u000b!i\u0000\n!2\ni+!c(\u0006!i\u0000!H)\u0001\n3!2\ni\u00072!i!0\u0000!c!A; (C13)\nF\u0006=1\n2!M(!2\ni\u0000!c!A)\n\u000b(!2\ni+!c(\u0006!i\u0000!H)):(C14)\nNote that for each resonance of \u001f+at!ithere is a cor-\nresponding resonance of \u001f\u0000at\u0000!i.\nThe magnitudes of the x- andy-components of nare\nrelated to mvia the susceptibility \u001f\u0006\nngiven in Eq. (C10).9\nFor high frequencies !we find\u001f\u0006\nn\u00190and for low fre-\nquencies\u001f\u0006\nn\u00191. Therefore, the main axis nis nearly\nstatic for the high-frequency mode, while for the low-\nfrequency modes nstays approximately parallel to m.\nThesusceptibility \u001f\u0006giveninEq.(C11)canberelated\nto the usual magnetic susceptibilities (\u000b;\f=x;y),\n\u001f\u000b\f=M\u000b\nh\f\f\f\f\f\nh?=0: (C15)\nDefining the symmetric and antisymmetric parts of the\nsusceptibility \u001f\u0006,\n\u001f\u0006=\u001fs\u0006\u001fa: (C16)\nwe find the relations\n\u001fxx=\u001fyy=\u001fs; (C17a)\n\u001fxy=\u0000\u001fyx=i\u001fa: (C17b)\nThe magnetization dynamics in terms of the magnetic\nsusceptibility reads\nRe\u0012\nmx(t)\nmy(t)\u0013\n= Re\u0014\u0012\n\u001fxx\u001fxy\n\u0000\u001fxy\u001fxx\u0013\u0012\nhx(t)=Ms\nhy(t)=Ms\u0013\u0015\n;\n(C18)\nwhere\u001fyy=\u001fxxand\u001fyx=\u0000\u001fxy. For linear polariza-\ntionhx(t) =jhxjei!tandhy(t) = 0,\nRe\u0012\nmx(t)\nmy(t)\u0013\n=jhxj\nMs\u0012\nRe\u001fxxcos(!t)\u0000Im\u001fxxsin(!t)\n\u0000Re\u001fxycos(!t) + Im\u001fxysin(!t)\u0013\n:\n(C19)\nAccording to Fig. 5, jRe\u001fxxj;jIm\u001fxyj \u001c j Re\u001fxyj \u0019\njIm\u001fxxj, and Im\u001fxx<0for both low-frequency modes\n!l1and!l2. The direction of the precession depends now\non the sign of Re\u001fxy, which is negative for !l1and posi-\ntive for!l2. The mode !l1is a clockwise precession,\nRe\u0012\nmx(t)\nmy(t)\u0013\n/\u0012\nsin(!l1t)\ncos(!l1t)\u0013\n; (C20)\nwhereas the mode !l2precesses counterclockwise:\nRe\u0012\nmx(t)\nmy(t)\u0013\n/\u0012\nsin(!l2t)\n\u0000cos(!l2t)\u0013\n:(C21)\nNote that\u001f\u0000(!)has a low-frequency peak only at !l1\nand\u001f+(!)only at!l2(for!>0).\n2. Easy-plane configuration\nHere, we consider an equilibrium magnetization nor-\nmal to the principal axis ( m?n) due to the shape\nanisotropy of an oblate spheroid. Linearizing for small\ndeviations from the equilibrium ( \u0012\u001c1,my\u00191,nz\u00191),\nthe equations of motion for the magnetization and me-\nchanical angular momentum read_mx=!Hmz\u0000!Mhz\nMs\u0000!A(mz+ny) +\u000b( _mz+ _ny);\n(C22a)\n_my= 0; (C22b)\n_mz=\u0000!Hmx+!Mhx\nMs\u0000\u000b_mx\u0000\u000b\nz; (C22c)\n_Lx=\u0000I?ny; (C23a)\n_Ly=I?nx; (C23b)\n_Lz=I3_\nz=VMs\n\r(\u0000\u000b_mx\u0000\u000b\nz);(C23c)\nwith\nnx= 0; (C24a)\nny=!2\nN(mz+ny)\u0000\u000b!c( _mz+ _ny);(C24b)\nnz= 0: (C24c)\nIn the presence of ac magnetic fields\nhx(t) =hx;0ei!t; hz(t) =hz;0ei!t;(C25)\nwe use the ansatz\nmx(t) =mx;0ei!t; mz(t) =mz;0ei!t; ny(t) =ny;0ei!t:\n(C26)\nFrom Eq. (C23c)\n\nz=\u0000!I!\u000bmx\n!\u0000i\u000b!I\u0019\u0000\u000b!Imx;(C27)\nwhere!I=VMs=(\rI3)and provided \u000b!Iis sufficiently\nsmaller than all the other relevant frequencies. We ap-\nproximate\u000b\nz=O(\u000b2)\u00190in Eq. (C22c). Due to\nthe reduced symmetry for m?n, we cannot simplify\nthe equations of motion by introducing chiral modes, but\nhave to calculate the Cartesian components of the mag-\nnetic susceptibility tensor \u001fas\n\u001fxx=!M\u0002\n!2(!A\u0000!H)\u0000i\u000b(!3\u0000!!c!H)\u0000!H!2\nN\u0003\n=\u001fd;\n(C28a)\n\u001fzz=\u0000!M(!H+i\u000b!)(!2+!2\nN\u0000i\u000b!c!)=\u001fd;(C28b)\n\u001fxz=i!!M(!2+!2\nN\u0000i\u000b!c!)=\u001fd; (C28c)\n\u001fzx=\u0000\u001fxz; (C28d)\nwhere the denominator\n\u001fd=!4(1 +\u000b2) +i\u000b!3(!A\u0000!c\u00002!H)\n+!2(!A!H\u0000!2\nH+!2\nN\u0000\u000b2!c!H)\n+i\u000b!!H(!c!H\u0000!2\nN)\u0000!2\nH!2\nN:(C29)\nThe singularities in \u001fmark the two resonance frequen-\ncies. For small damping ( \u000b\u001c1)\n!2\n1;2=\u00001\n2(!A!H\u0000!2\nH+!2\nN)\n\u00061\n2q\n(!A!H\u0000!2\nH+!2\nN)2+ 4!2\nH!2\nN:(C30)10\nFrom Eq. (C24b), we obtain the following relation be-\ntween the magnetic and mechanical motion\nny=\u0000!2\nN+i\u000b!c!\n!2+!2\nN\u0000i\u000b!c!mz: (C31)\nFor high frequencies ny\u00190and for low frequencies ny\u0019\n\u0000mz. This implies that for the high frequency mode\n!?=!1we recover the bulk FMR, while in the low-\nfrequency mode !l=!2the magnetization is locked to\nthe lattice.\nAppendix D: FMR absorption\nFMRabsorptionspectraareproportionaltotheenergy\ndissipated in the magnet [25]. The energy density of the\nmagnetic field is given by\nw(t) =1\n2H(t)\u0001B(t); (D1)\nwhere B=\u00160\u001fH. The absorbed microwave power by a\nmagnet of volume Vis\nP(t) =V_w(t) =VH(t)\u0001_B(t): (D2)\nThe average over one cycle T= 2\u0019=!,\nP\u0011hP(t)i=1\nTZT\n0dtP(t); (D3)can be calculated using the identity\n\nRe(Aei!t)\u0001Re(Bei!t)\u000b\n=1\n2Re (A\u0003\u0001B):(D4)\nWhen a monochromatic ac component of the magnetic\nfieldh?is normal to its dc component, the power reads\nP=\u0000\u00160V\n2!Im (h\u0003\n?\u0001M?); (D5)\nwhere M?is the transverse magnetization. When the\nmagnetization and static magnetic field are parallel to\nthe principal axis of the particle, we can write\nP=\u0000\u00160V\n2!h\n(jhxj2+jhyj2)Im\u001fs(!)\n\u00002Im(h\u0003\nxhy)Im\u001fa(!)]; (D6)\nwhere the symmetric and antisymmetric parts of the\nsusceptibility \u001f\u0006Eq. (C11) as defined by Eq. (C16)\nobeythesymmetryrelations Im\u001fs(\u0000!) =\u0000Im\u001fs(!)and\nIm\u001fa(\u0000!) = Im\u001fa(!). 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Adv.\n2, e1501286 (2016)."    },    {        "title": "2301.11027v3.Anisotropic_spin_current_spectroscopy_of_ferromagnetic_superconducting_gap_symmetries.pdf",        "content": "Anisotropic Spin-Current Spectroscopy of Ferromagnetic Superconducting Gap\nSymmetries\nHiroshi Funaki,1Ai Yamakage,2and Mamoru Matsuo1, 3, 4, 5\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n4RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n(Dated: May 9, 2023)\nWe develop a microscopic theory of tunneling spin transport at the magnetic interface between\na ferromagnetic insulator (FI) and a ferromagnetic superconductor (FSC) driven by ferromagnetic\nresonance. We show that the spin susceptibilities of the FSC can be extracted from the spin currents\nby tuning the easy axis of the FI, and thus the spin currents can be a probe for the symmetries of\nthe spin-triplet Cooper pairing. Our results will o\u000ber a route to exploiting the synergy of magnetism\nand superconductivities for spin devices.\nI. INTRODUCTION\nTunneling spin current in magnetic heterostructures\ndriven by magnetization dynamics using ferromagnetic\nresonance (FMR) has been studied intensively in spin-\ntronics. It is widely known as the spin pumping e\u000bect[1],\na versatile way to generate the spin current in nanohy-\nbrid systems from a ferromagnet into various conducting\nmaterials. Recently, the spin pumping has been recog-\nnized as a quantum probe to detect magnetic proper-\nties of thin \flms[2] because the generated spin current\ncan be measured very sensitively even for nano-scale thin\n\flms[3]. From a theoretical point of view, the spin cur-\nrent re\rects the spin susceptibility of adjacent materi-\nals [4]. This property has a signi\fcant impact on su-\nperconducting spintronics research[5], where a variety of\nconversions between Cooper pair supercurrents and spin\ncurrents have been intensively studied including triplet\nCooper pair currents[6]. In particular, the tunneling spin\ncurrent can be utilized as a direct probe of spin excita-\ntions in the Cooper pair symmetries of conventional SCs\n[7{20] and unconventional SCs [21{24].\nSymmetry of the Cooper pair characterizes the na-\nture of superconductors [25]. In particular, spin-triplet\nCooper pairs o\u000ber a fascinating state from the viewpoint\nof superconducting spintronics, because they can carry\nspin one as a supercurrent without dissipation.3He is\na well-established spin-triplet super\ruid with (breaking)\ntime-reversal symmetry in the B (A) phase. On the\nother hand, almost all existing superconductors are spin-\nsinglet rather than spin-triplet superconductors. Estab-\nlishing spin-triplet candidates in superconductors is an\nessential issue in condensed matter physics. Indeed, the\nsearch for spin-triplet superconductors continues vigor-\nously [26{30]. Among superconductors, it is also believed\nthat the spin-triplet pair is likely formed in the ferro-\nmagnetic superconducting state in uranium compounds\n[31{33], UGe 2[34], UIr [35], URhGe [36], and UCoGe\n[37]. However, despite many years of research, no de\fni-\n－ +＋\n－FIG. 1. (a) Schematic diagram of the spin pumping e\u000bect\nat a junction system of a ferromagnetic insulator (FI) and a\nferromagnetic superconductor (FSC). The tunneling spin cur-\nrentISis generated at the interface driven by magnetization\ndynamics due to microwave irradiation in FI. (b) The spin\npolarization of the generated spin current can be controlled\nby tuning the easy axis of the magnetization in FI. The z-\npolarized spin current Iz\nSand thex-polarized one Ix\nSre\rect\nthe magnetic properties of the FSC characterized by the spin\nsusceptibilities \u001f?and\u001fk. The transverse spin susceptibil-\nity\u001f?can be extracted from the z-polarized spin current Iz\nS\nwhile the longitudinal one \u001fkfrom 2Ix\nS\u0000Iz\nS, where\u001f?is the\ncorrelation between the majority spin and the minority spin\nand\u001fkconsists of the spin susceptibility of the majority spins\n\u001f\"and that of the minority spins \u001f#. (c) The superconducting\ngaps we consider in this paper.\ntive conclusions have been reached on its superconduct-\ning symmetry, such as the gap node and d-vector con-\n\fguration. A complete characterization of the properties\nof ferromagnetic superconductivity will provide the basis\nfor further understanding the physics of superconductiv-\nity and its development into anisotropic superconductingarXiv:2301.11027v3  [cond-mat.supr-con]  6 May 20232\nspintronics. Moreover, it has been proposed that these\nferromagnetic superconductors may exhibit gapless sur-\nface states characterized by the Z4topological invariant\nunder high pressure if a speci\fc superconducting symme-\ntry is realized [38]. Determining the symmetry of their\nparent state, ferromagnetic superconductivity, is one of\nthe most fundamental issues for studying topological ma-\nterials.\nIn this paper, we propose a spectroscopy of ferromag-\nnetic superconducting gap symmetries of FSC thin \flms\nby using the tunnel spin current at a magnetic interface\nexcited by FMR. To this end, we consider the tunnel-\ning spin transport at magnetic interface between a fer-\nromagnetic insulator (FI) and a ferromagnetic supercon-\nductor (FSC) driven by FMR as shown in Fig. 1. We\ndevelop a microscopic theory of the spin current gener-\nation, caused by the di\u000berences of the nonequilibrium\ndistribution functions of the FI and the FSC. We show\nthat we can generate several types of tunneling spin cur-\nrent, including Iz\nSandIx\nSby tuning the relative angle\nbetween the easy axis of the magnetization of the FI and\nthat of the FSC. We \fnd that the spin susceptibilities\nof the FSC can be extracted from the spin currents and\npropose a method to determine the symmetries of the\nspin-triplet Cooper pairing by using the inverse spin Hall\nvoltage measurements. Our results will o\u000ber a route to\nexploiting the synergy of magnetism and superconduc-\ntivities for spin devices.\nII. MODEL\nLet us consider a model of tunneling spin transport at\nthe magnetic interface consisting of a FI and a FSC aim-\ning at extracting the magnetic properties of the FSC from\nthe generated spin currents as shown in Fig. 1. The gen-\nerated spin current can be calculated by the spin tunnel-\ning Hamiltonian method [4, 8, 23, 24]. The total Hamil-\ntonianHconsists of the three terms:\nH=HFSC+HFI+Hex: (1)\nThe \frst termHFSCis the mean \feld Hamiltonian of the\nbulk FSC:\nHFSC=1\n2X\nkcy\nkHBdGck; (2)\nwhere the fermion operator is de\fned by ck=\n(ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)Tin the Nambu space for k=\n(kx;ky;kz). The Bogoliubov-de Gennes (BdG) Hamil-\ntonianHBdG for a spin-triplet superconductor with an\nequal-spin pairing is given by\nHBdG=0\nBB@\u0018k\" 0 \u0001 k\"\" 0\n0\u0018k# 0 \u0001 k##\n\u0000\u0001\u0003\n\u0000k\"\" 0\u0000\u0018\u0000k\" 0\n0\u0000\u0001\u0003\n\u0000k## 0\u0000\u0018\u0000k#1\nCCA;(3)where\u0018k\u001b=~2k2\n2m\u0000\"F\u0000\u001b\u0001FMis the energy dispersion\nof spin up [ \u001b=\"(+)] and down [ \u001b=#(\u0000)] electron in a\nferromagnet in the normal state, and \"Fand \u0001 FMare the\nFermi energy and the spin-splitting energy, respectively.\nHere we focus on a single-band superconductor in order\nto elucidate properties intrinsic to ferromagnetic super-\nconductivity, especially to gapless excitations of quasi-\nparticles and nonunitarity of breaking time-reversal sym-\nmetry. For simplicity we here neglect the collective ex-\ncitations in the FSC and assume that the spin-splitting\nenergy \u0001 FMis the constant and su\u000eciently larger than\nthe superconducting gap, where only the spin-up band\nis superconducting, i.e., \u0001 k##= 0. This assumption is\nappropriate for examining the key characteristics of non-\nunitary superconductivity, a leading candidate for the\norder parameter in FSC. In particular, we consider two\ntypes of the pair potential\n\u0001k\"\"=\u0000\u00010sin\u0012ke\u0000i\u001ek;for axial; (4)\n\u0001k\"\"=\u0000\u00010cos\u0012k;for polar; (5)\nas shown in Fig. 1 (c). Note that the above pair potentials\nare of generic form within the single-band p-wave equal-\nspin-pairing superconductivity. The axial gap, Auirrep\nof the magnetic point group 4 =mm0m0[39{43], has point\nnodes on the north and south poles while the polar gap,\nEuirrep, has a line node on the equator. This di\u000berence\nin the excitation gap is re\rected in the spin excitations\ninvolved in FMR.\nThe second term HFIdescribes the bulk FI given by\nHFI=\u0000JX\nhi;jiSi\u0001Sj\n\u0000~\rX\nih\nhdcSz\ni+hac(cos\ntSx\ni\u0000sin\ntSy\ni)i\n;(6)\nwhere Siis the magnetization at the site iin the FI,\nJis the exchange interaction, hdcis a static magnetic\n\feld,hacand \n are the amplitude and frequency of the\napplied microwave radiation, respectively, and \ris the\ngyromagnetic ratio.\nThe third termHexis the interfacial exchange coupling\nwhich describes the spin transfer between the FI and the\nFSC:\nHex=TX\nk;qSk\u0001sq; (7)\nwhereTis the tunneling amplitude between the magne-\ntization in the FI Skand the conduction electron spin in\nthe FSC sq. We assumed a constant tunneling amplitude\ncorresponding to a rough interface limit.\nIII. TUNNELING SPIN CURRENTS\nThe tunneling spin current at the interface driven by\nFMRh^Ii\nSiis calculated by the statistical average of the3\nV V\nFIG. 2. Schematic diagram of measuring spin currents Iz\nS\n(a) andIx\nS(b) by the inverse spin Hall e\u000bect (ISHE). The\nspin current is converted to a voltage and measured. The\nconversion results from the ISHE in a heavy metal (HM) with\nstrong spin-orbit interaction. The spin current at the interface\nbetween the FI and the FSC can be measured almost directly,\nespecially when the FSC is thin.\nspin current operator ^Ii\nSde\fned by\n^Ii\nS=\u0000~@t(si\ntot) =i[si\ntot;H] =\u0000X\nk;q\u000fijkTSj\nksk\nq;(8)\nwheresi\ntot=si\nq=0. We calculate the statistical average\nh^Ii\nSiusing the Schwinger-Keldysh approach. By taking\ninto account the second-order perturbation of the interfa-\ncial exchange coupling Hexand assuming that the Fermi\nenergy is su\u000eciently larger than the spin splitting energy\nin FSC, we obtain the relations between the generated\nspin currents and the dynamic spin susceptibilities of the\nFSC (see the Appendices A and B for the detailed deriva-\ntion):\nIz\nS:=h^Iz\nSi=T2\nSIm\u001fR;?\nloc;\n\u0001n\nImGR\n0;\n; (9)\nIx\nS:=h^Ix\nSi=T2\n2Sh\nIm\u001fR;?\nloc;\n+ Im\u001fR;k\nloc;\ni\n\u0001n\nImGR\n0;\n;\n(10)\nwhere\u001fR;?\nloc;\nand\u001fR;k\nloc;\nare transverse and longitudi-\nnal components of the local spin susceptibility in the\nFSC, respectively, and Im GR\n0;\nis spin susceptibility at\nk=0in the FI, and \u0001n\nis a change of magnon num-\nber due to microwave irradiation. Using sxand spin-\ndependent electron number n\u001b,\u001fR;?\nloc;\nand\u001fR;k\nloc;\nare\ndescribed as \u001fR;?\nloc;\n=iR\ndtP\nq\n[sx\nq(t);sx\n\u0000q(0)]\u000b\n\u0012(t)ei\nt\nand\u001fR;k\nloc;\n= (\u001fR;\"\nloc;\n+\u001fR;#\nloc;\n)=4 with\u001fR;\u001b\nloc;\n=\niR\ndtP\nqh[nq\u001b(t);n\u0000q\u001b(0)]i\u0012(t)ei\nt, respectively. Equa-\ntions (9) and (10) indicate that the local spin suscep-\ntibilities can be extracted from the spin currents as\nIm\u001fR;?\nloc/Iz\nSand Im\u001fR;k\nloc/2Ix\nS\u0000Iz\nS(see Fig.1 (b)). The\nspin polarization of the spin currents can be controlled\nby tuning the relative angle between the easy axis of the\nmagnetization of the FI and that of the FSC. Therefore,\nwe can systematically identify the spin susceptibilities of\nthe FSCs by measuring the spin currents and combining\ntheir frequency dependencies, as shown in Sec. IV.\nWe consider using the inversion spin Hall e\u000bect (ISHE)\nto measure the generated spin currents at the magneticinterface. Our measurement setup is shown in Fig. 2,\nwhere the heavy metal (HM) with strong spin-orbit in-\nteraction (SOI), such as Pt, is attached to the FSC thin\n\flm as the spin-current detector. Here we assume the\nFSC is su\u000eciently thin to avoid bulk spin scattering pro-\ncesses in the FSC for simplicity. The generated spin cur-\nrent between the FI and the FSC \rows into the HM and\nis converted into an inverse spin Hall voltage due to the\nstrong SOI. In particular, the z-polarized spin current Iz\nS\ngenerated when the easy axis of the FI is parallel to the\nz-axis can be measured by the setup (a) in Fig. 2, while\nthex-polarized spin current Ix\nSby (b). Note that we can\nobtain su\u000ecient information to identify the pairing sym-\nmetries of the FSC from Iz\nSandIx\nSas discussed below.\nNamely, the y-polarized spin current, which cannot be\nmeasured in our setup, is unnecessary to determine the\nsymmetries.\nIV. FREQUENCY DEPENDENCIES OF THE\nSPIN SUSCEPTIBILITIES\nFigure 3 shows the numerical results of the spin suscep-\ntibilities. We focus on the frequency dependence of the\nspin susceptibility when the temperature is lower than\nthe frequency of the applied microwave radiation, thus\nthe temperature can be approximated as zero. The fre-\nquency dependencies of the imaginary part of the local\nspin susceptibilities in the axial type FSC Im \u001fR;?\nloc(a) and\nIm\u001fR;k\nloc(b) and those in the polar type FSC (c, d) at\nzero temperature represented on a log-log scale, where\nthe spin susceptibilities are normalized as Im\u0016 \u001fR;\u000b\nloc=\nIm\u001fR;\u000b\nloc;\n=Im\u001fR;?\nloc;N;\n=\u0001 0=~(\u000b=?;k;\";#). Here Im\u001fR;?\nloc;N\nis the spin susceptibility in the normal state, and the fre-\nquency is normalized by \u0001 0. The susceptibility Im\u0016 \u001fR;\u000b\nloc\ndoes not depend on the spin-splitting energy \u0001 FMap-\nproximately because the Fermi energy \"Fis su\u000eciently\nlarger than \u0001 FM. Thus, the normalized spin susceptibil-\nity has only one parameter, ~\n=\u00010. It is remarkable that\nthe spin susceptibilities show the characteristic power-law\nfrequency dependencies. The transverse spin susceptibil-\nity in the axial superconducting state Im \u001fR;?\nlocis propor-\ntional to \n3while that in the polar state to \n2in the low\nfrequency region ~\n.\u00010, as indicated in the blue area\nin Figs. 2 (a) and (c). Such frequency dependencies can\nbe obtained from the analytical power expansions of the\nspin susceptibility (see the Appendix E for the detailed\nderivation). For instance, the power expansion of Im \u001fR;?\nloc\nin the axial superconducting state when ~\n.\u00010is given\nby\nIm\u0016\u001fR;?\nloc;\n\u0019(~\n)3\n3\u00013\n0: (11)\nIt should be noted that the dependence on \n3appears\nwhen~\n.\u00010, whereas it is proportional to \n when the\nfrequency becomes larger than the superconducting gap,\nwhere it no longer di\u000bers from the normal state.4\n－ +\n＋\n－\nFIG. 3. The frequency dependencies of the imaginary part of the local spin susceptibilities in the axial type FSC (a,b) and\nthose in the polar type FSC (c,d) at zero temperature. represented on a log-log scale. The characteristic power-law frequency\ndependencies (indicated in the blue area) originate from the symmetries of the superconducting gaps. Together with the fact\nthat Im\u001fR;?\nloc(a,c) is extracted from Iz\nSand Im\u001fR;k\nloc(b,d) from 2 Ix\nS\u0000Iz\nS(see also Fig. 1), we can identify the ferromagnetic\nsuperconducting gap symmetries from the tunneling spin currents.\nTABLE I. The characteristic power-law frequency dependen-\ncies of the transverse (Im \u001fR;?\nloc) and longitudinal (Im \u001fR;k\nloc=\n[Im\u001fR;\"\nloc+ Im\u001fR;#\nloc]=4) components of the imaginary part of\nthe local spin susceptibilities in various states. The suscepti-\nbilities in the anisotropic superconducting states indicate the\npower-law frequency dependencies in sharp contrast to the\nexponential dependence in the s-wave spin-singlet SC. In ad-\ndition to the axial and polar properties, the non-unitarity of\nthe FSC can be identi\fed from the frequency dependencies of\nthe susceptibilities.\nSC state Im \u001fR;?\nlocIm\u001fR;\"\nlocIm\u001fR;#\nloc\nFM p-wave (axial) non-unitary 3 5 1\nFM p-wave (polar) non-unitary 2 3 1\nFM p-wave (axial) unitary 5 5 5\nFM p-wave (polar) unitary 3 3 3\nd-wave (2D polar) singlet 3 3 3\ns-wave singlet exp. exp. exp.\nIn addition to the axial and polar features, the fre-\nquency dependencies in the low frequency region provide\ninformation on the non-unitarity of the FSCs. Due to\nthe anisotropy of the non-unitary pair, \u0001 \"\"6= 0 and\n\u0001##= 0, the spin excitations are also anisotropic, mak-\ning\u001fR;\"\nlocand\u001fR;#\nlocdi\u000berent. Moreover, \u001fR;?\nlocdi\u000bers from\n\u001fR;\"\nlocand\u001fR;#\nlocbecause it depends on both spin-up and\ndown bands. In particular, their exponents are also dif-ferent. They are given by Im \u001fR;?\nloc/\n3, Im\u001fR;\"\nloc/\n5,\nand Im\u001fR;#\nloc/\n for the axial non-unitary pair. On the\nother hand, no such anisotropy is observed for the uni-\ntary pair with \u0001 \"\"= \u0001##, and then all components of\nsusceptibility are proportional to the \ffth power of fre-\nquency. Thus, the measurement of spin excitations via\nspin currents by FMR is also helpful in measuring the\nnon-unitary nature of the Cooper pair.\nWe mention two points concerning the scope of our\nmodel. Firstly, our model does not account for the multi-\nband of FSC. Even when the multiband is considered,\nthe exponent of the frequency dependence remains un-\nchanged for intraband pairings because the node struc-\nture of the superconducting gap, which are the same as\nfor the single-band FSCs, primarily governs the expo-\nnent. When the interband paring is dominant, a di\u000ber-\nent exponent is expected to be observed. Secondly, our\nmodel does not consider the Andreev bound states be-\ncause our objective is to study the spin excitation in the\nbulk and not the Andreev bound states emerging on the\ninterface between the FM and FSC. They are gapless sur-\nface excitations in superconductors with gap nodes and\nhence can change the exponents discussed above. In the\ncase of line nodes, Andreev bound states form in the\nxy-plane and do not impact the exponents. In contrast,\npoint nodes create Andreev bound states along the xz\nplane, which in\ruences the exponents. In fact, a previ-\nous study has demonstrated that Andreev bound states5\nsigni\fcantly contribute to spin pumping in d-wave su-\nperconductors [44]. These issues will be left for future\nstudies.\nV. CONCLUSION\nIn this paper, we have developed a microscopic theory\nof the tunneling spin current at the magnetic interface\nof a FI and a FSC by microwave irradiation, aiming to\nidentify the ferromagnetic superconducting gap symme-\ntries. We obtained the relations between the tunneling\nspin currents and the dynamic spin susceptibilities of the\nFSCs, and found that the spin susceptibilities can be\nextracted from the spin currents by tuning the relative\nangle between the easy axis of the magnetization of the\nFI and that of the FSC. We revealed that the spin sus-\nceptibilities of the FSC indicate the characteristic power-\nlaw frequency dependencies re\recting the axial and polar\nproperties as well as the non-unitarity and unitarity of\nthe FSCs. Accordingly, the tunneling spin currents in our\nsetups can be a probe of the ferromagnetic superconduct-\ning gap symmetries by combining the tunability of their\nspin polarization and their frequency dependencies. Our\ntheory paves the way for ferromagnetic superconducting\nspintornics, where the synergy of magnetism and super-\nconductivities are exploited.\nACKNOWLEDGMENTS\nThe authors are grateful to Yuya Ominato for valu-\nable comments. This work was supported by the Pri-\nority Program of the Chinese Academy of Sciences un-\nder Grant No. XDB28000000, and by JSPS KAK-\nENHI for Grants (Nos. 20K03835, 20H04635, 20H01863,\n21H01800, 21H04565, and 23H01839) from MEXT,\nJapan.\nAppendix A: Spin currents at the interface of\nferromagnetic junctions\nIn this Section, we describe spin currents generated\nby spin pumping at the interface of ferromagnetic junc-\ntions. We derive the explicit expression of the spin\ncurrent of the second-order perturbation of the interfa-\ncial exchange coupling between a ferromagnetic insulator\n(FI) and a ferromagnetic superconductor (FSC) using the\nSchwinger-Keldysh approach.\nWe de\fne the spin operator in the FI, Sk, and that in\nthe FSC, sq, as\nSk=Sr\nker+S\u0012\nke\u0012+S\u001e\nke\u001e; (A1)\nS\u0006\nk=S\u0012\nk\u0006iS\u001e\nk; (A2)\nsq=sz\nqez+sx\nqex+sy\nqey; (A3)\ns\u0006\nq=sx\nq\u0006isy\nq; (A4)whereS\u0006\nkands\u0006\nq are ladder opera-\ntors, er = (sin\u0012cos\u001e;sin\u0012sin\u001e),e\u0012 =\n(cos\u0012cos\u001e;cos\u0012sin\u001e;\u0000sin\u0012),e\u001e= (\u0000sin\u001e;cos\u001e;0),\nez= (0;0;1),ex= (1;0;0), and ey= (0;1;0). Here,\neach quantization axis is chosen along each magnetic\neasy axis er= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012) for the FI\nandez= (0;0;1) for the FSC, respectively.\nThe interfacial exchange coupling between the FI and\nthe FSC is\nHex=TX\nk;qSk\u0001sq=TX\nk;qB\u000b\fS\u000b\nks\f\nq; (A5)\nwhereTis the strength of the interfacial exchange cou-\npling andB\u000b\f=e\u000b\u0001e\f. The spin-current operator is\nde\fned by\n^Ii\nS=\u0000~@t(si\ntot) =i[si\ntot;H] =\u0000X\nk;q\u000fijkTSj\nksk\nq\n=\u0000TX\nk;qAi\n\u000b\fS\u000b\nks\f\nq; (A6)\nwheresi\ntot=si\nq=0andAi\n\u000b\f=ei\u0001(e\u000b\u0002e\f).\nUsing the Schwinger-Keldysh approach and taking into\naccount the second-order perturbation of T, the statisti-\ncal average of the spin current is\nh^Ii\nSi=\u0000T\n2X\nk;qAi\n\u000b\fhTC[s\f\nqS\u000b\nk]Ki\n=\u0000T\n2X\nk;qh\nAi\n\u000b\fhS\u000b\nkihs\f\nqi\n+iTZd!\n2\u0019Ai\n\u000b\fB\u000b0\f0[\u001f\f\f0\nFSC;q;!\u001f\u000b\u000b0\nFI;k;!]Ki\n;(A7)\nwhereTCis the time-ordering operator on the Keldysh\ncontour and spin susceptibilities are de\fned as\n\u001f\u000b\u000b0\nFI;k;!=iZ\nCd(\u001c1\u0000\u001c2)\u0002\nhTC[S\u000b\nk(\u001c1)S\u000b0\n\u0000k(\u001c2)]i\n\u0000ihS\u000b\n0(\u001c1)ihS\u000b\n0(\u001c2)i\u0003\nei!(\u001c1\u0000\u001c2); (A8)\n\u001f\f\f0\nFSC;q;!=iZ\nCd(\u001c1\u0000\u001c2)\u0002\nhTC[s\f\nq(\u001c1)s\f0\n\u0000q(\u001c2)]i\n\u0000ihs\f\n0(\u001c1)ihs\f\n0(\u001c2)i\u0003\nei!(\u001c1\u0000\u001c2); (A9)\nwhereR\nCmeans integral on the Keldysh contour. The\nspin susceptibilities are detailed in Appendices C and D.\nThe integrand of the last term in Eq. (A7) can be ex-\npanded with the Langreth rule as\n[\u001f\f\f0\nFSC;q;!\u001f\u000b\u000b0\nFI;k;!]K=\u001fR;\f\f0\nFSC;q;!\u001fK;\u000b\u000b0\nFI;k;!\n+\u001fK;\f\f0\nFSC;q;!\u001fA;\u000b\u000b0\nFI;k;!; (A10)6\nwhere each component is de\fned as\n\u001fK;\f\f0\nFSC;q;!=iZ1\n\u00001dt\u0002\nhs\u000b\nq(t)s\u000b0\n\u0000q(0) +s\u000b0\n\u0000q(0)s\u000b\nq(t)i\n\u00002hs\f\n0(t)ihs\f0\n0(0)i\u0003\nei!t; (A11)\n\u001fR;\f\f0\nFSC;q;!=iZ1\n\u00001dth[s\u000b\nq(t);s\u000b0\n\u0000q(0)]i\u0012(t)ei!t; (A12)\n\u001fK;\u000b\u000b0\nFI;k;!=iZ1\n\u00001dt\u0002\nhS\u000b\nk(t)S\u000b0\n\u0000k(0) +S\u000b0\n\u0000k(0)S\u000b\nk(t)i\n\u00002hS\u000b\n0(t)ihS\u000b0\n0(0)i\u0003\nei!t; (A13)\n\u001fA;\u000b\u000b0\nFI;k;!=\u0000iZ1\n\u00001dth[S\u000b\nk(t);S\u000b0\n\u0000k(0)]i\u0012(\u0000t)ei!t:(A14)\nHere, the Keldysh component of the Green's function is\nde\fned by\nGK\nA;B(q;!) =\niZ1\n\u00001dt\u0002\nhAq(t)B\u0000q(0) +B\u0000q(0)Aq(t)i\u0003\nei!t;(A15)\nwhereAandBare arbitrary bosonic operators. Fur-\nthermore, the Green's function becomes pure imag-\ninary after integrating over k;q, and!because\n([\u001f\f\f0\nFSC;q;!\u001f\u000b\u000b0\nFI;k;!]K)\u0003=\u0000[\u001f\f\f0\nFSC;\u0000q;\u0000!\u001f\u000b\u000b0\nFI;\u0000k;\u0000!]K. Ac-\ncordingly, we can rewrite the spin current as\nh^ISi=\u0000TX\nk;qh\n(er\u0002ez)hSr\nkihsz\nqi\n\u0000T\n2Zd!\n2\u0019n\n(er\u0002ez)h\n(er\u0001ez)\nIm[(\u001fzz\nFSC;q;!\u0000\u001fxx\nFSC;q;!)(\u001frr\nFI\u0000\u001f\u0012\u0012\nFI;k;!)]K\n+ Im[\u001fxy\nFSC;q;!\u001f\u0012\u001e\nFI;k;!]Ki\n+fer\u0000(er\u0001ez)ezg\nIm[(\u001fzz\nFSC;q;!+\u001fxx\nFSC;q;!)\u001f\u0012\u001e\nFI;k;!]K\n+ 2(er\u0001ez)ezIm[\u001fxx\nFSC;q;!\u001f\u0012\u001e\nFI;k;!]K\n\u0000fez\u0000(ez\u0001er)erg\nIm[\u001fxy\nFSC;q;!(\u001frr\nFI;k;!+\u001f\u0012\u0012\nFI;k;!)]K\n\u00002(ez\u0001er)erIm[\u001fxy\nFSC;q;!\u001f\u0012\u0012\nFI;k;!]Koi\n;(A16)\nwhere we use equations originating from the spin conser-\nvation law:\n\u001fzy\nFSC;q;!=\u001fyz\nFSC;q;!=\u001fzx\nFSC;q;!=\u001fxz\nFSC;q;!= 0;(A17)\n\u001fzz\nFSC;q;!6=\u001fxx\nFSC;q;!=\u001fyy\nFSC;q;!; (A18)\n\u001fxy\nFSC;q;!=\u0000\u001fyx\nFSC;q;!; (A19)\n\u001fzy\nFI;k;!=\u001fyz\nFI;k;!=\u001fzx\nFI;k;!=\u001fxz\nFI;k;!= 0; (A20)\n\u001fzz\nFI;k;!6=\u001fxx\nFI;k;!=\u001fyy\nFI;k;!; (A21)\n\u001fxy\nFI;k;!=\u0000\u001fyx\nFI;k;!; (A22)In addition, we can approximate \u001fxy\nFSC\u00190 because the\nFermi energy in the FSC is usually su\u000eciently larger than\nthe considered frequency. Therefore, we obtain\nh^ISi\u0019\u0000TX\nk;qh\n(er\u0002ez)hSr\nkihsz\nqi\n\u0000T\n2Zd!\n2\u0019n\n[er\u0000(er\u0001ez)ez]\nIm[(\u001fzz\nFSC;q;!+\u001fxx\nFSC;q;!)\u001f\u0012\u001e\nFI;k;!]K\n+ 2(er\u0001ez)ezIm[\u001fxx\nFSC;q;!\u001f\u0012\u001e\nFI;k;!]Koi\n;(A23)\nwhere we retain only the lowest order of Tin each direc-\ntion component of the spin current.\nAppendix B: Spin susceptibilities extracted from the\nmeasured spin currents\nIn this section, we discuss how to extract the spin\nsusceptibilities of FSC from the measured spin current.\nFrom Eq. (A23), the z- andx-polarized spin currents are\nIz\nS=T2X\nk;qZd!\n2\u0019h\nIm\u001fR;xx\nFSCRe\u001fK;\u0012\u001e\nFI+ Im\u001fK;xx\nFSCRe\u001fA;\u0012\u001e\nFIi\n;\n(B1)\nIx\nS=T2X\nk;qZd!\n2\u0019h1\n2[Im\u001fR;xx\nFSC+ Im\u001fR;zz\nFSC]Re\u001fK;\u0012\u001e\nFI\n+1\n2[Im\u001fK;xx\nFSC+ Im\u001fK;zz\nFSC]Re\u001fA;\u0012\u001e\nFIi\n; (B2)\nwhere we use that \u001fK;\u0012\u001e\nFI is a real number as shown in\nAppendix C. Furthermore, rewriting the spin suscepti-\nbility of the FI in terms of magnon Green's function (See\nAppendix C for detail), we obtain\nIz\nS=T21\nS[Im\u001fR;?\nloc;\n]\u0001n\nImGR\n0;\n; (B3)\nIx\nS=T21\n2S[Im\u001fR;?\nloc;\n+ Im\u001fR;k\nloc;\n]\u0001n\nImGR\n0;\n;(B4)\nwhereGR\n0;\nand\u0001n\nare the retarded component of\nmagnon Green's function and the deviation in the num-\nber of magnons under the microwave radiation with fre-\nquency \n, respectively, and \u001fR;\f\f\nloc;\n=P\nq\u001fR;\f\f\nFSC;q;\nis the\nlocal spin susceptibility of the FSC. The transverse ?\nand the longitudinal kcomponents are equal to xxand\nzzcomponents, respectively. The spin susceptibilities of\nthe FSC normalized by the spin susceptibility of the nor-\nmal state are\nIm\u001fR;?\nloc;\nIm\u001fR;?\nloc;N;\n=Iz\nS\nIz\nS;N; (B5)\nIm\u001fR;k\nloc;\nIm\u001fR;?\nloc;N;\n=2Ix\nS\u0000Iz\nS\nIz\nS;N; (B6)7\nwhere Im\u001fR;?\nloc;N;\nandIS;Nare the spin susceptibility and\nthe spin current in the normal state, respectively. The\nspin susceptibility of FI disappears by the normalization\nbecause it is equal in the superconducting and normal\nstates. These equations tell us the relationships between\nthe spin susceptibilities of the FSC and the measured\nspin currents.\nAppendix C: Spin susceptibilities in ferromagnetic\ninsulators\nIn this section, we derive the spin susceptibilities\n\u001f\u0012\u001e\nFI;k;!in the FI under microwave radiation from the\nmagnon Green's function, and show the detailed calcula-\ntions for the spin currents in Eqs. (B3) and (B4).\nWe use the Holstein-Primako\u000b transformation to the\nspin operators in the FI and employ the spin-wave ap-\nproximation:\nSr\nk= (S\u0000ay\nkak); (C1)\nS+\nk=S\u0012\nk+iS\u001e\nk\u0019p\n2Sak; (C2)\nS\u0000\nk=S\u0012\nk\u0000iS\u001e\nk\u0019p\n2Say\nk; (C3)\nwhereay\nkandakare the boson creation and annihilation\noperators, respectively. The Hamiltonian in the FI is\nrepresented in terms of the boson operators as\nHFI=\u0000JX\nhi;jiSi\u0001Sj\n\u0000~\rX\nih\nhdcSz\ni+hac(cos\ntSx\ni\u0000sin\ntSy\ni)i\n\u0019X\nk(!k+!0)ay\nkak\u0000Vac(e\u0000i\ntay\nk=0+ei\ntak=0);\n(C4)\nwherehdcis static magnetic \feld, hacand \n are ampli-\ntude and frequency of applied microwave, respectively,\nandJis the exchange coupling constant, hi;jirepresents\nsummation over all nearest-neighbor sites, \ris the gyro-\nmagnetic ratio, and !0=~\rhdc; Vac=~\rhacq\nSN\n2with\nthe number of sites N. Here, we assume the parabolic\nmagnon dispersion: !k/k2. The magnon Green's func-\ntions are de\fned as\nGk;!:=\u0000iZ\nCd(\u001c1\u0000\u001c2)hTC[ak(\u001c1)ay\nk(\u001c2)]iei!(\u001c1\u0000\u001c2);\n(C5)\nGR\nk;!:=\u0000iZ1\n\u00001dth[ak(t);ay\nk(0)]i\u0012(t)ei!t; (C6)\nGK\nk;!:=\u0000iZ1\n\u00001dthak(t)ay\nk(0) +ay\nk(0)ak(t)iei!t:(C7)\nIntroducing a phenomenological lifetime with the Gilbert\ndamping constant \u000band considering the second-orderterm ofVacas self-energy, the magnon Green's functions\nare written as\nGR\nk;!=1\n~!\u0000(!k+!0) +i\u000b~!; (C8)\nGK\nk;!=GK\n0;k;!+ 2GR\nk;!h\n\u0000i\n~V2\nac\u000ek;02\u0019\u000e(!\u0000\n)i\nGA\nk;!\n=\u00002iImGA\nk;!h\n2n!+ 2\u000enk;!+ 1i\n; (C9)\nwhereGK\n0is the non-perturbative Keldysh component,\nn!is the Bose distribution function, and \u000enk;!is the\ndeviation in the Bose distribution function due to the\noscillating magnetic \feld. Note that the self-energy does\nnot a\u000bect the retarded and advanced components since\nVacis a c-number. Here, \u000enk;!is given by\n\u000enk;!=\u0001n!\u000ek;02\u0019\u000e(!\u0000\n); (C10)\nwhere\u0001n!=V2\nac=(2\u000b~2!). The spin susceptibility\n\u001f+\u0000\nFI;k;!is represented by\n\u001f+\u0000\nFI;k;!=\u00001\n2SGk;!; (C11)\nwhere\n\u001f+\u0000\nFI;k;!:=iZ\nCd(\u001c1\u0000\u001c2)hTC[s+\nk(\u001c1)s\u0000\n\u0000k(\u001c2)]iei!(\u001c1\u0000\u001c2):\n(C12)\nThe\u0012\u0012,\u001e\u001e,\u0012\u001eand\u001e\u0012components of the spin suscepti-\nbility are rewritten as\n\u001f\u0012\u0012\nFI;k;!=\u001f\u001e\u001e\nFI;k;!=1\n4(\u001f+\u0000\nFI;k;!+\u001f\u0000+\nFI;k;!); (C13)\n\u001f\u0012\u001e\nFI;k;!=\u0000\u001f\u001e\u0012\nFI;k;!=i\n4(\u001f+\u0000\nFI;k;!\u0000\u001f\u0000+\nFI;k;!); (C14)\nwhere\n\u001f\u0000+\nFI;k;!:=iZ\nCd(\u001c1\u0000\u001c2)hTC[s\u0000\nk(\u001c1)s+\n\u0000k(\u001c2)iei!(\u001c1\u0000\u001c2):\n(C15)\nThe +\u0000and\u0000+ components have relationships:\n\u001fR;\u0000+\nFI;k;!=\u001fA;+\u0000\nFI;\u0000k;\u0000!; (C16)\n\u001fK;\u0000+\nFI;k;!=\u001fK;+\u0000\nFI;k;!: (C17)\nTherefore, the \u0012\u001ecomponents of the spin susceptibility\nare\n\u001fR;\u0012\u001e\nFI;k;!=\u0000i1\n8S[GR\nk;!\u0000GA\n\u0000k;\u0000!]; (C18)\n\u001fK;\u0012\u001e\nFI;k;!=\u0000i1\n8S[GK\nk;!\u0000GK\n\u0000k;\u0000!]: (C19)\nThe Keldysh component \u001fK;\u0012\u001e\nFI;k;!is real because GK\nk;!is\npure imaginary. From Eqs. (C18), (C19) and (C9), we\nobtain\n\u001fK;\u0012\u001e\nFI;k;!=\u00002(2n!+ 1)Re\u001fA;\u0012\u001e\nFI;k;!\n\u00001\n2S[\u000enk;!\u0000\u000en\u0000k;\u0000!]ImGA\n0;\n: (C20)\nWith this equation, we can reproduce the spin currents\nin Eqs. (B3) and (B4).8\nAppendix D: Spin susceptibilities in ferromagnetic\nsuperconductors\nIn this section, we derive the spin susceptibilities in\nFSC, and obtain the \",#, +\u0000and\u0000+ components,\ntransform them into the ?andkcomponents required\nto calculate the spin currents in Eqs. (B3) and (B4).\nThe components \u001b(=\";#),?andkare de\fned as\n\u001f\u001b\nFSC;q;!=iZ\nCd(\u001c1\u0000\u001c2)hTC[nq\u001b(\u001c1)n\u0000q\u001b(\u001c2)]iei!(\u001c1\u0000\u001c2)\n(D1)\n\u001fk\nFSC;q;!=\u001fzz\nFSC;q;!; (D2)\n\u001f?\nFSC;q;!=\u001fxx\nFSC;q;!=\u001fyy\nFSC;q;!; (D3)\nwherenq\u001b=P\nkcy\nk\u001bck+q\u001bandck\u001bis the annihilation\noperator of the electron with spin \u001b.\nThe mean \feld Hamiltonian in the FSC is\nHFSC=1\n2X\nkcy\nkHBdGck; (D4)\nwhere ck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T, andHBdGis\nHBdG=0\nBBB@\u0018k\" 0 \u0001 k\"\" 0\n0\u0018k# 0 \u0001 k##\n\u0000\u0001\u0003\n\u0000k\"\" 0\u0000\u0018\u0000k\" 0\n0\u0000\u0001\u0003\n\u0000k## 0\u0000\u0018\u0000k#1\nCCCA;(D5)\nwhere\u0018k\u001b=\"k\u0000\"F\u0000\u001b\u0001FM, with kinetic energy\n\"k=~2\n2mk2, the Fermi energy \"F, the spin-splitting en-\nergy \u0001 FM, and \u0001 k\u001b\u001bis the superconducting gap. We\nconsider that the superconducting gap opens only in the\nspin-up band (i. e. \u0001 k##= 0). The retarded Green's\nfunctions are given by\n\u001fR;\"\nFSC;q;\n=i1\nN2X\nkk0Z1\n\u00001dt ei\nt\nD\n[cy\nk\"ck+q\"(t);cy\nk0\"ck0\u0000q\"(0)]E\n\u0012(t)\n=1\nNX\nkX\n\u0015=\u0006X\n\u00150=\u0006\n1\n4h\n1 +\u0018\"\u00180\n\"+E\"\u0015\u00180\n\"+E0\n\"\u0015\u0018\"+ \u0001k\"\"\u0001k+q\"\"\nE\"\u0015E0\n\"\u00150i\nf(E0\n\"\u00150)\u0000f(E\"\u0015)\n~\n +i\u000e\u0000E0\n\"\u00150+E\"\u0015; (D6)\n\u001fR;#\nFSC;q;\n=i1\nN2X\nkk0Z1\n\u00001dt ei\nt\nD\n[cy\nk#ck+q#(t);cy\nk0#ck0\u0000q#(0)]E\n\u0012(t)\n=1\nNX\nkf(\u0018k+q#)\u0000f(\u0018k#)\n~\n +i\u000e+\"k\u0000\"k+q; (D7)\u001fR;+\u0000\nFSC;q;\n=i1\nN2X\nkk0Z1\n\u00001dt ei\nt\nD\n[cy\nk\"ck+q#(t);cy\nk0#ck0\u0000q\"(0)]E\n\u0012(t)\n=1\nNX\nkX\n\u0015=\u00061\n2\u0010\n1 +\u00180\n\"\nE0\n\"\u0015\u0011f(E0\n\"\u0015)\u0000f(\u0018#)\n~\n +i\u000e\u0000E0\n\"\u0015+\u0018#;\n(D8)\n\u001fR;\u0000+\nFSC;q;\n=i1\nN2X\nkk0Z1\n\u00001dt ei\nt\nD\n[cy\nk#ck+q\"(t);cy\nk0\"ck0\u0000q#(0)]E\n\u0012(t)\n=1\nNX\nkX\n\u0015=\u00061\n2\u0010\n1 +\u0018\"\nE\"\u0015\u0011f(\u00180\n#)\u0000f(E\"\u0015)\n~\n +i\u000e\u0000\u00180\n#+E\"\u0015\n= [\u001fR;+\u0000\nFSC;\u0000q;\u0000\n]\u0003; (D9)\nwhere\u0018=\u0018k\u001b,\u00180=\u0018k+q\u001b,E\"\u0015=\u0015q\n\u00182\nk\"+ \u00012\nk\"\", and\nE0\n\"\u0015=\u0015q\n\u00182\nk+q\"+ \u00012\nk\"\", respectively. To calculate the\nspin currents in Eqs. (B3) and (B4), we only need the\nimaginary part of the local spin susceptibilities. Inte-\ngrating the wave number k,qwith the axial \u0001 k\"\"=\n\u0000\u00010sin\u0012ke\u0000i\u001ekor the polar \u0001 k\"\"=\u0000\u00010cos\u0012ktype\nsuperconducting gap and using1\nx+i\u000e=\u0000i\u0019\u000e(x) + P1\nx,\nthe imaginary part of the local spin susceptibilities are\nIm\u001fR;\"\nloc;\n\u0019\u0000\u0019Z1\n\u00001d\"DS;\"DS;\"+~\n[f\"+~\n\u0000f\"];(D10)\nIm\u001fR;#\nloc;\n\u0019\u0000\u0019Z1\n\u00001d\"D#\nFD#\nF[f\"+~\n\u0000f\"]\n\u0019\u0019D#\nF2~\n; (D11)\nIm\u001fR;+\u0000\nloc;\n\u0019\u0000\u0019Z1\n\u00001d\"D#\nFDS;\"+~\n[f\"+~\n\u0000f\"];(D12)\nIm\u001fR;\u0000+\nloc;\n\u0019\u0019Z1\n\u00001d\"D#\nFDS;\"\u0000~\n[f\"\u0000~\n\u0000f\"]\n=\u0000Im\u001fR+\u0000\nloc;\u0000\n= Im\u001fR+\u0000\nloc;\n: (D13)\nHere, we replaced the wave number summation with the\nenergy integral as\n1\nNX\nk(\u0001\u0001\u0001)!Z1\n\u00001d\"D#\nF(\u0001\u0001\u0001); (D14)\nfor the normal state and\n1\nNX\nk(\u0001\u0001\u0001)!D\"\nF\n4\u0019Z1\n\u00001dEZ\u0019\n0d\u0012Z2\u0019\n0d\u001e\njEjsin\u0012p\nE2\u0000\u00012(\u0012;\u001e)(\u0001\u0001\u0001)\n=Z1\n\u00001d\"DS;\"(\u0001\u0001\u0001); (D15)9\nfor the superconducting state, where D\u001b\nFis the spin-\ndependent density of states in the normal state at the\nFermi energy.\nThe?component of the spin susceptibility is rewritten\nas\n\u001f?\nFSC;q;!=1\n4(\u001f+\u0000\nFSC;q;!+\u001f\u0000+\nFSC;q;!); (D16)\nIn itinerant electron systems, the kcomponent of the spin\nsusceptibility can be rewritten as\n\u001fk\nFSC;q;!=1\n4(\u001f\"\nFSC;q;!+\u001f#\nFSC;q;!): (D17)\nTherefore, we obtain\nIm\u001fR;k\nloc;\n=\u0019\n4~\nD#\nF2\u0000\u0019\n4D\"\nF2\n\u0002Z1\n\u00001d\"\u0016DS;\"\u0016DS;\"+~\n[f\"+~\n\u0000f\"];(D18)\nIm\u001fR;?\nloc;\n=\u0000\u0019\n2D\"\nFD#\nFZ1\n\u00001d\"\u0016DS;\"+~\n[f\"+~\n\u0000f\"];\n(D19)\nwhere \u0016DS;\"=DS;\"=D\"\nF. Thus, we can calculate the spin\ncurrents in Eqs. (B3) and (B4) with these longitudinal\nand transverse components of the local spin susceptibil-\nity.\nAppendix E: Power series expansion for spin\nsusceptibilities of FSC\nIn this section, we show the power expansion of the\nspin susceptibility of the FSC. The spin susceptibilities\nof the superconducting state are normalized by those of\nthe normal state with the frequency \fxed to be equal to\n\u00010as\nIm\u0016\u001fR;?\nloc;\n=Im\u001fR;?\nloc;\nIm\u001fR;?\nloc;\n=\u0001 0=~;T>T c\n=\u00001\n\u00010Z1\n\u00001d\"\u0016DS;\"+~\n[f\"+~\n\u0000f\"];(E1)\nIm\u0016\u001fR;k\nloc;\n=1\n2\u00101\n2Im\u0016\u001fR;\"\nloc;\n+1\n2Im\u0016\u001fR;#\nloc;\n\u0011\n; (E2)\n1\n2Im\u0016\u001fR;\"\nloc;\n=1\n2Im\u001fR;\"\nloc;\nIm\u001fR;?\nloc;\n=\u0001 0=~;T>T c\n\u0019\u00001\n\u00010Z1\n\u00001d\"\u0016DS;\"\u0016DS;\"+~\n[f\"+~\n\u0000f\"];\n(E3)\n1\n2Im\u0016\u001fR;#\nloc;\n=1\n2Im\u001fR;#\nloc;\nIm\u001fR;?\nloc;\n=\u0001 0=~;T>T c\u0019~\n\u00010; (E4)where Im\u001fR;?\nloc;\n=\u0001 0=~;T>T c=\u0019\n2\u00010D\"\nFD#\nFand we use the\napproximation D\"\nF\u0019D#\nF, assuming that the spin split-\nting energy \u0001 FMis su\u000eciently smaller than the Fermi\nenergy\"F. In the axial superconductor, which has point\nnodes, the density of states is\n\u0016DS;\"=\"\n2\u00010ln\f\f\f\"+ \u0001 0\n\"\u0000\u00010\f\f\f: (E5)\nWhen\"\u001c\u00010, it becomes \u0016DS;\"\u0019\"2\n\u00012\n0. In the polar su-\nperconductor, which has line nodes, the density of states\nis\n\u0016DS;\"=(\n\"\n\u00010arcsin\u00010\n\"\u0000\u00010\n\"<1\u0001\n\u0019j\"j\n2\u00010\u0000\u00010\n\">1\u0001\n:(E6)\nWhen \n<\u00010, we can expand the spin susceptibility to\npowers of frequency. In the case of axial type, they are\n1\n2Im\u0016\u001fR;\"\nloc;\n\u00191\n30(~\n)5\n\u00015\n0; (E7)\nIm\u0016\u001fR;?\nloc;\n\u00191\n3(~\n)3\n\u00013\n0: (E8)\nIn the case of polar type, they are\n1\n2Im\u0016\u001fR;\"\nloc;\n\u0019\u00192\n24(~\n)3\n\u00013\n0; (E9)\nIm\u0016\u001fR;?\nloc;\n\u0019\u0019\n4j~\nj(~\n)\n\u00012\n0: (E10)\nHere, we expand the integral required to calculate the\nspin susceptibility as\nZ1\n\u00001d\"DA;\"+~\n=2DB;\"\u0000~\n=2[f\"+~\n=2\u0000f\"\u0000~\n=2]\n=Z1\n\u00001d\"DA;\"+~\n=2DB;\"\u0000~\n=21X\nn=0f(2n+1) \n2n+1\n22n(2n+ 1)!\n\u0019\u00001X\nn=0h@2n\n@\"2n(DA;\"+~\n=2DB;\"\u0000~\n=2)i\n\"=0\n2n+1\n22n(2n+ 1)!;\n(E11)\nwhereDA;Bis the density of states in the superconduct-\ning or normal state, and we use f0(\")\u0019\u0000\u000e(\") assuming\nzero temperature. Thus, we can analytically obtain the\npower exponent of the spin susceptibility of superconduc-\ntors.10\n[1] Y. Tserkovnyak, A. Brataas, G. E. Bauer, and B. 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B 107, 144504 (2023)."    },    {        "title": "1802.10221v1.Single_ferromagnetic_fluctuations_in_UCoGe_revealed_by_73Ge__and_59Co_NMR_studies.pdf",        "content": "Single ferromagnetic fluctuations in UCoGe revealed by73Ge- and59Co-NMR studies\nMasahiro Manago∗and Kenji Ishida\nDepartment of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan\nDai Aoki\nIMR, Tohoku University, Oarai, Ibaraki 311-1313, Japan and\nINAC/PHELIQS, CEA-Grenoble, 38054 Grenoble, France\n73Ge and59Co nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR)\nmeasurements have been performed on a73Ge-enriched single-crystalline sample of the ferromagnetic\nsuperconductor UCoGe in the paramagnetic state. The73Ge NQR parameters deduced from NQR\nand NMR are close to those of another isostructural ferromagnetic superconductor URhGe. The\nKnight shifts of the Ge and Co sites are well scaled to each other when the magnetic field is parallel\nto theborcaxis. The hyperfine coupling constants of Ge are estimated to be close to those of\nCo. The large difference of spin susceptibilities between the aandbaxes could lead to the different\nresponse of the superconductivity and ferromagnetism with the field parallel to these directions. The\ntemperature dependence of the nuclear spin-lattice relaxation rates 1 /T1at the two sites is similar\nto each other above 5 K. These results indicate that the itinerant U-5 felectrons are responsible\nfor the ferromagnetism in this compound, consistent with previous studies. The similarities and\ndifferences in the three ferromagnetic superconductors are discussed.\nI. INTRODUCTION\nUranium-based ferromagnetic (FM) super-\nconductors1–5have attracted much attention because\nof the intimate relationship between ferromagnetism\nand superconductivity. In these systems, the supercon-\nducting (SC) phases are inside the FM state, and both\nof the ordered states are attributed to itinerant U-5 f\nelectrons. The pairing state and SC mechanism have\nbeen considered to be different from the ordinary s-wave\npairing mediated by the electron-phonon coupling.\nUCoGe is a member of the FM superconductors, and\nhas a Curie temperature TCurie/similarequal3 K and SC transition\ntemperature TSC/similarequal0.7 K.6We have shown the relation-\nship between FM fluctuations with the Ising anisotropy\nand superconductivity from the measurements of the\nfield-angle and field-magnitude dependencies, and sug-\ngested that the FM fluctuations induce spin-triplet su-\nperconductivity in UCoGe.7,8Recently, Wu et al. showed\nthat this scenario explains the macroscopic properties of\nsuperconductivity in UCoGe quantitatively.9An interest-\ning and important question in the FM superconductors is\nwhether this scenario is applicable for UGe 2and URhGe.\nTo answer the question, we need to know similarities and\ndifferences of the magnetic properties of three FM super-\nconductors.\nThe superconductivity of UCoGe is enhanced at the\nFM critical pressure Pc/similarequal1 GPa, and the SC phase\npersists even if the ferromagnetism is suppressed by the\npressure10,11, and this phase disappears at P/similarequal4 GPa.12\nIn contrast, the SC state in UGe 2terminates at the FM\ncritical pressure.13In addition, it seems that the relation-\nship between magnetic properties and superconductivity\nis also different between URhGe and UCoGe with respect\nto the enhancement of superconductivity by the field par-\nallel to the baxis in the orthorhombic structure.14,15A\nreorientation of the magnetic moment occurs in URhGeatµ0H∼12 T,14which is most likely related to the\nstrong FM fluctuations parallel to the baxis,16and the\nre-entrant SC phase is observed in the limited field re-\ngion around 12 T. In contrast, the above picture that the\nFM moment rotates under the field does not apply to the\nenhancement of superconductivity at ∼12 T in UCoGe,\nsince such a moment polarization occurs at ∼50 T along\nthebaxis.17Magnetic characters of three FM supercon-\nductors have some differences against the pressure and\nfield responses, although they also have intimate simi-\nlarities, for instance, that their ferromagnetism is in the\nitinerant regime from the angle resolved photoelectron\nspectroscopy (ARPES).18,19To understand such similar-\nities and differences more precisely, it is crucial to investi-\ngate magnetic characters from the same probes through-\nout these systems. The73Ge-nuclear magnetic reso-\nnance (NMR) and nuclear quadrupole resonance (NQR)\nmeasurements are valuable since Ge are included in all\nthe three compounds, and make it possible to compare\nthese FM superconductors from the microscopic point of\nview. The73Ge NMR and NQR have been performed on\nUGe 220–22and URhGe23in the previous studies, and in\nthis paper, we report first73Ge NMR and NQR results\non UCoGe.\nWe also performed59Co NMR in the same sample of\nUCoGe for clarifying how the U-5 felectrons of UCoGe\ninteract with the59Co and73Ge nuclei. Previously, we\nhave shown from59Co NMR and band calculation on\nUCoGe as well as a reference compound YCoGe that the\nCo-3dstate is not in the Fermi level, and is in nonmag-\nnetic state.24However, there are several reports that the\nhybridization between U-5 fand Co-3dis strong, and the\nCo-3dcontributes the magnetic moments.19,25We con-\nsider that the comparison between the hyperfine field at\nthe Co and Ge sites gives valuable information about the\nhybridization between U-5 fand Co-3d.\nIn this paper, we show from the comparison betweenarXiv:1802.10221v1  [cond-mat.str-el]  28 Feb 20182\nabcU\nCoGe\na (y= 1/4)cU\nCo Ge(a) (b)\nFIG. 1. (Color online) (a) A crystal structure of UCoGe. The\nlargest, middle, and the smallest circles indicate U, Ge, Co\natoms, respectively. (b) Atomic positions at y= 1/4 plane.\nthe59Co and73Ge NMR and NQR that the magnetic\nfluctuations are governed by the single-component fluc-\ntuations from the U-5 felectrons, implying that those\narising from the Co-3 delectrons are negligibly small be-\nlow 3 T. These results are in good agreement with previ-\nous studies.24We also found that the spin susceptibility\nalong theaaxis is substantially smaller than that along\nthebaxis. This large difference may have some relation\nto the different field responses of superconductivity and\nmagnetism along these directions.15,17In addition, we\nshow from the comparison of73Ge-NQR results in the\nthree FM superconductors that their magnetic fluctua-\ntions are similar, but the itinerant degree of U-5 felec-\ntron is different, which is consistent with the transport\nbehavior and magnitude of the ordered moment.1,2\nII. EXPERIMENT\nCrystal structure of UCoGe is shown in Fig. 1. UCoGe\nhas a crystal structure of Pnma space group (#62,\nD16\n2h),26and the local symmetry of Co and Ge (and also\nU) atoms are expressed by [ .m.].\nTABLE I. The data of59Co and73Ge nuclei; the nuclear spin\nI, the nuclear gyromagnetic ratio γn, the nuclear quadrupolar\nmomentQ, and the natural abundance (N.A.).\nI γ n/2π(MHz/T)aQ(10−28m2)bN.A. (%)c\n59Co 7/2 10.03 +0 .42(3) 100\n73Ge 9/2 1.4852 −0.196(1) 7.76(8)\naRef. 27\nbRef. 28\ncRef. 29\nWe used a73Ge-enriched single-crystalline UCoGe\nsample in this study. The resistivity shows a broad hump\nat around 1.3 K, suggestive of a FM transition. From re-\nsistivity and susceptibility measurements, this sample ex-\nhibits a SC transition at TSC/similarequal0.48 K. These transition\ntemperatures are lower than those of the previous higher\nquality samples [for instance, TSC/similarequal0.57 K (Ref. 30) and\nTCurie/similarequal3 K (Ref. 6)]. The lower TSCandTCurie are due\nto the quality of the enriched Ge ingredient.\n 1.5  2  2.5  3  3.5  4UCoGe  73Ge NQR\nνQ = 0.968 MHz, η  = 0.10 T = 4.2 K\n±3/2 ↔ ±5/2 ±5/2 ↔ ±7/2 ±7/2 ↔ ±9/2Intensity (arb. units)\nf (MHz)FIG. 2. (Color online)73Ge NQR spectrum in UCoGe under\nzero magnetic field at 4.2 K (paramagnetic state). Three out\nof four peaks expected in I= 9/2 are observed. The NQR\nparameters are deduced as νQ= (0.968±0.001) MHz and\nη= 0.10±0.01. The arrows represent the calculated peak\npositions with these parameters.\nWe have performed NMR and NQR measurements at\nthe Ge and Co sites down to 1.5 K to investigate spin\nsusceptibility and FM fluctuations, and their anisotropy\nin the paramagnetic state. The nuclear parameters of\n59Co and73Ge nuclei, for which NMR and NQR are pos-\nsible, are listed in Table I. The NMR measurement was\nperformed with a split-coil SC magnet with a single-axis\nrotator. The field was applied parallel to the crystallo-\ngraphica,b, andcaxes, and the directions were deter-\nmined from the59Co NMR spectra. The details of the\nalignment of the single-crystalline sample were described\nin a previous paper.31NMR and NQR spectra depend on\nthe electric field gradient (EFG) at the nuclear sites, and\nthe EFG tensor has three principal axes. Usually, the\nEFG are represented with the NQR frequency νQ∝Vzz\nand asymmetric parameter η≡|(Vxx−Vyy)/Vzz|, where\nVii(i=x,y, andz) is an eigenvalue of the EFG ten-\nsor along the principal axis iand|Vzz|≥|Vyy|≥|Vxx|.\nThe direction zis referred as “maximum principal axis”\nhereafter. In the case of Co and Ge in UCoGe, one of\nthe principal axes is parallel to the baxis from the local\nsymmetry, but a degree of freedom remains in the other\nprincipal-axis directions.\nIII. RESULTS AND DISCUSSION\nFigure 2 shows the73Ge-NQR spectrum in UCoGe at\n4.2 K at zero field.73Ge hasI= 9/2 nuclear spin, and the\nobserved three peaks correspond to the ±3/2↔± 5/2,\n±5/2↔ ± 7/2, and±7/2↔ ± 9/2 transitions. The3\nTABLE II. The73Ge and59Co NQR parameters in UCoGe\nand URhGe; the NQR frequency νQ, asymmetric parameter η,\nand the angle θzzbetween the crystal aaxis and the maximum\nprincipal axis. The maximum principal axis lies in the ac\nplane at all the sites.\nνQ(MHz) η θ zz(°) Ref.\nUCoGe73Ge 0.968±0.001 0.10±0.01 15.5 this work\n59Co 2.85 0.52 10 30\nURhGe73Ge 1.06 0.09 9.8 23\n 0.94 0.96 0.98\n 0 0.2 0.4(a) UCoGe  73Ge\nνQ (MHz)\nνQη = |νxx − νyy| (MHz)\n 14 15 16 17\n 0  50  100  150  200(b)\nθzz (° )\nT (K)\nFIG. 3. (Color online) Temperature dependence of EFG pa-\nrameters of73Ge. (a)νQandνQη=|νxx−νyy|, (b) the\nangleθzzbetween the maximum principal axis of EFG and\nthe crystallographic aaxis in theacplane are shown. νQand\nηare determined from the NQR spectra at H= 0, whileθzz\nis deduced from the field-swept NMR spectra.\nlowest peak corresponding to ±1/2↔± 3/2 transitions\ncould not be detected due to the low frequency beyond\nthe range of our NMR receiver. The73Ge quadrupole fre-\nquencyνQand the asymmetric parameter ηof UCoGe\natT= 4.2 K are determined from the experiments as\nshown in Table II. These parameters are close to those\nof URhGe,23butηis much smaller than that at the Co\nsite in UCoGe.30\nThe temperature dependence of νQandνQηare shown\nin Fig. 3 (a), and νQslightly decreases as temperature in-\ncreases. At finite temperature, NQR frequency in metals\nis empirically expressed as32νQ(T) =νQ(0)(1−αT3/2),\nwhereαis a positive value. Although the present result is\nnot so accurate to distinguish whether this relation holds,\nthe monotonic decrease of νQwith increasing tempera-\nture is a conventional behavior. No large temperature\nvariation of the anisotropic parameter was detected in\nthis system.\nFigure 4 shows the field-swept spectra of73Ge along\nthe three directions. The principal axes of EFG at the\nUCoGe  73Ge NMR\nT = 20 K, f = 4.8 MHzH || a\nθ = 15.5° , φ = 0° \nH || b\nθ = 90° , φ = 90° Intensity (arb. units)\n 0.5  1 1.5  2 2.5  3 3.5  4 4.5H || c\nθ = 74.5° , φ = 0° \nµ0H (T)FIG. 4. (Color online) Field-swept73Ge NMR spectra in\nUCoGe with a fixed frequency f= 4.8 MHz at 20 K for\nthree field directions. The vertical lines indicate the calcu-\nlated peak positions (see text). The polar angle θand the az-\nimuthal angle φrepresent the field directions with respect to\nthe coordinate of the electric field gradient [( θ,φ) = (90 °,0°)\ncorresponds to Vxxdirection].\nGe site are deduced from these spectra. The best fit to\nthe experimental data was obtained when the maximum\nprincipal axis is in the acplane and tilts θzz= 15.5°\nfrom theaaxis at 20 K, and the second principal axis\nis parallel to the baxis. The temperature dependence of\nθzzis shown in Fig. 3 (b). This change seems to be tiny,\nbut it cannot be neglected for extracting the accurate\nKnight shift of73Ge owing to the small gyromagnetic\nratio. Thus, the Knight shift is calculated after subtract-\ning the temperature-dependent EFG for the73Ge site. It\nis interesting that the maximum principal axes of EFG\nat the Co33and Ge sites in UCoGe and the Ge site in\nURhGe23are roughly parallel to the aaxis. This feature\nmay originate from the crystal structure because that\nof UCoGe and URhGe can be regarded as a deformed\nhexagonal AlB 2-type structure, and the hexagonal caxis,\nwhich is the maximum principal axis, corresponds to the\northorhombic aaxis in UCoGe and URhGe.\nFigure 5 shows the73Ge and59Co Knight shifts of\nthree directions with a fixed field of 3 T at the central\nline (1/2↔−1/2). The Knight shift in UCoGe is highly\nanisotropic with the cdirection being an easy axis, re-\nsulting from the strong Ising anisotropy.\nThe Knight shift along the idirection (i=a,b, andc)\nat themsite (m= 73 and 59) is described as\nmKi=mAiχspin,i+mKorb,i, (1)\nwheremAiis the hyperfine coupling constant, χspin,iis4\n 0 2 4 6 8\n 0  50  100  150  200UCoGe\nµ0H = 3.00 T73Ge Ka73Ge Kb73Ge Kc59Co Ka59Co Kb59Co KcK (%)\nT (K)01020\n0100 20073Ge59Co\nFIG. 5. (Color online)73Ge (closed symbol) and59Co (open\nsymbol) Knight shifts measured at the central line (1 /2↔\n−1/2) with the field of 3 T parallel to the a(squares),b\n(circles), and c(triangles) axes. The inset shows the result of\ncdirection with a different scale.\nthe spin susceptibility, andmKorb,iis the orbital part\nof the Knight shift. The latter part is usually tem-\nperature independent in d-electron systems, because a\ncrystal electric field (CEF) splitting is much larger than\nroom temperature, while it is temperature dependent in\nf-electron systems, where the CEF is not so large as\nin thed-electron systems. We also note that the first\nterm of the right hand side of Eq. (1) is no longer pure\nspin in the f-electron system because of the strong spin-\norbit interaction, and this indicates the susceptibility of\nquasiparticles.34Nevertheless, we use the term “spin sus-\nceptibility” for simplicity.\nFigure 6 shows the relation of Knight shifts between\nthe Ge and Co sites, where temperature is an implicit pa-\nrameter. When the field is parallel to the borcaxis, good\nlinear relations hold between two sites in a wide tem-\nperature range. This result indicates that the dominant\ntemperature dependence of the Knight shift is attributed\nto the single component of the spin susceptibility from\nthe U-5felectrons, which is in good agreement with the\nprevious reports that the magnetism is carried by U.25,35\nThe good linearity also implies that the simple treatment\nof the Knight shifts described as Eq. (1) is valid even in\nthe 5felectron systems since the system has a large spin\nsusceptibility and the temperature dependence of Korb\nis relatively small. The hyperfine coupling constants of\n73Ge are estimated from the slopes of the lines showing\nin Fig. 6 and the hyperfine coupling constants of59Co\nreported previously,33and they are73Ab/similarequal(4.3±0.1)\nT/µBand73Ac/similarequal(4.2±0.1) T/µBalong thebandc\ndirections, respectively. These values are 0.8–0.9 times\nthose at59Co, suggesting that the U-5 felectrons couple\n 0 1 2 3 4 5\n 0  1  2  3  4  5UCoGe\nµ0H = 3.00 TKaKb\nKc/559Co K (%)\n73Ge K (%)FIG. 6. (Color online)59Co Knight shifts versus those of73Ge\nof three directions with the temperature being an implicit\nparameter. The Knight shifts along the caxis are scaled to\n1/5. The solid lines are the best fit of the linear relation of\nthe Knight shifts for H/bardblbandc.\nto the59Co and73Ge nuclei almost equally. In addition,\nif we assume that73Korb,i∼0.2%, which is a typical\nvalue of the orbital shift of Ge and similar p-electron\natoms such as Ga and As36and is an order of magnitude\nsmaller than59Korb,i, then59Korb,band59Korb,care es-\ntimated to be 2.2 and 1.7%, respectively. These values\nare similar to59Korb/similarequal1.6% in a non-magnetic metal\nYCoGe.24\nIn contrast, when the field is parallel to the aaxis, the\ntemperature dependence of the Knight shifts is relatively\nsmall, and the linear relation is not seen between two\nsites. These results suggest that the spin susceptibility\nalong theaaxis is much smaller than that of bandcaxis\nsincemAiis considered to be isotropic in this system.33A\npossible origin of the anomalous Kais the temperature-\ndependent Korbowing to the small CEF splitting, as\nmentioned before. The59Co Knight shift of the adi-\nrection has a broad maximum at T∗∼40 K as59Kof\nthebdirection, while the73Ge Knight shift monotonically\ndecreases with decreasing temperature with a broad kink\naroundT∗. These anomalies may be related to the broad\nmaximum of the Knight shifts and bulk susceptibility17\ninH/bardblb. As discussed later, the anomaly around T∗\nis also recognized in the nuclear spin-lattice relaxation\nrate 1/T1, which suggests that the system becomes more\nitinerant below T∗.\nFigure 7 shows 1 /T1divided by temperature at73Ge\nand59Co sites under zero field and fields of 3 T parallel to\nthea,b, andcaxes. The dashed line shows 1 /T1Tmea-\nsured by the59Co NQR in the previous single-crystalline\nsample.30Contrary to the previous result, the peak of\n1/T1showing the FM transition was not detected down5\n10−210−1100101102103\n 1  10  100UCoGe1/T1T (s−1 K−1)\nT (K)73Ge\nH = 0\nµ0H = 3 T\nH || a\nH || b\nH || c59Co\nH = 0\nµ0H = 3 T\nH || a\nH || b\nH || c\nFIG. 7. (Color online)73Ge (closed symbol) and59Co (open\nsymbol) nuclear spin-lattice relaxation rates 1 /T1divided by\nTmeasured with NQR ( H= 0, diamond) and NMR ( µ0H=\n3 T) with the field parallel to the a(squares),b(circles), and c\n(triangles) axes. The dashed line indicates the previous59Co\nNQR result with the different single-crystalline sample at the\nparamagnetic signal.30\nto 1.5 K in this sample. When the field is parallel to the\naorbaxis, 1/T1Tis close to that at zero field and is\nenhanced at low temperatures, while it is strongly sup-\npressed with H/bardblc. This field-direction dependence of\n1/T1is consistent with the previous results, which indi-\ncate that the FM fluctuations are strongly anisotropic33\nand are suppressed by H/bardblc.7\nAs shown in Fig. 8, the behavior of 1 /T1at two sites is\nessentially similar to each other above 5 K in any direc-\ntion of the field, although the deviation was observed in\nthe lower temperatures, where the FM fluctuations de-\nvelop. As discussed in the previous paper, 1 /T1Tat the\nmsite measured with H/bardblαis expressed in terms of the\nimaginary part of the dynamic susceptibility along the β\nandγdirectionsχ/prime/prime\nβ,γ(q,ω0), perpendicular to α, as33\nm/parenleftbigg1\nT1T/parenrightbigg\nα=mγ2\nnkB\n(γe~)2/summationdisplay\nq/bracketleftbigg\n|mAβ|2χ/prime/prime\nβ(q,ω0)\nω0\n+|mAγ|2χ/prime/prime\nγ(q,ω0)\nω0/bracketrightbigg\n,\nwheremγnandω0are the gyromagnetic ratio at the\nmsite and NMR frequency, respectively. The ratio\nof 1/T1between59Co and73Ge site is expected to be\n(59γn/73γn)2·(59A/73A)2/similarequal59, shown by the dashed\nline in Fig. 8, if the magnetic fluctuations consist of a sin-\ngle component. It is noted that the ratio estimated from\nthe NMR measurements is close to the expected one in\nany field direction, indicating that single magnetic fluc-\ntuations arising from the U-5 felectrons are dominant at\n100101102103(a)\n73Ge NQRTCurieTCurie\nT*73Ge 1/ T1 (s−1)UGe2 (P = 1.2 GPa, Kotegawa et al. )\nURhGe (Kotegawa et al. )\nUCoGe\n 0 100 200 300\n 1  10  100(b)\nUCoGeURhGe\nJ || cTCurie\nTCurieρ (µΩ cm)\nT (K)FIG. 8. (Color online) 1 /T1at59Co divided by that at73Ge\nmeasured by NQR and NMR. The dashed line is the expected\nrelation (see text).\nboth sites. However, the ratio from the NQR measure-\nments is slightly larger than the expected value. This is\nconsidered to be due the difference of the EFG parame-\ntersηandθzzas shown in Table II. The relaxation curve\nin NQR is affected by η,37and the angle difference of the\nmaximum principal axis gives rise to the different weight\nof the FM fluctuations in 1 /T1.\nBelow 5 K, the ratio becomes smaller with decreasing\ntemperature, which is due to the suppression of the in-\ncrease of 1/T1Tat59Co compared with that at73Ge. As\nfor the 1/T1measurements with NQR, H/bardbla, andH/bardblb,\nrf-pulse fields ( H1) were applied along the caxis, which\nis parallel to the direction of the Ising FM fluctuations.\nIn this case, we found that the value of 1 /T1nearTCurie\ndepends on the intensity of H1, the smaller H1gives the\nlarger 1/T1. Therefore, the deviation from the expected\nratio would be due to the difference of the effect of H1\nfor the NMR measurements between two nuclear sites.\nWhen we compare 1 /T1of73Ge with that of59Co, we\nshould be careful for the presence of such differences.\nFigure 9 shows the temperature dependence of 1 /T1\nof73Ge NQR in the present sample of UCoGe, along\nwith that of UGe 2(P= 1.2 GPa)20and URhGe23at\nH= 0. 1/T1of73Ge shows similar temperature de-\npendence in three FM superconductors, and, particu-\nlarly, 1/T1of UCoGe and URhGe becomes constant with\nthe similar values at higher temperatures. This implies\nthat the U-5 fis a localized state at higher tempera-6\n 0 20 40 60 80 100 120\n 0  20  40  60  80UCoGe(59Co 1/ T1) / (73Ge 1/ T1)\nT (K)NQR ( H = 0)\nµ0H = 3 T\nH || a\nH || b\nH || c\nFIG. 9. (Color online) (a) Temperature dependence of 1 /T1of\n73Ge in the present sample of UCoGe (closed circle), UGe 2at\nP= 1.2 GPa (closed square),20and URhGe (open circle)23\nunder zero field. The arrows indicate TCurie for UGe 2and\nURhGe, and T∗for UCoGe. (b) Temperature dependence of\nthe electric resistivity in J/bardblcin the present sample of UCoGe\nand a single-crystal of URhGe.38\ntures since in general 1 /T1is temperature independent\nin compounds with local moments.39However, we note\nthat the development of the FM fluctuations and FM\nordering in UCoGe occur after the gradual decrease of\n1/T1belowT∗, where the magnetic susceptibility χde-\nviates from the Curie-Weiss behavior4and the electri-\ncal resistivity along the c-axis shows metallic behavior\nas shown in Fig. 9 (b). As pointed out in the previous\nstudy, these behaviors are quite different from those of\nURhGe.23In URhGe, the development of the FM fluctu-\nations and FM ordering occurs where the most U-5 fis\nstill in the localized state, which is known from the 1 /T1\nand the electrical-resistivity behaviors shown in Fig. 9.\nThus, these results clearly indicate that the itinerant de-\ngree of the U-5 fis different in UCoGe and URhGe, as\ndiscussed in Ref. 4, although the quasiparticle bands with\nlarge contributions from U-5 fstate were observed from\nARPES.18,19We suggest that the difference of the itin-\nerant degree is one of the key factors to understand the\ndifferences of the superconductivity and ferromagnetism\nof these compounds. Another interesting difference be-\ntween UCoGe and URhGe is the anisotropy of the spin\nsusceptibility at low temperatures. As shown in Fig. 5,\nthe strong Ising-type anisotropy with the caxis being the\neasy axis was observed in UCoGe, but the susceptibili-\nties along the bandcaxes are comparable above TCurie\nin URhGe.23These differences are important to under-\nstand the differences of the metamagnetic behavior and\nfield-enhanced superconductivity observed in URhGe14\nand UCoGe.15,17The anisotropy of the spin susceptibility perpendic-\nular to the Ising axis is considered to be an origin of\nthe different field responses between the aandbaxes\nin the superconductivity and magnetism of UCoGe.15,17\nThe FM phase is suppressed by the field parallel to the\nbaxis as well as the SC phase are reinforced at around\nµ0H∼12 T, while the TCurie and theAcoefficient of the\nresistivity are hardly changed by the field parallel to the\naaxis. Because the spin susceptibility along the baxis is\nmuch larger than that along the aaxis, it is expected that\nthe field along the former axis affects the ferromagnetism\nmore seriously. Such anisotropic field responses are also\nreported in the re-entrant SC region of URhGe: the re-\norientation of the FM moment occurs and re-entrant SC\nphase arises along the baxis, and these anomalies are in-\nsensitive to the field along the aaxis.14,40Thus, UCoGe\nand URhGe have some similarity concerning the field de-\npendencies of the SC and FM phases. In addition, the\nanisotropy of the spin susceptibility should be taken into\naccount when determining the dvector in the spin-triplet\nSC state of these FM superconductors. Spin components\nof the Cooper pairs are active and perpendicular to the\ndvector in spin-triplet superconductors.41Thus, it is a\nfuture task to reveal how the anisotropy of the spin sus-\nceptibility in the normal state affects the structure of the\norder parameter of the SC state in these systems.\nFinally, we comment on the relation between the FM\nand SC phases in UCoGe. The present sample exhibits\nthe FM transition at around TCurie∼1.3 K, which\nis much lower than ∼3 K reported previously.6Since\nTSC= 0.48 K is comparable to the previous results\n[TSC= 0.57 K (Ref. 30)], it is considered that the FM\nphase is more sensitive to the quality of the sample than\nthe SC phase, as pointed out in a previous study.42It was\nreported that the SC phase can exist even without the\nstatic FM ordering.43Since the SC phase without the FM\nphase is also induced by the hydrostatic pressure,10,11the\nFM ordering is not a necessary condition of the SC phase\nin UCoGe but the FM fluctuations are, as discussed in\npreviously.7,9This is also inferred from the pressure de-\npendence of TSC, which exhibits no discontinuity across\nthe FM transition line.10,11In this sense, UCoGe seems\nto be a typical example of the system where the SC phase\nis induced by the FM quantum fluctuations. Thus, the\norder of the FM transition appears to be the second or-\nder, although the first-order-like behavior was observed\nin the previous NQR spectrum.30Further studies are still\nneeded to uncover the nature of the FM quantum transi-\ntion in UCoGe, and the detailed comparison of the NMR\nresults obtained in the different quality of samples will\nbe summarized in a separated paper.\nIV. SUMMARY\nWe performed the73Ge and59Co NMR and NQR\nmeasurements on the paramagnetic state of UCoGe, and\nfound that the electric field gradient of the Ge site in7\nUCoGe is close to that of the isostructural compound\nURhGe. It was revealed that the static and dynamic\nspin susceptibilities at these sites are essentially similar\nto each other, but the spin susceptibility along the aaxis\nis extremely small. This result indicates that the U-5 f\nelectrons are the dominant origin of the ferromagnetism\nin this system and couple to the73Ge and59Co nuclei\nalmost equally. In addition, it was found that the con-\ntribution of Co 3 delectrons probed with59Co NMR and\nNQR is negligibly small. Therefore, we can safely say\nthat the73Ge NMR and NQR give the essentially the\nsame information about 5 felectrons as those of59Co in\nUCoGe.The authors would like to thank S. Kitagawa, Y. Toku-\nnaga, T. Hattori, H. Kotegawa, Y. Maeno, S. Yonezawa,\nN. K. Sato, J.-P. Brison, D. Braithwaite, A. Pourret, C.\nBerthier, A. de Visser, and Y. Kitaoka for valuable dis-\ncussions and contribution to experiments. One of the\nauthors (MM) is a Research Fellow of Japan Society for\nthe Promotion of Science (JSPS). 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Lebedev Physical Institute of the RAS, 119991, Moscow,\nRussia;4National Research Nuclear University MEPhI (Moscow Engine ering Physics Institute),\n31 Kashirskoye Shosse, 115409 Moscow, Russia;5Center for Energy Science and Technology,\nSkolkovo Institute of Science and Technology, Nobel str. 3, 121205 Moscow,\nRussia;6Dukhov Research Institute of Automatics (VNIIA),\n127055 Moscow, Russia;7Skobeltsyn Institute of Nuclear Physics, MSU,\nMoscow, 119991, Russia;8HSE University, 101000 Moscow, Russia.\nIn this work, we report ferromagnetic resonance spectrosco py of EuFe 2As2single crystals. We\nobserve ferromagnetic resonance responses, which are attr ibuted to antiferromagnetic resonances of\nEu sub-lattice with orthorhombic crystal structure and wit h different orientations of twin domains\nrelative to the external field. We confirm validity of the rece ntly-proposed spin Hamiltonian with\nanisotropic Eu-Eu exchange interaction and biquadratic Eu -Fe exchange interaction.\nI. INTRODUCTION\nThe discovery of the iron pnictide high temperature\nsuperconductors has been one of the most striking dis-\ncoveries in recent years in condensed-matter physics. In\nthis classof materials, intriguing properties wererecently\nfound in the subclass of compounds originating from the\nparent EuFe 2As2[1–8]. Superconductivity in EuFe 2As2-\nbased ferromagnetic superconductors emerges by doping\nit with phosphorus [9–14], or substituting europium lay-\ners with rubidium [15–21]. In these materials the Fe\n3d orbitals exhibit a dual itinerant-localized magnetism,\nand, simultaneously, participatein superconductingpair-\ning. When it comes to the interplay between the su-\nperconductivity and the ferromagnetism, the studies are\nmainly focused on their superconducting properties and\non the physical origin behind the emergence of super-\nconductivity. In case of EuFeAs-based ferromagnetic\nsuperconductors the coexistence is considered between\nmagnetic ordering of Eu2+ions with large spin number\nS= 7/2andsuperconductingorderingofFe-3delectrons.\nThe EuFe 2As2compound itself is rich in various mag-\nnetic phenomena. At about 190 K the crystal structure\nof EuFe 2As2undergoes the tetragonal-to-orthorhombic\nphase transition [4, 7, 8] accompaniedby the spin density\nwave antiferromagnetic transition in Fe sub-lattice. The\ndirection of spins in the spin density wave is locked along\nthe longer acrystal axis of the orthorhombic structure\n(see Fig. 1). At about 20 K the Eu subsystem undergoes\nthe A-type antiferromagnetic transition [5, 7, 8]. The\northorhombic crystal structure is naturally subjected to\ntwinning, but also to magnetostriction due to the move-\nment of twin boundaries in response to changes of the\nmagnetic field and magnetization of Eu sub-lattice [6–8].\nThe latter indicates Eu-Fe exchange interaction. This\nrelocation of twin boundaries is one of the main ob-\nstructions for studies of magnetic ordering in EuFe 2As2with magnetization measurements: upon magnetization\nthe variation in fraction of differently oriented twin do-\nmains additionally impacts the magnetization. Only re-\ncently an adequate microscopicform of magnetic interac-\ntions in EuFe 2As2was established [7, 8]. It includes the\nanisotropicEu-Euexchangeinteraction,bi-quadraticEu-\nFe exchange interaction, and implies the spin-flip transi-\ntion in A-type antiferromagnetic Eu sub-lattice when the\nmagnetic field is applied along the acrystal axis. Meta-\nmagnetic transitions in EuFe 2As2[3, 7, 8] are developed\ndue to magnetization of twin domains with the π/2 dif-\nference in their mutual orientation.\nFIG. 1. Magnetic crystal structure of EuFe 2As2(made us-\ning VESTA software). The crystal structure of EuFe 2As2\nis orthorhombic with the space group Fmmm [4]. Fe spin\nsub-lattice is in the spin density wave antiferromagnetic s tate\naligned with the aaxis. Eu spin sub-lattice is in the A-type\nantiferromagnetic state.\nIn this work, we consider magnetization dynamics in\nsingle crystal EuFe 2As2. In general, ferromagnetic res-\nonance (FMR) measurements are an ultimate tool for\ntesting interlayer exchange interactions in various anti-\nferromagnets [22–24]. In addition, FMR studies are free2\nfrom the twinning problem, since the ratio of differently\noriented twin domains impact the intensity but not the\nposition of resonance lines. By observing and analysing\nantiferromagnetic resonance spectral lines we have found\nthat the FMR spectrum confirms the validity of the 3D-\nexpanded version of the spin Hamiltonian proposed in\nRef. [8].\nII. EXPERIMENTAL DETAILS\nEuFe2As2single crystalswere grown using the self-flux\nmethod, byanalogywithpreviousworks[25,26]. Theini-\ntial high purity components of phase homogeneous EuAs\n(99.95% Eu + 99.9999% As) obtained by high-pressure\ntechnique, and preliminary synthesized precursor Fe 2As\n(99.98% Fe+99.9999% As) were mixed with a 1:6 molar\nratio. The mixture in an alumina crucible was sealed\nin a niobium tube under residual argon pressure. The\nsealed container was loaded into a tube furnace with an\nargon atmosphere to prevent niobium oxidation. Then,\nthe furnace was heated up to 1250◦C, held at this tem-\nperature for 24 h to homogenize melting, cooled down\nto 900◦C at a rate of 2◦C/h, and then cooled down\nto room temperature inside the furnace. Finally, crys-\ntals were collected from the crucible in an argon glove\nbox. Visually, as-grown bulk crystals demonstrate well-\ndefined layered structure and their pliability for cleavage\nand exfoliation along the layering direction only. XRD\nstudies confirm alignment of abcrystal planes within the\nlayers, and orientation of c crystal axis across the lay-\ners. Samples for ferromagnetic resonance spectroscopy\nwere obtained by cleavage of as-synthesized bulk crys-\ntals. Cleaved EuFe 2As2samples were of a few mm in size\nalongabcrystal planes, and about 50 µm thick along the\nccrystal axis, which ensured the “thin film” geometry\nwith defined crystal orientation.\nFerromagnetic resonance spectroscopy was performed\nusing the VNA-FMR flip-chip approach[27, 28]. Cleaved\nEuFe2As2sample was glued on top of the transmis-\nsion line of coplanar waveguide. The waveguide with\nimpedance50Ohmandthewidthofthetransmissionline\n0.5 mm is patterned on a Arlon AD1000 copper board\nand is equipped with SMP rf connectors. The board\nwith the sample is installed in a brass sample holder. A\nthermometer and a heater are attached directly to the\nholder for precise temperature control. The holder is\nplaced in a home-made superconducting solenoid inside\na closed-cycle cryostat (Oxford Instruments Triton, base\ntemperature 1.2 K). Magnetic field is applied in-plane\nalong the direction of the waveguide. The response of\nexperimental samples is studied by analysing the trans-\nmitted microwave signal S21(f,H) with the VNA Rohde\n& Schwarz ZVB20. The setup allows to perform spec-\ntroscopy in the temperature range from 2 K up to 30 K,\nin the field range up to 1 T, and in the frequency range\nup to 26.5 GHz.III. RESULTS AND DISCUSSIONS\nFigure2shows FMR absorption spectra of EuFe 2As2\nsample measuredat magnetic field applied in-plane along\nablayersat T= 5 K and 20 K. At 5 K (Fig. 2a) the spec-\ntrum contains three absorption features (encircled with\nreddashedlines): (i)aresonancelineat0 < µ0H/lessorsimilar0.5T\nand 10< f <23 GHz with a negative-slope linear field\ndependence, referred to as line I; (ii) a weaker resonance\nline atµ0H/greaterorsimilar0.7 T and 18 < f <22 GHz, which also\nhas approximately linear field dependence with a nega-\ntive slope, referred to as line II; and (iii) an absorption\n“patch” at 0 .55/lessorsimilarµ0H/lessorsimilar0.7 T and f >20 GHz, re-\nferred to as the absorption feature III. A spectral line\nwith the negative field-slope is commonly attributed to\nan antiferromagnetic resonance response. Cross-sections\nS21(H)ofthespectrumat5Kandatselectedfrequencies\nare shown in Fig. 2b. TheS21(H) curves clearly indicate\nabsorption peaks of resonance lines I and II. Derivation\nof FMR frequency dependence on applied magnetic field\nfr(H) was performed by fitting S21(H) curves with the\ncomplex resonant susceptibility [28].\nIn contrast, at temperatures above 19 K the spectrum\ncontains a single resonance line with approximately lin-\near dependence of the resonance frequency on the mag-\nnetic field, corresponding to paramagnetic resonance of\nEu spins. Figure 2c shows the representative example of\nthe spectrum at T= 20 K. The field slope of the para-\nmagnetic resonance at 20 K fr/µ0H≈36.6 GHz/T is\nslightly higher than the gyromagnetic ratio of free elec-\ntrons 28 GHz/T. In addition, the resonance line shows\nsome deviation from the linear behaviour. These effects\nare attributed to the residual susceptibility of the para-\nmagnetic phase in vicinity to the Curie temperature [29]\nand should be understood as follows. The magnetiza-\ntion of the paramagnetic phase is proportional to the\napplied field M=χ(T,H)H. In the thin-film geometry\ntheresonancefrequencyofthe paramagneticphaseisstill\nprovided by the Kittel formula 2 πf=γ/radicalbig\nH(H+M) =\nγH/radicalbig\n1+χ(T,H). At temperatures slightly above the\nCurie temperature, magnetization χ(T,H)His compa-\nrable to the saturation magnetization at high fields and\nshow nonlinear dependence on H. These factors result\nin larger slope of the linear fit and in some nonlinear-\nity of the resonance line, which are observed in Fig. 2c.\nAt highertemperatures χ(T,H) is graduallyreduced and\nthe resonance line approachesthe conventional paramag-\nnetic one, which is observed at 25 K and 30 K (see sup-\nplementary). Figure 2d shows the cross-section of the\nspectrum S21(H) atf= 17.5 GHz, which contains a\nsingle resonance peak, and its fit with the complex sus-\nceptibility.\nThe spin configuration of Eu and Fe subsystems in\nEuFe2As2and the twinning problem were studied ex-\ntensively in a number of previous works with neutron\nscattering and XMCD measurements. As a consensus\n[8], it is shown that at low temperature Fe sub-lattice is\nin the spin density wave antiferromagnetic state aligned3\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s73/s73/s73\n/s73/s73\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/s49/s46/s48/s49/s46/s48/s73\n/s49/s46/s48/s48/s53\n/s48/s46/s57/s56/s53\n/s40/s97/s41/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\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/s49/s46/s48/s49/s46/s48 /s49/s46/s48/s48/s53\n/s48/s46/s57/s53/s53\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s57/s57/s50/s48/s46/s57/s57/s52/s48/s46/s57/s57/s54/s48/s46/s57/s57/s56/s49/s46/s48/s48/s48\n/s40/s98/s41/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110/s32 /s83\n/s50/s49/s44/s32/s97/s98/s115\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84/s32/s102 /s32/s61/s32/s49/s51/s46/s53/s32/s71/s72/s122\n/s32/s108/s105/s110/s101/s32/s102/s105/s116/s32/s97/s116/s32 /s102/s32/s61/s32/s49/s51/s46/s53/s32/s71/s72/s122\n/s32/s102/s32/s61/s32/s49/s57/s46/s48/s32/s71/s72/s122\n/s32/s108/s105/s110/s101/s32/s102/s105/s116/s32/s97/s116/s32 /s102/s32/s61/s32/s49/s57/s32/s71/s72/s122/s73 /s73 \n/s73 /s73 /s73 \n/s73/s73\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s57/s54/s48/s46/s57/s55/s48/s46/s57/s56/s48/s46/s57/s57/s49/s46/s48/s48\n/s40/s100/s41/s32/s102/s32/s61/s32/s49/s55/s46/s53/s32/s71/s72/s122\n/s32/s108/s105/s110/s101/s32/s102/s105/s116/s32/s97/s116/s32 /s102/s32/s61/s32/s49/s51/s46/s53/s32/s71/s72/s122/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110/s32 /s83\n/s50/s49/s44/s32/s97/s98/s115\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84\nFIG. 2. a,c) FMR absorption spectra S21(f,H) of EuFe 2As2measured at T= 5 K (a) and 20 K (c). Magnetic field is applied\nin-plane along abcrystal planes. Circular red dashed lines in (a) highlight t hree absorption features. Red line in (c) corresponds\nto the linear fit of the spectral line with the slope 36.6 GHz/T . (b,d) Cross-sections S21(H) of spectra at T= 5 K (b) and 20 K\n(d) at specified frequencies. Solid lines in (b,d) show the fit of resonance lines, while arrows indicate corresponding ab sorption\nfeatures.\nwith the longer side of orthorhombic lattice, while Eu\nsub-lattice has the A-type antiferromagnetic order (see\nFig.1). Importantly, Eu-Fe exchange interaction results\nin anisotropic Eu-Eu exchange interaction and in devel-\nopment of the easy axis along the direction of the spin\ndensity wave (a-direction) due to bi-quadratic Eu-Fe ex-\nchange. The total free energy of the spin configuration\nin a unit cell of Eu layers is [8]\nF= 2(J+W)e1xe2x+2(J−W)e1ye2y+2Je1ze2z+\n−8Ka2/summationdisplay\ni=1e2\nix+Ku2/summationdisplay\ni=1e2\niz−Ms2/summationdisplay\ni=1/vector ei/vectorH,\n(1)\nwhere (eix,eiy,eiz) is the unit vector of ferromagnetic\nmoment of the Eu atomic layer in spherical coordinates\n/vector e= sinθcosφˆx+ sinθsinφˆy+ cosθˆz, [ˆx,ˆy,ˆz] axes are\naligned with [ a,b,c] crystal axes in Fig. 1, respectively,\nthe first three terms are the exchange interaction terms,\nwhich areanisotropicin xandydirections by the param-\neterW,theforthtermisthebi-quadraticEu-Feexchangeinteraction in a form of the x-axis uniaxial anisotropy,\nthe fifth term is the z-axis uniaxial anisotropy, and the\nlast term is the Zeeman energy with the external field\n/vectorH, which is applied in abplane at the angle φHwith re-\nspect to the a-axis. In comparison to Ref. [8], two terms\n[2Je1ze2z] and [Ku/summationtext2\ni=1e2\niz] are added to complete the\n3D representation of the free energy, while the Eu-Fe\nexchange interaction is redefined in the x-axis uniaxial\nform.\nFollowing Ref. [8], exchange and anisotropy parame-\nters of the free energy can be derived from saturation\nfields of the canted spin state and of the spin-flip transi-\ntion as follows. When magnetic field is applied along the\nbcrystal axis, magnetization of Eu occurs via spin cant-\ning (i.e., via the spin-flop phase) and the saturation field\nof the spin-flop phase is Hsat\nb= (4J+16Ka)/Ms. When\nmagnetic field is applied along the acrystal axis, mag-\nnetization saturation occurs via abrupt spin-flip transi-\ntion and the saturation field (i.e., the spin-flip field) is\nHsat\na= 2(J+W)/Ms. Notice that counter-intuitively4\nHsat\naandHsat\nbdo not match each other even in the\nisotropic case of W= 0,Ka= 0. This is the consequence\nof the spin-flip as the dominating magnetization process\nfor the corresponding magnetic orientation [8]. The con-\ndition for the spin-flip transition is J/(8Ka+W)<1.\nWhen magnetic field is applied at 45◦with respect to a\norbdirection, the saturation field is Hsat\n45= 4J/Ms. By\nexpanding the treatment to the 3D case, the saturation\nfield of the canted spin state for field orientation along\nthecaxis isHsat\nc= (4J+2W+16Ka+2Ku)/Ms.\nThe dependence of orientations of Eu magnetic mo-\nments on the magnetic field can be derived numeri-\ncally by minimising the energy in Eq. 1with predefined\nanisotropy and exchange parameters. By knowing ori-\nentations of Eu magnetic moments, ferromagnetic reso-\nnance of Eu can be derived using the Suhl-Smit-Beljers\napproach[30, 31] extended for the case of magnetization\ndynamics in coupled magnetic multilayers [24, 32–34].\nWith this approach the following set of equations of mo-\ntion for magnetization vector in each Eu layer defines the\ncollective dispersion of the spin system with orientation\nalongabplanes (θi=π/2):\niωMs\nγ\nδθ1\nδθ2\nδφ1\nδφ2\n=\n0 0 Fφ1φ1Fφ1φ2\n0 0 Fφ1φ2Fφ2φ2\n−Fθ1θ1−Fθ1θ20 0\n−Fθ1θ2−Fθ2θ20 0\n\nδθ1\nδθ2\nδφ1\nδφ2\n,\n(2)\nwhereδθiandδφiare components of small deviations of\nmagnetization vector in spherical coordinates, Fθiθjand\nFφiφjare corresponding second-order partial derivatives\nof the free energy (Eq. 1),ωis the eigen-frequency of\nmagnetizationprecession,and γ= 28GHz/Tisthegyro-\nmagnetic ratio. Diagonal terms Fθiφj= 0 in Eq. 2due to\nthe in-plane configuration of magnetization ( θi=π/2).\nThe expression/summationtextcos(φi−φH)δφicorresponds to the\ndynamic susceptibility of the resonance.\nFigure3collects experimental and theoretical depen-\ndencies of FMR frequencies on magnetic field fr(H).\nIn calculations we consider the in-plane magnetic field\naligned with abcrystal planes, with the angle φHrela-\ntivetothe acrystalaxis. Inaccordancewiththetwinning\ndomain concept [6–8], the sample also contains domains\nwhere the orientation of the magnetic field is π/2−φH\nrelative to the acrystal axis.\nIn general, our calculations showed that the value φH\nis close to 0, which indicates that the sample consists of\ndomains whose a-axis is aligned with the magnetic field\nand domains whose a-axis is perpendicular to the mag-\nnetic field (A and B-domains, correspondingly, see illus-\ntrations in Fig. 3). For A-domains at H < Hsat\nathe spec-\ntrum consistsoftwoantiferromagneticspectrallineswith\nlinear-in-field increasing (decreasing) resonant frequency,\nattributed to individual resonances of oppositely-aligned\nEuspinsub-lattices. At H > Hsat\nathespin fliptransition\ninA-domainsoccursandthespectrumconsistsoftwocol-\nlective modes: the higher-frequency acoustic mode and\nthe lower-frequency optical mode, both with linear field\ndependence. Antiferromagnetic interaction between lay-/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/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/s32\n/s72 /s61/s48 /s32\n/s72 /s61/s48 /s44/s32/s111/s112/s116/s105/s99/s32/s109/s111/s100/s101\n/s32\n/s72 /s61/s57/s48 /s97/s99/s111/s117/s115/s116/s105/s99/s32/s109/s111/s100/s101/s59/s32 /s32\n/s72 /s61/s57/s48 /s111/s112/s116/s105/s99/s32/s109/s111/s100/s101\n/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116/s97/s108/s32 /s102\n/s114/s40/s72 /s41/s32/s108/s105/s110/s101/s115/s32/s102/s114/s111/s109/s32/s116/s104/s101/s32/s115/s112/s101/s99/s116/s114/s117/s109/s32/s97/s116/s32/s53/s32/s75\n/s72\n/s97/s115/s97/s116/s72\n/s98/s115/s97/s116\n/s73/s73/s73/s73/s73/s73\nFIG. 3. The plot shows experimental and theoretical de-\npendencies of the ferromagnetic resonance frequency on the\nmagnetic field fr(H). Black lines show theoretical data for\nthe A-domain with φH= 0. Red lines show theoretical data\nfor the B-domain with φH=π/2. Solid lines show either indi-\nvidual resonances of Eu sub-lattices, as in case of φH= 0 and\nµ0H <0.5, or the collective acoustic response. Dashed lines\nshow the collective optical response. Arrows indicate tran si-\ntion fields Hsat\naandHsat\nb. Black squared dots show experi-\nmentalfr(H) lines derived from the spectrum at 5 K. Error\nbars indicate the line-width of experimental resonance lin es\n∆H≈0.17 T and ∆ f≈4.5 GHz for line I, and ∆ H≈0.13 T\nand ∆f≈4 GHz for line II. Blue short-dashed line indicates\nthe accessible range of the experimental setup. Pictograms\nillustrate spin configurations in twinned domains at differ-\nent fields. A-domains (shown with blue) correspond to do-\nmains with the spin density wave axis ( a-axis) aligned with\nthe magnetic field. B-domains (shown with red) correspond\nto domains with the spin density wave axis ( a-axis) aligned\nperpendicular to the magnetic field.\ners, which are magnetized to saturation, results in higher\nresonance frequency for the acoustic mode in compari-\nson to the optical mode [34, 35]. For B-domains with\ntheb-axis aligned with the magnetic field the spectrum\nalso consists of two lines. At H < Hsat\nbthe spectrum of\nB-domains contains collective modes: higher-frequency\nacoustic mode with positive frequency dependence on\nmagnetic field, and the lower-frequency optical mode\nwith negative frequency dependence on magnetic field.\nAtH > Hsat\nbthe spin-flip transition (saturation) occurs\nin B-domains manifested by a kink on both curves and\nboth collective modes show positive dependence of fre-\nquency on magnetic field.\nA rough numerical optimisation of magnetic param-\neters in Eqs. 1and2yields the following set of param-\neters, consistent with Ref. [8]: 4 J/Ms≈0.8−0.9 T;5\n2W/Ms≈0.1−0.2 T;Hsat\na≈0.45−0.55 T;Hsat\nb=\n(4J+16Ka)/Ms≈1.15−1.25 T; 2Ku/Ms≈0.2−0.3 T;\n|φH|<5◦. The large width of resonance lines and lim-\nited experiential range did not allow to perform more ac-\ncurateoptimisationofparameters. Solidanddashedlines\nin Fig.3showfr(H) obtained using Eqs. 1and2and the\nfollowing set of parameters: 4 J/Ms≈0.8 T; 2W/Ms≈\n0.1 T;Hsat\na≈0.5 T;Hsat\nb= (4J+16Ka)/Ms≈1.2 T;\n2Ku/Ms≈0.25 T;|φH|<5◦. According to Fig. 3, the\nspectral line I corresponds to the resonance of Eu spins\naligned against the external field in the domain having\nthe angle φH= 0 with the external field. The spectral\nline II corresponds to the optical antiferromagnetic re-\nsponse in the domain having the angle φH=π/2 with\nthe external field. Its optical origin explains the low in-\ntensity in comparison with line I.\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s40/s97/s41/s73/s73/s73/s73/s73\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/s49/s46/s48/s49/s46/s48\n/s73/s49/s46/s48/s48/s53\n/s48/s46/s57/s55/s53\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s57/s56/s48/s46/s57/s57/s49/s46/s48/s48/s49/s46/s48/s49\n/s40/s98/s41/s32/s102/s32/s61/s32/s50/s53/s32/s71/s72/s122\n/s32/s108/s105/s110/s101/s32/s102/s105/s116/s32/s97/s116/s32 /s102/s32/s61/s32/s50/s53/s32/s71/s72/s122/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110/s32 /s83\n/s50/s49/s44/s32/s97/s98/s115\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84/s73/s73/s73\nFIG. 4. a) FMR absorption spectra S21(f,H) of EuFe 2As2\nmeasured at T= 13 K. Magnetic field is applied in-plane\nalong ab crystal planes. Red dots show experimental res-\nonance curves fr(H) obtained by fitting spectral lines. b)\nCross-section S21(H) of the spectrum at T= 13 K at\nf= 25 GHz. Solid lines in (b) show the fit of the resonant\nfeature.\nThe spectral feature III in Figs. 2a and3may be\nattributed to the optical antiferromagnetic response inthe same B-domain. In this case the change in the line\nintensity is the signature of the twin domain wall relo-\ncation when the fraction of these domains reduces with\nthe magnetic field. Alternatively, the spectral feature III\nmaybeatraceoftheacousticmodeoftheA-domainwith\nφH= 0 andH > Hsat\na. The origin of the spectral feature\nIII can be established by studying temperature depen-\ndenceofthespectrum. Uponincreasingthemeasurement\ntemperature all spectral lines shift to lower frequencies,\nwhile the transition fields decrease (see supplementary).\nFigure4a shows FMR absorption spectrum of EuFe 2As2\nsample measuredat magnetic field applied in-plane along\nablayers at T= 13 K. The spectrum contains the same\nthreeabsorptionfeatures(experimental fr(H) dependen-\ncies are shown with red dots). The resonance line I at\n0.2< µ0H/lessorsimilar0.4 T and 10 < f <15 GHz corresponds to\nresonanceofEu spins alignedagainst the externalfield in\nthe A-domain having the angle φH= 0 with the external\nfield (see schematic images in Fig. 3). A weak resonance\nline II at 0 .7/lessorsimilarµ0H/lessorsimilar0.8 T and 12 < f <18 GHz\ncorresponds to the optical antiferromagnetic response\nin the B-domain having the angle φH=π/2 with the\nexternal field. The spectral feature III at T= 5 K\n(Fig.2a) is transformed at T= 13 K into a clearly dis-\ntinguishable resonance line at 0 .45/lessorsimilarµ0H/lessorsimilar0.7 T and\n18< f <26 GHz with positive-slope linear dependence\nfr(H), thus, manifesting the acousticmode ofA-domains\nwithφH= 0 andH > Hsat\na. Lines I and III indicate that\nthe spin-flip field at T= 13 K is reduced to Hsat\na≈0.4 T\nas compared to Hsat\na≈0.5 T at 5 K. Figure 4b shows\nthe cross-section of the spectrum S21(H) atf= 25 GHz.\nThe cross-section indicates a drop of the transmission\natµ0H≈0.4 T. This drop is attributed to the spin-\nflip transition of A-domains but is not related to a spin\nresonance process. At µ0H >0.4 T magnetization of\nA-domains is changed step-wise, which result in corre-\nsponding changes of the impedance of the transmission\nline and, consequently, in its transmission characteristics\nregardless a resonance process. In addition, the curve\nS21(H) shows a resonance peak III at µ0H≈0.58 T.\nThe fit of resonance peaks with the complex susceptibil-\nity allowed to derive the resonance line fr(H), which is\nshown in Fig. 4a with red dots.\nThe overall correspondence between experimental and\ntheoretical lines in Fig. 3and established nature of all\nlines in Figs. 3and4confirm the validity of the free\nenergy relation in Eq. 1with the anisotropic Eu-Eu ex-\nchange interaction and bi-quadratic Eu-Fe exchange in-\nteraction for EuFe 2As2compound together with the do-\nmain twinning concept of its orthorhombic crystal struc-\nture.\nOncethevalidityoftheHamiltonian 1andtheoriginof\nall three lines are established it is instructive to consider\nresonance properties of EuFe 2As2in more details. First,\nit can be noticed that at zero field in Fig. 3resonance\nlines are split by ∆ fr≈0.5 GHz. In general, for two-\nsublatticeantiferromagnetsthezero-filedsplitoccursdue\nto the two-axes anisotropy. From Eq. 2atH= 0 it6\n/s48 /s49 /s50 /s51 /s52/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/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/s32\n/s72 /s61/s48\n/s32\n/s72 /s61/s48 /s111/s112/s116/s105/s99/s97/s108\n/s32\n/s72 /s61/s48 /s44/s32/s69/s113/s46/s32/s51\n/s32\n/s72 /s61/s57/s48 /s97/s99/s111/s117/s115/s116/s105/s99\n/s32\n/s72 /s61/s57/s48 /s111/s112/s116/s105/s99/s97/s108\n/s32\n/s72 /s61/s48 /s69/s113/s46/s32/s52\nFIG. 5. Theoretical dependencies of the ferromagnetic reso -\nnance frequency on the magnetic field fr(H) obtained numer-\nically with Eqs. 1,2 using the same parameters as in Fig. 3\n(dots), and analytically with Eqs. 4,5 (solid lines).\ncan be derived that the split in FMR frequency is set\nby the difference between two eigen frequencies given by\nexpressions\n/parenleftbigg2πfr\nµ0γ/parenrightbigg2\n= (2W+16Ka+2Ku)(4J+16Ka),\n/parenleftbigg2πfr\nµ0γ/parenrightbigg2\n= (4J+2W+16Ka+2Ku)(4W+16Ka).\n(3)\nAs follows from Eq. 3, in the case of EuFe 2As2the width\nof the split ∆ fris also affected by the anisotropy in ex-\nchange interaction. In the isotropic case, W= 0 and\nKu= 0, both expressions relax to the known textbook\nsplit-less expression [23, 35]. However, it should be no-\nticed that ∆ fris by far smaller that the linewidth of\nresonance lines, and, thus, can not be verified directly\nfor EuFe 2As2.\nNext, we consider the resonance of magnetically-\nsaturatedA- andB-domains. In termsofEq. 1bysetting\n/vector e1=/vector e2the Suhl-Smit-Beljers approachyields for a singlemagnetic layer\n/parenleftbigg2πfr\nµ0γ/parenrightbigg2\n= (H+16Ka−4W)(H+16Ka−2W+2Ku)\n(4)\nfor the higher-frequency acoustic mode of the A-domain\natH > Hsat\na, and\n/parenleftbigg2πfr\nµ0γ/parenrightbigg2\n= (H−16Ka+4W)(H+2W+2Ku) (5)\nfor the higher-frequency acoustic mode of the B-domain\natH > Hsat\nb. Figure 5shows theoretical dependencies\nof the ferromagnetic resonance frequency on the mag-\nnetic field fr(H) obtained numerically with Eqs. 1,2us-\ning the same parameters as in Fig. 3, and analytically\nwith Eqs. 4,5. The consistency between corresponding\nresonance lines confirms the validity of numerical stud-\nies. Interestingly, the exchange anisotropy parameter W\nenters both expressions, which can be used additionally\nin FMR determination of magnetic properties of complex\nantiferromagnets.\nIV. CONCLUSION\nIn conclusion, in this work we report ferromag-\nnetic resonance spectroscopy of EuFe 2As2single crys-\ntals. The spectrum reveals several resonant features\nattributed to antiferromagnetic resonances of Eu sub-\nlattice. By employing the recently proposed spin Hamil-\ntonian with anisotropic Eu-Eu exhange interaction and\nbi-quadraticEu-Feexchangeinteraction,thespectralfea-\ntures have been identified and attributed to antiferro-\nmagnetic and collective resonances of Eu layers in or-\nthorhombic twinned crystal with different orientation of\ntwin domains with respect to the external field. The ob-\ntained magnetic parameters are quantitatively consistent\nwith thosereportedpreviously,thus, confirmingthe com-\nplex biquadratic Hamiltonian for Eu spins in EuFe 2As2\nproposed earlier.\nV. ACKNOWLEDGMENTS\nThis workwassupported bythe RussianScience Foun-\ndation and by the Ministry of Science and Higher Edu-\ncation of the Russian Federation. 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Melkov, eds., Magnetization\nOscillations and Waves (CRC Press, 1996)."    },    {        "title": "1007.1041v1.Three_attractively_interacting_fermions_in_a_harmonic_trap__Exact_solution__ferromagnetism__and_high_temperature_thermodynamics.pdf",        "content": "arXiv:1007.1041v1  [cond-mat.quant-gas]  7 Jul 2010Three attractively interacting fermions in a harmonic trap :\nExact solution, ferromagnetism, and high-temperature the rmodynamics\nXia-Ji Liu1,∗Hui Hu1,†and Peter D. Drummond1‡\n1ARC Centre of Excellence for Quantum-Atom Optics,\nCentre for Atom Optics and Ultrafast Spectroscopy,\nSwinburne University of Technology, Melbourne 3122, Austr alia\n(Dated: November 3, 2018)\nThree fermions with strongly repulsive interactions in a sp herical harmonic trap, constitute the\nsimplest nontrivial system that can exhibit the onset of iti nerant ferromagnetism. Here, we present\nexact solutions for three trapped, attractively interacti ng fermions near a Feshbach resonance. We\nanalyze energy levels on the upper branch of the resonance wh ere the atomic interaction is effec-\ntively repulsive. When the s-wave scattering length ais sufficiently positive, three fully polarized\nfermions are energetically stable against a single spin-fli p, indicating the possibility of itinerant\nferromagnetism, as inferred in the recent experiment. We al so investigate the high-temperature\nthermodynamics of a strongly repulsive or attractive Fermi gas using a quantum virial expansion.\nThe second and third virial coefficients are calculated. The r esulting equations of state can be tested\nin future quantitative experimental measurements at high t emperatures and can provide a useful\nbenchmark for quantum Monte Carlo simulations.\nI. INTRODUCTION\nFew-particle systems have become increasingly crucial\nto the physics of strongly interacting ultracold quantum\ngases [1–3]. Because of large interaction parameters, con-\nventional perturbation theory approaches to quantum\ngases such as mean-field theory simply break down [2–4].\nA small ensemble of a few fermions and/or bosons, which\nis either exactly solvable or numerically tractable, is mor e\namenable to nonperturbative quantal calculations. Al-\nthough challenging experimentally, such ensembles bene-\nfit from the same unprecedented controllability and tun-\nability as in a mesoscopic system containing a hundred\nthousand particles. The atomic species, the quantum\nstatistics, the s-wave and higher partial wave interac-\ntions [5], and the external trapping environment can all\nbe controlled experimentally. The study of few-particle\nsystems can therefore give valuable insights into the more\ncomplicated mesoscopic many-body physics of a strongly\ninteracting quantum gas. In addition to qualitative in-\nsights, these solutions have already proved invaluable in\ndeveloping high-temperature quantum virial or cluster\nexpansions for larger systems [6], which have been re-\ncently verified experimentally [7].\nThe purpose of this paper is to add a further milestone\nin this direction. By exactly solving the eigenfunctions\nof three attractively interacting fermions in a spherical\nharmonic trap, we aim to give a few-body perspective of\nitinerant ferromagnetism in an effectively repulsive Fermi\ngas, which was observed as a transient phenomenon in a\nrecent measurement at MIT [8]. This is possible because\nthe quantum three-body problem with s-wave interac-\n∗Electronic address: xiajiliu@swin.edu.au\n†Electronic address: hhu@swin.edu.au\n‡Electronic address: pdrummond@swin.edu.autions is exactly soluble in three dimensions. It is inter-\nesting to recall that the corresponding classical three-\nbody problem is notoriously insoluble. The reason for\nthis unexpectedly docile quantum behaviour is that the\ns-wave interaction Hamiltonian applicable to ultra-cold\nBose and Fermi gases is essentially just a boundary condi-\ntion on an otherwise non-interacting quantum gas. Thus,\nwe have an unusual situation where quantum mechanics\nactually simplifies an intractable classical problem.\nFor Bose-Einstein condensates (BEC) in the strongly\ninteracting regime, three trapped bosonic atoms with\nlarges-wave scattering length were already investigated\ntheoretically as a minimum prototype [9] of this few-\nbody physics. To understand the fascinating crossover\nfrom a BEC to a Bardeen-Cooper-Schrieffer (BCS) super-\nfluid, two spin-up and two spin-down fermions in a trap\nwere also simulated numerically, constituting the sim-\nplest model of the BEC-BCS crossover problem [10, 11].\nMoreover, knowledge of few-particle processes such as\nthree-body recombination is primarily responsible for\ncontrolling the loss rate or lifetime of ultracold atomic\ngases, which, in many cases, imposes severe limitations\non experiments. Important examples in this context in-\nclude the confirmation of stability of dimers in the BEC-\nBCS crossover [12] and the discovery of the celebrated\nEfimov state (i.e., a bound state of three resonantly inter-\nacting bosons) as well as the related universal four-body\nbound state [13].\nWhether an itinerant Fermi gas with repulsive inter-\nactions exhibits ferromagnetism is a long-standing prob-\nlem in condensed matter physics [14]. It has recently at-\ntracted increasing attention in the cold-atom community\n[15–24]. The answer depends on a competition between\nthe repulsive interaction energy and the cost of kinetic\nenergy arising from Pauli exclusion. A strong repulsive\ninteraction can induce polarization or ferromagnetism,\nsince fermions with the same spin orientation are pro-\ntected from local interactions by the exclusion principle.2\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s53/s48/s53/s49/s48/s49/s53/s32\n/s32/s100 /s47/s97/s69\n/s114/s101/s108/s47/s40 /s126 /s41/s32\n/s50\n/s49\nFigure 1: (Color online) Energy spectrum of the relative mo-\ntion of a trapped two-fermion system near a Feshbach res-\nonance (i.e, d/a= 0, wheredis the characteristic harmonic\noscillator length). For a positive scattering length a >0in the\nright part of the figure, the ground state is a molecule with si ze\na, whose energy diverges as Erel≃ −/planckover2pi12/(ma2). The excited\nstates or the upper branch of the resonance may be viewed\nas the Hilbert space of a “repulsive” Fermi gas with the same\nscattering length a. In this two-body picture, the level from\nthe point 2 to 3 is the ground state energy level of the repul-\nsive two-fermion sub-space, whose energy initially increa ses\nlinearly with increasing afrom1.5/planckover2pi1ωat the point 2 and fi-\nnally saturates towards 2.5/planckover2pi1ωat the resonance point 3. For\ncomparison, we illustrate as well the ground state energy le vel\nin the case of a negative scattering length and show how the\nenergy increases with increased scattering length from poi nt\n1 to 2.\nThis, however, increases the Fermi energy, as all fermions\nmust now occupy the same band. The difficulty in finding\nthe transition point is that quantum correlations change\nthe interaction energy in a way that is difficult to calcu-\nlate in general.\nAs early as the 1930’s, Stoner showed with a sim-\nple model using mean-field theory that ferromagnetism\nin a homogeneous Fermi gas will always take place\n[14]. This model, however, gives the unphysical result\nthat the interaction energy within the mean-field ap-\nproximation scales linearly with the s-wave scattering\nlengthaand therefore could be infinitely large. The\npredicted critical interaction strength at zero tempera-\nture,(kFa)c=π/2, wherekFis the Fermi wave-vector\nis also too large in the mean-field picture. An im-\nproved prediction from second-order perturbation the-\nory,(kFa)c≃1.054, suffers from similar doubts about\nits validity [15, 18]. Most recently, three independent\nab-initio quantum Monte Carlo simulations conclusively\nreported a zero-temperature ferromagnetic phase transi-\ntion at(kFa)c≃0.8−0.9[19, 22, 23].\nSeveral important issues are still open, including the\nnature of transition at finite temperatures. The unitarity\nlimited interaction energy at infinitely large scattering\nlength (a→ ∞) is also to be determined.The exciting experiment at MIT of6Li atoms realized\nin some sense the textbook Stoner model [8]. A crucial\naspect in this realization is that the interatomic interac-\ntions are very different from the conventional model of\nhard-sphere interactions [15, 22, 23]. In the experiment,\nthe atoms are on the upper branch of a Feshbach reso-\nnance with a positive scattering length a>0and a neg-\nligible effective interaction range. The properties of the\natoms are therefore universal, independent of the details\nof the interactions [25, 26]. This universality, however,\ncomes at a price: the underlying two-body interaction\nis always attractive, so that the ground state is a gas of\nmolecules of size a. The experiment thus suffers from\nconsiderable atom loss and has to be carried out under\nnonequilibirum conditions. This is clearly explained in\nFig. 1, which shows the relative energy spectrum of a pair\nof fermions in a harmonic trap with frequency ωacross a\nFeshbach resonance [27]. For a>0, the whole spectrum\nconsists of two distinct parts, the lowest ground-state\nbranch diverging as Erel≃ −/planckover2pi12/(ma2)and the regular\nupper branch having a finite energy. In the context of\nthistwo-body picture, an s-wave “repulsive” Fermi gas\nis realized, provided that there are no pairs of fermions\noccupying the ground state branch of molecules. How-\never, as far as the many-particle aspect is concerned, it\nis not clear to what extent this two-body picture of a\n“repulsive” Fermi gas will persist. In other words, can we\nprove rigorously that the whole Hilbert space of an at-\ntractive many-fermion system with a positive scattering\nlength consists exactly of a sub-Hilbert space of a repul-\nsive Fermi gas with the same scattering length, together\nwith an orthogonalized subspace of molecules?\nThis paper addresses the problem of itinerant ferro-\nmagnetism in an attractive Fermi gas using a few-particle\nperspective, by examining the exact solutions for the en-\nergy spectrum of three trapped, attractively interacting\nfermions in their upper branch of the Feshbach resonance.\nOur main results may be summarized as follows.\n•First, we present an elegant and physically trans-\nparent way to exactly solve the Hamiltonian of\nthree interacting fermions in a harmonic trap. The\nmethod may easily be generalized to treat other\nsystems with different types of atomic species, ge-\nometries, and interactions.\n•Secondly, we observe clearly from the whole relative\nenergy spectrum of three attractive fermions (see\nFig. 3) that, there are indeed two branches of the\nspectrum on the side of positive scattering length.\nAs the scattering length goes to an infinitely small\npositive value, the lower branch diverges in energy\nto−∞, while another upper branch always con-\nverges to the non-interacting limit. The latter may\nbe interpreted as the energy spectrum of three “re-\npulsively” interacting fermions. However, close to\nthe Feshbach resonance, there are many nontriv-\nial avoided crossings between two types of spec-\ntrum, making it difficult to unambiguously iden-3\ntify a repulsive Fermi system. These avoided cross-\nings are expected in more general cases, and lead to\nnontrivial consequences in a time-dependent field-\nsweep experiment passing from the weakly inter-\nacting regime at a= 0+to the unitarity limit at\na= +∞.\n•Thirdly, we show exactly that near the Feshbach\nresonance, three “repulsively” interacting fermions\nin a spherical harmonic trap, say, in a two spin-up\nand one spin-down configuration, are higher in to-\ntal energy than three fully polarized fermions (see\nthe ground state energy in Fig. 3b). Thus, there\nmust be a ferromagnetic transition occurring at a\ncertain critical scattering length. Note that fer-\nromagnetism cannot be obtained in a two-fermion\nsystem. As shown in Fig. 1, even at resonance\nthe total ground state energy of a repulsively in-\nteracting pair, Epair= 4/planckover2pi1ω, which is the sum\nof the relative ground state energy Erel= 2.5/planckover2pi1ω\nand the zero-point energy of center-of-mass mo-\ntionEcm= 1.5/planckover2pi1ω, cannot be larger than the total\nground state energy of two fully polarized fermions,\ni.e.,E↑↑= 4/planckover2pi1ω.\n•Last but most importantly, we obtain the high tem-\nperature equations of state of strongly interacting\nFermi gases (see Figs. 4, 5, 6, and 7), within a quan-\ntum virial expansion theory, which was developed\nrecently by the present authors [6]. The second and\nthird virial (expansion) coefficients of both attrac-\ntive and repulsive Fermi gases can be calculated,\nusing the full energy spectrum of three interacting\nfermions. In the unitarity limit, we find that in the\nhigh temperature regime where our quantum virial\nexpansion is reliable, the itinerant ferromagnetism\ndisappears.\nThe paper is organized as follows. In the next section,\nwe outline the theoretical model for a few fermions with\ns-wave interactions in a spherical harmonic trap. In Sec.\nIII, we explain how to construct the exact wavefunctions\nfor three interacting fermions and discuss in detail the\nwhole energy spectrum. In Sec. IV we develop a quan-\ntum virial expansion for thermodynamics and calculate\nthe second and third virial coefficients, based on the full\nenergy spectrum of two-fermion and three-fermion sys-\ntems, respectively. The high temperature equations of\nstate of strongly interacting Fermi gases are then calcu-\nlated and discussed in Sec. V. Finally, Sec. VI is devoted\nto conclusions and some final remarks. The appendix\nshows the numerical details of the exact solutions.\nII. MODELS\nConsider a few fermions in a three-dimensional (3D)\nisotropic harmonic trap V(x) =mω2x2/2with the same\nmassmand trapping frequency ω, occupying two dif-\nferent hyperfine states or two spin states. The fermionswithunlike spins attract each other via a short-range\ns-wave contact interaction. It is convenient to use the\nBethe-Peierls boundary condition to replace the s-wave\npseudopotential. That is, when any particles iandjwith\nunlike spins close to each other, rij=|xi−xj| →0, the\nfew-particle wave function ψ(x1,x2,...,xN)with proper\nsymmetry should satisfy [28–30],\nψ=Aij(Xij=xi+xj\n2,{xk/negationslash=i,j})/parenleftbigg1\nrij−1\na/parenrightbigg\n,(1)\nwhereAij(Xij,{xk/negationslash=i,j})is a function independent of rij,\nandaiss-wave scattering length. This boundary condi-\ntion can be equivalently written as,\nlim\nrij→0∂(rijψ)\n∂rij=−rijψ\na. (2)\nOtherwise, the wave function ψobeys a non-interacting\nSchrödinger equation,\nN/summationdisplay\ni=1/bracketleftbigg\n−/planckover2pi12\n2m∇2\nxi+1\n2mω2x2\ni/bracketrightbigg\nψ=Eψ. (3)\nWe now describe how to solve all the wave functions with\nenergy level Efor a two- or three-fermion system.\nIII. METHOD\nIn a harmonic trap, it is useful to separate the center-\nof-mass motion and relative motion. We thus define the\nfollowing center-of-mass coordinate Rand relative coor-\ndinatesri(i≥2) forNfermions in a harmonic trap\n[29, 30],\nR= (x1+···+xN)/N, (4)\nand\nri=/radicalbigg\ni−1\ni/parenleftBigg\nxi−1\ni−1i−1/summationdisplay\nk=1xk/parenrightBigg\n, (5)\nrespectively. In this Jacobi coordinate, the Hamiltonian\nof the non-interacting Schrödinger equation takes the\nformH0=Hcm+Hrel, where,\nHcm=−/planckover2pi12\n2M∇2\nR+1\n2Mω2R2, (6)\nand\nHrel=N/summationdisplay\ni=2/bracketleftbigg\n−/planckover2pi12\n2m∇2\nri+1\n2mω2r2\ni/bracketrightbigg\n. (7)\nThe center-of-mass motion is simply that of a harmoni-\ncally trapped particle of mass M=Nm, with well-known\nwave functions and spectrum Ecm= (ncm+ 3/2)/planckover2pi1ω,\nwherencm= 0,1,2...is a non-negative integer. In the\npresence of interactions, the relative Hamiltonian should\nbe solved in conjunction with the Bethe-Peierls boundary\ncondition, Eq. (2).4\nA. Two fermions in a 3D harmonic trap\nLet us first briefly revisit the two-fermion problem in a\nharmonic trap, where the relative Schrödinger equation\nbecomes\n/bracketleftbigg\n−/planckover2pi12\n2µ∇2\nr+1\n2µω2r2/bracketrightbigg\nψrel\n2b(r) =Erelψrel\n2b(r),(8)\nwhere two fermions with unlike spins do not stay at the\nsame position ( r>0). Here, we have re-defined r=√\n2r2\nand have used a reduced mass µ=m/2. It is clear that\nonly thel= 0subspace of the relative wave function\nis affected by the s-wave contact interaction. According\nto the Bethe-Peierls boundary condition, as r→0the\nrelative wave function should take the form, ψrel\n2b(r)→\n(1/r−1/a), or satisfy, ∂/parenleftbig\nrψrel\n2b/parenrightbig\n/∂r=−/parenleftbig\nrψrel\n2b/parenrightbig\n/a. The\ntwo-fermion problem in a harmonic trap was first solved\nby Busch and coworkers [27]. Here, we present a simple\nphysical interpretation of the solution.\nThe key point is that, regardless of the boundary con-\ndition, there are twotypes of general solutions of the\nrelative Schrödinger equation (8) in the l= 0subspace,\nψrel\n2b(r)∝exp(−r2/2d2)f(r/d). Here the function f(x)\ncan either be the first kind of Kummer confluent hyper-\ngeometric function 1F1or the second kind of Kummer\nconfluent hypergeometric function U. We have taken\nd=/radicalbig\n/planckover2pi1/µωas the characteristic length scale of the trap.\nIn the absence of interactions, the first Kummer func-\ntion gives rise to the standard wave function of 3D har-\nmonic oscillators. With interactions, however, we have\nto choose the second Kummer function U, since it di-\nverges as 1/rat origin and thus satisfies the Bethe-Peierls\nboundary condition.\nTherefore, the (un-normalized) relative wave function\nand relative energy should be rewritten as,\nψrel\n2b(r;ν) = Γ(−ν)U(−ν,3\n2,r2\nd2)exp(−r2\n2d2),(9)\nand\nErel= (2ν+3\n2)/planckover2pi1ω, (10)\nrespectively. Here, Γis the Gamma function, the\nreal number νplays the role of a quantum number\nand should be determined by the boundary condition,\nlimr→0∂/parenleftbig\nrψrel\n2b/parenrightbig\n/∂r=−/parenleftbig\nrψrel\n2b/parenrightbig\n/a. By examining the\nshort range behavior of the second Kummer function\nU(−ν,3/2,x), this leads to the familiar equation for en-\nergy levels,\n2Γ(−ν)\nΓ(−ν−1/2)=d\na. (11)\nIn Fig. 1, we give the resulting energy spectrum as a\nfunction of the dimensionless interaction strength d/a.\nThe spectrum is easy to understand. At infinitely\nsmall scattering length a→0−,ν(a= 0−) =nrel\nFigure 2: (Color online) Configuration of three interacting\nfermions, two spin-up and one spin-down.\n(nrel= 0,1,2...), which recovers the spectrum in the non-\ninteracting limit. With increasingly attractive interac-\ntions, the energies decrease. In the unitarity (resonance)\nlimit where the scattering length diverges, a→ ±∞ , we\nfind thatν(a=±∞) =nrel−1/2. As the attraction\nincreases further, the scattering length becomes positive\nand decreases in magnitude. We then observe two dis-\ntinct types of behavior: the ground state is a molecule of\nsizea, whose energy diverges asymptotically as −/planckover2pi12/ma2\nasa→0+, while the excited states may be viewed as two\nrepulsively interacting fermions with the same scattering\nlengtha. Their energies decrease to the non-interacting\nvalues asa→0+.\nIn this two-body picture, a universal repulsively inter-\nacting Fermi gas with zero-range interaction potentials\nmay be realized on the positive scattering length side of\na Feshbach resonance for an attractive interaction po-\ntential, provided that all two fermions with unlike spins\noccupy the exited states or the upper branch of the two-\nbody energy spectrum.\nB. Three fermions in a 3D harmonic trap\nLet us turn to the three fermion case by considering\ntwo spin-up fermions and one spin-down fermion, i.e., the\n↑↓↑configuration shown in Fig. 2. The relative Hamil-\ntonian can be written as [29, 30],\nHrel=/planckover2pi12\n2µ/parenleftbig\n∇2\nr+∇2\nρ/parenrightbig\n+1\n2µω2/parenleftbig\nr2+ρ2/parenrightbig\n, (12)\nwhere we have redefined the Jacobi coordinates r=√\n2r2\nandρ=√\n2r3, which measure the distance between the\nparticle 1 and 2 (i.e., pair), and the distance from the\nparticle 3 to the center-of-mass of the pair, respectively.5\n1. General exact solutions\nInspired by the two-fermion solution, it is readily seen\nthat the relative wave function of the Hamiltonian (12)\nmay be expanded into products of two Kummer conflu-\nent hypergeometric functions. Intuitively, we may write\ndown the following ansatz [6],\nψrel\n3b(r,ρ) = (1−P13)χ(r,ρ), (13)\nwhere,\nχ(r,ρ) =/summationdisplay\nnanψrel\n2b(r;νl,n)Rnl(ρ)Ym\nl(ˆρ).(14)\nThe two-body relative wave function ψrel\n2b(r;νl,n)with en-\nergy(2νl,n+ 3/2)/planckover2pi1ωdescribes the motion of the paired\nparticles 1 and 2, and the wave function Rnl(ρ)Ym\nl(ˆρ)\nwith energy (2n+l+3/2)/planckover2pi1ωgives the motion of particle\n3 relative to the pair. Here, Rnl(ρ)is the standard radial\nwave function of a 3D harmonic oscillator and Ym\nl(ˆρ)is\nthe spherical harmonic. Owing to the rotational symme-\ntry of the relative Hamiltonian (12), it is easy to see that\nthe relative angular momenta landmare good quantum\nnumbers. The value of νl,nis uniquely determined from\nenergy conservation,\nErel= [(2νl,n+3/2)+(2n+l+3/2)]/planckover2pi1ω, (15)\nfor a given relative energy Erel. It varies with the index\nnat a given angular momentum l. Finally, P13is an\nexchange operator for particles 1 and 3, which ensures\nthe correct exchange symmetry of the relative wave func-\ntion due to Fermi exclusion principle, i.e., P13χ(r,ρ) =\nχ/parenleftbig\nr/2+√\n3ρ/2,√\n3r/2−ρ/2/parenrightbig\n. The relative energy Erel\ntogether with the expansion coefficient anshould be de-\ntermined by the Bethe-Peierls boundary condition, i.e.,\nlimr→0[∂rψrel\n3b(r,ρ)]/∂r=−[rψrel\n3b(r,ρ)]/a. We note\nthat the second Bethe-Peierls boundary condition in case\nof particle 2 approaching particle 3 is satisfied automati-\ncally due to the exchange operator acting on the relative\nwave function.\nBy writing χ(r,ρ) =φ(r,ρ)Ym\nl(ˆρ), the Bethe-Peierls\nboundary condition takes the form ( r→0),\n−1\na[rφ(r,ρ)] =∂[rφ(r,ρ)]\n∂r−(−1)lφ(√\n3ρ\n2,ρ\n2).(16)\nUsing the asymptotic behavior of the second kind of\nKummer function, limx→0Γ(−νl,n)U(−νl,n,3/2,x2) =√π/x−2√πΓ(−νl,n)/Γ(−νl,n−1/2), it is easy to show\nthat in the limit of r→0,\n−1\na[rφ(r,ρ)] =−√π\na/summationdisplay\nnanRnl(ρ), (17)\nand\n∂[rφ(r,ρ)]\n∂r=−√π/summationdisplay\nnanRnl(ρ)2Γ(−νl,n)\nΓ(−νl,n−1/2).(18)Thus, the Bethe-Peierls boundary condition becomes,\n/summationdisplay\nnan/bracketleftBigg\nBnRnl(ρ)−Rnl/parenleftBigρ\n2/parenrightBig\nψrel\n2b(√\n3ρ\n2;νl,n)/bracketrightBigg\n= 0,\n(19)\nwhere\nBn= (−1)l√π/bracketleftbiggd\na−2Γ(−νl,n)\nΓ(−νl,n−1/2)/bracketrightbigg\n.(20)\nProjecting onto the orthogonal and complete set of basis\nfunctionsRnl(ρ), we find that a secular equation,\n2Γ(−νl,n)\nΓ(−νl,n−1/2)an+(−1)l\n√π/summationdisplay\nn′Cnn′an′=/parenleftbiggd\na/parenrightbigg\nan,(21)\nwhere we have defined the matrix coefficient,\nCnn′≡∞ˆ\n0ρ2dρRnl(ρ)Rn′l/parenleftBigρ\n2/parenrightBig\nψrel\n2b(√\n3ρ\n2;νl,n′),(22)\nwhich arises from the exchange effect due to the operator\nP13. In the absence of Cnn′, the above secular equation\ndescribes a three-fermion problem of a pair and a sin-\ngle particle, un-correlated to each other. It then simply\nreduces to Eq. (11), as expected.\nThe secular equation (21) was first obtained by Kestner\nand Duan by solving the three-particle scattering prob-\nlem using Green function [31]. To solve it, for a given\nscattering length we may try different values of relative\nenergyErel, implicit via νl,n. However, it turns out to be\nmore convenient to diagonalize the matrix A={Ann′}\nfor a given relative energy, where\nAnn′=2Γ(−νl,n)\nΓ(−νl,n−1/2)δnn′+(−1)l\n√πCnn′. (23)\nThe eigenvalues of the matrix Athen gives all the pos-\nsible values of d/afor a particular relative energy. We\nfinally invert a(Erel)to obtain the relative energy as a\nfunction of the scattering length. Numerically, we find\nthat the matrix Ais symmetric and thus the standard\ndiagonalization algorithm can be used. We outline the\ndetails of the numerical calculation of Eq. (23) in the\nAppendix A.\n2. Exact solutions in the unitarity limit\nIn the unitarity limit with infinitely large scatter-\ning length, a→ ∞ , we may obtain more physical so-\nlutions using hyperspherical coordinates, as shown by\nWerner and Castin [28, 30]. By defining a hyperradius\nR=/radicalbig\n(r2+ρ2)/2and hyperangles /vectorΩ = (α,ˆr,ˆρ), where\nα= arctan(r/ρ)andˆrandˆρare respectively the unit\nvector along randρ, we may write [28, 30],\nψrel\n3b/parenleftBig\nR,/vectorΩ/parenrightBig\n=F(R)\nR(1−P13)ϕ(α)\nsin(2α)Ym\nl(ˆρ),(24)6\nto decouple the motion in the hyperradius and hyperan-\ngles for given relative angular momenta landm. It leads\nto the following decoupled Schrödinger equations [30],\n−F′′−1\nRF′+/parenleftBigg\ns2\nl,n\nR2+ω2R2/parenrightBigg\nF= 2ErelF, (25)\nand\n−ϕ′′(α)+l(l+1)\ncos2αϕ(α) =s2\nl,nϕ(α), (26)\nwheres2\nl,nis the eigenvalue for the n-th wave function of\nthe hyperangle equation.\nFor three-fermions, s2\nl,nis always positive. There-\nfore, the hyperradius equation (25) can be interpreted\nas a Schrödinger equation for a fictitious particle of mass\nunity moving in two dimensions in an effective potential\n(s2\nl,n/R2+ω2R2)with a bounded wave function F(R).\nThe resulting spectrum is [28, 30]\nErel= (2q+sl,n+1)/planckover2pi1ω, (27)\nwhere the good quantum number qlabels the number of\nnodes in the hyperradius wave function.\nThe eigenvalue sl,nshould be determined by the Bethe-\nPeierls boundary condition, which in hyperspherical co-\nordinates takes the from [28, 30],\nϕ′(0)−(−1)l4√\n3ϕ/parenleftBigπ\n3/parenrightBig\n= 0. (28)\nIn addition, we need to impose the boundary condition\nϕ(π/2) = 0 , since the relative wave function (24) should\nnot be singular at α=π/2. The general solution of the\nhyperangle equation (26) satisfying ϕ(π/2) = 0 is given\nby,\nϕ∝xl+12F1/parenleftbiggl+1−sl,n\n2,l+1+sl,n\n2,l+3\n2;x2/parenrightbigg\n,\n(29)\nwherex= cos(α)and2F1is the hypergeometric function.\nIn the absence of interactions, the Bethe-Peierls bound-\nary condition (28) should be replaced by ϕ(0) = 0 , since\nthe relative wave function (24) should not be singular\natα= 0either. As ϕ(0) = Γ(l+ 3/2)Γ(1/2)/[Γ((l+\n2+sl,n)/2)Γ((l+2−sl,n)/2)], this boundary condition\nleads to [l+ 2−s(1)\nl,n]/2 =−n, ors(1)\nl,n= 2n+l+ 2,\nwheren= 0,1,2,...is a non-negative integer and we have\nused the superscript “ 1” to denote a non-interacting sys-\ntem. However, a spurious solution occurs when l= 0and\nn= 0, for whichs(1)\nl,n= 2,ϕ(α) = sin(2α)/2and thus, the\nsymmetry operator (1− P13)gives a vanishing relative\nwave function in Eq. (24) that should be discarded [30].\nWe conclude that for three non-interacting fermions,\ns(1)\nl,n=/braceleftbigg\n2n+4, l= 0\n2n+l+2, l>0. (30)For three interacting fermions, we need to determine sl,n\nby substituting the general solution (29) into the Bethe-\nPeierls boundary condition (28). In the Appendix B, we\ndescribe how to accurately calculate sl,n. In the bound-\nary condition Eq. (28), the leading effect of interactions\nis carried by ϕ′(0)and therefore, ϕ′(0) = 0 determines\nthe asymptotic values of sl,nat large momentum lorn.\nThis gives rise to (l+1−¯sl,n)/2 =−n, or,\n¯sl,n=/braceleftbigg\n2n+3, l= 0\n2n+l+1, l>0, (31)\nwhere we have used a bar to indicate the asymptotic re-\nsults. By comparing Eqs. (30) and (31), asymptotically\nthe attractive interaction will reduce sl,nby a unity.\n3. Energy spectrum of three interacting fermions\nWe have numerically solved both the general exact so-\nlution (13) along the BEC-BCS crossover and the exact\nsolution (24) in the unitarity limit. In the latter uni-\ntary case, the accuracy of results can be improved to\narbitrary precision by using suitable mathematical soft-\nware, described in Appendix B. Fig. 3 reports the energy\nspectrum of three interacting fermions with increasingly\nattractive interaction strength at three total relative an -\ngular momenta, l= 0,1, and2. For a given scattering\nlength, we typically calculate several ten thousand en-\nergy levels (i.e., Erel<(l+ 256)/planckover2pi1ω) in each different\nsubspace. To construct the matrix A, Eq. (23), we have\nkept a maximum value of nmax= 128 in the functions\nRnl(ρ). Using the accurate spectrum in the unitarity\nlimit as a benchmark, we estimate that the typical rela-\ntive numerical error of energy levels is less than 10−6. We\nhave found a number of nontrivial features in the energy\nspectrum.\nThe spectrum on the BCS side is relatively simple.\nIt can be understood as a non-interacting spectrum at\nd/a→ −∞ , in whichErel= (2Q+ 3)/planckover2pi1ωatl= 0and\nErel= (2Q+l+ 1)/planckover2pi1ωatl≥1, with a positive integer\nQ= 1,2,3,...that denotes also the degeneracy of the\nenergy levels. The attractive interactions reduce the en-\nergies and at the same time lift the degeneracy. Above\nthe resonance or unitary point of d/a= 0, however, the\nspectrum becomes much more complicated.\nThere are a group of nearly vertical energy levels that\ndiverge towards the BEC limit of d/a→+∞. From the\ntwo-body relative energy spectrum in Fig. 1, we may\nidentify these as energy states containing a molecule of\nsizeaand a fermion. For a given scattering length, these\nnearly vertical energy level differ by about 2/planckover2pi1ω, resulting\nfrom the motion of the fermion relative to the molecule.\nIn addition to the nearly vertical energy levels, most in-\nterestingly, we observe also some nearly horizontal energy\nlevels, which converge to the non-interacting spectrum in\nthe BEC limit. In analogy with the two-body case, we\nmay identify these horizontal levels as the energy spec-\ntrum of three repulsively interacting fermions. We show7\n/s50/s52/s54/s56/s49/s48\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48\n/s32\n/s100 /s47/s97/s69\n/s114/s101/s108/s47/s40 /s126 /s41\n/s61/s48/s40/s97 /s41\n/s50/s52/s54/s56/s49/s48\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48\n/s32\n/s100 /s47/s97/s69\n/s114/s101/s108/s47/s40 /s126 /s41\n/s61/s49/s40/s98 /s41\n/s50/s52/s54/s56/s49/s48\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48\n/s32\n/s100 /s47/s97/s69\n/s114/s101/s108/s47/s40 /s126 /s41\n/s61/s50/s40/s99 /s41\nFigure 3: (Color online) Relative energy spectrum of three\ninteracting fermions at different subspaces or relative ang ular\nmomenta l. On the positive scattering length (BEC) side of\nthe resonance, there are two types of energy levels: one (is\nvertical and) diverges with decreasing the scattering leng tha\nand the other (is horizontal) converges to the non-interact ing\nspectrum. The latter may be viewed as the energy spectrum\nof three repulsively interacting fermions. In analogy with the\ntwo-fermion case, we show in the ground state subspace ( l=\n1) the ground state energy level of the repulsive three-fermi on\nsystem (i.e, from point 2 to 3), as well as the ground state\nenergy level of the attractive three-fermion system for a <0\n(i.e., from the point 1 to 2). In the unitarity limit, we show\nby the circles the energy levels that should be excluded when\nwe identify the energy spectrum for infinitely large repulsi ve\ninteractions.explicitly in Fig. 3b the ground state level of three re-\npulsively interacting fermions, which increases in energy\nfrom the point 2 to 3 with increasing scattering length\nfroma= 0+toa= +∞. For comparison, we also show\nthe ground state level of three attractively interacting\nfermions at a negative scattering length, which decreases\nin energy from the point 2 to 1 with increasing absolute\nvalue ofa.\nThis identification of energy spectrum for repulsive in-\nteractions, however, is not as rigorous as in the two-body\ncase. There are many apparent avoided crossings be-\ntween the vertical and horizontal energy levels. There-\nfore, by changing a positive scattering length from the\nBEC limit to the unitarity limit, three fermions initially\nat the horizontal level may finally transition into a verti-\ncal level, provided that the sweep of scattering length is\nsufficiently slow and adiabatic. This leads to the conver-\nsion of fermionic atoms to bosonic molecules. A detailed\nanalysis of the loss rate of fermionic atoms as a func-\ntion of sweep rate may be straightforward obtained by\napplying the Landau-Zener tunnelling model.\nLet us now focus on the resonance case of most signifi-\ncant interest. In Fig. 3, we show explicitly by green dots\nthe vertical energy levels in the unitarity limit. These\nlevels should be excluded if we are interested in the spec-\ntrum of repulsively interacting fermions. Amazingly, for\neach given angular momentum, these energy levels form\na regular ladder with an exact energy spacing 2/planckover2pi1ω[29].\nUsing the exact solution in the unitarity limit, Eq. (27),\nwe may identify unambiguously that the energy ladder is\ngiven by,\nErel= (2q+sl,0+1)/planckover2pi1ω. (32)\nTherefore, in the unitarity limit the lowest-order solutio n\nof the hyperangle equation gives rise to the relative wave\nfunction of a molecule and a fermion. Thus, it should be\ndiscarded when considering three resonantly interacting\nfermions with an effective repulsive interaction.\nThis observation immediately leads to the ground state\nenergy of three repulsively interacting fermions,\nE↑↓↑\ngs= (s1,1+1)/planckover2pi1ω+1.5/planckover2pi1ω≃6.858249309 /planckover2pi1ω,(33)\nincluding the zero-point energy of the center-of-mass mo-\ntion,1.5/planckover2pi1ω. This ground state energy is higher than that\nof three fully polarized fermions, which is,\nE↑↑↑\ngs= 1.5/planckover2pi1ω+2.5/planckover2pi1ω+2.5/planckover2pi1ω= 6.5/planckover2pi1ω. (34)\nThus, in the presence of repulsive interactions, the\nground state of three fully polarized fermions is unsta-\nble with respect to a single spin-flip, suggestive of an\nitinerant ferromagnetic transition at a certain scatterin g\nlength for three fermions.8\nIV. QUANTUM VIRIAL EXPANSION FOR\nTHERMODYNAMICS\nThe few-particle solutions presented above can provide\ninformation about the high temperature thermodynamics\nof many-body systems, through a quantum virial expan-\nsion of the grand thermodynamic potential [6, 32]. In the\ngrand canonical ensemble, the thermodynamic potential\nis given by,\nΩ =−kBTlnZ, (35)\nwherekBis the Boltzmann constant and\nZ=Trexp[−(H−µN)/kBT] (36)\nis the grand partition function. We may rewrite this in\nterms of the partition function of clusters,\nQn=Trn[exp(−H/kBT)], (37)\nwhere the integer ndenotes the number of particles in the\ncluster and the trace Tr nis taken over n-particle states\nwith a proper symmetry. The partition function of clus-\ntersQncan be calculated using the complete solutions\nof an-particles system. The grand partition function is\nthen written as\nZ= 1+zQ1+z2Q2+···, (38)\nwherez= exp(µ/kBT)is the fugacity. At high tem-\nperatures, it is well-known that the chemical potential\nµdiverges to −∞, so the fugacity would be very small,\nz≪1. We can then expand the high-temperature ther-\nmodynamic potential in powers of the small parameter\nz,\nΩ =−kBTQ1/bracketleftbig\nz+b2z2+···+bnzn+···/bracketrightbig\n,(39)\nwherebnmay be referred to as the n-th (virial) expansion\ncoefficient. It is readily seen that,\nb2=/parenleftbig\nQ2−Q2\n1/2/parenrightbig\n/Q1, (40)\nb3=/parenleftbig\nQ3−Q1Q2+Q3\n1/3/parenrightbig\n/Q1, etc. (41)\nThese equations present a general definition of the quan-\ntum virial expansion and are applicable to both homo-\ngeneous and trapped systems. The determination of the\nn-th virial coefficient requires knowledge of up to the n-\nbody problem.\nIn practice, it is convenient to concentrate on the in-\nteraction effects only. We thus consider the difference\n∆bn≡bn−b(1)\nnand∆Qn≡Qn−Q(1)\nn, where the super-\nscript “1” denotes the non-interacting systems. For the\nsecond and third virial coefficient, we shall calculate\n∆b2= ∆Q2/Q1 (42)\nand\n∆b3= ∆Q3/Q1−∆Q2. (43)A. Non-interacting virial coefficients\nThe background non-interacting virial coefficients\ncan be conveniently determined by the non-interacting\nthermodynamic potential. For a homogeneous two-\ncomponent Fermi gas, this takes the form,\nΩ(1)\nhom=−V2kBT\nλ32√π∞ˆ\n0t1/2ln/parenleftbig\n1+ze−t/parenrightbig\ndt, (44)\nwhereλ≡[2π/planckover2pi12/(mkBT)]1/2is the thermal wavelength\nandQ1,hom= 2V/λ3. This leads to\nb(1)\nn,hom=(−1)n+1\nn5/2. (45)\nHereafter, we use the subscript “hom” to denote the quan-\ntity in the homogeneous case, otherwise, by default we\nrefer to a trapped system.\nFor a harmonically trapped Fermi gas, the non-\ninteracting thermodynamic potential in the semiclassical\nlimit (neglecting the discreteness of the energy spectrum)\nis,\nΩ(1)=−2(kBT)4\n(/planckover2pi1ω)31\n2∞ˆ\n0t2ln/parenleftbig\n1+ze−t/parenrightbig\ndt, (46)\nwhereQ1= 2(kBT)3/(/planckover2pi1ω)3. Taylor-expanding in pow-\ners ofzgives rise to\nb(1)\nn=(−1)n+1\nn4. (47)\nWe note that the non-interacting virial coefficients in the\nhomogeneous case and trapped case are related by,\nb(1)\nn=b(1)\nn,hom\nn3/2. (48)\nB. Second virial coefficient in a harmonic trap\nWe now calculate the second virial coefficient of a\ntrapped interacting Fermi gas. In a harmonic trap, the\noscillator length dprovides a large length scale, compared\nto the thermal wavelength λ. Alternatively, we may use\n˜ω=/planckover2pi1ω/kBT≪1to characterize the intrinsic length\nscale relative to the trap. All the virial coefficients and\ncluster partition functions in harmonic traps therefore\ndepend on the small parameter ˜ω. We shall be interested\nin a universal regime with vanishing ˜ω, in accord with\nthe large number of atoms in a real experiment.\nTo obtain ∆b2, we consider separately ∆Q2andQ1.\nThe single-particle partition function Q1is determined\nby the single-particle spectrum of a 3D harmonic os-\ncillator,Enl= (2n+l+ 3/2)/planckover2pi1ω. We find that Q1=\n2/[exp(+˜ω/2)−exp(−˜ω/2)]3≃2(kBT)3/(/planckover2pi1ω)3. The9\nprefactor of two accounts for the two possible spin states\nof a single fermion. In the calculation of ∆Q2, it is easy\nto see that the summation over the center-of-mass energy\ngives exactly Q1/2. Using Eq. (10), we find that,\n∆batt\n2=1\n2/summationdisplay\nνn/bracketleftBig\ne−(2νn+3/2)˜ω−e−(2ν(1)\nn+3/2)˜ω/bracketrightBig\n,(49)\nwhere the non-interacting ν(1)\nn=n(n= 0,1,2,...) and\nthe superscript “att” (or “rep”) means the coefficient of an\nattractively (or repulsively) interacting Fermi gas. The\nsecond virial coefficient of a trapped attractive Fermi gas\nin the BEC-BCS crossover was given in Fig. 3a of Ref.\n[6].\nTo consider the second virial coefficient of a repulsively\ninteracting Fermi gas, we shall restrict ourselves to a posi -\ntive scattering length and exclude the lowest ground state\nenergy level in the summation of the first term in Eq.\n(49), which corresponds to a bound molecule.\n1. Unitarity limit\nAt resonance with an infinitely large scattering length,\nthe spectrum is known exactly: νn,∞=n−1/2, giving\nrise to,\n∆batt\n2,∞=1\n2exp(−˜ω/2)\n[1+exp( −˜ω)]= +1\n4−1\n32˜ω2+···.(50)\nFor a repulsive Fermi gas in the unitarity limit, we shall\ndiscard the lowest ‘molecular’ state with ν0,∞=−1/2\nand therefore,\n∆brep\n2,∞=1\n2exp(−˜ω/2)\n[1+exp(+˜ω)]=−1\n4+˜ω\n4+···.(51)\nThe term ˜ω2or˜ωin Eqs. (50) and (51) is nonuniversal\nand is negligibly small for a cloud with a large number\nof atoms. We therefore obtain the universal second virial\ncoefficients: ∆batt\n2,∞= +1/4and∆brep\n2,∞=−1/4, which\nare temperature independent.\nC. Third virial coefficient in a harmonic trap\nThe calculation of the third virial coefficient, which is\ngiven by ∆b3= ∆Q3/Q1−∆Q2, is more complicated.\nEither the term ∆Q3/Q1or∆Q2diverges as ˜ω→0, but\nthe leading divergences cancel with each other. In the\nnumerical calculation, we have to carefully separate the\nleading divergent term and calculate them analytically.\nIt is readily seen that the spin states of ↑↓↑and↓↑↓\nconfigurations contribute equally to Q3. The term Q1in\nthe denominators is canceled exactly by the summation\nover the center-of-mass energy. We thus have\n∆Q3/Q1= [/summationdisplay\nexp(−Erel/kBT)−/summationdisplay\nexp(−E(1)\nrel/kBT)].\n(52)To proceed, it is important to analyze analytically the\nbehavior of Erelat high energies. For this purpose, we\nintroduce a relative energy ¯Erel, which is the solution of\nEq. (23) in the absence of the exchange term Cnm, and\ncan be constructed directly from the two-body relative\nenergy. In the subspace with a total relative momentum\nl, it takes the form,\n¯Erel= (2n+l+3/2)/planckover2pi1ω+(2ν+3/2)/planckover2pi1ω, (53)\nwhereνis the solution of the two-body spectrum of Eq.\n(11). At high energies the full spectrum Erelapproaches\nasymptotically to ¯Erelas the exchange effect becomes\nincreasingly insignificant. There is an important excep-\ntion, however, occurring at zero total relative momentum\nl= 0. As mentioned earlier, the solution of ¯Erelatn= 0\nandl= 0is spurious and does not match any solution of\nErel. Therefore, for the l= 0subspace, we require n≥1\nin Eq. (53).\nIt is easy to see that if we keep the spurious solution in\nthel= 0subspace, the difference [/summationtextexp(−¯Erel/kBT)−/summationtextexp(−E(1)\nrel/kBT)]is exactly equal to ∆Q2, since in\nEq. (53) the first part of spectrum is exactly identical\nto the spectrum of center-of-mass motion. The spurious\nsolution gives a contribution,\n/summationdisplay\nνn/bracketleftBig\ne−(2νn+3)˜ω−e−(2ν(1)\nn+3)˜ω/bracketrightBig\n≡2e−3˜ω/2∆batt\n2,(54)\nwhich should be subtracted. Keeping this in mind, we\nfinally arrive at the following expression for the third\nvirial coefficient of a trapped Fermi gas with repulsive\ninteractions:\n∆batt\n3=/summationdisplay/bracketleftbigg\ne−Erel\nkBT−e−¯Erel\nkBT/bracketrightbigg\n−2e−3˜ω/2∆batt\n2.(55)\nThe summation is over all the possible relative energy\nlevelsEreland their asymptotic values ¯Erel. It is well-\nbehaved and converges at any scattering length. The\nthird virial coefficient of a trapped attractive Fermi gas\nin the BEC-BCS crossover was given in Fig. 3b of Ref.\n[6].\n1. Unitarity limit\nIn the unitarity limit, it is more convenient to use the\nexact spectrum given by Eq. (27), where sl,ncan be\nobtained numerically to arbitrary accuracy and the non-\ninteracting s(1)\nl,nis given by Eq. (30). To control the\ndivergence problem, we shall use the same strategy as\nbefore and to approach sl,nby using its asymptotic value\n¯sl,ngiven in Eq. (31).\nIntegrating out the qdegree of freedom and using Eq.\n(50) to calculate ∆Q2, we find that,\n∆batt\n3,∞=e−˜ω\n1−e−2˜ω\n/summationdisplay\nl,n/parenleftbig\ne−˜ωsl,n−e−˜ω¯sl,n/parenrightbig\n+A\n,(56)10\nwhereAis given by\nA=/summationdisplay\nl,n/parenleftBig\ne−˜ω¯sl,n−e−˜ωs(1)\nl,n/parenrightBig\n−e−˜ω\n(1−e−˜ω)2.(57)\nWe note that for the summation, implicitly there is a\nprefactor (2l+1), accounting for the degeneracy of each\nsubspace. The value of Acan then be calculated analyt-\nically, leading to,\nA=−e−˜ω/parenleftbig\n1−e−˜ω/parenrightbig\n. (58)\nWe have calculated numerically/summationtext\nl,n(e−˜ωsl,n−e−˜ω¯sl,n)\nby imposing the cut-offs of n < n max= 512 andl <\nlmax= 512 . We find that,\n∆batt\n3,∞≃ −0.06833960+0 .038867˜ω2+···.(59)\nThe numerical accuracy can be further improved by suit-\nably enlarging nmaxandlmax. For a Fermi gas with in-\nfinitely large repulsions, we need to exclude the states in-\nvolving a molecule. Thus, in the calculation of ∆Q3/Q1,\nwe exclude the energy levels associated with sl,n=0, as\ngiven by Eq. (32). In the calculation of ∆Q2, we shall\nremove the lowest two-body state with ν0,∞=−1/2. In\nthe end, we find that,\n∆brep\n3,∞≃0.34976−0.77607˜ω+···. (60)\nBy neglecting the dependence on ˜ωin the thermodynamic\nlimit, we obtain the universal third virial coefficients:\n∆batt\n3,∞≃ −0.06833960, (61)\n∆brep\n3,∞≃0.34976. (62)\nD. Unitary virial coefficients in homogeneous space\nWe have so far studied the virial coefficients in a har-\nmonic trap. In the unitarity limit, there is a simple re-\nlation between the trapped and homogeneous virial co-\nefficient, as inspired by relation (48). This stems from\nthe universal temperature independence of all virial co-\nefficients in the unitarity limit. In the thermodynamic\nlimit, let us consider the thermodynamic potential of a\nharmonic trapped Fermi gas in the local density approx-\nimationΩ =´drΩ(r), whereΩ(r)is the local thermo-\ndynamic potential\nΩ(r)∝z(r)+b2,∞,homz2(r)+···. (63)\nHere, the local fugacity z(r) =zexp[−V(r)/kBT]is de-\ntermined by the local chemical potential µ(r) =µ−V(r).\nOn spatial integration, it is readily seen that the univer-\nsal (temperature independent) part of the trapped virial\ncoefficient is,\nbn,∞=bn,∞,hom\nn3/2. (64)We therefore immediately obtain that the homogeneous\nsecond virial coefficients in the unitarity limit are:\n∆batt\n2,∞,hom= +1√\n2, (65)\n△brep\n2,∞,hom=−1√\n2, (66)\nand the homogeneous third virial coefficients are:\n∆batt\n3,∞,hom≃ −0.35501298, (67)\n∆brep\n3,∞,hom≃+1.8174. (68)\nThe homogeneous virial coefficients are therefore signifi-\ncantly larger than their trapped counterparts. The factor\nofn3/2is clearly due to the higher density of states in a\nharmonically trapped geometry.\nV. HIGH- TEQUATION OF STATE OF A\nSTRONGLY INTERACTING FERMI GAS\nWe are now ready to calculate the equation of states\nin the high temperature regime, by using the thermody-\nnamic potential\nΩhom= Ω(1)\nhom−V2kBT\nλ3/parenleftbig\n∆b2,homz2+···/parenrightbig\n(69)\nand\nΩ = Ω(1)−2(kBT)4\n(/planckover2pi1ω)3/parenleftbig\n∆b2z2+∆b3z3+···/parenrightbig\n,(70)\nrespectively, for a homogeneous or a harmonically\ntrapped Fermi gas. Here, the non-interacting thermo-\ndynamic potentials are given by Eqs. (44) and (46).\nAll the other thermodynamic quantities can be derived\nfrom the thermodynamic potential by the standard ther-\nmodynamic relations, for example, N=−∂Ω/∂µ,S=\n−∂Ω/∂T, and thenE= Ω+TS+µN.\nAs an concrete example, let us focus on the unitarity\nlimit in the thermodynamic limit, which is of the great-\nest interest. The equations of state are easy to calculate\nbecause of the temperature independence of virial coef-\nficients. It is also easy to check the well-known scaling\nrelation in the unitarity limit: E=−3Ω/2for a homo-\ngeneous Fermi gas [32] and E=−3Ωfor a harmonically\ntrapped Fermi gas [4]. The difference of the factor of two\narises from the fact that according to the virial theorem,\nin harmonic traps the internal energy is exactly equal to\nthe trapping potential energy.\nTo be dimensionless, we take the Fermi temperature\nTFor Fermi energy ( EF=kBTF) as the units for tem-\nperature and energy. For a homogeneous or a harmon-\nically trapped Fermi gas, the Fermi energy is given by\nEF=/planckover2pi12(3π2N/V)2/3/2mandEF= (3N)1/3/planckover2pi1ω, respec-\ntively. In the actual calculations, we determine the num-\nber of atoms N, the total entropy S, and the total energy11\nEat given fugacity and a fixed temperature (i.e., T= 1),\nand consequently obtain the Fermi temperature TFand\nFermi energy EF. We then plot the energy or energy per\nparticle,E/(NEF)andS/(NkB), as a function of the\nreduced temperature T/TF.\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s50/s52/s54/s56\n/s32/s86/s69/s51/s32/s40 /s114/s101/s112 /s46/s41\n/s32/s86/s69/s50/s32/s40 /s114/s101/s112 /s46/s41\n/s32/s86/s69/s51/s32/s40 /s97/s116/s116 /s46/s41\n/s32/s86/s69/s50/s32/s40 /s97/s116/s116 /s46/s41\n/s73/s100/s101/s97/s108/s32/s70/s101/s114/s109/s105/s32/s71/s97/s115\n/s32/s32/s69 /s47/s40 /s78/s69\n/s70/s41\n/s84 /s47/s84\n/s70\nFigure 4: (Color online) Energy per particle E/(NEF)as\na function of reduce temperature T/TFfor a homogeneous\nFermi gas with infinitely attractive and repulsive interact ions.\nThe predictions of quantum virial expansion up to the second -\nand third-order are shown by solid line and dashed line, re-\nspectively. For comparison, we plot the ideal gas result by\nthe dot-dashed line.\nA. Homogeneous equation of state\nWe report in Figs. 4 and 5 the temperature depen-\ndence of energy and entropy of a strongly attractively or\nrepulsively interacting homogeneous Fermi gas. The solid\nline and dashed line are the predictions of the quantum\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53\n/s32/s86/s69/s51/s32/s40 /s97/s116/s116 /s46/s41\n/s32/s86/s69/s50/s32/s40 /s97/s116/s116 /s46/s41/s32/s86/s69/s51/s32/s40 /s114/s101/s112 /s46/s41\n/s32/s86/s69/s50/s32/s40 /s114/s101/s112 /s46/s41\n/s32/s32/s83 /s47/s40 /s78/s107\n/s66/s41\n/s84 /s47/s84\n/s70/s71/s97/s115\n/s70/s101/s114/s109/s105\n/s73/s100/s101/s97/s108\nFigure 5: (Color online) Entropy per particle S/(NEF)as\na function of reduce temperature T/TFfor a homogeneous\nFermi gas with infinitely attractive and repulsive interact ions.\nThe others are the same as in Fig. 4./s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48\n/s71/s97/s115\n/s70/s101/s114/s109/s105/s32/s86/s69/s51/s32/s40 /s114/s101/s112 /s46/s41\n/s32/s86/s69/s50/s32/s40 /s114/s101/s112 /s46/s41\n/s32/s69/s78/s83/s32 /s101/s120/s112/s116/s46\n/s32/s86/s69/s51/s32/s40 /s97/s116/s116 /s46/s41\n/s32/s86/s69/s50/s32/s40 /s97/s116/s116 /s46/s41/s73/s100/s101/s97/s108\n/s32/s32/s69 /s47/s40 /s78/s69\n/s70/s41\n/s84 /s47/s84\n/s70\nFigure 6: (Color online) Energy per particle E/(NEF)as a\nfunction of reduced temperature T/TFfor a trapped Fermi\ngas with infinitely attractive and repulsive interactions. The\npredictions of quantum virial expansion up to the second- an d\nthird-order are shown by solid line and dashed line, respec-\ntively. For comparison, we plot the ideal gas result by the\ndot-dashed line. We show also the experimental data mea-\nsured at ENS by empty squares for an attractive Fermi gas at\nunitarity [4, 7], which agree extremely well with the predic tion\nfrom quantum virial expansion.\nvirial expansion up to the third order (VE3) and second\norder (VE2), respectively. For comparison, we also show\nthe ideal gas result by the thin dot-dashed line.\nFor a strongly attractively interacting Fermi gas, we\nobserve that the quantum virial expansion is valid down\nto the degeneracy temperature TF, where the predictions\nusing the second-order or third-order expansion do not\ngreatly differ. We note that our prediction of the third\nvirial coefficient of a unitarity Fermi gas, ∆batt\n3,∞,hom≃\n−0.35501298 , was experimentally confirmed to within 5%\nrelative accuracy in the most recent thermodynamic mea-\nsurement at ENS by Nascimbène and co-workers [7].\nHowever, for a strongly repulsively interacting Fermi\ngas, the applicability of the quantum virial expansion\nis severely reduced: it seems to be applicable only for\nT >5TF. Below this characteristic temperature, the dif-\nference between the second-order and third-order predic-\ntion becomes very significant. This is partly due to the\nlarge absolute value of the third virial coefficient, sug-\ngesting that in this case the virial expansion converges\nvery slowly.\nB. Harmonically trapped equation of states\nWe finally present in Figs. 6 and 7 the high-\ntemperature expansion prediction for the equation of\nstate of a harmonically trapped Fermi gas in the strongly\ninteracting regime. Due to the significantly reduced virial\ncoefficients, the virial expansion in a trap has much\nbroader applicability. For a strongly attractively inter-12\n/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s52/s53/s54\n/s32/s69/s78/s83/s32 /s101/s120/s112/s116/s46\n/s32/s86/s69/s51/s32/s40 /s97/s116/s116 /s46/s41\n/s32/s86/s69/s50/s32/s40 /s97/s116/s116 /s46/s41/s32/s86/s69/s51/s32/s40 /s114/s101/s112 /s46/s41\n/s32/s86/s69/s50/s32/s40 /s114/s101/s112 /s46/s41\n/s32/s32/s83 /s47/s40 /s78/s107\n/s66/s41\n/s84 /s47/s84\n/s70/s71/s97/s115\n/s70/s101/s114/s109/s105\n/s73/s100/s101/s97/s108\nFigure 7: (Color online) Entropy per particle S/(NEF)as\na function of reduce temperature T/TFfor a trapped Fermi\ngas with infinitely attractive and repulsive interactions. The\nothers are the same as in Fig. 6.\nacting Fermi gas, it is now quantitatively applicable down\nto0.5TF, as confirmed by the precise experimental mea-\nsurement at ENS (empty squares) [4, 7]. At the same\ntime, the virial expansion for a strongly repulsively in-\nteracting gas seems to be qualitatively valid at T >T F.\nAt this temperature, the energy of the repulsively inter-\nacting Fermi gas is only marginally higher than the ideal,\nnon-interacting energy. Considering the large energy dif-\nference between a fully polarized Fermi gas and a non-\npolarized Fermi gas (i.e., at the order of NEF), we con-\njecture that a strongly repulsively interacting Fermi gas\ndoes not have itinerant ferromagnetism in the tempera-\nture regime where the quantum virial expansion theory\nis applicable.\nVI. CONCLUSIONS AND OUTLOOK\nIn conclusion, we have presented a complete set of ex-\nact solutions for three attractively interacting fermions in\na harmonic trap, with either positive or negative scatter-\ning lengths. Firstly, we have outlined the details of our\nprevious studies on the quantum virial expansion [6], in\nparticular the method for calculating the third virial co-\nefficient which was recently confirmed experimentally. In\naddition, we have opened up the previously unexplored\nrepulsively interacting regime, and have presented a few-\nbody perspective of itinerant ferromagnetism. We have\nalso studied the high-temperature thermodynamics of a\nstrongly repulsively interacting Fermi gas, by calculatin g\nits second and third virial coefficients in the unitarity\nlimit.\nOn the positive scattering length side of a Feshbach res-\nonance, a repulsively interacting Fermi gas is thought to\noccur by excluding all the many-body states which con-\ntain a molecule-like bound state for any two atoms with\nunlike spins. Strictly speaking, this is a conjecture whichstems from a two-body picture. We have examined this\nconjecture using the exact three-fermion energy spectrum\nnear the resonance. We have found some horizontal en-\nergy levels that may be identified as the energy spectrum\nof three “repulsively” interacting fermions, as well as som e\nvertical energy levels involving a tightly-bound molecule .\nHowever, many avoided crossings between horizontal and\nvertical levels make it difficult to unambiguously identify\nthe energy spectrum of a repulsive Fermi system.\nFor three “repulsively” interacting fermions in a har-\nmonic trap, we have shown that close to the resonance,\nthe ground state energy is higher than that of three fully\npolarized fermions. This is an indication of the existence\nof itinerant ferromagnetism in a trapped strongly repul-\nsively interacting Fermi gas. We have also considered\nthe possibility of itinerant ferromagnetism at high tem-\nperatures. We have found that it does not exist in the\nregime where a quantum virial expansion is applicable.\nThis gives an upper bound ( ∼TF) for the critical ferro-\nmagnetic transition temperature.\nOur high-temperature equations of state of a strongly\nrepulsively interacting Fermi gas have a number of po-\ntential applications. We anticipate that these results\ncan provide an unbiased benchmark for future quantum\nMonte Carlo simulations of strongly repulsively interact-\ning Fermi gases at high temperatures [33–35], using either\nhard-sphere interatomic potentials or resonance interac-\ntions. These results are also directly testable in future ex -\nperimental measurements, as inspired by the most recent\nthermodynamics measurement at ENS that have already\nconfirmed our predicted second and third virial coeffi-\ncients for strongly attractively interacting fermions [7] .\nOur exact three-fermion solutions in 3D harmonic traps\nwill also be useful for understanding the dynamical prop-\nerties of strongly interacting Fermi gases at high temper-\natures, by applying a similar quantum virial expansion\nfor the dynamic structure factors [36] and single-particle\nspectral functions [37].\nThese exact solutions of three interacting particles can\nbe generalized to other dimensions, by adopting a suit-\nable Bethe-Peierls boundary condition for the contact in-\nteractions. Of particular interest is the case of two di-\nmensions, where the reduction of the spatial dimension-\nality increases the role of fluctuations and therefore im-\nposes severe challenges for theoretical studies. The three -\nbody solutions in 2D and the resulting high-temperature\nequations of state of strongly interacting systems will\nbe given elsewhere, and provide a useful starting point\nto understanding more sophisticated collective phenom-\nena such as the Berezinsky-Kosterlitz-Thouless transitio n\nand non-Fermi-liquid behavior.\nNote added : On finishing this manuscript, we are\naware of a very recent work by Daily and Blume [38], in\nwhich the energy spectrum of three and four fermions has\nbeen calculated using hyperspherical coordinates with a\nstochastic variational appoach. Our exact results are in\nexcellent agreement with theirs when there is an overlap.13\nAcknowledgments\nThis work was supported in part by the ARC Centre\nof Excellence, ARC Discovery Project No. DP0984522\nand No. DP0984637, NSFC Grant No. 10774190, and\nNFRPC (Chinese 973) Grant No. 2006CB921404 and\nNo. 2006CB921306.\nAppendix A: Calculation of Cnn′\nIn this appendix, we outline the details of how to con-\nstruct the matrix element Cnn′in Eq. (23), which is\ngiven by,\nCnn′≡∞ˆ\n0ρ2dρRnl(ρ)Rn′l/parenleftBigρ\n2/parenrightBig\nψrel\n2b(√\n3\n2ρ;νl,n′),(A1)\nwhere\nRnl(ρ) =/radicalBigg\n2n!\nΓ(n+l+3/2)ρle−ρ2/2L(l+1/2)\nn/parenleftbig\nρ2/parenrightbig\n,(A2)\nis the radial wave function of an isotropic 3D harmonic\noscillator and the two-body relative wave function is\nψrel\n2b= Γ(−νl,n′)U(−νl,n′,3\n2,3\n4ρ2)exp(−3\n8ρ2).(A3)\nHere, for convenience we have set d= 1as the unit of\nlength.L(l+1/2)\nn is the generalized Laguerre polynomial\nandUis the second Kummer confluent hypergeometric\nfunction. A direct integration for Cnn′is difficult, since\nthe second Kummer function has a singularity at the ori-\ngin. The need to integrate for different values of νl,n′also\ncauses additional complications.\nIt turns out that a better strategy for the numerical\ncalculations is to write,\nψrel\n2b=∞/summationdisplay\nk=01\nk−νl,n′/radicalbigg\nΓ(k+3/2)\n2k!Rk0/parenleftBigg√\n3\n2ρ/parenrightBigg\n,(A4)\nby using the exact identity,\nΓ(−ν)U(−ν,3\n2,x2) =∞/summationdisplay\nk=0L1/2\nk/parenleftbig\nx2/parenrightbig\nk−ν. (A5)\nTherefore, we find that\nCnn′=∞/summationdisplay\nk=01\nk−νl,n′/radicalbigg\nΓ(k+3/2)\n2k!Cl\nnn′k, (A6)where\nCl\nnn′k≡∞ˆ\n0ρ2dρRnl(ρ)Rn′l/parenleftBigρ\n2/parenrightBig\nRk0/parenleftBigg√\n3\n2ρ/parenrightBigg\n(A7)\ncan be calculated to high accuracy with an appropriate\nintegration algorithm. In checking convergence of the\nsummation over k, we find numerically that for a cut-off\nnmax(i.e.,n,n′< nmax),Cl\nnn′kvanishes for a sufficient\nlargek>kmax∼4nmax.\nIn practical calculations, we tabulate Cl\nnn′kfor a given\ntotal relative angular momentum. The calculation of\nCnn′for different values of νl,n′then reduces to a simple\nsummation over k, which is very efficient. Numerically,\nwe have confirmed that the matrix Cnn′is symmetric,\ni.e.,Cnn′=Cn′n.\nAppendix B: Calculation of sl,n\nThe calculation of sl,nseems straightforward by using\nthe Bethe-Peierls boundary condition in hyperspherical\ncoordinates (28). However, we find that numerical accu-\nracy is low for large nandldue to the difficulty of calcu-\nlating the hypergeometric function 2F1accurately using\nIEEE standard precision arithmetic. We have therefore\nutilized MATHEMATICA software that can perform an-\nalytical calculations with unlimited accuracy. For this\npurpose, we introduce ∆sl,n=sl,n−¯sl,n. 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A 81, 053615\n(2010); also avaialbe as arXiv: 1006.0769 (2010)."    },    {        "title": "2101.02440v1.Abnormal_Critical_Fluctuations_Revealed_by_Magnetic_Resonance_in_the_Two_Dimensional_Ferromagnetic_Insulators.pdf",        "content": "Abnormal Critical Fluctuations Revealed by Magnetic Resonance in the\nTwo-Dimensional Ferromagnetic Insulators\nZefang Li,1, 2Dong-Hong Xu,1, 2Xue Li,1, 2Hai-Jun Liao,1, 3Xuekui Xi,1Yi-Cong Yu,1, 4,∗and Wenhong Wang1, 3,†\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2University of Chinese Academy of Sciences, Beijing 100049, China\n3Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China\n4State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,\nWuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology,\nChinese Academy of Sciences, Wuhan 430071, China\n(Dated: January 8, 2021)\nPhase transitions and critical phenomena, which are dominated by fluctuations and correlations,\nare one of the fields replete with physical paradigms and unexpected discoveries. Especially for\ntwo-dimensional magnetism, the limitation of the Ginzburg criterion leads to enhanced fluctuations\nbreaking down the mean-field theory near a critical point. Here, by means of magnetic resonance,\nwe investigate the behavior of critical fluctuations in the two-dimensional ferromagnetic insulators\nCrXTe 3(X = Si,Ge). After deriving the classical and quantum models of magnetic resonance, we\ndeem the dramatic anisotropic shift of the measured gfactor to originate from fluctuations with\nanisotropic interactions. The deduction of the gfactor behind the fluctuations is consistent with the\nspin-only state ( g≈2.050(10) for CrSiTe 3and 2.039(10) for CrGeTe 3). Furthermore, the abnormal\nenhancement of gshift, supplemented by specific heat and magnetometry measurements, suggests\nthat CrSiTe 3exhibits a more typical two-dimensional nature than CrGeTe 3and may be closer to\nthe quantum critical point.\nFluctuations and correlations drive abundant phase\ntransitions and critical phenomena. Regardless of the\nclassical or quantum regime or of the order parameter\nand symmetry, they are universal in nature and follow\nstatistical laws [1]. Among them, a particularly fasci-\nnating aspect of two-dimensional (2D) magnetism asso-\nciated with strong intrinsic magnetization fluctuations\nhas introduced rich physical paradigms: the quantum\nspin liquid (QSL) state of the Kitaev model, Berezinskii-\nKosterlitz-Thouless (BKT) transition of the XY model,\nMermin-Wagner theorem of the isotropic Heisenberg\nmodel, Ising transition, etc. [2]. Herein, recent discover-\nies of magnetic van der Waals (vdW) materials provide\nthe ideal platform for exploring intrinsic 2D magnetism\ndown to the 2D limit and potential opportunities for new\nspin-related applications [3].\nNotably, 2D magnetism is particularly susceptible to\nfluctuations. The Ginzburg criterion indicates that fluc-\ntuations become much more relevant with decreasing di-\nmensions, leading to the failure of mean-field theory [4].\nThe Mermin-Wagner theorem recognizes that no long-\nrange order can survive thermal fluctuations at finite\ntemperature in a 2D system with continuous symme-\ntry [5]. However, by breaking the continuous symmetry,\nanisotropy in the exchange interaction will open up the\nspin wave excitation gap to resist thermal agitations of\nmagnons at finite temperature. Such notable examples of\nmagnetic order in single atomic layers have been discov-\nered in CrI 3[6], CrGeTe 3[7], Fe 3GeTe 2[8] and VSe 2[9].\n∗ycyu@wipm.ac.cn\n†wenhong.wang@iphy.ac.cnMoreover, 2D magnetism is associated with strong intrin-\nsic competition between quantum fluctuations and ther-\nmal fluctuations [10]. In the ground state where thermal\nfluctuations vanish, the quantum fluctuations demanded\nby Heisenberg’s uncertainty principle will dominate the\nquantum phase transition (QPT), which is driven by\nsome nonthermal external parameters such as the mag-\nnetic field, pressure, or chemical doping [11]. At finite\ntemperature, the energy of a system and the enthalpy\nof its thermal fluctuations compete, resulting in a clas-\nsical phase transition (CPT). Although fluctuations play\na crucial role in 2D magnetism, most of the theoreti-\ncal predictions by ab initio methods are based on zero\ntemperature and ignore fluctuations. Obtaining phase\nboundary information of fluctuations and critical points\nthrough experimental detection is very important.\nHere, we demonstrate magnetization-fluctuation-\ninduced effective gfactor anisotropy in the 2D ferromag-\nnetic insulators CrXTe 3(X = Si,Ge) by means of ferro-\nmagnetic resonance (FMR) and electron paramagnetic\nresonance (ESR). In general, the dominant critical fluc-\ntuations occur at the critical temperature Tcand decay\nexponentially when deviating from Tc. Compared with\nCrGeTe 3(CGT), the observation of critical fluctuations\nwith enhanced intensity and broad temperature range\nin CrSiTe 3(CST) is abnormal, which is associated with\nthe 2D nature even in the bulk counterparts. Although\nthe critical behavior can be indirectly characterized by\nneutron scattering [12, 13], magnetic susceptibility mea-\nsurement [14, 15], specific heat measurement [16], nuclear\nmagnetic resonance [17] and the dynamic magnetoelec-\ntric coupling technique [18] and directly characterized by\nreal-time magneto-optical imaging technology [19], accu-arXiv:2101.02440v1  [cond-mat.mes-hall]  7 Jan 20212\nrately estimating the temperature dependence of magne-\ntization fluctuations by means of magnetic resonance is\nvery exciting.\nFIG. 1. (a) Crystalline structure of bulk rhombohedral\nCrXTe 3(X = Si,Ge) with ABC vdW stacking. (b) Schematic\nof different coordinate systems from the top view of the ab\nplane: crystallographic axes a, b, and c and FMR coordinate\naxes x, y, and z. (c) Magnetic ion Cr3+surrounded by a\ndistorted octahedral crystal field with an out-of-plane arrow\nrepresenting its easy magnetization direction. Split d levels\nin the e gand t 2gmanifolds, with 3 unpaired electrons in the\ndxy,dyz, anddxzorbitals.\n2D vdW CST and CGT are ferromagnetic insulators\nand belong to the family of layered vdW transition metal\ntrichalcogenides (TMTCs), which are crystallized in the\nR¯3 (148) rhombohedral structure. Fig. 1 shows the hon-\neycomb ABC layers stacked by a cleavable vdW gap. In\neach layer, the magnetic Cr3+ions and Ge3+/Si3+pairs\nare located at Te2−octahedral sites in a distorted D3d\nlocal symmetry, with a crystal field splitting the Cr-3 d3\norbitals and sustaining Cr-Te-Cr ferromagnetic superex-\nchange. As shown in Fig. 1(c), three unpaired electrons\nof Cr3+are accommodated in the lower t2gtriplet or-\nbital, thus resulting in a quenched orbital moment for\nspinsJ=S= 3/2 and agfactor near 2.0023 for free\nelectrons. Especially for CST, the strong Coulomb in-\nteractions from the narrow d bands and the half-filled\ncondition favor a Mott transition [20]. As further shown\nin Fig. 1(b), these octahedra arrange in edge-bond-\nsharing networks in the ab plane and form a magnetic\nhoneycomb lattice. The interplay of spin-orbit coupling\nand the crystal field currently explains the uniaxial mag-\nnetic anisotropy with an easy axis perpendicular to the\nab plane [21]. However, controversy about the uncer-\ntainty among the Kitaev, Ising, Heisenberg and single-\nion anisotropy terms remains. CGT has been demon-\nstrated to be well described by the Heisenberg behav-\nior with a single-ion anisotropy term, which has been\nproven to exhibit ferromagnetic order in the monolayer\n[7, 14]. In contrast, CST with giant magnetic anisotropy\nis determined to be consistent with the 2D Ising behav-\nior [12, 15], for which the ground state of the monolayerstill lacks experimental confirmation. Moreover, in struc-\nturally related CrI 3[22, 23],α−RuCl 3[24] and Na 2IrO3\n[25], Kitaev anisotropic exchange interactions are found\nin competition with Heisenberg interactions, which are\nassociated with a possibly QSL state. Recently, first-\nprinciples-based simulations predicted the possible Ki-\ntaev QSL state in epitaxially strained CST monolayers\n[26, 27]. After comprehensive consideration, we consider\nan XXZ Hamiltonian with single-ion anisotropy:\nH=−1\n2/summationdisplay\n/angbracketleftj,l/angbracketright(JSj·Sl+ ΛSz\njSz\nl)\n−/summationdisplay\njASz\njSz\nj−µBH·g·/summationdisplay\njSj.(1)\nThe first term corresponds to the Heisenberg isotropic\nexchangeJand the anisotropic symmetric exchange Λ.\nThe second term is the additional single-ion anisotropy\nterm, and the last term corresponds to the Zeeman en-\nergy.J > 0 favors ferromagnetic coupling, and A > 0\nfavors the out-of-plane easy axis. Setting Λ /Jto infinity\nrecovers the 2D Ising model, while the isotropic Heisen-\nberg model is recovered for Λ ≈0 andA≈0.\nTABLE I. Physical quantities extracted from magnetometry\nand specific heat measurements [28].\nCrSiTe 3 CrGeTe 3\nSpace group R¯3(148)R¯3(148)\nCritical temp. 34.15 68.15 Derived MT\nTc(K) 32.50 65.50 Arrott plot\n32.68 64.90 Specific heat\nCurie-Weiss temp. 57.16 101.46 H/bardblc\nΘ (K) 53.86 100.24 H/bardblab\nFrustration param. 1.67 1.49 H/bardblc\nf 1.58 1.47 H/bardblab\nEffective mag.a4.00(5) 4.03(4) H/bardblc\nµeff(µB) 3.97(6) 3.99(6) H/bardblab\nSaturation mag.b3.00 3.08 H/bardblc\nMs(µB/f.u.) 2.84 3.02 H/bardblab\nMag. entropyc3.91 0.86\nδS(J/mol K) 48.67% 38.85% above Tc\nCritical exponentd0.171(6) 0.221(2) β\n1.461(11) 1.416(45) γ\n9.684(13) 7.287(12) δ\naExpected value µeff=g/radicalbig\n(J(J+ 1))µB= 3.87µB\nbExpected value Ms=gJµB= 3.00µB/f.u.\ncExpected value Rln(2J+ 1) = 11.53 J/mol K\ndCST close to 2D Ising model ( β= 0.125,γ= 1.75);\nCGT close to tricritical mean-field model ( β= 0.25,γ= 1.0)\nThe presence of critical fluctuations near the critical\npoint can be evidenced by magnetometry and specific\nheat measurements [28]. As shown in Table I, Curie-\nWeiss behavior is observed at high temperatures, and\nan estimate of the effective moment gives µeff≈4µB,\nconsistent with the spin-only magnetic moment µeff=\ng/radicalbig\n(J(J+ 1))µB= 3.87µBforJ=S= 3/2 and\ng= 2.0023. A deviation from the Curie-Weiss fit be-3\nFIG. 2. (a) Coplanar waveguide with a rectangular single\ncrystal placed parallel to the ab plane. The domain wall is\nrepresented by a blue solid line, and the angle with the y axis\nisα. (b) Illustration of three different resonant modes from\nthe side view. (c) Frequency- and field-dependent FMR spec-\ntra (in-plane, β=π/2) for CrSiTe 3at 5 K. The solid lines\nrepresent fitting for the single domain mode (fitted with Eq.\n2) and multidomain mode (fitted with Eq. 3). (d) Typical mi-\ncrowave transmission at 28 GHz sliced from (c). The resonant\npeak of the multidomain mode is formed by the superposition\nof a series of peaks with different angles α.\nlow 150 K can be recognized in the M(T) curves, result-\ning in a higher Weiss temperature Θ than the critical\ntemperature Tc. The ratio is defined as the frustration\nparameterf=|Θ|/Tc, which corresponds to short-range\nferromagnetic correlations persisting in the paramagnetic\nstate. Moreover, the magnetic entropy S(T) aboveTc, es-\ntimated from the magnetic specific heat Cm(T), recovers\nto a value as large as nearly 48 .67% (CST) and 38 .85%\n(CGT) of the total spin entropy. Such short-range corre-\nlations were indeed directly detected by elastic neutron\nscattering measurements in CST, as the measured corre-\nlation length remained larger than the nearest-neighbor\ndistance up to 250 K [13]. Furthermore, the critical expo-nents are obtained from modified Arrott plots extracted\nfrom the isotherm M(H) curves, which well match the\n2D Ising model for CST and the tricritical mean-field\nmodel for CGT. The above evidence implies that CST\nhas a stronger critical fluctuation and an exchange inter-\naction closer to a 2D nature, but further verification is\nneeded.\nOn the basis of the correspondence principle, the classi-\ncal and quantum mechanical descriptions of the magnetic\nresonance are identical. Therefore, concrete expressions\nfor the free energy and Hamiltonian are necessary to re-\nveal the physics behind an observable. Here, we recover\nthe resonance solutions of the Larmor equation (classical)\nand Heisenberg equation of motion (quantum) for a gen-\neral case (more details are provided in the supplementary\nmaterials) [28–30]. We conclude that the anomaly of the\nmeasured effective g factor, as well as the anomaly of the\nrelationship between magnetocrystalline anisotropy Ku\nand saturation magnetization Ms, can be explained by\nthe specificity of fluctuations and correlations, which are\nunsettled in previously reported FMR measurements of\nCrI3[22], CrCl 3[31, 32], and CrGeTe 3[33–35].\nOne of the important correspondences is the spectro-\nscopicgfactor, which can be determined precisely by gy-\nromagnetic ratio fitting using FMR and ESR spectra. As\na classical description, shown in Fig. 2(a), a rectangular-\nshaped single crystal is placed in a coplanar waveguide\nwhere it is acted upon by an alternating magnetic field\nHrf. In response to scanning of a strong homogeneous\nmagnetic field Hextat right angles, resonance absorption\nsignals can be detected in the case that ωres=γHeff,\nwhereγ=gµB//planckover2pi1is the gyromagnetic ratio and Heffis\nthe effective internal field. When resonance occurs, the\nsaturation magnetization Msinduces Larmor precession\nalong the effective field direction. In consideration of\nthe Zeeman splitting energy, crystallographic anisotropy,\ndemagnetizing field, and Bloch domain structure, the so-\nlutions to the Larmor equations are given by the Smit-\nBeljers approach [28, 29]:\n/parenleftbiggωres\nγ/parenrightbigg2\n={H−[HA−(Nz−Ny)Ms]}{H−(Ny−Nx)Ms}, (2)\nfor single-domain mode, ϑ0=ϕ0=π\n2, H≥HA+NyMs.\n/parenleftbiggωres\nγ/parenrightbigg2\n= (HA+NxMs)(HA+Mssin2α)−(HA+Mssin2α−NzMs)(HA+NxMs)\n(HA+NyMs)2H2, (3)\nfor multidomain mode, ϕ10=ϕ20=π\n2,sinϑ10= sinϑ20=H\nHA+NyMs, H≤HA+NyMs.\nwhereHA= 2K/M sis the anisotropic field, Nis the\ndemagnetization factor, and αis the angle between thedomain wall and the external magnetic field. When the\nmagnetic field Hextis applied parallel to the ab plane,4\nwe observe three different resonant modes (illustrated in\nFig. 2(b)), which are plotted as a function of excitation\nfrequency and applied magnetic field in Fig. 2(c). The\ndomain wall resonance peaks excited under weak fields\nare much smaller in amplitude than the FMR peaks.\nTheir dependence on the resonance frequency versus the\nin-plane field corresponds to the conventional theory for\nthe Bloch wall model [36]. Remarkably, the multido-\nmain mode has a continuously changing angle α, result-\ning in asymmetric peak shapes (Fig. 2(d)). The cross-\ning point of the multidomain mode and single-domain\nmode indicates the saturation field of the domain struc-\nture (H=HA+NyMs), which exists below the critical\ntemperature Tc. Moreover, when the magnetic field Hext\nis applied parallel to the c axis, the resonance frequency\ncan be determined by the sum of the external field, the\nequivalent anisotropy field, and the demagnetizing field:\nωres\nγ=H+HA−NzMs,forϑ= 0. (4)\nFIG. 3. (a,b) Temperature dependence of the effective gfactor\nwith the field applied both in-plane and out-of-plane for CST\nand CGT. The real gfactor is calculated with Eq. 6. (c)\nReduced anisotropy constant and magnetization at different\ntemperatures shown on the logarithmic scale. The red line\nindicates an exponent of 3 for the Callen-Callen power law.\n(d) Schematic phase diagram showing the paramagnetic and\nferromagnetic phases. The vertical paths for CST and CGT\nrepresent the CPT.\nHerein, we extract the spectroscopic gfactor by fitting\nthe in-plane and out-of-plane FMR and ESR data with\nthe above equations (more details are provided in the\nsupplementary materials) [28]. As shown in Fig. 3(a,b),\nthe temperature dependence of the effective gfactor for\nthe in-plane ( H/bardblab) and out-of-plane ( H/bardblc) orien-\ntations has a contrasting behavior. A downwards (in-\nplane) or upwards (out-of-plane) shift of the gfactor\nis observed as the temperature increases, with a maxi-\nmum value at Tc. Notably, the deviation of the gfac-\ntor is beyond an orbital contribution, which is almostcompletely quenched due to the crystal field [37]. Such\na temperature-dependent shift in the gfactor has been\nfound in ESR measurements of low-dimensional metal al-\nloys, metal complexes, or purely organic compounds [38].\nBased on Nagata’s theory in the ESR case [30, 39–41], we\nstrictly solve a general solution in the FMR case and con-\nclude that magnetization fluctuations with anisotropic\ninteractions are responsible for the gshift (more details\nare provided in the supplementary materials) [28]. To\nbe more specific, the Hamiltonian is substituted into the\nprecession motion equation:\n/planckover2pi1ω=/angbracketleft[S−,[S+,H]]/angbracketright\n2/angbracketleftSz/angbracketright. (5)\nAfter calculating the thermodynamic average, we find\nthat the isotropic Heisenberg term Jdoes not contribute\nto thegshift, whereas the anisotropic symmetric ex-\nchange Λ and single-ion anisotropy term Ado. The ab-\nsolute value of the gshift along the easy axis of mag-\nnetization ∆ gcis twice that along the orthogonal hard\nplane ∆gab. By taking the value-weighted average, we\ncan obtain the real gfactor of the sample:\ng=1\n3×gab+2\n3×gc. (6)\nAs shown in Fig. 3(a,b), the calculated gfactor is a con-\nstant value independent of temperature. After averaging\nthegfactors over the entire temperature range, we obtain\nthe averaged gfactors as 2.050(10) for CST and 2.039(10)\nfor CGT, which are consistent with the orbital-quenched\ngfactors.\nIn addition, thermal fluctuations lead to an effec-\ntive reduction in both the saturation magnetization and\nmagnetocrystalline anisotropy, which can be represented\nby the Callen-Callen power law based on the single-ion\nanisotropy model:\nKu(T)\nKu(0)=/bracketleftbiggMs(T)\nMs(0)/bracketrightbiggl(l+1)/2\n, (7)\nwherelis the order of spherical harmonics and depends\non the symmetry of the crystal. In the case of uniaxial\nanisotropy for CST and CST, l= 2 and an exponent of\n3 are expected. However, as shown in Fig. 3(c), CGT\nshows little agreement with the power law for exponents\nof 2.37(2) (FMR) and 2.51(3) (SQUID). In contrast, CST\nexhibits nonlinear behavior, which obviously violates the\npower law. Hence, the departure from the Callen-Callen\npower law suggests that the consideration of thermal fluc-\ntuations for single-ion anisotropy is incomplete. This is\nconsistent with the fact that the single-ion anisotropy for\nCr3+is sufficiently small due to the weak spin-orbit cou-\npling (ξL·S) with quenched orbital angular momentum\n(L≈0).\nTo illustrate the concepts, we consider the schematic\nphase diagram shown in Fig. 3(d), where Tis the temper-\nature andgis the strength of the ferromagnetic exchange5\ncoupling. On the one hand, the curve of the FM and PM\nphase boundary corresponds to the critical temperature\nTc. The CPT occurs by varying the temperature through\nTc. In the classical critical region, the correlation length\ntends to infinity, and critical fluctuations are dominant.\nOn the other hand, changing gin the ground state will\nlead to a QPT at the quantum critical point gc, where\nthe quantum fluctuations are the strongest. According to\nthe results of our experiment, the critical temperature of\nCST is relatively low, and the fluctuations observed are\nmuch stronger than those for CGT. Therefore, we can\nreasonably indicate that CST is closer to the quantum\ncritical point gc, which is dominated by both classical\nand quantum critical behavior. This inference is also sup-\nported by a recent report on pressure-induced supercon-\nductivity in CST [42]. We believe that doping, pressure,\ncleavage, and electrical regulation can achieve a QPT in\nCST, but more experimental verification is needed.\nIn summary, we have combined magnetic resonance,\nspecific heat and magnetometry measurements to inves-\ntigate the behavior of critical fluctuations in bulk CST\nand CGT single crystals. Although fluctuations near the\ncritical temperature are natural in magnetic materials,\nthe observation of such anisotropic shifts of resonancepeaks in low-dimensional systems is unique because of\nthe Ginzburg criterion. Despite the structural and elec-\ntronic similarities, CST and CGT show strong contrasts\nin critical behavior. Our work implies the presence of\nshort-range correlation far above Tcand a signally 2D\nnature even in bulk counterparts of CST. Although CST\nshows a stronger magnetic anisotropy, the absence of fer-\nromagnetic order in the monolayer should be attributed\nto the enhanced fluctuations. Last but not least, such\nunignorable magnetization fluctuations in 2D magnetic\nmaterials will interact with the spins of scatterers (X-\nrays, neutron beams, spin currents, etc.) and enhance\nthe scattering effect. 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B 102,\n144525 (2020).Supplementary Materials for\nAbnormal Critical Fluctuations Revealed by Magnetic Resonance in Two Dimensional\nFerromagnetic Insulators\nZefang Li,1, 2Dong-Hong Xu,1, 2Xue Li,1, 2Hai-Jun Liao,1, 3Xuekui Xi,1Yi-Cong Yu,1, 4,∗and Wenhong Wang1, 3,†\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2University of Chinese Academy of Sciences, Beijing 100049, China\n3Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China\n4State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,\nWuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology,\nChinese Academy of Sciences, Wuhan 430071, China\n(Dated: January 8, 2021)\nCONTENTS\nI. Crystal growth, Magnetometry and Specific heat 1\nA. Methods 1\nB. M(T) and M(H) curves 2\nC. Heat capacity 3\nD. Arrott plot 4\nII. Derivation of Ferromagnetic Resonance Model and Experimental Fitting 5\nA. Single-domain mode 5\nB. Multi-domain mode 6\nC. Measured FMR spectra and experimental fitting 7\nIII. Quantum explanation of fluctuations induced g-factor anisotropy 8\nA. Solution for a general model applicable for FMR and ESR case 8\nB. Solution for the XXZ model with single ion anisotropy 12\nReferences 12\nI. CRYSTAL GROWTH, MAGNETOMETRY AND SPECIFIC HEAT\nA. Methods\nHigh quality CrSiTe 3(CST) and CrGeTe 3(CGT) single crystals were grown by the self-flux method [1, 2]. The\nmixture of pure elements (Cr : Si : Te = 1 : 2 : 6 ,Cr : Ge : Te = 10 : 13 .5 : 76.5) were mounted in an alumina\ncrucible, which was sealed inside a quartz tube under high vacuum ( <10−4Pa). Then the quartz tube is heated up\nto 1100◦C in a tube furnace and slowly cooled down to 700◦C. Excessive molten flux was centrifuged quickly before\nsolidification. The final hexagonal flakes were shiny and soft, and were easy to peel off.\nEnergy dispersive X-ray spectroscopy (EDXS, equipped in Hitachi S-4800 microscope) was used for element analy-\nsis. As well as the X-ray diffraction (XRD, Bruker D8 Anvance) proved the stoichiometric ratio and high quality of\nCGT and CST single crystals. We also measured the heat capacity at zero field (PPMS-9T, Quantum Design physi-\ncal properties measurement system) and characterized the temperature and field dependent magnetization (MPMS,\nQuantum Design magnetic property measurement system). In consideration of the demagnetization effect, it should\nbe noted that the external applied field has been corrected for the internal magnetic field as Hint=Hext−NM,\nwhereNis the demagnetization factor [3] and Mis the measured magnetization.\n∗ycyu@wipm.ac.cn\n†wenhong.wang@iphy.ac.cnarXiv:2101.02440v1  [cond-mat.mes-hall]  7 Jan 20212\nFIG. 1. (a), (b) Temperature dependence of the magnetization and its derivative measured in ZFC, FC and FW modes with a\nfield of 100 Oe. The inset shows the enlarged picture of splitting between H/bardblabandH/bardblc. (c), (d) Inverse of the magnetic\nsusceptibility for FC curve. The black solid lines indicate the linear fits in paramagnetic region with Curie-Weiss law. (e), (f)\nMagnetization as function of field at ferromagnetic and paramagnetic state respectively.\nB. M(T) and M(H) curves\nThe magnetometry results are listed in Fig. 1. Temperature dependent zero field cooled (ZFC), field cooled (FC),\nand field warmming (FW) with an external field of H= 100 Oe both parallel to c axis and ab plane are shown in\nFig. 1(a) and Fig. 1(b). The paramagnetic-ferromagnetic (PM-FM) transitions occurs at the Curie temperature\n(34.15 K for CST and 68 .15 K for CGT) are determined by the derivative of M(T) curve, which are the highest among\npreviously reported values [1, 2, 4–6] and indicating a better sample quality. These two samples both exist a “thermo-3\nhysteresis” between FC and FW curves ( H/bardblc) belowTc. As reported in CrGeTe 3[7] and Fe 3GeTe 2[8], this anomalous\nbehavior is related to the existence of skyrmions, but has not been confirmed in CrSiTe 3. And then Fig. 1(c) and\nFig. 1(d) display the temperature dependent inverse susceptibility H/M under FC, parallel to c axis and ab plane\nrespectively. A linear fit of Curie-Weiss law at PM state yields the Weiss temperature Θ and effective moment µeff.\nAbove 150 K, the fitted effective magnetic moment is close to the expected value µeff=g/radicalbig\n(J(J+ 1))µB= 3.87µB.\nBelow 150 K, however, the H/M curves deviate from the linear fit, resulting a higher Weiss temperature Θ than\ncritical temperature Tc. And the ratio of them is defined as frustration parameter f=|Θ|/Tc. This suggests that\nshort-range magnetic correlations persist in PM state above Tc. Fig. 1(e) and Fig. 1(f) show the field dependence of\nmagnetization measured at FM and PM state. The M(H) curves splitting at 5K confirm the easy c-axis and larger\nmagnetocrystalline anisotropy in CST than CGT. And the saturation magnetization fitted by the law of approaching\nsaturation is very close to the expected value Ms=gJµ B= 3.00µB/f.u..\nC. Heat capacity\nFIG. 2. (a), (b) Temperature dependence of zero-field specific heat for CrSiTe 3and CrGeTe 3. The blue line is the fitting of\nlattice specific heat by Thirring model. The inset represents the magnetic contribution of Cp/T versus T and the integration\nof magnetic entropy S after subtracting the lattice contribution.\nFig. 2 shows specific heat data at zero field. As expected, the λ-shaped anomaly is observed at PM-FM transition.\nIn order to compare the entropy change associated with magnetism, we subtract the non-magnetic contributions due\nto lattice vibrations. For this purpose, the lattice specific heat is estimated with the Thirring model [9]:\nClattice = 3NR\n1 +∞/summationdisplay\nn=1bn/parenleftBigg/parenleftbigg2πT\nθD/parenrightbigg2\n+ 1/parenrightBigg−n\n. (1)\nwhereNis the number of atoms in the unit cell, Ris the ideal gas constant, θDis the Debye temperature. We\nusenup to 4 for fitting and obtain a reasonable accuracy as shown by the blue solid line. After subtracting the\nlattice contribution, the inset shows the magnetic contribution Cp/TversusTand the magnetic entropy obtained\nfrom numerical integration. In general, the molecular field results in the second-order phase transition with a jump\nin specific heat. However, the fractions of the magnetic entropies gained above Tcare 48.67% for CST and 38 .85%\nfor CGT. This behavior is attributed to short-range magnetic correlations between the moments of nearest-neighbor\natoms.\nAt Zero field, the 2 J+ 1 energy states of Nnon-interacting magnetic moments consist of W= (2J+ 1)Navailable\nstates. Therefore the corresponding entropy can be calculated by Boltzman’s theory:\nS=klnW=Nkln(2J+ 1) =Rln(2J+ 1) = 11.53J/mol K for J=S= 3/2.\nDue to short-range magnetic correlations, the measured entropy changes (3 .91 J/mol K for CST and 0 .86 J/mol K\nfor CGT) are relatively small.4\nD. Arrott plot\nFIG. 3. : (a), (d) Modified Arrott plot of isotherms. The dashed line is the linear fit of isotherm at critical temperature. (b),\n(e) Temperature dependence of Msandχ. The Tc and critical exponents are obtained from the fitting of Eq. 2 and 3. (c), (f)\nIsothermal MH atTc. The inset shows the fitting of Eq. 4 and the extracted exponents.\nThe second-order PM-FM phase transition near the critical point can be characterized by a set of critical exponents\nβ,γ, andδ. The spontaneous magnetization Ms(T) belowTc, the inverse initial susceptibility χ−1\n0(T) aboveTcand\nthe measured magnetization M(H) atTcfollow the power law:\nMs(T)∝(Tc−T)β, T <T c (2)5\nχ−1\n0(T)∝(T−Tc)γ, T >T c (3)\nM∝H1/δ, T =Tc (4)\nIt is known that the critical exponents are not independent of each other but follow the scaling relation:\nδ= 1 +γ\nβ(5)\nTherefore, we apply a modified Arrott plot by self-consistent iteration to determine the critical exponents and phase\ntransition temperature Tc[10]. As shown in Fig. 3(a) and (b), isotherms plotted in form of M1/βversus (H/M )1γ\nconstitute a set of parallel straight lines, and the isotherm at Tcwill pass through the origin. The relationship is given\nby the following equation:\n/parenleftbiggH\nM/parenrightbigg1/γ\n=aT−Tc\nT+bM1/β(6)\nwhereaandbare constants. Linear fitting of the isotherms at high field region gives ( Ms)1/βand (χ−1\n0)1/γas an\nintercept on M1/βand (H/M )1/γ. According to Eq. 2 and 3, linear fitting of log[ Ms(T)] versus log( Tc−T) and\nlog[χ−1\n0(T)] versus log( T−Tc) give new values of βandγ, while free parameter Tcis adjusted for best fitting. And\nthen we substitute the new values into the Arrott plot and iterate repeatedly to get the optimal solution.\nIn order to test the accuracy of the final solution, we have analyzed the ( Ms)1/βand (χ−1\n0)1/γdata by Kouvel-\nFisher (KF) plot [11]. As shown in Fig. 3(c) and (d), the values obtained are within the error accuracy. Furthermore,\naccording to Eq. 4, linear fitting of log[ M(H)] versus log( H) gives a straight line with slope 1 /δ. And it is noteworthy\nthat these obtained β,γ, andδare related in the scaling relation of Eq. 5.\nThese obtained critical exponents for CST ( β= 0.171(6),γ= 1.461(11)) can be explained by the 2D Ising model\n(β= 0.125,γ= 1.75), while CGT ( β= 0.221(2),γ= 1.416(45)) can be explained by tricritical mean-field model\n(β= 0.25,γ= 1.0). Our conclusions are consistent with with previous reports of neutron [12, 13] and magnetic\nmeasurements [2, 4–6].\nII. DERIVATION OF FERROMAGNETIC RESONANCE MODEL AND EXPERIMENTAL FITTING\nA. Single-domain mode\nConsidering a regular shaped magnetic sample in the uniform magnetic field, the motion of magnetization vector\nMfollows the Larmor equation:\ndM\ndt=−γ[µ0M×Heff]. (7)\nThe effective magnetizing field Heffis determined by the free energy Fper unit volume. In single-domain case, it can\nbe represented in the sum of Zeeman energy, the demagnetization energy, and magnetocrystalline anisotropy energy:\nF=−µ0MsHsinϑsinϕ+Ksin2θ+1\n2µ0M2\ns(Nxsin2ϑcos2ϕ+Nysin2ϑsin2ϕ+Nzcos2θ). (8)\nIn whichKis the magneto-crystalline anisotropy, and Nis demagnetization factor. The equilibrium orientation of the\nmagnetization vector Ms(ϑ0,ϕ0) are determined by ∂F/∂ϑ = 0,∂F/∂ϕ = 0. When the vector Msdeviates slightly\nfrom the equilibrium position, the total free energy can be Taylor expanded as F(ϑ0+δϑ,ϕ 0+δϕ) =F(ϑ0,ϕ0) +\n1\n2(Fϑϑδϑ2+2Fϑϕδϑδϕ +Fϕϕδϑ2). In this case, the internal effective field are defined as Hϑ=−(µ0Ms)−1∂F/∂ϑ,H ϕ=\n−(µ0Mssinϑ0)−1∂F/∂ϕ . Considering that Larmor precession has periodic solution δϑ,δϕ∝eiωt, we obtain the\nequations of motion in the polar coordinate system as the following matrix form:\n/parenleftbigg\nFϕϑ+iωγ−1µ0Mssinϑ0 Fϕϕ\nFϑϑ Fϑϕ−iωγ−1µ0Mssinϑ0/parenrightbigg/parenleftbigg\nδϑ\nδϕ/parenrightbigg\n=/parenleftbigg\n0\n0/parenrightbigg\n. (9)\nWhereFϑϕ=∂2F/∂ϑ∂ϕ . The equation of motion has a non-zero solution only when the determinant is zero. Defining\nthe anisotropic field HA= 2K/Ms, then we can get the resonance frequency as6\n/parenleftbiggωres\nγ/parenrightbigg2\n=/braceleftbig\n[HA−(Nz−Ny)Ms]2−H2/bracerightbigHA−(Ny−Nx)Ms\nHA−(Nz−Ny)Ms,\nfor sinϑ0=H\nHA+ (Ny−Nz)Ms,ϕ0=π\n2,H <HA+ (Ny−Nz)Ms.(10)\n/parenleftbiggωres\nγ/parenrightbigg2\n={H−[HA−(Nz−Ny)Ms]}{H−(Ny−Nx)Ms},\nforϑ0=ϕ0=π\n2,H >HA+ (Ny−Nz)Ms.(11)\nB. Multi-domain mode\nFor CST and CGT single crystals with hexagonal lattice and uniaxial anisotropy, the Bloch magnetic domain\nstructure was observed experimentally below the saturation field [7, 14]. It should be pointed out that Smit and\nBeljers first derived the multi-domain FMR model in BaFe 12O19for a square-shaped sample. But a mistake in the\nderivation led to an unexplainable item in the result [15]. Here we recover the derivation and generalize to samples\nof arbitrary shape.\nFor simplicity, we assume that the magnetic domains of adjacent domains have two kinds of magnetization\nM1(ϑ1,ϕ1) and M2(ϑ2,ϕ2). The demagnetization energy of adjacent domains with infinitely thin thickness is equal\nto\n1\n2/parenleftbigg(M1−M2)·en\n2/parenrightbigg2\n. (12)\nIn considering the multi-domain case, the angle between the domain wall and the external magnetic field is α, and\nthe expression of free energy is equal to:\nF=−1\n2µ0MsH(sinϑ1sinϕ1+ sinϑ2sinϕ2) +1\n2K(sin2ϑ1+ sin2ϑ2)\n+1\n2µ0M2\ns/bracketleftbiggNx\n4(sinϑ1cosϕ1+ sinϑ2cosϕ2)2+Ny\n4(sinϑ1sinϕ1+ sinϑ2sinϕ2)2\n+Nz\n4(cosϑ1+ cosϑ2)2+1\n4(sinϑ1cos(ϕ1−α)−sinϑ2cos(ϕ2−α))2/bracketrightbigg\n.(13)\nSimilarly, near the equilibrium position Ms(ϑ10,ϕ10,ϑ20,ϕ20), we can get the equation of motion in matrix form:\n\nF11F12−iΩ/2F13F14\nF21+iΩ/2F22F23F24\nF31F32F33F34−iΩ/2\nF41F42F43+iΩ/2F44\n\nδϑ1\nδϕ1\nδϑ2\nδϕ2\n=µ0MsH/2\ncosβcosϑ10\nsinβsinϑ10\ncosβcosϑ20\nsinβsinϑ20\n. (14)\nWhere Ω = ωγ−1µ0Mssinϑ10,F12=∂2F/(∂ϑ10∂ϕ10), andβis the angle between HrfandHext. In considering of\nthe symmetry, we have F11=F33,F22=F44,F12=−F14=−F23=F34. For convenience we put ∆ ϑ±=δϑ1±δϑ2,\n∆ϕ±=δϕ1±δϕ2. Then we get\n\nF11+F13−iΩ/2 0 0\niΩ/2F22+F24 0 0\n0 0 F11−F132F12−iΩ/2\n0 0 2 F12+iΩ/2F22−F24\n\n∆ϑ+\n∆ϕ+\n∆ϑ−\n∆ϕ−\n=µ0MsH\n0\nsinβsinϑ10\ncosβcosϑ20\n0\n. (15)\nTherefore, the 4×4 matrix can be split into two 2 ×2 matrices, corresponding to two situations respectively:\n(a) whenHrf⊥Hext (β=π/2)\n/parenleftbiggωres\nγ/parenrightbigg2\n= (HA+NxMs)(HA+Mssin2α)−(HA+Mssin2α−NzMs)(HA+NxMs)\n(HA+NyMs)2H2,\nfor sinϑ10= sinϑ20=H\nHA+NyMs,ϕ10=ϕ20=π\n2.(16)7\n(b) whenHrf/bardblHext (β= 0)\n/parenleftbiggωres\nγ/parenrightbigg2\n= (HA+NyMs)(HA+Mscos2α)−HA+Mscos2α\nHA+NyMsH2−M2\nssin2αcos2α/parenleftbigg\n1−H2\n(HA+NyMs)2/parenrightbigg\n,\nfor sinϑ10= sinϑ20=H\nHA+NyMs,ϕ10=ϕ20=π\n2.(17)\nC. Measured FMR spectra and experimental fitting\nFIG. 4. : (a-m) Frequency and field dependent ferromagnetic resonance spectra for CrSiTe 3. The color maps show the in-plane\nresonance spectra, while the white squares are the out-of-plane resonant peaks added afterwards. (n) Optical photograph of\nthe single crystal used in the resonance experiment.\nAs shown in Fig. 4 and 5, broadband ferromagnetic resonance experiments were carried out on a home-made\ncoplanar waveguide (CPW) sample rod, which was adapted to the magnetic field and temperature control system of\nPhysical Property Measuring System (PPMS, Quantum Design). FMR spectra was recorded with a vector network\nanalyzer (ZVA 40, Rohde & Schwarz) in transmission mode (S12) over a frequency range of 1-40 GHz, in response to\nthe scanning of magnetic field with the rate of 50 Oe per second. The resonance field was determined by Lorentzian\nfit to the spectra.\nWhen the external magnetic field Hextis applied in ab plane, we use Eq. 11 to fit the data above the saturation\nfield (H >HA+NyMs) and Eq. 17 to fit the data below the saturation field ( H <HA+NyMs). ForH=HA+NyMs,\nthe two equations 11 and 17 get consistent results:8\nFIG. 5. : (a-h) Frequency and field dependent ferromagnetic resonance spectra for CrGeTe 3. The color maps show the in-plane\nresonance spectra, while the white squares are the out-of-plane resonant peaks added afterwards. (i) Optical photograph of the\nsingle crystal used in the resonance experiment.\n/parenleftbiggωres\nγ/parenrightbigg2\n= (HA+NxMs)NzMs. (18)\nIn order to determine these two free variables γ,Ms, we use equations 16 and 18 as two constraints to fit the data\nabove the saturated field. Therefore, we can obtain the spectroscopic splitting factor gfrom the formula γ=gµB//planckover2pi1,\nwhereµB=e/planckover2pi1/(2me) is the Bohr magneton.\nWhen the external magnetic field Hextis applied in out-of-plane direction, the internal effective field Heffacting\nthe spins is equal to the sum of the external field Hext, the equivalent anisotropy field HA= 2K/Ms, and the\ndemagnetizing field HD=−NzMs, whilst the resonant frequency can be determined from the well-known formula\nωres\nγ=H+HA−NzMs. (19)\nIn the paramagnetic regime, usually the demagnetizing effects can be neglected as the Msis small. But for CST\nand CGT, obvious resonance peak shifts causes ωres−Hcurves to not exceed the zero point. Therefore, the g factor\ncannot be directly fitted by the formula of paramagnetic resonance ωres=γH. In order to eliminate the influence\nof the demagnetizing field, it is reasonable to fit the experimental data with equations 16 and 19. Because when\nHA= 0,Ms= 0, andNx=Ny=Nzare satisfied with a total paramagnetic state, equations 16 and 19 degenerate\nintoωres=γH. Therefore, our fitted g factor has excluded the influence of shape anisotropy.\nIII. QUANTUM EXPLANATION OF FLUCTUATIONS INDUCED G-FACTOR ANISOTROPY\nA. Solution for a general model applicable for FMR and ESR case\nConsider a quasi-Heisenberg magnet consisting of similar spins with a common g-tensor. The Hamiltonian for such\na spin system is expressed as [16]9\nH=−2/summationdisplay\n/angbracketleftj,l/angbracketrightJjlSj·Sl−/summationdisplay\n/angbracketleftj,l/angbracketrightSj·Kjl·Sl−µBH·g·/summationdisplay\njSj−A/summationdisplay\nj(Sz\nj)2. (20)\nRemark of the Hamiltonian (20): (i) the first term is the usual Heisenberg exchanging coupling; (ii) the second term\ndescribes the magnetic dipole-dipole interaction, where the Kis a tensor and can contain non-diagonal elements; (iii)\nthe third term is the Zeeman energy, gis the generalized g-factor which is also a tensor; (iv) the last term corresponds\nto the additional single-ion anisotropy [17].\nThe main technique to calculate the g-shift, which was developed by Nagata [18]:\n/planckover2pi1ω=/angbracketleft[S−,[S+,H]]/angbracketright\n2/angbracketleftSz/angbracketright. (21)\nHere theωis the frequency of the precession motion. Notice that in (21) it is demanded that the direction of the\neffective magnetic field His alongz-axis (or saying, the axis of the precession motion is the z-axis). Substituting the\nHamiltonian (20) into the formula (21), the frequency of precession can be evaluated one term by one term.\nZeeman term First we evaluate the Zeeman term HZeeman =−µBH·g·/summationtext\njSj. Notice that this term has spatial\ntranslational symmetry, thus we only need to evaluate the single site, for this purpose we calculate\n[S−,[S+,Sz]] =/planckover2pi13\n8[X−iY,[X+ iY,Z]] = 2 /planckover2pi12Sz. (22)\nIn the calculation above we take Sx=/planckover2pi1\n2X,Sy=/planckover2pi1\n2Y,Sz=/planckover2pi1\n2Zfor convenience. Similarly\n[S−,[S+,Sx]] =/planckover2pi12(Sx−iSy)\n[S−,[S+,Sy]] = i /planckover2pi12(Sx−iSy). (23)\nThe (23) terms are necessary when the off-diagonal gis take into consideration, although in all the Nagata’s articles\nthegwas always a diagonal tensor. For the off-diagonal case, we have\ng/defines\ngxxgxygxz\ngyxgyygyz\ngzxgzygzz\n. (24)\nThe external magnetic field vector is H= [Hx,Hy,Hz], then according to (22) and (23),\n−µBH·g·S=−µB(gxxHx+gyxHy+gzxHz)Sx\n−µB(gxyHx+gyyHy+gzyHz)Sy\n−µB(gxzHx+gyzHy+gzzHz)Sz, (25)\nthen the contribution of the Zeeman term can be evaluated by (21)\n/planckover2pi1ωZeeman =/angbracketleft[S−,[S+,−µBH·g·S]]/angbracketright\n2/angbracketleftSz/angbracketright\n=−µB/planckover2pi12(gxzHx+gyzHy+gzzHz)−µB/planckover2pi12\n2[(gxxHx+gyxHy+gzxHz) + i(gxyHx+gyyHy+gzyHz)]/angbracketleftSx−iSy/angbracketright\n/angbracketleftSz/angbracketright\n(26)\nNotice that this term is different from the formula (6) in Nagata’s article [16] with extra terms. Because in the\nHamiltonian, the Zeeman term is negative defined, we have −µBHz/planckover2pi12gzz=µB|gH|, however, the/angbracketleftSx/angbracketright,/angbracketleftSy/angbracketrightand\n/angbracketleftSz/angbracketrightin the extra term are hard to be determined. This difficulty comes from the off-diagonal element in the g\ntensor. The reason is, according to the original derivation in Nagata’s article [18], the formula (21) was arrived with\nassumption that “S+andS−are good normal modes” , which seems to be not true if the gis not diagonal.10\nHowever, if the magnetic field was tuned above the saturation field in experiments, the inner effective magnetic\nfield is parallel to the external field, then in this case the formula (25) and (26) can be simplified by Hx=Hy= 0,\nand moreover/angbracketleftSx/angbracketright=/angbracketleftSy/angbracketright= 0 because the magnetic precession is along the z-direction. In this respect we claim\n/planckover2pi1ωZeeman =−µB/planckover2pi12gzzHz. (27)\nHeisenberg term HHeisenberg =−2/summationtext\n/angbracketleftj,l/angbracketrightJjlSj·Sldescribes the isotropy exchanging interaction between near\nneighbour spins, and it is invariant under spatial rotations of the frame. The numerator of the (21) is zero. To proof\nthis, we evaluate\n[HHeisenberg,/summationdisplay\niSx\ni+ iSy\ni]\n=−/planckover2pi13\n4/summationdisplay\nα,i,/angbracketleftj,l/angbracketrightJjl[σα\njσα\nl,σx\ni+ iσy\ni] =−/planckover2pi13\n4/summationdisplay\nα,/angbracketleftj,l/angbracketrightJjl[σα\njσα\nl,σx\nj+ iσy\nj] +Jjl[σα\njσα\nl,σx\nl+ iσy\nl]\n=−/planckover2pi13\n2/summationdisplay\nα,β,/angbracketleftj,l/angbracketright/parenleftBig\nJjl(i/epsilon1αxβσβ\njσα\nl+ i/epsilon1αxβσα\njσβ\nl)−Jjl(/epsilon1αyβσβ\njσα\nl+/epsilon1αyβσα\njσβ\nl)/parenrightBig\n= 0. (28)\nIn (28) the /epsilon1is the antisymmetry tensor satisfying\n/epsilon1ijk=\n\n1 if (ijk) = (xyz) or (yzx) or (zxy),\n−1 if (ijk) = (xzy) or (zyx) or (yzx),\n0 otherwise.(29)\nThe antisymmetry property (29) leads to the last equal sign in the derivation (28). As a result, we have\n/planckover2pi1ωHeisenberg = 0. (30)\nDipole-dipole interaction term The dipole-dipole interaction\nHdipole =−/summationdisplay\n/angbracketleftj,l/angbracketrightSj·Kjl·Sl=−/summationdisplay\n/angbracketleftj,l/angbracketright,α,βKαβ\njlSα\njSβ\nl(31)\nwhich was proposed in [16] is the generalization of the cases discussed in [17–19]. Here the Kis not necessary diagonal,\nalthough the diagonal case (XXZ,XYZ model) is regarded to be quite general to describe the spin-spin interactions.\nThe evaluation of the contribution of this term to the frequency is straight forward: substitute Hdipole into (21), after\ntedious calculation we arrive at the result\n/planckover2pi1ωdipole =/summationdisplay\n/angbracketleftj,l/angbracketright,α,βTαβ\njl/angbracketleftSα\njSβ\nl/angbracketright\n2/angbracketleftSz/angbracketright, (32)\nand the form of the tensor Tαβ\njlcan be arranged into matrices\nTjl=−/planckover2pi12\n−2Kxx\njl+ 2Kzz\njl−2iKxy\njl−2Kxy\njl+ iKxx\njl−iKyy\njl−5Kxz\njl−iKyz\njl\n−2Kxy\njl+ iKxx\njl−iKyy\njl−2Kyy\njl+ 2Kzz\njl+ 2iKxy\njl−5Kyz\njl+ iKxz\njl\n−5Kxz\njl−iKyz\njl−5Kyz\njl+ iKxz\njl 2Kxx\njl+ 2Kyy\njl−4Kzz\njl\n (33)\nRemark: When the external field is above the saturaction filed, the precession of the magnetics are along the z-axis\nand we could argue /angbracketleftSxSy/angbracketright=/angbracketleftSySz/angbracketright=/angbracketleftSzSx/angbracketright≈0, and under this assumption we can safely drop the off-diagonal\nterms in (32). Then we have\n/planckover2pi1ωdipole =/summationdisplay\n/angbracketleftj,l/angbracketright/planckover2pi12\n/angbracketleftSz/angbracketright/parenleftBig\n(2Kzz\njl−Kxx\njl−Kyy\njl)/angbracketleftSz\njSz\nl/angbracketright+ (Kxx\njl−Kzz\njl)/angbracketleftSx\njSx\nl/angbracketright+ (Kyy\njl−Kzz\njl)/angbracketleftSy\njSy\nl/angbracketright/parenrightBig\n, (34)11\nthis corresponds with [16]. Notice that in the above derivation we ignore the term Kxy\njlfor the isotropy in the crystal\nplane. From (34) it can be read that\n/planckover2pi1ω/bardbl\ndipole=/summationdisplay\n/angbracketleftj,l/angbracketright/planckover2pi12\n/angbracketleftSz/angbracketright(2Kzz\njl−Kxx\njl−Kyy\njl)/angbracketleftSz\njSz\nl−Sx\njSx\nl/angbracketright\n/planckover2pi1ω⊥\ndipole =/summationdisplay\n/angbracketleftj,l/angbracketright/planckover2pi12\n/angbracketleftSz/angbracketright(Kxx\njl−Kzz\njl)/angbracketleftSz\njSz\nl−Sx\njSx\nl/angbracketright (35)\nSingle-ion anisotropy term The single-ion anisotropy term Hani=A/summationtext\nj(Sz\nj)2can be evaluated by\n[S−,[S+,/summationdisplay\nj(Sz\nj)2]] = (/planckover2pi1\n2)4/summationdisplay\nj[Xj−iYj,[Xj+ iYj,Z2\nj]] =/summationdisplay\nj/planckover2pi12/parenleftbig\n4(Sz\nj)2−2(Sx\nj)2−2(Sy\nj)2/parenrightbig\n[S−,[S+,/summationdisplay\nj(Sx\nj)2]] = (/planckover2pi1\n2)4/summationdisplay\nj[Xj−iYj,[Xj+ iYj,X2\nj]] =/summationdisplay\nj/planckover2pi12/parenleftbig\n2(Sx\nj)2−2(Sz\nj)2/parenrightbig\n(36)\nThen for the case that the external magnetic field is along z-axis, we arrive\n/planckover2pi1ω/bardbl\nani=/planckover2pi12A/summationdisplay\nj/angbracketleft2(Sz\nj)2−(Sx\nj)2−(Sy\nj)2/angbracketright\n/angbracketleftSz/angbracketright(37)\nand when the external magnetic field is along x-axis\n/planckover2pi1ω⊥\nani=/planckover2pi12A/summationdisplay\nj/angbracketleft(Sx\nj)2−(Sz\nj)2/angbracketright\n/angbracketleftSz/angbracketright(38)\nThe thermodynamic average Consider the finite-temperature case β=1\nkBTin which the mean values of the\noperators can be evaluated by the partition function Z= e−βH\n/angbracketleftSz/angbracketright=Tr[e−βH0−βHfSz]\nTr[e−βH0−βHf]/similarequalTr[e−βH0(1−βHf)Sz]\nTr[e−βH0](39)\nwhereHf.=gµBHSzis the external magnetic filed, and H=H0+Hf. Then we have\n/angbracketleftSz/angbracketright=−βgµBH/angbracketleft/angbracketleftSzSz/angbracketright/angbracketright (40)\nwhere “/angbracketleft/angbracketleft /angbracketright/angbracketright ” means average in zero filed. Similarly\n/angbracketleftSz\njSz\nl/angbracketright=β2g2µ2\nBH2/angbracketleft/angbracketleft(Sz)2Sz\njSz\nl/angbracketright/angbracketright=/summationdisplay\nkmβ2g2µ2\nBH2/angbracketleft/angbracketleftSz\nkSz\nmSz\njSz\nl/angbracketright/angbracketright\n/angbracketleftSx\njSx\nl/angbracketright=β2g2µ2\nBH2/angbracketleft/angbracketleft(Sz)2Sx\njSx\nl/angbracketright/angbracketright=/summationdisplay\nkmβ2g2µ2\nBH2/angbracketleft/angbracketleftSz\nkSz\nmSx\njSx\nl/angbracketright/angbracketright (41)\nThen according to formula (35) we arrive at the expression\n∆/planckover2pi1ω/bardbl\ndipole=g/bardblµBH\n2kBT/angbracketleft/angbracketleftSzSz/angbracketright/angbracketright/summationdisplay\njlkm(Kzz\njl−Kxx\njl)/angbracketleft/angbracketleft2(Sz\njSz\nl−Sx\njSx\nl)Sz\nkSz\nm/angbracketright/angbracketright\n∆/planckover2pi1ω⊥\ndipole =g⊥µBH\n2kBT/angbracketleft/angbracketleftSzSz/angbracketright/angbracketright/summationdisplay\njlkm(Kzz\njl−Kxx\njl)/angbracketleft/angbracketleft(−Sz\njSz\nl+Sx\njSx\nl)Sz\nkSz\nm/angbracketright/angbracketright (42)12\nand from the (42) we can obtain directly that\n∆g/bardbl\ndipole\ng/bardbl\ndipole=−2∆g⊥\ndipole\ng⊥\ndipole. (43)\nThe similar argument can also arise for the results (37) and (38), it can be proved\n∆g/bardbl\nani\ng/bardbl\nani=−2∆g⊥\nani\ng⊥\nani. (44)\nBecause the g-shift comes from the Hdipole andHani, and the other terms in Hdo not contribute to the ∆ g, thus\naccording to the formulas (43) and (44), and we arrive\n∆g/bardbl\ng/bardbl=−2∆g⊥\ng⊥. (45)\nWe further assume that the gfactor is isotropic g/bardbl=g⊥=g, and obtain ∆ g/bardbl=−2∆g⊥, which is equivalent to\ng=1\n3×gab+2\n3×gc, (46)\nwheregab=g+ ∆g⊥andgc=g+ ∆g/bardblare measured gfactor along the ab plane and c axis.\nB. Solution for the XXZ model with single ion anisotropy\nConsider a XXZ Hamiltonian with single ion anisotropy:\nH=−1\n2/summationdisplay\n/angbracketleftj,l/angbracketright(JSj·Sl+ ΛSz\njSz\nl)−/summationdisplay\njA(Sz\nj)2−µBH·g·/summationdisplay\njSj. (47)\nBased on the above derivation, we can directly draw conclusions: (i) the Heisenberg isotropic exchange Jhas no\ncontribution; (ii) the anisotropic symmetric exchange Λ, as well as the single-ion anisotropy Acontribute the g-shift.\n[1] L. D. Casto, A. J. Clune, M. O. Yokosuk, J. L. Musfeldt, T. J. Williams, H. L. Zhuang, M.-W. Lin, K. Xiao, R. G. Hennig,\nB. C. Sales, J.-Q. Yan, and D. Mandrus, APL Mater. 3, 041515 (2015).\n[2] Y. Liu and C. Petrovic, Phys. Rev. B 96, 054406 (2017).\n[3] A. Aharoni, J. Appl. Phys. 83, 3432 (1998).\n[4] B. Liu, Y. Zou, L. Zhang, S. Zhou, Z. Wang, W. Wang, Z. Qu, and Y. Zhang, Sci. Rep. 6, 33873 (2016).\n[5] G. T. Lin, H. L. Zhuang, X. Luo, B. J. Liu, F. C. Chen, J. Yan, Y. Sun, J. Zhou, W. J. Lu, P. Tong, Z. G. Sheng, Z. Qu,\nW. H. Song, X. B. Zhu, and Y. P. Sun, Phys. Rev. B 95, 245212 (2017).\n[6] W. Liu, Y. Dai, Y.-E. Yang, J. Fan, L. Pi, L. Zhang, and Y. Zhang, Phys. Rev. B 98, 214420 (2018).\n[7] M.-G. Han, J. A. Garlow, Y. Liu, H. Zhang, J. Li, D. DiMarzio, M. W. Knight, C. Petrovic, D. Jariwala, and Y. Zhu,\nNano Lett. 19, 7859 (2019).\n[8] B. Ding, Z. Li, G. Xu, H. Li, Z. Hou, E. Liu, X. Xi, F. Xu, Y. Yao, and W. Wang, Nano Lett. 20, 868 (2020).\n[9] J. G. Cheng, Y. Sui, X. J. Wang, Z. G. Liu, J. P. Miao, X. Q. Huang, Z. L¨ u, Z. N. Qian, and W. H. Su, J. Phys. Condens.\nMater. 17, 5869 (2005).\n[10] A. Arrott and J. E. Noakes, Phys. Rev. Lett. 19, 786 (1967).\n[11] J. S. Kouvel and M. E. Fisher, Phys. Rev. 136, A1626 (1964).\n[12] V. Carteaux, F. Moussa, and M. Spiesser, Europhys. Lett. 29, 251 (1995).\n[13] T. J. Williams, A. A. Aczel, M. D. Lumsden, S. E. Nagler, M. B. Stone, J.-Q. Yan, and D. Mandrus, Phys. Rev. B 92,\n144404 (2015).\n[14] S. Wu, L. Wang, B. Gao, Y. Wang, Y. S. Oh, S.-W. Cheong, J. Hong, and X. Wang, AIP Adv. 8, 055016 (2018).13\n[15] H. Beljers and J. Smit, Phil. Res. Rep. 10(1955).\n[16] K. Nagata, I. Yamamoto, H. Takano, and Y. Yokozawa, J. Phys. Soc. Jpn. 43, 857 (1977).\n[17] K. Nagata, Physica B+C 86-88 , 1283 (1977).\n[18] K. Nagata and Y. Tazuke, J. Phys. Soc. Jpn. 32, 337 (1972).\n[19] K. Nagata, Y. Tazuke, and K. Tsushima, J. Phys. Soc. Jpn. 32, 1486 (1972)."    },    {        "title": "2211.08130v1.Experimental_Observation_of_a_Magnetic_Interfacial_Effect.pdf",        "content": "Experimental Observation of a Magnetic Interfacial\nE\u000bect\nWibson W. G. Silva, S\u0013 ergio V. B. Degiorgi and Jos\u0013 e Holanda\u0003\nPrograma de P\u0013 os-Gradua\u0018 c~ ao em Engenharia F\u0013 \u0010sica, Universidade Federal Rural de\nPernambuco, 54518-430, Cabo de Santo Agostinho, Pernambuco, Brazil.\nAbstract\nWe observed a magnetic interfacial e\u000bect due to the coupling between two inter-\nfaces of di\u000berent materials. The interface is compoust of an antiferromagnetic\nand other quasi-ferromagnetic material. This e\u000bect we measured through the\nferromagnetic resonance technique without and with electric current.\nKeywords: Magnetic, non-collinear, uncompensated, interface\nAntiferromagnetic materials exhibiting topologically protected states in their\nmomentum-space bands as well as those exhibiting topologically non-trivial real-\nspace at the interface, inspire the production of new devices in antiferromagnetic\nspintronics [1-6]. A non-collinear antiferromagnet that has a giant magnetocrys-\ntalline anisotropy energy of 10 meV per unit cell is the IrMn 3[7-12]. The IrMn 3\nhas a giant spin Hall angle (up to \u00190.35) in the direction [001] and crystallizes\nin a face-centered cubic lattice [10]. The local ferromagnetism from uncompen-\nsated spins can induce antiferromagnetic symmetry in the adjacent material [4,\n13-15]. Platinum (Pt) can be an excellent adjacent material for this coupling\nbecause Pt is quasi-ferromagnetic according to Stoner's criterion [16]. Thus,\nIrMn 3and Pt are promissory candidates for the detection of resonance signals\ndue to the ferromagnetism at the interface. The detection of these signals can\n\u0003Corresponding author: joseholanda.silvajunior@ufrpe.br\nPreprint submitted to ArXiv November 16, 2022arXiv:2211.08130v1  [cond-mat.mtrl-sci]  15 Nov 2022be performed using the ferromagnetic resonance (FMR) technique without and\nwith electric current [17-29]. In this Letter, we observe ferromagnetic resonance\nsignals at the sample of IrMn 3/Pt bilayer without and with electric current.\nFurthermore, we analyzed the change of the interface damping as a function of\nthe electric current.\nAll IrMn 3\flms were epitaxially grown in a magnetron sputtering system\nwith pressures of base approximately 1 \u000210\u00007torr and of argon of 3 mtorr at\na temperature of 843 K on the (100)MgO substrates. We used one Ti(2 nm)\nlayer to protect the surface properties of IrMn 3or Pt layer. We deposited the\nPt layer at 373 K temperature to di\u000busion on IrMn 3and the Ti layer at room\ntemperature. Fig. 1 (a) shows an X-ray di\u000braction pattern for IrMn 3grown\non a (100)MgO substrate. We found that IrMn 3has a lattice constant of (0.377\n\u00060.001) nm, which is consistent with previous literature values [30]. The X-ray\ndi\u000braction pattern of Fig. 1 (b) shows that the Pt has grown on the IrMn 3\nsurface, showing di\u000berent phases. The quality of samples also were analyzed\nby transmission electron microscopy (TEM), as shown in the micrograph of the\nIrMn 3(10 nm)/Pt(6 nm)/Ti(2 nm) sample in Fig. 1 (c) .\nAs it is well known, an exchange bias can be stabilized at room temperature\nfor IrMn 3/Py bilayer, and thus IrMn 3is antiferromagnetically ordered at room\ntemperature [2, 4, 10]. In the same way, the ferromagnetism in the antiferro-\nmagnetic and quasi-ferromagnetic interface is stabilized [28]. For the deposition\nof samples, we used a metallic mask to avoid any type of contamination on the\nsurface of MgO, and we utilized the laser write technique for the production of\nthe coplanar waveguide on the sample surface, as shown in the Fig. 1 (d) . We\nmeasured resonance signals by the \rip-chip vector network analyzer ferromag-\nnetic resonance (VNA-FMR) technique. We obtained via Lorentz \ftting the\nfrequency swept linewidths (\u0001 fV NA). Detailed steps, including the conversion\nfrom \u0001fV NA to the \feld linewidths \u0001 H, are shown in ref. [28]. In the process of\nVNA-FMR, we measured the transmission coe\u000ecient by sweeping the frequency\nat every \fxed magnetic \feld [4].\nIn the Fig. 2 (a) , we show the unit cell of IrMn 3, where the Mn atoms\n2ISi (500 nm)/Au(500 nm)Au(20 nm)30 40 50 60100102104106108\nPt (111) Pt (200) \nIrMn3(200) MgO(200) Intensity (a. u.)\n2 (degrees)30 40 50 60100102104106108\nIrMn3(200) MgO(200) Intensity (a. u.)\n2 (degrees)IrMn3PtTi(a) (b)\n(c)\nMgO RF\nIdc\n[010][100]\n[001]Au(20 nm)\nG G S(d)\nH2 mm\n5 mmFigure 1: (Color online) X-ray di\u000braction pattern measured for a 10 nm thick IrMn 3layer\nwithout or with a 6 nm thick Pt layer capped with a 2-nm Ti layer: (a)IrMn 3(10 nm)/Ti\n(2 nm) and (b)IrMn 3(10 nm)/Pt(6 nm)/Ti(2 nm). (c)Transmission Electron Microscopy\n(TEM) micrograph of IrMn 3(10 nm)/Pt(6 nm)/Ti(2 nm). (d)Shows the coplanar waveguide\n(CPW) deposited by Electron Beam Evaporation at the up of the samples for measures of\nferromagnetic resonance without and with electric current.\nare on thef111gplanes and their spins are aligned along the <112>direc-\ntions. As it is known, spins-in and spins-out con\fgurations are nonequivalent\nground states and are chiral images of each other [3, 4, 23]. For analyses of\nthe spins-in or spins-out con\fgurations, a transversal section can be made in\nthe unit cell of IrMn 3shown from a kagome lattice in the (111) plane with\neither pointing outwards in each triangular Mn arrangement (see the Fig. 2\n(b)). This con\fguration represents the interface con\fguration of the spins-out\nfor the \flm. In this case, the thin layer of Pt deposited on top of the IrMn 3\ngrows with [100] and [111] crystallographic directions (see the Fig. 1 (b) ).\n3The net chirality at the IrMn 3/Pt interface in the resonance condition can be\n[100][010][001][111]\n3 IrMn\nTransversal section [100][010][001]\n[111]\nPtPtH\n[100] [010][001] [111]\nPt\n3 IrMnPt\nj\nj\nSpins -out\n[100][010][001]\n[111]H\n[100][010][001]\n[111]\nPtPtH\nMagnetic Interfacial Effect\n)j (j2ejz\nIrMn x,3xyz\ndcIH\n(a) (b) (c)\n(d) (e) (f)\n[011]\nFigure 2: (Color online) (a)Show the unit cell of IrMn 3whose moments Mn are parallel to\nthe planesf111gand aligned with the directions <112>.(b)The transversal section in\nwhich the interface con\fguration of the spins-out for IrMn 3.(c)Con\fguration of resonance\ncondition of the rotation of spins of IrMn 3and Pt on a magnetic \feld. (d)Arrangement\nfrom the magnetic interfacial e\u000bect of IrMn 3in interface resonance condition, where a spin\naccumulation is created in the direction from Pt. (e)Arrangement from spin Hall e\u000bect of Pt\nin interface resonance condition, where a spin accumulation is created in the direction from\nIrMn 3.(f)Coordinates system of the spin currents of the antiferromagnet IrMn 3.\nunderstood by the rotation of uncompensated spins of IrMn 3, and the spins at\nthe surface of Pt, as shown in Fig. 2 (c) . The coupling of the noncollinear\nspins of IrMn 3, and the quasi-ferromagnetic spins of the Pt represent the ideal\ncondition to explore the magnetic interface. [3, 4, 23, 29-35]. In resonance con-\ndition using the Hall e\u000bects of the IrMn 3(seeFig. 2 (d) ) and of the Pt (see\nFig. 2 (e) ), a spin accumulation \feld is created as two spin currents \rowing\nat directions of the interface [4, 23]. The polarization of the spin accumulation\n4\feld de\fned by the magnetic \feld is transversal at the magnetic interface, as\nshown in Figs. 2 (d) and(e). The spin current of the IrMn 3is de\fned as\njz\nIrMn 3=jz\ny;SSS = (j\"+j#)\u0016h=2e, which depend of the spins-in and spins-out\ncon\fgurations, as shows the Fig. 2 (f) .\nThe e\u000eciency from the FMR signals is due to the net chirality of the spin\nstructures, which provides more complexity compared to non-magnetic materi-\nals [29-36]. Fig. 3 (a) shows the FMR signals obtained with a VNA for one fre-\nquency of 10 GHz, magnetic \feld of 10.3 kOe, and di\u000berent electric currents -1, 0,\nand +1 mA. Two observations need highlight: \frst, the electric current produces\na signi\fcant variation in frequency swept linewidths (\u0001 fV NA), and second, a\nshift is caused by the accumulation \feld ( HAc). InFig. 3 (b) we show that the\naccumulation \feld (H Ac) due to the electric current produces a modi\fcation in\nthe resonance \feld. This is clear evidence of a magnetic interfacial e\u000bect due to\nthe strong coupling at interface IrMn 3/Pt interface, similar to the IrMn 3/Py bi-\nlayer [4, 12, 19]. The solid curve represents the \ft from the experimental data to\nthe Kittel equation, f=\r[(HR)(HR+ 4\u0019Meff\u0006HAc)]1=2, where the gyromag-\nnetic ratio is ( \rIrMn 3=Pt)/\rIrMn 3=Py= 5%, the spectroscopic splitting factor\ncalculated considering the Stoner`s criterion [16] is gIrMn 3=Pt/gIrMn 3=Py= 5%,\n\u0016Bis the Bohr magneton, \u0016 his the reduced Planck constant, and 4 \u0019Meff=\n4\u0019MS+HASis the e\u000bective magnetization that is much larger than the satura-\ntion magnetization 4 \u0019MSdue to the e\u000bect of the surface anisotropy \feld HAS.\nUsing the \ft with the Kittel equation to zero electric currents, we obtained for\ne\u000bective magnetization 4 \u0019Meff= (523:78\u00060:005) kOe, which is smaller than\nthat obtained for IrMn 3=Py bilayer [4, 10, 12, 19].\nThe properties of noncollinear antiferromagnetic materials with magnetic\ntopological states yield large changes [31-35], this also is a characteristic of\nuncompensated spins that induces magnetism. In the Fig. 4 (a) , we represent\nthe change from resonance frequency as a function of the spin accumulation\n\feld (HAc) for electric current \u00061 mA. The spin accumulation \feld increase by\nobeying a Kittel equation describes by greens lines. In this case, the spins-in and\nspins-out con\fgurations exhibit the same energies, and both exist spontaneously\n5(a)(b)\n0 4 8 12 1603691215\n+1.0 \n  0.0 \n -1.0 Idc (mA)Frequ\u001e ency (GHz)\nMagnetic Field (kOe)\n8 10 12-1.2-0.9-0.6-0.30.0\nIdc (mA)S12 (dB)/|SMax\n12 (dB)|\nFrequency (GHz) +1.0\n   0.0\n  -1.0Figure 3: (Color online) (a)Ferromagnetic resonance (FMR) signals were obtained using a\nVNA for one frequency of 10 GHz, magnetic \feld of 10.3 kOe, and di\u000berent electric currents -1,\n0, and +1 mA. (b)FMR frequency as a function of the magnetic \feld, which the accumulation\n\feld (H Ac) due to the electric current produces a change in the resonance \feld. The \fts are\nperformed with the Kittel equation, where ( \rIrMn 3=Pt)/\rIrMn 3=Py= 5% and 4 \u0019Meff=\n(523:78\u00060:005) kOe.\nin the material [31]. On the other hand, in the Fig. 4 (b) we show the spin\naccumulation \feld as a function of electric current in the resonance frequency of\n10 GHz. A small onset of saturation of the spin accumulation \feld is observed\nfor electric currents Idc<-1mA andIdc>+ 1mA, which is also observed in\nother bilayers [4, 10, 11, 27]. The polarization from the spin accumulation \feld\nobeys the electric current con\frming the process from the manipulation of the\nmagnetic interfacial e\u000bect in the IrMn 3/Pt bilayer, similar to the manipulation\nof the exchange bias at the antiferromagnetic/ferromagnetic bilayer [4, 12, 19].\nIn the Fig. 4 (c) , the \ft is made with the expression \u0001 H= (\u000b=\r)f[33], where\n\u000bis the magnetic Gilbert damping of the interface.\nIn the Fig. 4 (d) , we show the change of the damping of the IrMn 3/Pt\nbilayer as a function of the electric current. Damping starts to increase for\nelectric currents below -1 mA, as well as decreases for electric currents above\n+1 mA. This behavior agrees with the data of the Fig. 4 (b) , and is due\nto the spins-in and spins-out con\fgurations in the IrMn 3/Pt bilayer that loses\ntheir dynamic characteristics due to the saturation of the spin accumulation\n\feld and the fact that the interface presents a local temperature change [12,\n6-2 -1 0 1 2-5.0-2.50.02.55.0HAc (kOe)\nIdc (mA)f  = 10 GHz\n-4.2 -2.8 -1.4 0.0 1.4 2.8 4.2036912 Frequ\u001e ency (GHz)\nAc (kOe) +1 mA\n -1 mA\n     fAc\n-2 -1 0 1 20246Damping (x 10-3)\nIdc (mA)\n0 3 6 9 120100200300400\nIdc (mA)\n-1.0\n  0.0\n+1.0Linewidth H (Oe)\nFrequency (GHz)(a)(b)\n(c)\n(d)Figure 4: (Color online) (a)Ferromagnetic resonance (FMR) frequency as a function of the\nspin accumulation \feld H Ac. The \fts are performed with the Kittel equation. (b)Spin\naccumulation \feld as a function of the electric current to the frequency of 10 GHz. (c)Shows\nthe linewidth change as a function of the FMR frequency to three values of electric current\n-1, 0, +1 mA. (d)Damping change as a function of electric current.\n19]. In summary, we experimentally observed a magnetic interfacial e\u000bect in\nIrMn 3/Pt bilayer, and we manipulated it through the FMR technique without\nand with electric current.\nAcknowledgements\nThis research was supported by Conselho Nacional de Desenvolvimento Cien-\nt\u0013 \u0010\fco e Tecnol\u0013 ogico (CNPq), Coordena\u0018 c~ ao de Aperfei\u0018 coamento de Pessoal de\nN\u0013 \u0010vel Superior (CAPES), Financiadora de Estudos e Projetos (FINEP), and\nFunda\u0018 c~ ao de Amparo \u0012 a Ci^ encia e Tecnologia do Estado de Pernambuco (FACEPE).\n7References\n[1] J. Zelezny, Y. Zhang, C. Felser, and B. Yan. Spin-polarized current in\nnoncollinear antiferromagnets. Phys. Rev. Lett. 119, 187204 (2017).\n[2] S. Jenkins, R. W. Chantrell, T. J. Klemmer, and R. F. L. Evans. Magnetic\nanisotropy of the noncollinear antiferromagnet IrMn 3. Phys. Rev. B 100 ,\n220405(R) (2019).\n[3] V. T. N. Huyen, M.-T. Suzuki, K. 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Magn. 40, 3443 (2004).\n12"    },    {        "title": "2011.01583v1.Symmetry_breaking_induced_magnon_magnon_coupling_in_synthetic_antiferromagnets.pdf",        "content": "arXiv:2011.01583v1  [cond-mat.mes-hall]  3 Nov 2020Symmetry breaking induced magnon-magnon coupling in synth etic antiferromagnets\nJie Lu,1Mei Li,2,∗and Wei He3,†\n1College of Physics and Hebei Advanced Thin Films Laboratory ,\nHebei Normal University, Shijiazhuang 050024, People’s Re public of China\n2Physics Department, Shijiazhuang University, Shijiazhua ng, Hebei 050035, People’s Republic of China\n3State Key Laboratory of Magnetism and Beijing National Labo ratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China\n(Dated: November 4, 2020)\nWe propose a general theory of microwave absorption spectro scopy for symmetry-breaking synthetic anti-\nferromagnets (SAFs). Generally, inhomogeneity or differe nt thickness of the two ferromagnetic sublayers of a\nSAF results in the intrinsic symmetry breaking, while out-o f-plane components of dc magnetic fields lead to the\nextrinsic one. The broken symmetry of SAFs excludes the orig inal symmetry-protected crossing between pure\nin-phase and out-of-phase resonance modes with opposite pa rity. Alternatively, new frequency branches become\nhybridization of original bare modes in terms of symmetry-b reaking-induced magnon-magnon coupling, which\nresults in an indirect gap in ferromagnetic resonance frequ encies. Also, the dependence of gap width on the de-\ngree of symmetry breaking for several typical cases are pres ented and compared with existing experiments. Our\ntheory provides a simple but physical understanding on the r ich structure of ferromagnetic resonance spectra for\nasymmetric SAFs.\nSynthetic antiferromagnets (SAFs) are magnetic multilay-\ners with two ferromagnetic (FM) sublayers coupled anti-\nferromagnetically through a nonmagnetic metallic spacer[ 1].\nThey have attracted tremendous interest in the past decades\ndue to their potential for developing the “SAF spintronics”\nand wide range of applications in magnetic nanodevices [ 2–\n21]. Compared with the strong exchange coupling in gen-\nuine antiferromagnetic (AFM) materials with terahertz int rin-\nsic frequencies[ 22,23], the relatively weak interlayer cou-\npling in SAFs mainly comes from the Ruderman-Kittel-\nKasuya-Yosida (RKKY) interaction[ 24–26] and thus realizes\ngigahertz FM resonance (FMR) frequencies that mature mi-\ncrowave electronics can match. Similar behaviors have also\nbeen observed in layered crystals[ 27,28] and compensated\nferrimagnets[ 29] with AFM inter-layer and inter-sublattice\ncouplings, respectively. More interestingly, for symmetr ical\nNiFe/Ru/NiFe[ 2,16] and FeCoB/Ru/FeCoB SAFs[ 18–21], or\nlayered crystal CrCl 3[27], symmetry-protected mode cross-\nings between in-phase and out-of-phase branches of FMR\nspectra have been observed under in-plane external dc mag-\nnetic fields. This indicates the absence of coupling between\nmagnons with opposite parity, which is, on the contrary, ver y\ncommon in yttrium iron garnet/ferromagnet bilayers[ 30–32].\nIn fact, this mode crossing can be eliminated in several\nways. For symmetrical SAFs or layered crystals, extrinsica lly\nexerting an out-of-plane dc field will lift the system’s rota tion-\nsymmetry axis away from the SAF plane, thus breaks the rota-\ntion symmetry of the hard axis (normal of SAF plane) from the\nmagnetostatic interaction. This introduces a magnon-magn on\ncoupling between the original uncoupled modes with opposit e\nparity, hence hybridizes the two modes and generate an anti-\ncrossing gap[ 20,27]. Very recently, strong magnon-magnon\ncoupling under in-plane dc fields is also proposed by the dy-\nnamic dipolar interaction (nonuniform precession) in symm et-\nrical FeCoB/Ru/FeCoB SAFs[ 18,19], which provides an al-\nternative way of extrinsic symmetry breaking (SB). The othe rstrategy is to break the intrinsic symmetry between the two\nFM sublayers in SAFs. In most existing experiments the two\nsublayers are prepared from the same FM materials but with\ndifferent thickness. A frequency gap can be observed even un -\nder in-plane dc fields[ 16,17]. In addition, intrinsic asymme-\ntry should also appear when the two sublayers are made from\ndifferent FM materials. However, to our knowledge the corre -\nsponding FMR measurements are not yet in the press, which\nmainly comes from the difficulty in sample preparation.\nA lot of theoretical work has been performed to understand\nthe rich structure of FMR spectra in SAFs[ 4,14–18,20,21,\n27,29]. Representatively, in 2014 a discrete-lattice approach\nhas been raised for asymmetric NiFe/Ru/NiFe SAFs[ 17],\nwhere the RKKY interaction, biquadratic exchange coupling s\nand the uniaxial anisotropy at the NiFe/Ru interfaces are al l\nconsidered, however a clear and simple physical picture is s till\nlacking. In 2019, based on “macrospin” assumption MacNeill\net. al. proposed a systematic analysis for the gap induction\nby extrinsic SB from out-of-plane dc fields in which only the\nbilinear RKKY interaction is included[ 27]. However, they did\nnot consider the intrinsic SB since the sublayers of their sy s-\ntem are always symmetrical. In this Letter, we demonstrate\nthat intrinsic and extrinsic SB in SAFs can induce magnon-\nmagnon couplings via different mechanisms, thus further le ad\nto indirect gaps in their FMR spectra. In brief, a general the -\nory including both intrinsic and extrinsic SB (from tilting dc\nfields) will be presented first and then followed by several ex -\namples that are easy to compare with experiments.\nThe SAF system under consideration is sketched in Fig. 1\n(a). The saturation magnetization and thickness of the up-\nper (lower) FM sublayer are MA,0\nsanddA(MB,0\nsanddB), re-\nspectively. The crystalline anisotropy in both sublayers a re\nneglected for simplicity. In typical SAFs, the FM intralaye r\nnearest-neighbor exchange is much stronger than the inter-\nlayer AFM coupling. Therefore the “macrospin” approxima-\ntion generally provides enough information about spin-wav e2\nx\nyz\nxcyczc\neq \nAmeq \nBm\nIc\nH\n\\Ad\nBd,0 A\nA s A M M m \nM\n,0 B\nB s B M M m (a) \nIntrinsic + \nextrinsic: \n(b) \nPure \nextrinsic: \naZ2A y B G G c  m m \u0010\noZ2A y B G G c  \u0010 m m \u0010\n2y z z cze e \u0010\n(c) \nPure \nintrinsic: \naZ2A y B G G c  m m \u0010\noZ2A y B G G c  \u0010 m m \u0010\n,0 eq ,0 eq \n2\neq eq \n2 and/or  A B \ny s A s B \nA B \ny s A s B M M \nM M c\ncz\nzm m \nm m \u0010\n\u0010yc ycxx c{ee\nzz c{ee\nyc yc\nFIG. 1. (a) Sketch of a typical SAF with saturation magnetiza tion\nMA(B),0\ns and thickness dA(B)in its two FM sublayers. The macrospins\n(mAandmB) of the two sublayers are coupled antiferromagnetically\nand tilted at equilibrium ( meq\nAandmeq\nB) under both extrinsic and in-\ntrinsic SB. Generally, the new x′y′z′coordinate system (see definition\nin the main text) is totally different with the initial xyzone. The de-\nviation of z′(x′,y′) from the z(x,y) axis comes from the extrinsic\n(intrinsic) SB. (b) Sketch of magnon-magnon coupling betwe en the\nin-phase ( δmA=C2y′δmB) and out-of-phase ( δmA=−C2y′δmB)\nmodes with opposite parity under pure extrinsic SB with ex≡ex′.\nThis coupling comes from the breaking rotation symmetry of m ag-\nnetostatics under C2y′(C2y′ez/negationslash=ez). (c) Sketch of magnon-magnon\ncoupling between in-phase and out-of-phase modes under pur e in-\ntrinsic SB with ez≡ez′. This coupling comes from the breaking ro-\ntation symmetry of magnetostatics and/or magnetization un derC2y′\n(C2y′MA,0\nsmeq\nA/negationslash=MB,0\nsmeq\nBand/or C2y′MAsmeq\nA/negationslash=MBsmeq\nB).\nbehaviors. Now the magnetization within each sublayer can\nbe viewed as uniform and denoted as mAandmB, respec-\ntively. The interlayer exchange energy is then modeled as\nµ0λES(dA+dB)MA,0\nsMB,0\nsmA·mB/2, where λE>0 is a di-\nmensionless coefficient and Sis the SAF cross-section area.\nIn literatures, this term is also expressed as the “interlay er ex-\nchange energy per unit area” JIEC[6,7,10,12,15,18,19,21].\nHence λE=2JIEC/[µ0MA,0\nsMB,0\ns(dA+dB)]. By neglecting the\nGilbert damping, a coupled Landau-Lifshitz-Gilbert (LLG)equation reads\n˙mA=−γmA×/bracketleftbig\nH−λEMB\nsmB−MA,0\ns(mA·n)n/bracketrightbig\n+τA,\n˙mB=−γmB×/bracketleftbig\nH−λEMA\nsmA−MB,0\ns(mB·n)n/bracketrightbig\n+τB,(1)\nwhere an overdot means d/dt,nis the sublayer normal and\nMA\ns=dA+dB\n2dBMA,0\ns,MB\ns=dA+dB\n2dAMB,0\ns (2)\nare the “thickness-modified saturation magnetization” of t he\nrespective sublayers. γ=µ0γewith µ0andγebeing the vac-\nuum permeability and electron gyromagnetic ratio, respec-\ntively. At last, τA(B)is the torque on mA(B)which arises from\nthe rf excitation field of coplanar waveguides.\nIn the following we present a general theory describing the\ncombining effects of intrinsic and extrinsic SB on FMR spec-\ntroscopy. As preparation, we calculate the equilibrium mag -\nnetization orientations under a general dc magnetic field H\nwith strength Hand tilting angle ψ(0≤ψ<π/2). First we\nconstruct the Cartesian coordinate system (ex,ey,ez):ez≡n\nthus the out-of-plane field component is Hsinψ,eyis paral-\nlel to the in-plane field component H−Hsinψez, and ex=\ney×ez. Due to the AFM interlayer coupling, without Hthe\nunit magnetization vectors in the two sublayers orients opp o-\nsitely in a certain in-plane direction in the absence of crys -\ntalline anisotropy. When His applied, in principle they are\npulled out of xyplane and their final equilibrium states are\ndenoted as meq\nAandmeq\nB, respectively. We then set θA(B)as\ntheir respective polar angles, and φA(φB) being the angle that\nthe in-plane component of meq\nA(meq\nB) rotates anticlockwise\n(clockwise) with respect to ex(−ex). These four angles can\nbe explicitly solved from the static LLG equation (see Sup-\nplement Material).\nNext we define a new x′y′z′coordinate system based on\nmeq\nAandmeq\nB:ex′/bardblmeq\nA−meq\nB,ey′/bardblmeq\nA+meq\nB, and ez′/bardbl\nmeq\nA×meq\nB. Note that to ensure the noncollinearity of meq\nA\nandmeq\nB, the dc field strength Hshould be limited within the\nrange HAFM<|H|<HFM(see Supplement Material for their\ngeneral definitions). For |H|≤HAFM (≥HFM),meq\nAandmeq\nB\npoint in the opposite (same) direction thus the SAF falls int o\nAFM (FM) state. Then we denote C2y′as the rotation oper-\nator which rotates vectors around the y′axis by 180◦. Ob-\nviously, C2\n2y′=1 and C2y′meq\nA(B)=meq\nB(A). However, since\nMA,0\ns/negationslash=MB,0\nsand/or dA/negationslash=dB, the joint operation of C2y′and\nA-B sublayer exchange is no longer a symmetric operation of\nthe entire SAF. When a rf field with frequency f=ω/2πis\nexcited, the magnetization vectors deviate slightly from t heir\nequilibrium orientations and begin to vibrate. We then ex-\npand mA(B)=meq\nA(B)+δmA(B)eiωtandτA(B)=τ0\nA(B)eiωt. In\naddition, we introduce δm±≡δmA±C2y′δmBas the mag-\nnetization responses with even and odd parity (under C2y′) to\nthe torque terms τ±=τ0\nA±C2y′τ0\nB. By putting them into Eq.\n(1) and keeping up to linear-order terms, we get our central\nvectorial equation for the magnetization excitations δm±,3\niΩδm±=meq\nA×/braceleftbiggλE\n2/parenleftbig\nMA\ns+MB\ns/parenrightbig/parenleftbig\nδm±±C2y′δm±/parenrightbig\n+λE\n2/parenleftbig\nMA\ns−MB\ns/parenrightbig/parenleftbig\nδm∓±C2y′δm∓/parenrightbig\n+MA,0\ns±MB,0\ns\n4/bracketleftbig\nezez·+C2y′ez/parenleftbig\nC2y′ez/parenrightbig\n·/bracketrightbig\nδm++MA,0\ns±MB,0\ns\n4/bracketleftbig\nezez·−C2y′ez/parenleftbig\nC2y′ez/parenrightbig\n·/bracketrightbig\nδm−\n+MA,0\ns∓MB,0\ns\n4/bracketleftbig\nezez·+C2y′ez/parenleftbig\nC2y′ez/parenrightbig\n·/bracketrightbig\nδm−+MA,0\ns∓MB,0\ns\n4/bracketleftbig\nezez·−C2y′ez/parenleftbig\nC2y′ez/parenrightbig\n·/bracketrightbig\nδm+/bracerightBigg\n+τ±,(3)\nwhere Ω=ω/γ. Now δm+andδm−are coupled by both\nintrinsic (proportional to MA\ns−MB\nsandMA,0\ns−MB,0\ns) and ex-\ntrinsic (proportional to C2y′ez) SB terms, thus result in strong\nmagnon-magnon interaction between the bare modes with\neven and odd parities.\nNext we calculate the FMR spectroscopy. After definingthe average bare and “thickness-modified” saturation magne t-\nziation as M0\ns=(MA,0\ns+MB,0\ns)/2 and Ms=(MA\ns+MB\ns)/2, as\nwell as their deviations κ0= (MA,0\ns−MB,0\ns)/(MA,0\ns+MB,0\ns)\nand κ= (MA\ns−MB\ns)/(MA\ns+MB\ns)describing the degree of\nasymmetry between the two FM sublayers, the secular equa-\ntion for Eq. ( 3) is\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep1−iΩ p2 p3 p4\nq1+2λEMscos2φ′−p1−iΩ q2+2λEMsκcos2φ′−p3\np3 p4−2λEMsκ p1−iΩ p2−2λEMs\nq2+2λEMsκsin2φ′−p3 q1+2λEMssin2φ′−p1−iΩ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=0. (4)\nin which p1=M0\ns(cosαsinφ′−κ0cosβcosφ′)cosη,p2=\n−M0\nscos2η,p3=M0\ns(κ0cosαsinφ′−cosβcosφ′)cosη,\np4=−M0\nsκ0cos2η,q1= [( p1−p3)2(p4+p2)−(p1+\np3)2(p4−p2)]/[2(p2\n4−p2\n2)], and q2= [−(p1−p3)2(p4+\np2)−(p1+p3)2(p4−p2)]/[2(p2\n4−p2\n2)]. In addition,\ncosα, cos βand cos ηare the three cosines of ezin\nthe new x′y′z′coordinate system: cos α= (cosθA−\ncosθB)/(2cos φ′), cos β= (cosθA+cosθB)/(2sin φ′), and\ncosη=/radicalBig\nsin2θAsin2θB−(cosθAcosθB+cos2 φ′)2/sin2φ′.\nAt last, φ′is the angle between meq\nAand ex′which satis-\nfies cos2 φ′=−cosθAcosθB+[(MA\nssinθA)2+(MB\nssinθB)2−\n(Hcosψ/λE)2]/(2MA\nsMB\ns).\nIn the most general case where intrinsic and extrinsic SB\ncoexist ( κ0/negationslash=0,κ/negationslash=0,ψ>0), Eq. ( 4) can hardly be solved\nanalytically. However, based on the above results numerica l\ncalculation can always be performed once we have knowledge\nonMA(B),0\ns ,dA(B),λEandψ. By first fixing Hand solving\nthe above secular equation then further sweeping H, the en-\ntire FMR spectrum in the “ ω∼H” space can be obtained. In\nFig. 2 the dimensionless FMR spectra under different config-\nurations are provided based on Eqs. ( 3) and ( 4). In all cal-\nculations, saturation magnetizations and magnetic fields a re\nin the unit of MA,0\nsandωis in the unit of γMA,0\ns. In addi-\ntion, λE=0.1. In the absence of any SB, a mode crossing is\nclearly seen. When any single type of asymmetry appears, an\nanticrossing gap emerges. While all kinds of SB coexist, the\ngap width is greatly enlarged.\nThe mechanisms of inducing magnon-magnon coupling by\nextrinsic and intrinsic SB are different. We have already de -/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s32/s32/s47/s40 /s77\n/s115/s65/s44/s48\n/s41\n/s72/s47/s77\n/s115/s65/s44/s48/s32/s83/s121/s109/s109/s101/s116/s114/s105/s99\n/s32 /s61/s48/s46/s50\n/s32/s77\n/s115/s66/s44/s48\n/s61/s40/s53/s47/s54/s41/s77\n/s115/s65/s44/s48\n/s32/s100\n/s66/s61/s40/s51/s47/s52/s41/s100\n/s65\n/s32/s65/s108/s108/s32/s97/s115/s121/s109/s109/s101/s116/s114/s105/s99\n/s69/s61/s48/s46/s49\nFIG. 2. (Color online) Anticrossing gap opening when extrin sic\nand/or intrinsic symmetry breaking emerge. In all calculat ions based\non Eqs. ( 3) and ( 4),λE=0.1 and MA,0\nsis taken as the unit of satura-\ntion magnetizations and magnetic fields. ωis in the unit of µ0γMA,0\ns.\nThe tilting angle ( ψ) of dc fields and the ratios of bare saturation\nmagnetization ( MB,0\ns/MA,0\ns) as well as sublayer thickness ( dB/dA)\nare three sources of symmetry breaking. Each has been first in de-\npendently shifted away from the symmetrical configuration ( red, blue\nand magenta curves), and then together (green curve).\nscribed that for extrinsic SB, the breaking rotation symmet ry\nof hard axis n(from magnetostatic interaction) with respect to\nthe pulled up rotation-symmetry axis (due to out-of-plane d c\nfields) is the basic reason. While for intrinsic ones ( ψ=0), the\nnew rotation axis is still in the SAF plane but the entire mag-4\nnetic layout (including magnetization and magnetostatics ) is\nno longer unchanged under the two-fold rotation. The consid -\nerable entanglement between magnetization vibrations wit h\nopposite parity results in the strong magnon-magnon coupli ng\nthus greatly affects the FMR spectroscopy of SAFs.\nTo acquire clearer physics meantime provide more conve-\nnient fitting tools for experiments, in the following we focu s\non a few special cases and present more details about the anti -\ncrossing gap. In the first class, only extrinsic SB is present\nso that MA,0\ns=MB,0\ns,dA=dBandψ>0 [see Fig. 1(b)].\nThis is just the case similar to layered crystal CrCl 3and have\nbeen systematically investigated already, so we won’t repe at\nit here. The second class includes the cases where only in-\ntrinsic SB is involved, which means only in-plane dc fields\nare applied ( ψ=0) as depicted in Fig. 1(c). As a result,\nez′≡ezthus the new x′y′plane is identical to the old xyone.\nNow the AFM and FM critical fields take much simpler forms:\nHAFM=λE|MA\ns−MB\ns|andHFM=λE(MA\ns+MB\ns). Conse-\nquently, the secular equation ( 4) is simplified to\n˜ω4−/parenleftbigg\n˜h2+µ1+κ0κ\n2λE/parenrightbigg\n˜ω2+2λE+ν\n4λ2\nE/parenleftbig\n1−˜h2/parenrightbig/parenleftbig˜h2−κ2/parenrightbig\n=0,\n(5)\nwith ˜ω≡Ω/HFM,κ<˜h≡H/HFM<1,µ= (MA,0\ns+\nMB,0\ns)/(MA\ns+MB\ns)andν= (MA,0\ns/MA\ns)·(MB,0\ns/MB\ns). Gen-\nerally, a gap appears as long as κ2\n0+κ2/negationslash=0.\nIn particular, this gap can be further analyzed for two spe-\ncial cases. In the first case, the two FM sublayers have the\nsame thickness ( dA=dB) but are made of different FM mate-\nrials ( MA,0\ns/negationslash=MB,0\ns). Then MA(B)\ns=MA(B),0\ns , thus µ=ν=1\nandκ0=κ/negationslash=0. Similar to MacNeill et. al. in 2019[ 27], Now\nthe secular equation ( 5) can be rewritten into the eigenvalue\nproblem of a 2 ×2 matrix as\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜ω2\na(˜h)−˜ω2 ˜∆2\n˜∆2˜ω2\no(˜h)−˜ω2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=0. (6)\nHere ˜ωa=/radicalbig\n1+(2λE)−1˜hand ˜ωo= [( 1+κ2)(1−\n˜h2)/(2λE)+( κ2˜h2)/(2λE)]1/2are the bare in-phase (acous-\ntic) and out-of-phase (optical) mode frequencies, respect ively.\nMeantime ˜∆=[(2λE+1)κ2/(4λ2\nE)]1/4describes the dc-field-\nindependent magnon-magnon coupling strength. When |κ|≪\n1 (nearlly symmetric), ˜∆is negligible thus the solution of\nEq. ( 6) are approximately ˜ω≈˜ωaand ˜ω≈˜ωo. For finite\nκ, when ˜his close to κor 1, the solutions of Eq. ( 6) only\ndeviate slightly from ˜ωaand ˜ωo. When the optical and acous-\ntic modes get closer in frequency, they will be hybridized by\n˜∆term and thus a gap is opened. Direct calculations yield\nthat this gap is an indirect gap (see Supplement Material):\nthe minimum ˜ωmin\nup(maximum ˜ωmax\ndown) of the “up (down)”\nbranch takes place at ˜hmin\nup(˜hmax\ndown), where ˜hmin\nup/negationslash=˜hmax\ndownfor\nnozero κ. The corresponding dimensionless gap width reads\nδ˜ω=|κ|/radicalbig\n(2λE+1)/[λE(λE+1)]. Interestingly, the real\ngap width δ˜ωis linear to |κ|(degree of asymmetry between\nthe two FM sublayers), which is different from the square-ro ot\ndependence of the coupling ˜∆onκ. By recalling the defini-tions of ˜ωandκ, we then get the dimensional gap width\nδf=/radicalBigg\nλE(2λE+1)\nλE+1γ\n2π|MA,0\ns−MB,0\ns|, (7)\nwhich should be more useful for experimental physicists. Ob -\nviously, the AFM interlayer coupling and the inhomogeneity\nof sublayers are both crucial for the gap opening.\nAnother interesting case is that the two FM sublayers are\nmade of the same material ( MA,0\ns=MB,0\ns) but have different\nthickness ( dA/negationslash=dB), which is the most common choice in real\nexperiments. Accordingly κ0=0,κ= (dA−dB)/(dA+dB)\nandµ=ν=1−κ2. Now Eq. ( 5) can not be reorganized\nto the 2×2 matrix form as that in Eq. ( 6) if we require the\ncoupling term to be ˜h-independent. However, similar calculus\nshows that now the gap is also indirect. The new extremum,\nextremum locations and the gap width become more compli-\ncated (see Supplement Material), and we then focus on the\nsituation where |κ| ≪1. After standard linearization opera-\ntion, the gap width is approximated to another linear depen-\ndence on |κ|as|κ|[(2λE+1)/(λE+1)]3/2/(2√λE). Back to\ndimensional form, we have\nδf′≈√\nλE\n2/parenleftbigg2λE+1\nλE+1/parenrightbigg3\n2|1−(dB/dA)2|\n2(dB/dA)γ\n2πMA,0\ns. (8)\nEq. ( 8) can be directly compared with existing ex-\nperimental measurements for asymmetrical NiFe/Ru/NiFe\nSAFs[ 16,17]. For NiFe(13.6 nm)/Ru( tRu)/NiFe(27.2 nm)\nSAFs in Ref. [ 16], we choose tRu=4.7˚A as an ex-\nample. By taking MA,0\ns=860 kA m−1and JIEC≈ |J1|=\n286µJ m−2, we get λE≈0.015 and δf′≈1.4 GHz. While\nfor Ni 80Fe20(200 ˚A)/Ru(tRu)/Ni 80Fe20(100 ˚A)SAFs in Ref.\n[17], the magnetic parameters for tRu=3.3˚A are MA,0\ns=\n720 kA m−1andJIEC≈154 µJ m−2. These lead to λE≈\n0.016 and δf′≈1.2 GHz. For other Ru thickness, similar cal-\nculations can be performed and all results show good agree-\nment between analytics and experimental data.\nInterestingly, in FeCoB/Ru/FeCoB SAFs we can acquire\nlarger λE, although nearly all existing published experiments\nare performed in symmetrical cases[ 18–21]. For symmet-\nrical FeCoB thickness being 15 nm[ 18,19], 3 nm[ 20] and\n5 nm[ 21], the respective λEare estimated to be 0 .033, 0.093\nand 0.119(0.141)(two samples therein). Combined with\nlarger saturation magnetization of FeCoB, asymmetrical Fe -\nCoB/Ru/FeCoB SAFs are expected to open larger indirect\ngaps. In view of this, we have performed related measure-\nments in asymmetrical CoFeB/Ir/CoFeB SAFs (experimen-\ntal details will be published elsewhere). In the “CoFeB(10\nnm)/Ir(0.6 nm)/CoFeB(12 nm)” SAF, direction measurements\nprovide MA,0\ns=1400 kA m−1andHex≡2JIEC/[µ0MA,0\ns(dA+\ndB)]=549 Oe, thus λE≈0.0312 and δf′≈0.80 GHz, which\nis in good agreement with experimental observation of gap\nwidth (0.74 GHz). In addition, the gap is observed to take\nplace around 760 Oe, which is also in accordance with theo-\nretical prediction (770 Oe). We are looking forward to more5\nexperimental measurements in the near future. In very re-\ncent symmetrical CoFeB/Ru/CoFeB SAFs[ 20], out-of-plane\ndc fields are needed to open gaps up to 1 GHz. Our results pro-\nvide the possibility that by appropriately designing the th ick-\nness ratio of two CoFeB sublayers, greater gap can be opened\nwhich indicates stronger magnon-magnon coupling. On the\nother hand, we know that it is very difficult to prepare inho-\nmogeneous SAFs ( MA,0\ns/negationslash=MB,0\ns) experimentally, however this\nissue is indeed worth exploring further and Eq. ( 7) should help\nto reveal interesting physics.\nAt the end of this Letter, several points need to be clari-\nfied. First, in this work the crystalline anisotropy has been\nneglected because of two reasons: (i) in most existing exper i-\nmental setups, the FM sublayers of SAFs are made from soft\nmagnetic materials which can be viewed as isotropic; (ii) th e\nexplicit orientations of meq\nAandmeq\nBcan hardly be obtained\nanalytically if the in-plane crystalline anisotropy is con sid-\nered (even for the simplest uniaxial case), but they are cruc ial\nfor obtaining the vectorial LLG equation and then the secu-\nlar equations for FMR frequencies. For the above reasons, in\nthis work we choose the isotropic case for simplicity, but it can\ncover the vast majority of experimental data. Note that our a n-\nalytics also holds for perpendicular-magnetic-anisotrop y case\nas long as we change MA(B),0\ns toMA(B),0\ns−HA(B)\nKin Eq. ( 1)\nwhere HA(B)\nKis the out-of-plane anisotropic field in the respec-\ntive sublayer. Another neglected term is the Gilbert dampin g\nterm. In most investigations of spin wave, the damping term i s\ndropped off when only the resonance spectrum is under con-\nsideration. However when the linewidth is also of interest,\nthen the damping process should be included in Eq. ( 3) by\na respective term i ωαmeq\nA×δm±with αbeing the Gilbert\ndamping coefficient[ 27].\nIn summary, we have proposed a simple but revealing the-\nory for understanding the rich structure of FMR spectroscop y\nin asymmetrical SAFs under tilting dc magnetic fields. Both\nintrinsic and extrinsic SB cause entanglement between mag-\nnetization vibrations with opposite parity, thus excite st rong\nmagnon-magnon coupling between the bare in-phase and out-\nof-phase modes, then eventually result in the anticrossing gap\nin microwave absorption spectroscopy. In addition, for pur e\nintrinsic SB the two-fold rotation symmetry of both the mag-\nnetization and magnetostatics around meq\nA+meq\nBfails. While\nfor pure extrinsic SB, only that of magnetostatics fails. Th is\nnew picture helps to understand the rich experimental data o f\nFMR spectra for existing SAFs and future measurements on\nother SAF or SAF-like systems.\nJ.L. acknowledges support from the Natural Science Foun-\ndation for Distinguished Young Scholars of Hebei Province o f\nChina (A2019205310). M.L. is funded by the National Nat-\nural Science Foundation of China (Grant No. 11947023), the\nProject of Hebei Province Higher Educational Science and\nTechnology Program (QN2019309) and the PhD Research\nStartup Foundation of Shijiazhuang University (20BS022).\nW.H. is supported by the National Natural Science Founda-\ntion of China (Grant No. 51871235).∗limeijim@163.com\n†hewei@iphy.ac.cn\n[1]R. A. Duine, K.-J. Lee, S. S. P. Parkin, and M. D. 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The piezoelectric actuator\ninduces a voltage-controllable strain along di \u000berent crystalline directions of the ferromagnetic thin film, resulting in modifications\nof its magnetization by converse magnetoelastic e \u000bects. We quantify the magnetization changes in the hybrids via ferromagnetic\nresonance spectroscopy and superconducting quantum interference device magnetometry. These measurements demonstrate a sig-\nnificant strain-induced change of the magnetization, limited by an ine \u000ecient strain transfer and domain formation in the particular\nsystem studied. To overcome these obstacles, we address practicable engineering concepts and use a model to demonstrate that a\nstrain-controlled, nonvolatile magnetization switching should be possible in appropriately engineered ferromagnetic /piezoelectric\nactuator hybrids.\nKeywords: A. Multiferroic Hybrids, D. Magnetostriction, E. Ferromagnetic Resonance, E. SQUID magnetometry\n1. Introduction\nIn magnetoelectric multiferroics, where the ferromagnetic\nand ferroelectric order parameters are coupled, an electric-\nfield control of the magnetic properties becomes possi-\nble.1,2,3This opens the way for appealing novel magnetiza-\ntion control schemes in future spintronic devices.4Unfortu-\nnately, single-phase multiferroics with strong magnetoelec-\ntric coupling remain rare.5,2Attractive alternatives are com-\nposite material systems made from ferroelectric and ferro-\nmagnetic compounds.6,7,8,9In such systems, an electric-field\ncontrol of magnetism can be realized using electric field ef-\nfects in carrier-mediated ferromagnets,10,11or exchange cou-\npling at ferromagnetic /multiferroic interfaces.12,13A third,\npowerful approach relies on strain-mediated, indirect magne-\ntoelectric coupling in ferromagnetic /ferroelectric hybrid sys-\ntems. In recent years, these hybrids were mostly fabricated\nby depositing ferromagnetic thin films on ferroelectric sub-\nstrates.14,15,16,17,18,19,20,21,22,23,24,25Another approach to realize\na strain-mediated control of the magnetization is to fabricate\nferromagnetic thin film /piezoelectric actuator hybrids by either\ndepositing or cementing ferromagnetic thin films onto commer-\ncially available Pb (ZrxTi1\u0000x)O3(PZT) multilayer piezoelectric\nactuator stacks [cf. Fig. 1(a)].26,27,28,29,30,31In these hybrids, the\napplication of a voltage to the piezoelectric actuator results in\na deformation, which is transferred to the overlaying ferromag-\nnetic thin film, changing its magnetic anisotropy due to the con-\nverse magnetoelastic e \u000bect.\n\u0003Corresponding author\nEmail address: stephan.gepraegs@wmi.badw.de (S. Gepr ¨ags)In this paper, we report on two di \u000berent experimental\napproaches towards a strain-mediated, nonvolatile, voltage-\ncontrolled magnetization switching in the complete absence\nof magnetic fields. They are based on ferromagnetic thin\nfilm/piezoelectric actuator hybrids using Fe 3O4as the ferro-\nmagnet. Our experiments show that a significant modification\nof the magnetic anisotropy is possible via voltage-controlled\nstrain. This work extends our previous studies on ferromag-\nnetic/ferroelectric hybrids,26,27,30,20where we achieved a re-\nversible reorientation of the magnetization by up to 90\u000ein Ni\nbased hybrids. However, a true switching of the magnetization\nbetween two (or more) remanent states solely by means of an\nelectric field induced strain has not been realized experimen-\ntally up to now.32,33,34,35,36,37\n2. The spin-mechanics concept\nThe orientation of a well-defined homogeneous magnetiza-\ntion in a ferromagnet depends on external mechanical stress\ndue to magnetostriction.38,39We exploit this so-called spin-\nmechanics scheme to control the magnetic anisotropy in Fe 3O4\nthin films cemented on Pb (ZrxTi1\u0000x)O3(PZT) multilayer\npiezoelectric actuator stacks [cf. Fig. 1(a)]. In particular, we\ncompare di \u000berent hybrids fabricated by cementing Fe 3O4thin\nfilms with di \u000berent angles \u000bbetween the crystallographic axes\nfx;ygof the film and the principal elongation axes fx0;y0gof the\nactuator with zkz0.\nThe application of a voltage Vp>0\u0010\nVp<0\u0011\nto the piezoelec-\ntric actuator causes an elongation \u000f0\n2>0 (contraction \u000f0\n2<0)\nalong the actuator’s dominant elongation axis y0, which is due\nto elasticity accompanied by a contraction (elongation) along\nPreprint submitted to Solid State Communications August 28, 2018arXiv:1307.2433v1  [cond-mat.mtrl-sci]  9 Jul 2013x’y’\nz’\nz\n[001]x\n[100]y\n[010]\nα\nx\n[100]y\n[010]\nz\n[001]θ\nφ\nΦM\nHΘ(a) (b)\nVpdominant\nelongation\naxisFigure 1: (a) Schematic illustration of a Fe 3O4thin film /piezoelectric actuator\nhybrid. The coordinate system of the thin film and the actuator enclosing an\nangle\u000bare denoted byfx;y;zgandfx0;y0;z0g, respectively. (b) Orientation of\nthe magnetic field H(H;\u0012;\u001e)and the magnetization M(Ms;\u0002;\b)with respect\nto the crystallographic axes h100iof the Fe 3O4thin film.\nthe two orthogonal directions x0andz0[cf. Fig. 1(a)]. This leads\nto a change of the strain state \u000fof the Fe 3O4thin film elastically\nclamped onto the piezoelectric actuator. This causes a modifi-\ncation of the magnetic anisotropy, and thus alters the direction\nof the magnetization M. In a macrospin model, the magneti-\nzation Mof the Fe 3O4thin film described by M(Ms;\u0002;\b)=\nMsm(\u0002;\b)aligns in such a way that the free energy density F\ntakes its minimum value in equilibrium. Here, mx=sin\u0002sin\b,\nmy=cos\u0002, and mz=sin\u0002cos\b[cf. Fig. 1(b)] are directional\ncosines and Msthe saturation magnetization. The orientation\nof the magnetization m(\u0002;\b)can be calculated in the frame-\nwork of a single domain model by using a phenomenological\nthermodynamic model based on the free energy density\nF=FZeeman +F001\nu;e\u000b+Fmc+Fel+Fme (1)\nwith the Zeeman energy density FZeeman =\n\u0000\u00160Msm(\u0002;\b)Hh(\u0012;\u001e), the e \u000bective uniaxial anisotropy\ncontribution F001\nu;e\u000b=1\n2\u00160M2\nsm2\nz+K001\num2\nz, which comprises\nthe demagnetization contribution and the uniaxial contribution\nK001\nuresulting from the pseudomorphic growth of the ferromag-\nnetic thin film, the first-order magnetocrystalline anisotropy\ncontribution Fmc=Kc\u0010\nm2\nxm2\ny+m2\nym2\nz+m2\nzm2\nx\u0011\nwith the cu-\nbic anisotropy constant Kc, the elastic energy density40Fel=\n1\n2c11\u0010\n\u000f2\n1+\u000f2\n2+\u000f2\n3\u0011\n+c12(\u000f1\u000f2+\u000f2\u000f3+\u000f1\u000f3)+1\n2c44\u0010\n\u000f2\n4+\u000f2\n5+\u000f2\n6\u0011\n,\nand the magnetoelastic contribution\nFme=B1\"\n\u000f1 \nm2\nx\u00001\n3!\n+\u000f2 \nm2\ny\u00001\n3!\n+\u000f3 \nm2\nz\u00001\n3!#\n+B2\u0010\n\u000f4mymz+\u000f5mxmz+\u000f6mxmy\u0011\n: (2)\nThe magnetoelastic coupling coe \u000ecients B1andB2can be writ-\nten as a function of the magnetostrictive constants \u0015100and\u0015111,\nwhich yields B1=\u00003\n2\u0015100(c11\u0000c12)andB2=\u00003\u0015111c44.\nHere we use bulk values for the magnetostrictive constants\n(\u0015100=\u000019:5\u000210\u00006and\u0015111= +77:6\u000210\u00006) as well as\nfor the elastic sti \u000bness constants ci j(c11=27:2\u00021010N/m2,\nc12=17:8\u00021010N/m2, and c44=6:1\u00021010N/m2).41,42,43To determine the modification of the magnetic anisotropy\ncaused by strain e \u000bects induced by the piezoelectric actuator,\nwe first derive the strain tensor \u000fof the Fe 3O4thin film. In\nthefx0;y0;z0gcoordinate system, the strain components \u000f0\n4,\u000f0\n5,\nand\u000f0\n6vanish, since no shear strains are present. Furthermore,\n0° 90° 180°\nMorientationΘ000\n-εmax\n+εmax\n0° 90° 180°\nMorientationΘ000\n-εmax\n+εmax\nF(arb.units)\n0° 90° 180°F(arb.units)\nMorientationΘ0° 90° 180°F(arb.units)\nMorientationΘF(arb.units)Vp > 0 (ε’2>0)x\n[100]y\n[010]\nVp < 0 (ε’2<0)Vp > 0 (ε’2>0)x\n[100]y\n[010]\n45°\nVp < 0 (ε’2<0)(e)\n(f)x’y’\n(g)AC\nB\nA\nC\nD(a)(b)\nC\n(c)A\nB\nC\nDε’2=+εmax\nε’2=+εmax/2\nε’2=0\nε’2=-εmax/2ε’2=-εmax\nε’2=0\n(d) (h)\nC\nABD\nC\nABDAε’2=+εmax\nε’2=0\nε’2=-εmax\nε’2=0\nC\nFigure 2: (a)–(d) Stress applied to a cubic thin film along the in-plane crys-\ntallographich100iaxes (\u000b=0\u000e). (b)–(d) Corresponding free energy density\ncontours F\u0010\n\u0002;\u000f0\n2\u0011\n, with capital letters indicating the equilibrium magnetization\norientations. The full yellow line in (d) traces the minimum of F. Forward\nswitching occurs (A!C), while a back switching (C!A)is not possible.\n(e)–(h) Deformation of a cubic crystal along the in-plane h110iaxes (\u000b=45\u000e).\nBoth a forward and a back switching is feasible. The green arrows illustrate the\ndiscontinuous change of the magnetization orientation.\nas the thin film is clamped to the piezoelectric actuator, the in-\nplane strains \u000f0\n1and\u000f0\n2are not independent. Due to the actua-\ntor’s elastic properties, these strain components are related via\nthe Poisson ratio \u0017=0:45 according to \u000f0\n1=\u0000\u0017\u000f0\n2. To obtain the\nstrain tensor \u000fin the coordinate system of the Fe 3O4thin film\nfx;y;zg, we apply a tensor transformation as described in de-\ntail in Refs.38,44. The strain components \u000fi(i=3;4;5)can then\nbe deduced according to the mechanical equilibrium condition\n\u001bi=@F=@\u000fi=0(i=3;4;5). With this relation, we finally\n2obtain\u000fas a function of \u000f0\n2, neglecting comparably small mag-\nnetoelastic terms:\n\u000f=0BBBBBBBBBBBBBBBBBBBBBBB@\u00001\n2[\u00001+\u0017+(1+\u0017)cos(2\u000b)]\u000f0\n21\n2[1\u0000\u0017+(1+\u0017)cos(2\u000b)]\u000f0\n2\n\u0000c12\nc11(1\u0000\u0017)\u000f0\n2\n0\n0\n(1+\u0017)sin(2\u000b)\u000f0\n21CCCCCCCCCCCCCCCCCCCCCCCA: (3)\nNow we are in a position to derive the magnetization orien-\ntation m(\u0002;\b)by tracing the minimum of the total free en-\nergy density Fas a function of \u000f0\n2, which can be controlled by\nVp. The corresponding evolution is calculated by minimizing\nEq. (1) with respect to the orientation of the magnetization \u0002.\nSince the strain induced in the ferromagnetic thin film is of the\norder of 10\u00003in our hybrid structures, the magnetoelastic en-\nergy contribution Fmewill not overcome the demagnetization\nenergy in Fe 3O4thin films. Thus, the magnetization remains in-\nplane in case of zero magnetic field, which results in \b = 90\u000e.\nTo illustrate the concept of a strain-induced, nonvolatile mag-\nnetization switching in zero magnetic and electric fields, Fig. 2\nexemplary shows free energy density F(\u0002;\u000f0\n2) contours within\nthe film plane for \u000b=0\u000e[Fig. 2(b)–(d)] and \u000b=45\u000e[Fig. 2(f)–\n(h)]. In both cases, the induced uniaxial strain is symmetric\nwith respect to the crystallographic axes of the cubic ferro-\nmagnetic thin film. This results in two energetically equiva-\nlent minima in the free energy density F, which forces domain\nformation. To lift this degeneracy a small uniaxial magnetic\nanisotropy contribution in the film plane is introduced in the\nsimulations given by Fip\nu=Kip\nu\u0010\nmxsin\u0002u+mycos\u0002u\u00112with\nthe uniaxial anisotropy constant Kip\nu. For illustration purposes,\nwe here use \u0002u=10\u000eandKip\nu>0 with\f\f\f\fKip\nu=Kc\f\f\f\f=1=15.\nTo meet the experimental conditions of Fe 3O4thin films, we\nchoose K001\nu>0 and Kc<0.\nIn case of \u000b=0\u000e, the ferromagnetic thin film is elon-\ngated and contracted along the cubic axes ( x0jjxandy0jjy)\n[cf. Fig. 2(a)] and thus no shear strains appear ( \u000f6=0)\n[cf. Eq. (3)]. Starting at \u000f0\n2=0 (Vp=0 V) [cf. black line in\nFig. 2(b)], the magnetization orientation \u0002is aligned along a\nmagnetically easy axis, e.g., at \u0002 = 47\u000e(point A). Upon in-\ncreasing\u000f0\n2(Vp>0 V), the magnetically easy axis and thus\nthe magnetization orientation \u0002continuously rotates towards\n\u0002 = 98\u000e(point B). The corresponding free energy density con-\ntour for\u000f0\n2= +\u000fmax[cf. red line in Fig. 2(b)] is calculated as-\nsuming B1\u000fmax=Kc=3=5, which corresponds to \u000fmax=1\u000210\u00003\nin case of Fe 3O4thin films. By decreasing \u000f0\n2back to 0, the\nmagnetization orientation continuously rotates to the energet-\nically stable direction \u0002 = 133\u000e(point C) at \u000f0\n2=0. This\ndemonstrates that a reorientation of the magnetization by about\n86\u000eis feasible. To check the possibility to reorient the mag-\nnetization orientation to the initial configuration (point A), \u000f0\n2\nis inverted. Figure 2(c) discloses that the easy axis gradually\nrotates from \u0002 = 133\u000e(point C) to \u0002 = 165\u000e(point D) by\ninducing\u000f0\n2=\u0000\u000fmax. However, upon reducing \u000f0\n2back to 0,\nthe easy axis rotates back to \u0002 = 133\u000e(point C). Thus, the\nmagnetization remains in point C and a further strain-inducedswitching process is not possible. Consequently, the configu-\nration\u000b=0\u000eallows for a single, irreversible, and nonvolatile\nmagnetization switching. The whole reorientation process of\nthe magnetization for \u000b=0\u000eis shown in Fig. 2(d), which dis-\nplays the free energy density surface F(\u0002;\u000f0\n2). The full yel-\nlow line traces the minimum of F. By applying the sequence\n0!+\u000fmax!0!\u0000\u000fmax!0 for\u000f0\n2, the remanent magnetiza-\ntion aligns along A !B!C!D!C.\nIn contrast to \u000b=0\u000e, the configuration with \u000b=45\u000eleads to\na finite shear strain component \u000f6,0, since the piezoelectric\nactuator exerts stress along the in-plane h110idirections of the\nferromagnetic thin film [cf. Fig. 2(e)]. For simplicity, we as-\nsumejB1\u000fmax=Kcj=jB2\u000fmax=Kcj. At the beginning ( \u000f0\n2=0) [cf.\nblack line in 2(f)], the magnetization orientation \u0002is aligned\nalong 133\u000e(point A). While increasing \u000f0\n2, the easy axis basi-\ncally retains its initial orientation. However, the free energy\ndensity minimum gradually transforms into a maximum. Upon\na certain critical induced strain \u000fsw, the easy axis changes dis-\ncontinuously to \u0002 = 46\u000e(point B), indicating an abrupt mag-\nnetization switching [cf. green arrow in 2(f)]. The orientation\nof the easy axis essentially stays along \u0002 = 46\u000ewhile reduc-\ning\u000f0\n2back to 0 (point C). Subsequently, we continuously in-\ncrease the inverted induced strain \u000f0\n2<0 [Fig. 2(g)]. Starting\nfrom point C the easy axis abruptly rotates to \u0002 = 133\u000e(point\nD). This magnetization orientation remains unchanged, while\nincreasing\u000f0\n2back to zero again. Thus, in case of \u000b=45\u000e,\na reorientation of the magnetization back to the initial state\nis possible, which demonstrates that a reversible, nonvolatile\nmagnetization switching in the absence of a magnetic field is\npossible. The switching of the magnetization from point A to\npoint C and back to point A upon applying the strain sequence\n0!+\u000fmax!0!\u0000\u000fmax!0 is further illustrated in Fig. 2(h).\n3. Towards a nonvolatile magnetization switching via strain\nin experiment\nAs described in the previous section, a nonvolatile mag-\nnetization switching is theoretically possible in Fe 3O4thin\nfilm/piezoelectric actuator hybrid structures. In the following,\nwe discuss two hybrids corresponding to the configurations dis-\ncussed in Section 2. The hybrids are based on the same (001)-\noriented, 44 nm thick Fe 3O4film grown on a MgO (001) sub-\nstrate by laser-MBE. After the deposition the thin film sample\nwas cut into two pieces, which were cemented onto the piezo-\nelectric actuators in such a way that stress is either exerted along\ntheh100icrystal axes ( \u000b=0\u000e) or alongh110i(\u000b=45\u000e).\nThe fabrication process of the thin film /piezoelectric actuator\nhybrid structure is described in detail in Ref.26. The samples\nthus obtained are referred to as hybrid h100iand hybridh110i,\nrespectively. The magnetic anisotropy of the Fe 3O4thin film\nwas determined by angular-dependent ferromagnetic resonance\n(FMR) spectroscopy at constant actuator voltages Vpwith the\nmagnetic field applied in the film plane h(\u0012;\u001e=90\u000e)at room\ntemperature.26\nFor the hybridh100i(\u000b=0\u000e), the evolution of the obtained\nFMR fields \u00160Hres(\u0012)as a function of the external magnetic\nfield orientation \u0012reveals a superposition of a cubic magnetic\n30° 90° 180°250300µ0Hres(mT)\nHorientationθ250300µ0Hres(mT)\n04F/Ms(mT)\n0° 90° 180°-40F/Ms(mT)\nMorientationΘ(a)-30V\n0V+90V\n(b)\nhybrid <110>\n-30V\n+150V0V\n+60Vhybrid <100>\n(c) (d)hybrid <100>\nhybrid <110>-30V\n0V+90V\n+60V\n+150V-30V\n0VFigure 3: FMR fields \u00160Hresfor a rotation of the magnetic field in the film plane\nh(\u0012;\u001e=90\u000e)as a function of Vp(symbols) for the hybrid h100i(a) and the\nhybridh110i(c). The lines depict the simulated FMR fields. (b), (d) Calculated\nfree energy density contours as a function of the magnetization orientation \u0002in\nthe film plane at zero external magnetic field.\nanisotropy with a uniaxial one [cf. Fig. 3(a)].26For a quanti-\ntative simulation of the experimental data, the FMR angular\ndependence is simulated according to Eq. (1).45,26. The best\nagreement between the FMR fields for Vp=0 V observed\nin experiment [cf. black symbols in Fig. 3(a)] and simulation\n[cf. black line in Fig. 3(a)] was obtained by using the voltage-\nindependent anisotropy fields K001\nu;e\u000b=Ms=80:2 mT, Kc=Ms=\n\u000014:9 mT, and Kip\nu=Ms=3:2 mT. An additional uniaxial contri-\nbution in the film plane Fip\nuwith\u0012u=0\u000e, which is not observed\nin the as-grown Fe 3O4thin film and caused by an anisotropic\nthermal expansion during the curing process, has to be included\nin the free energy density F. For Vp,0 V , i.e.,\u000f0\n2,0,\nthe FMR fields \u00160Hresare modeled by using \u000f0\n2as fit param-\neter [cf. red and blue lines in Fig. 3(a)]. The derived strain\n\u0001\u000f0\n2=\u000f0\n2(+90 V)\u0000\u000f0\n2(\u000030 V) =0:23\u000210\u00003induced in the\nferromagnetic thin film amounts to only about 27% of the nom-\ninal stroke of \u0001\u000fideal\n2=0:87\u000210\u00003of the piezoelectric actua-\ntor.46This is most likely caused by an imperfect strain transmis-\nsion between the piezoelectric actuator and the Fe 3O4thin film,\nwhich can be described by \u0001\u000f0\n2=\u001f100\u0001\u000fideal\n2with\u001f100=0:27.\nThe corresponding free energy contours within the film plane\nF=Ms(\u0002;\b = 90\u000e) in the absence of an external magnetic field\nare shown in Fig. 3(b). In agreement with Figs. 2(b)–(d), the\ncontour for Vp=0 V (\u000f0\n2=0) exhibits a fourfold symmetry with\na superimposed magnetic hard axis, which is found to be along\n\u0002u=0\u000ein the experiment. Upon the application of Vp,0 we\nobserve a change of the relative strength of the magnetic hard\naxes, as evident from the di \u000berent magnitudes of the maxima\nof the free energy, while they retain their orientation. The easy\naxes—i.e., the free energy density minima—almost retain their\nstrength, but the orientation \u0002of the easy axes clearly is depen-\ndent on Vp. This is the basis for the continuous and reversible\nrotation of Min the spin-mechanics scheme and confirms thesimulations of Fig. 2. However, the free energy density con-\ntours in Fig. 3(a) reveal a rotation of the easy axes by \u0001\u0002 = 6\u000e\nfor -30 V\u0014Vp\u0014+90 V at room temperature. Hence, a continu-\nous and reversible voltage control of magnetization orientation\nis possible, but a voltage-controlled magnetization switching is\nout of reach in the present hybrid, since the induced strain \u000f0\n2is\nmuch lower than the nominal strain of the piezoelectric actua-\ntor.\nIn case of\u000b=45\u000e, the experimentally obtained and simu-\nlated FMR fields \u00160Hres(\u0012)are shown in Fig. 3(c). In analogy\nto the configuration \u000b=0\u000e, the total free energy density Ffor\nthe present sample is composed of Eq. (1), with the additional,\nthermally induced uniaxial anisotropy Fip\nuin the film plane\nalong \u0002u=5\u000e. The solid lines in Fig. 3(b) represent the nu-\nmerically simulated FMR fields using the voltage-independent\nanisotropy fields K001\nu;e\u000b=Ms=75:3 mT, Kc=Ms=\u000014:5 mT,\nKip\nu=Ms=1:1 mT, and the strain \u000f0\n2as fit parameter. From these\nvalues a non-ideal strain transfer of \u001f110=0:09 can be inferred.\nThe corresponding calculated free energy density curves in the\nfilm plane F=Ms(\u0002;\b = 90\u000e) are depicted in Fig. 3(d). Ac-\ncording to Section 2, upon the application of a voltage Vp, the\nenergy minima mainly retain their orientation. More impor-\ntantly, the relative strengths of the magnetic easy axes consid-\nerably change, as illustrated for the energy density minimum at\n\u0002 = 133\u000e, which remarkably loses depth for Vp= +150 V and\napproaches transforming into a maximum. Due to the low \u001f110\nvalue the strain-induced anisotropy is unfortunately not large\nenough to cause an abrupt magnetization switching as shown in\nFig. 2(f) and (g). However, by optimizing the strain transmis-\nsion e \u000eciency, magnetization switching should be possible for\n\u000b=45\u000e.\nFigure 3 demonstrates that angular-dependent FMR mea-\nsurements allow to quantitatively determine the contributions\nto the total free energy density F. However, it does not di-\nrectly measure the remanent magnetization orientation. More-\nover, Eq. (1) is applicable only to homogeneously magnetized\nsamples. This is valid for FMR measurements, since the ap-\nplied external field su \u000eces to fully saturate the magnetization\nfor the present hybrids. As we are particularly aiming at a\nmagnetization switching at vanishing external magnetic field,\nmagnetic domain formation might be important. Therefore,\nin the following, we utilize superconducting quantum interfer-\nence device (SQUID) magnetometry measurements as a func-\ntion of the in-plane magnetic field orientation \u0012to directly mea-\nsure the remanent magnetization as a function of Vp. For these\nangular-dependent magnetization measurements, we magne-\ntized the hybrid along a magnetically easy axis by applying\n\u00160Hprep= +1 T and then swept the magnetic field to \u00160H=0 T\nat a fixed strain state, i.e., at a fixed voltage Vp. After the prepa-\nration of the magnetization, we recorded the projection of the\nmagnetization on the magnetic field direction as a function of\nthe magnetic field orientation \u0012.\nIn case of the hybrid h100i, the preparation of the magnetiza-\ntion was carried out at Vp=\u000030 V with the external magnetic\nfield oriented along \u0012prep=50\u000e, which corresponds to a mini-\nmum of the free energy density F[cf. Fig. 3(b)]. The results ob-\n4-2000200M(kA/m)\n50°60°\n12θmaxMmax(102kA/m)\n0° 180° 360°-2000200M(kA/m)\nHorientationθ0 100120°130°\n12θmax\nVp(V)Mmax(102kA/m)\n360°hybrid <100>\nhybrid <110>-30V\n+150V+60V0V-30V\n+90V0V(a) (b)\n(c) (d)Figure 4: SQUID magnetometry measurements as a function of \u0012with\u001e=90\u000e\nat di\u000berent voltages Vpusing the hybridh100i(a), (b) and the hybrid h110i\n(c), (d). All measurements were carried out at \u00160H=0 mT. The symbols\nrepresent the experimental data and the lines denote fits to cosine functions.\n(b), (d) Orientation \u0012max, which denotes the angle of the maximum value of\nM(\u0012)(black squares), and the corresponding magnitude Mmax(green circles)\nas a function of Vp.\ntained by carrying out angular-dependent magnetometry mea-\nsurements are shown in Fig. 4(a). Since in the absence of an ex-\nternal magnetic field the magnetization preferably aligns along\na magnetic easy axis, maxima in the M(\u0012)curves correspond\nto minima in the F(\u0002)contours. The respective maxima of the\nM(\u0012)curves are evaluated in Fig. 4(b), regarding their orien-\ntation\u0012max(black squares) and magnitude Mmax(green circles)\nas a function of Vp.Mmaxchanges by only 1% in the volt-\nage range -30 V\u0014Vp\u0014+90 V and thus is almost independent\nofVp. This demonstrates that domain formation plays only a\nnegligible role in case of \u000b=0\u000e. However, \u0012maxchanges by\nabout 9\u000e. This proves that the macroscopic, homogeneous re-\nmanent magnetization Mrotates by about 9\u000ein the film plane\nfor -30 V\u0014Vp\u0014+90 V , which confirms the results obtained by\nFMR measurements [cf. Fig. 3(b)].\nWe now turn to the hybrid h110i. In a first set of exper-\niments, the preparation field was applied along \u0012prep=133\u000e\n[Fig. 4(c)]. As the free energy density at this orientation con-\ntinuously evolves from a deep minimum towards a shallow one\nwith increasing \u000f0\n2, i.e., Vp[cf. Fig. 3(d)], this minimum will\nbe referred to as local minimum in the following. The angle-\ndependent SQUID measurements reveal a qualitatively di \u000berent\nbehavior compared to the measurements on the hybrid h100i.\nIn case of the hybrid h110i, both the magnitude Mmaxas well\nas the orientation \u0012maxof the maximum significantly change\nas a function of Vp[cf. Fig. 4(d)]. Upon increasing Vpfrom\n-30 V to +150 V , Mmaxdecreases by 49% of its initial value,\nwhile the orientation of Mmaxrotates by 20\u000etowards the free\nenergy density minimum at 47\u000e[cf. Fig. 2(f)]. The reduction\nofMmaxelucidates magnetic domain formation with increas-\ningVp. After the magnetic preparation at Vp=\u000030 V , the\nFe3O4thin film exhibits a single-domain state with the homo-geneous magnetization oriented along \u0002 = 133\u000e. As the ap-\nplied voltage Vpincreases, the local energy density minimum\nat\u0002 = 133\u000elooses depth and thus magnetically hardens, while\nthe global minimum of the energy density at \u0002 = 47\u000emagnet-\nically softens, favoring domain formation [cf. Fig. 3(c)]. Thus,\nthe angular-dependent magnetization measurements shown in\nFigs. 4(c) and (d) are not consistent with the single-domain free\nenergy density approach used to calculated the energy density\ncontours in Fig. 3(d).\nIn a second set of experiments, we repeated the SQUID mea-\nsurements with the magnetic preparation field Hprepapplied\nalong\u0012prep=43\u000e, close to the global minimum of the free\nenergy density at \u0002 = 47\u000e. The experimental data coincide\nin good approximation for di \u000berent applied voltages Vp[not\nshown here]. The magnetization Mretains its orientation at\n\u0002 = 47\u000eindependent of Vp, while the magnitude of the mag-\nnetization at this orientation Mmaxchanges by only 5% within\nthe full voltage range. Considering the free energy density con-\ntours [cf. Fig. 3(d)], the energy barrier for domain formation is\nmuch larger in this case, such that the Fe 3O4thin film remains\nin a magnetically single domain state and can be described by\nEq. (1).\n4. Impact of di \u000berent strain orientations\nThe experimental results discussed above show that an align-\nment of the strain axes along crystallographic axes might favor\nmagnetic domain formation. Therefore, we now discuss con-\nfigurations with an angle \u000bin between 0\u000eand 45\u000e. The cor-\nresponding concept is exemplarily illustrated for \u000b=20\u000ein\nFigs. 5(a), (b), and (c).\nStarting at\u000f0\n2=0, we assume an initial magnetization ori-\nentation along \u0002 = 135\u000e[point A in Fig. 5(a)]. Upon increas-\ning\u000f0\n2>0 in the thin film, the easy axis continuously rotates,\nuntil it switches discontinuously to point B at a certain criti-\ncal strain\u000f0;crit\n2=\u000fsw(\u000b)[green arrow in Fig. 5(a)]. When the\nstrain\u000f0\n2is reduced back to 0, the easy axis rotates to point C at\n\u0002 = 45\u000e. Upon subsequently increasing the strain \u000f0\n2<0 with\nopposite sign [Fig. 5(b)], the situation appears qualitatively dif-\nferent from the situation illustrated in Fig. 2(g), as we do not\nobserve a back switching to the initial orientation (point A), but\na further switching process along the original direction of ro-\ntation via point D to point E at \u0002 =\u000045\u000e[Fig. 5(b)]. Hence,\niteratively applying \u000fswwith alternating sign provides a concept\nto discontinuously rotate the equilibrium magnetization orienta-\ntion by 90\u000evia nonvolatile switching processes.37Such magne-\ntization switching processes are “quasi-reversible”, since four\nconsecutive switching processes (in a ferromagnet with cubic\nsymmetry) evidently restore the initial magnetization orienta-\ntion state [Fig. 5(c)]. Hence, this constitutes a very elegant\nvoltage-control scheme of magnetization orientation.\nAssuming a perfect strain transmission between the piezo-\nelectric actuator and the ferromagnetic thin film ( \u001f=1),\u000fsw\ncan be derived using the free energy density Fgiven in Eq. (1)\nwith a cubic anisotropy field Kc=Ms=\u000014:7 mT, which is\nthe averaged value of the cubic anisotropy measured in hy-\nbridh100iand hybridh110i, as well as the elastic and mag-\n50 510152025303540450.000.020.040.060.080.10εsw(%)\nα(°)020406080100120140\nVsw\np(V)F(arb.units)\n0° 90°F(arb.units)\nMorientationΘ-90°0°90°180°270°\nMorientationΘ\nχ=0.27\nχ=1.0AE\nBD\nCα=20°(a)\n(d)C\nA\nB\n(b)\nC\nDE(c)\n000\n-εmax\n+εmax00\n-εmax\n+εmaxε’2=+εmax\nε’2=+εmax/2\nε’2=0\nε’2=-εmax/2ε’2=-εmax\nε’2=0α=20°\nα=20°Figure 5: Approach to a nonvolatile, all-voltage controlled magnetization\nswitching. (a)–(c) Calculated free energy density contours F\u0010\n\u0002;\u000f0\n2\u0011\nfor\u000b=\n20\u000e. The full yellow line in (c) traces the minimum of F. Discontinuous switch-\ning processes of the magnetization by 90\u000efrom A!C!E while applying\n\u000fswwith alternating sign are visible. (d) Calculated critical strain \u000fsw(\u000b)values\nand corresponding switching voltages Vsw\np(\u000b)at which a discontinuous switch-\ning of the magnetization, which is illustrated by the green arrows in (a) and\n(b), occurs. The blue curve depicts perfect strain transmission ( \u001f=1:0) and\nthe black curve represents the experimentally realized strain in the hybrid h100i\n(\u001f=0:27).\nnetoelastic constants of Fe 3O4. Figure 5(d) shows that \u000fsw\nrequired to induce a magnetization switching process signifi-\ncantly decreases with increasing angle \u000b, exhibits a minimum\nat\u000b=33\u000e, and finally slightly increases with \u000bapproaching\n45\u000e. Overall,\u000fswhas comparatively moderate values lower than\n10\u00003, which are experimentally achievable using the concept\ndescribed in Fig. 1(a). These values correspond to switching\nvoltages Vsw\np(\u000b)<150 V in our hybrid concept, which are ex-\nperimentally accessible [cf. Fig. 5(d)].\nTo furthermore lower the switching strain \u000fsw, the properties\nof the ferromagnetic film itself must be fine-tuned, as \u000fswlin-\nearly depends on the cubic anisotropy constant Kcand inversely\ndepends on the magnetostriction constants \u0015100and\u0015111. Most\npromising candidates regarding the realization of a magnetiza-\ntion switching therefore evidently are materials with a small\ncubic anisotropy and high magnetostriction constants.5. Conclusion\nIn summary, we have investigated concepts for a voltage-\ncontrolled, nonvolatile 90\u000eswitching of the remanent magneti-\nzation in Fe 3O4thin film /piezoelectric actuator hybrids at room\ntemperature. The possibility to induce strain along di \u000berent di-\nrections in the film plane with respect to the crystallographic\naxes depending on the cementing procedure, allows to inves-\ntigate the switching behavior and particularly to take advan-\ntage of the magnetostriction constants \u0015along di \u000berent crys-\ntalline orientations. We have discussed the qualitatively di \u000ber-\nent switching behavior for two di \u000berent configurations, namely\nstrain exerted along the in-plane crystalline Fe 3O4h100iand\nalong the in-plane h110idirections. The free energy density\nof the ferromagnetic thin films was determined by FMR spec-\ntroscopy, which allows to infer the equilibrium magnetization\norientation in a Stoner-Wohlfarth model. The results show a ro-\ntation of the easy axes by a few degrees and a significant mod-\nification of the relative strength of the easy axes, respectively.\nHowever, in combination with SQUID magnetometry measure-\nments we find that the angle of magnetization reorientation is\nnot large enough to induce a magnetization switching in the for-\nmer, and magnetic domain formation impedes a coherent mag-\nnetization switching in the latter approach. This shows that\nine\u000ecient strain transfer and magnetic domain formation are\nmajor obstacles towards a non-volatile strain-controlled mag-\nnetization switching. Using the experimental free energy and\nstrain transfer parameters, we find in simulations that skillful\nalignment of the strain within the films should reduce the strain\nvalues required to switch the magnetization and impede domain\nformation. 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Zwierlein4, Eugene Demler1\n1Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA\n2Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA\n3Department of Physics and Astronomy, Aarhus University, Aarhus, DK-8000, Denmark\n4MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics,\nand Department of Physics, Cambridge, MA 02139, USA\nWe study the quench dynamics of a two-component ultracold Fermi gas from the weak into the strong inter-\naction regime, where the short time dynamics are governed by the exponential growth rate of unstable collective\nmodes. We obtain an effective interaction that takes into account both Pauli blocking and the energy depen-\ndence of the scattering amplitude near a Feshbach resonance. Using this interaction we analyze the competing\ninstabilities towards Stoner ferromagnetism and pairing.\nFerromagnetism in itinerant Fermions is a prime example\nof a strongly interacting system. Most theoretical treatments\nrely on a mean-field Stoner criterion [1], but whether this ar-\ngument applies beyond mean-field remains an open problem.\nIt is known that the existence of the Stoner instability is very\nsensitive to the details of band structure and interactions [2–4],\nhowever how to account for these details in realistic systems\nremains poorly understood. Exploring the Stoner instability\nwith ultracold atoms has recently attracted considerable atten-\ntion. Following theoretical proposals [5], the MIT group made\nuse of the tunability [6] and slow time scales [7–10] of ultra-\ncold atom systems to study the Stoner instability [11]. Sig-\nnatures compatible with ferromagnetism, as understood from\nmean-field theory [12], were observed in experiments: a max-\nimum in cloud size, a minimum in kinetic energy and a maxi-\nmum in atomic losses at the transition. However, no magnetic\ndomains were resolved.\nAn important aspect of the MIT experiments is that they\nwere done dynamically: the Fermi gas was originally prepared\nwith weak interactions and then the interactions were ramped\nto the strongly (repulsive) regime. Dynamic rather than adi-\nabatic preparation was used in order to avoid production of\nmolecules. This raises the question of what are the dominant\ninstabilities of the Fermi gas in the vicinity of a Feshbach res-\nonance.\nNaively, one would expect that on the BEC-side, molecule\nproduction is slow, as it requires a three-body process. There-\nfore, instability towards Stoner ferromagnetism would dom-\ninate over the instability toward molecule production. In\nthis picture, one would expect that quenches to the attrac-\ntive (BCS) regime always yield an instability towards pairing,\nwhereas quenches to the repulsive (BEC) regime an instability\ntowards ferromagnetism for sufficiently strong interactions.\nIn this Letter, we argue that this picture, which was used to\ninterpret the MIT experiments, is incomplete. Near the Fes-\nhbach resonance, even on the BEC side, pair production re-\nmains a fast two-body process as long as the Fermi sea can\nabsorb the molecular binding energy. As a result, near the\nFeshbach resonance, both on the BEC and the BCS side, the\npairing and the Stoner instabilities compete directly. We now\n-2 000.40.8growth rate  Δmax/εF Pairing\nStoner\nRPA Stoner\n-1 1\n1/kFa FIG. 1: Growth rate of the pairing and Stoner ferromagnetic insta-\nbilities after a quench as a function of the final interaction strength\n1=kFa. Final interactions with negative (positive) values of 1=kFa\ncorrespond to the BCS (BEC) side of the Feshbach resonance. The\nStoner instability simultaneously occurs in multiple channels. The\nmost unstable channel is indicated by the solid red line, the others\nby dashed red lines. The “RPA Stoner” instability corresponds to the\nRPA result with bare as opposed to Cooperon-mediated interaction\n(see text and Ref. [13]).\nInset : Schematic diagram of the pair creation process showing the\nbinding energy (spring) being absorbed by the Fermi sea (arrows).\ndiscuss these instabilities and their competition in detail.\nWe start by describing the inter-atomic interactions. A Fes-\nhbach resonance enables tunable interactions between ultra-\ncold atoms by coupling the collision partners to a molecular\nstate with different magnetic moment. For broad resonances,\nwhere the coupling is much larger than the Fermi energy, this\ncan be modelled by a single collision channel that supports\none shallow bound state [14]. An often used, but patholog-\nical choice, is to describe repulsive interactions with a hard-\nsphere pseudo-potential. Instead, one should use the full T-\nmatrix that includes the molecular bound state [15]. Although\nat low energies the scattering amplitude from the hard-sphere\npotential and the T-matrix match, at higher energies compara-\nble to the molecular binding energy, they do not. Specifically,\nin the strong interaction regime where the Stoner instability\nis expected to occur, the Fermi energy is comparable to the\nbinding energy of a molecule in vacuum, precluding the use\nof the hard-sphere potential.arXiv:1005.2366v1  [cond-mat.quant-gas]  13 May 20102\nIn light of this remark, we study the initial dynamics of\nthe collective modes of a Fermionic system after a sudden\nquench, taking the Cooperon (full T-matrix and Pauli block-\ning) into account. We focus on the case of a sudden quench,\nas it is simpler and captures the essential physics of the insta-\nbility of the Fermi surface. Extensions to finite rate quenches\nare discussed in Ref. [13]. Our main findings are summa-\nrized in Fig. 1 and are: (1) We find that with the full T-matrix\nthe Stoner instability survives with a finite growth rate in the\nrange\u00000:2.kFa.1, whereais the scattering length and\nkFis the Fermi momentum. In contrast, using bare interac-\ntions [13] results in an unphysical divergence of the growth\nrate at unitarity and no magnetic instabilities on the BCS side\n(see Fig. 1). (2) The pairing instability persists on the BEC\nside, where it competes with the Stoner instability. (3) Within\nour approximations, the pairing instability is stronger.\nAt first sight, the survival of the pairing and Stoner instabil-\nities on the wrong side of the resonance is quite remarkable.\nHowever, both can be understood by taking into account the\npresence of the Fermi sea. On the BEC side, due to Pauli\nblocking, the binding energy of the pair-like molecule can be\nabsorbed by the two holes that are left behind (see the inset\nof Fig. 1). Thus, the two-body pairing process becomes for-\nbidden when the binding energy \u00181=ma2exceeds the maxi-\nmum energy that can be absorbed by the holes \u0018k2\nF=m(mis\nthe Fermion mass, athe scattering length, kFthe Fermi mo-\nmentum, and throughout this Letter we use the units in which\n~= 1). On the BCS side, although interactions at low ener-\ngies are indeed attractive, the same is not true at high energies.\nAs the Stoner instability involves all scattering energies up to\nthe Fermi energy, it is natural that it can persist around unitar-\nity, even on the BCS side.\nFormalism – We consider a system of interacting Fermions\ndescribed by the Hamiltonian:\nH=X\nk;\u001b\u0018k\u001bcy\nk\u001bck\u001b+Z\nddrU(t;r\u0000r0)cy\nr\"cy\nr0#cr0#cr\";(1)\nwherecy\n\u001b(c\u001b)are the Fermion creation (annihilation) opera-\ntors with spin \u001b,\u0018k\u001b=k2=2m\u0000\u0016\u001b,\u0016\u001bare the chemical\npotentials, and U(t;r\u0000r0)is the time dependent pseudo-\npotential that describes the inter-atomic interaction. We fo-\ncus on the instantaneous quench limit, in which the coupling\nUchanges from a negligible initial value Uito a final value\nUfat timet= 0. In this limit, we can describe short time\ndynamics of a collective mode at momentum qusing the cor-\nresponding susceptibility, \u001fq(!q;Uf), evaluated with final in-\nteractions but initial Fermionic configuration [13, 16]. In par-\nticular, if\u001fq(!q;Uf)has poles at !q= \n q+i\u0001qin the\nupper half of the complex plane, then fluctuations that occur\nafter the quench will grow exponentially in time. Next, we\nobtain a universal description of interactions created by the\npseudo-potential U(r)by modifying the T-matrix formalism\nto take into account Pauli blocking, and apply these ideas to\nthe Stoner and BCS instabilities.\nCooperon – In this section, we obtain the Cooperon, C, i.e.\na-1 0 11/kFa\n-3-2-10\nVacuum\nFermi seaΩq=0/εF\n02468\nkFa00.20.40.60.8pairing rate KE rate b cΔq=0/εF\nd〈KE/εF〉/dt 0\n.75\n0246810\nkFa-1-0.50FIG. 2: Pairing instability. (a) “Binding energy” of a Feshbach\nmolecule in vacuum and in the presence of a Fermi sea (relative\nto2\u000fF) as a function of interaction strength, corresponding to the\nreal part of the T-matrix pole frequency \nq=0=Re[!q=0]. Pauli\nblocking by the Fermi sea results in stronger binding across the res-\nonance. The kink occurs when the pair becomes stable. (b) Pairing\nrate and (c) rate of change of Kinetic Energy as a function of interac-\ntion strength on the BEC side for various temperatures [ T= 0(pur-\nple, solid), 0.12, 0.22, 0.5, 0.66, 0:75TF(red, dashed)]. Tempera-\nture is more effective at suppressing pair production at larger values\nofkfaas the binding energy is smaller, thus the peaks in (b) and (c)\nbecome sharper at higher temperatures. The peaks in growth rate and\nkinetic energy rate qualitatively match experiments [11]. Sharp onset\natkFa\u00191:1is expected to be smoothed by three-body processes.\nthe T-matrix that takes into account Pauli blocking of states\nby the Fermi sea (see Fig. 3), which is needed for an accurate\nstudy of instabilities near a Feshbach resonance.\nIn the center of mass frame in vacuum, the scattering of a\npair of particles with identical masses mnear a wide Feshbach\nresonance is described by the T-matrix (scattering amplitude)\n\u001c(E) =m\n4\u0019\u00121\na+ip\nmE\u0013\u00001\n: (2)\nHere,Eis the energy of the scattered particles and the pseudo-\npotentialU(r\u0000r0)that appears in Eq. 1 is related to the\nT-matrix via the Lippmann-Schwinger equation. To cor-\nrectly renormalize the Cooperon, we compare the Lippmann-\nSchwinger equations in a Fermi sea and in vacuum. For the\nCooperon we can not just use the center of mass frame, as the\nFermi sea breaks translational invariance. Therefore, we use\nthe laboratory frame for both to obtain\nC\u00001(E;q) =\u001c\u00001\u0000\nE+ 2\u000ff\u0000q2=4m\u0001\n+Zd3k\n(2\u0019)3nF(q\n2+k) +nF(q\n2\u0000k)\nE\u0000\u000fq\n2+k\u0000\u000fq\n2\u0000k:(3)\nHere,Eandqare the center of mass frequency and momen-\ntum of the pair, \u000ffis the Fermi energy, nF(k)is the Fermi\nfunction, and \u000fk=k2=2m\u0000\u000ff. Our approach is analogous3\nto the one used for the Fermi-polaron problem [17].\nPairing instability – The Cooperon enters the pairing sus-\nceptibility via\n\u001fpair(~ q) =Z\nd~k1d~k2G(~k1)G(~ q\u0000~k1)C(~ q)\u0002\n\u0002G(~k2)G(~ q\u0000~k2); (4)\nwhere~ qstands for the external frequency and momentum\nvectorfE;qg,d~k1stands ford!1dk1=(2\u0019)4, andG(~k1) =\nG(!1;k1)is the bare Fermionic Green function in the non-\ninteracting Fermi sea corresponding to the initial state. The\npoles of the pairing susceptibility correspond thus to the poles\nof the Cooperon, whose structure we now explain.\nWe begin our analysis with the T-matrix in vacuum. For\neach value of the scattering length, the T-matrix has a line of\npoles on the BEC side located at !q= \nq+i\u0001q=\u00001=ma2+\nmq2=4, corresponding to the binding energy of a Feshbach\nmolecule with center of mass momentum q. As a consequence\nof energy and momentum conservation the pole frequency is\nstrictly real ( \u0001q= 0), indicating that a two-body process in\nvacuum cannot produce a Feshbach molecule.\nIn the presence of a Fermi sea, the states below the Fermi\nsurface are Pauli-blocked, shifting the poles of the Cooperon\nrelative to the T-matrix in vacuum in two important ways.\nFirst, the real part of the pole \nq, which would correspond\nto the binding energy of a pair in the absence of an imaginary\npart, uniformly shifts down (see Fig. 2a). This shift is a result\nof Pauli blocking [18], and indicates an appearance of a paired\nstate on the BCS side as well as stronger binding of the pair on\nthe BEC side. Second, in the range \u00001<1=kFa.1:1the\npole acquires a positive imaginary part \u0001qthat corresponds to\nthe growth rate of the pairing instability. As depicted in Fig. 1,\nthe growth rate of the pairing instability increases exponen-\ntially \u0001q=0\u00198\u000fFe\u0019=2kFa\u00002as one approaches the Fesh-\nbach resonance from the BCS side, i.e.the growth rate of the\nBCS pairing is equal to the BCS gap at equilibrium [18]. On\nthe BEC side, the growth rate continues to increase, reach-\ning a maximum at kFa\u00192, and finally decreasing to zero at\nkFa\u00191:1, at which point the Fermi sea can no longer absorb\nthe energy of the Feshbach molecule in a two-body process.\nPairing deeper in the BEC regime takes place via the more\nconventional three-body process and would round the pairing\ninstability curve near kFa\u00191:1in Fig. 1.\nSo far, we have concentrated on the instability at q= 0.\nWe find that q= 0 is indeed the most unstable wavevector\nthroughout the Feshbach resonance, but the growth rate re-\nmains finite up to q=qcut. Throughout the resonance the\napproximation qcut\u0019(p\n3=2)(\u0001q=0=\u000fF)kFworks reason-\nably well except in the vicinity of kfa\u00182whereqcutreaches\nthe maximal value for a two-body process of 2kf.\nStoner instability – One can expect that a rapid quench to\nthe BEC side of the resonance, where interactions are strongly\nrepulsive, results in an instability towards Stoner ferromag-\nnetism. We shall assume that right after the quench, the atoms\nare still in the free Fermi sea initial state and the Stoner insta-\n= +\n= +Γq(k1)qq+k1\nk1\nq-k1q-k2\nk1k2C(q)FIG. 3: Diagrams for the vertex function \u0000and Cooperon C. Solid\nlines represent bare fermion propagators, dashed lines interactions,\nand wavy lines external sources. External legs, represented by gray\nlines, are shown for clarity.\n0 0.5 11/kFa\n00.250.5\nqmax/kF\nΔmax/εF\n1u1\n10-2\n10-4\n10-6\n10-210-4qmax/kF\nΔmax/εF\n00.20.40.60.8 1\nq/kF00.020.04Δq/εFa b\ncqmax\nΔmax\nqmax\nΔmax\nFIG. 4: Properties of growing collective modes in the Stoner insta-\nbility in 3D. (a) Growth rate of the most unstable mode \u0001qas a\nfunction of wavevector qforT= 0 and1=kFa= 0:85(top line),\n0:86,0:87, ...,0:93(bottom line). (b) The most unstable wavevec-\ntorqmax(blue) and the corresponding growth rate \u0001max(red) vs.\n1=kFa. A fit to the mean-field critical theory ( \u0017= 1=2,z= 3)\nis shown with black lines [19]. (c) Details of the critical behavior\nofqmaxand\u0001maxas a function of distance from the transition point\nu= (1=kFa)c\u0000(1=kFa),(1=kFa)c\u00190:94.\nbility is competing with the pairing instability. Our goal is to\ncompute the ferromagnetic susceptibility using the Cooperon\nto describe effective inter-atomic interactions. Using the full\nCooperon allows us to include three important aspects of the\nproblem: energy dependence of the scattering amplitude near\nthe Feshbach resonance; Pauli blocking, which renormalizes\nthe energy of the virtual two particle bound states involved in\nscattering; and Kanamori-like many-body screening [4].\nTechnically, we compute the vertex function \u0000!;q(!1;k1),\nwhich is related to the susceptibility via\n\u001fFM(~ q) =Z\nd~k1G(~ q+~k1)G(~k1) \u0000~ q(~k1): (5)\nWe note, that the poles of the susceptibility and the vertex\nfunction coincide. Replacing the point contact interaction ver-\ntex by the Cooperon in an RPA type resummation of the vertex\nfunction (see Fig. 3 and Ref. [4]) we obtain\n\u0000~ q(~k1) = 1\n+Z\nd~k2\u0000~ q(~k2)C(~k1+~k2+~ q)G(~k2+~ q)G(~k2):(6)4\nTo compute the vertex function, a number of approxima-\ntions are unavoidable. First, we assume that qand!are\nboth small, which is valid in the vicinity of the Stoner tran-\nsition. Second, in the spirit of Fermi liquid theory, we as-\nsume that the most important poles come from the Green func-\ntions, and thus we replace G(k2+q;!2+!)G(k2;!)!\n2\u0019\nvFq\u0001k2\nm!\u0000q\u0001k2\u000e(!)\u000e(jk2j\u0000kF)[18]. We then obtain\n\u0000q;!(^k1)=1+Zd^k2\n4\u0019\u0000q;!(^k2)C(^k1+^k2;!)Iq;!(^k2);(7)\nwhere\nIq;!(^k2) =Zk2\n2dk2\n2\u00192nF(k2\u0000q=2)\u0000nF(k2+q=2)\n!\u0000\u000fk2\u0000q=2+\u000fk2+q=2;(8)\nand^kindicates a vector on the Fermi surface. We thus make\nthe approximation that we can replace k1andk2by^k1and^k2\nwhen we evaluate the value of the Cooperon. In other words,\nwe assume that the Cooperon changes slowly compared to\nthe Green functions. The approximation is fully justified for\nweak interactions, where the Cooperon is momentum and fre-\nquency independent, and the vertex function matches the RPA\nresult [13]. For strong interactions, the Stoner instability is not\ndriven by the pole of the Cooperon, and we therefore believe\nthat our approximation captures the essential physics.\nIn the range\u00000:2.1=kFa.1:0, there is one or more\nlines of complex poles with a positive imaginary part \u0001q,\nwhich corresponds to the Stoner instability in different chan-\nnels (a combination of momentum and orbital moment). As\nq!0, the different instabilities can be identified as different\nangular momentum channels. Since magnetization is a con-\nserved order parameter, in each channel \u0001qgrows linearly\nfor smallq. At largeqthe cost of bending the order parame-\nter results in the vanishing of \u0001qforq > q cut. In between,\n\u0001qreaches its maximum value \u0001maxat a wave-vector qmax\nwhich corresponds to the fastest growing mode (see Fig. 4).\nDiscussion – The growth rates of the pairing instability\n\u0001BCS\nq=0and the ferromagnetic instabilities in the various chan-\nnels\u0001FM\nmax are compared across the Feshbach resonance in\nFig. 1. We have also included the naive RPA estimate of the\ngrowth rate of the Stoner instability in which we have replaced\nthe Cooperon by 4\u0019a=m . From the comparison, we see that\n(1) the Cooperon suppresses the growth rate of the ferromag-\nnetic instability but does not eliminate it, (2) the pairing and\nferromagnetic instabilities compete over a wide range of in-\nteraction strength on both sides of the resonance, and (3) the\npairing instability is always dominant. Our results suggest that\neven if there is a metastable ferromagnetic state [15], it prob-\nably cannot be reached via dynamic tuning of the interaction\nstarting from a balanced gas. However, for short timescales\n\u0018\u0000\n\u0001FM\nmax\u0001\u00001\u0018\u0000\n\u0001BCS\nq=0\u0001\u00001, both pairing and magnetic corre-\nlations will develop and may be detectable experimentally.\nComparison with experiment – The maximum of the pair-\ning instability in the vicinity of kFa\u00192closely matches the\nlocation of the transition found experimentally [11]. To inves-tigate further, we plot the pairing rate as a function of inter-\naction strength for several different temperatures in Fig. 2b.\nThe shape of the pairing rate curve, especially at higher tem-\nperatures looks qualitatively similar to the atom loss rate (into\npairs) found experimentally.\nA fast rampdown of the magnetic field was used to convert\nweakly bound molecules into strongly bound molecules, and\nthe kinetic energy of the remaining atoms was measured. It\nwas found to have a minimum at kFa\u00192[11]. We show that\nthis minimum can be qualitatively understood within our anal-\nysis of the pairing instability. The energy of each molecule\nproduced is given by \u0018Re[!q](see Fig. 2a). The molecular\nenergy corresponds to the kinetic energy of the Fermions re-\nmoved from the Fermi sea, measured with respect to the Fermi\nenergy. Thus the rate of kinetic energy change of “unpaired”\natoms is\u0018(Re[!q]\u00002\u000fF)\u0002Im[!q=0](see Fig. 2c). We\nfind that the kinetic energy minimum is in the vicinity of the\nmaximum of the pairing rate, in qualitative agreement with\nRef. [11].\nAcknowledgements – It is our pleasure to thank E. Alt-\nman, A. Chubukov, D. Huse, M. Lukin, S. Stringari, I. Caru-\nsotto, A. Georges and especially W. Ketterle for useful discus-\nsions. The authors acknowledge support from a grant from the\nArmy Research Office with funding from the DARPA OLE\nprogram, CUA, the Swiss national Science Foundation, NSF\nGrant No. DMR-07-05472 NSF and PHY-06-53514, AFOSR-\nMURI, and the Alfred P. Sloan Foundation.\n[1] E. Stoner, Phil. Mag. 15, 1018 (1933).\n[2] J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).\n[3] A. Tanaka, H. Tasaki, Phys. Rev. Lett. 98, 116402 (2007).\n[4] L. Chen, C. Bourbonnais, T. Li, and A.-M. S. Tremblay, Phys.\nRev. Lett. 66, 369 (1991).\n[5] M. Houbiers, et al., Phys. Rev. A 56, 4864 (1997); Y . Zhang and\nS. Das Sarma, Phys. Rev. B 72, 115317 (2005); R. A. Duine\nand A. H. MacDonald, Phys. Rev. Lett. 95, 230403 (2005); G.\nJ. Conduit and B. D. Simons, Phys. Rev. A 79, 053606 (2009);\nG. J. Conduit, A. G. Green, and B. D. Simons, Phys. Rev. Lett.\n103, 207201 (2009); G. J. Conduit and B. D. Simons, Phys.\nRev. Lett. 103, 200403 (2009).\n[6] I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 80, 885\n(2008).\n[7] H. J. Miesner et. al , Phys. Rev. Lett. 82, 2228 (1999); H.\nSchmaljohann, Phys. Rev. Lett. 92, 040402 (2004); L. E. Sadler\net. al., Nature 443, 312 (2006).\n[8] J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. Donley, E. A.\nCornell, and C. E. Wieman, Phys. Rev. Lett. 86, 4211 (2001).\n[9] M. Greiner, O. Mandel, T.W. H ¨ansch, I. Bloch, Nature 419, 51\n(2002).\n[10] N. Strohmaier et. al , Phys. Rev. Lett. 104, 080401 (2010).\n[11] G.-B. Jo et. al , Science 325, 1521 (2009).\n[12] L. J. LeBlanc, J. H. Thywissen, A. A. Burkov, and A.\nParamekanti, Phys. Rev. A 80, 013607 (2009).\n[13] M. Babadi, D. Pekker, R. Sensarma, A. Georges, and E. Dem-\nler, arXiv:0908.3483.\n[14] W. Ketterle and M. W. Zwierlein, in Proceedings of the In-\nternational School of Physics “Enrico Fermi”, Course CLXIV ,5\nVarenna, 20-30 June 2006, edited by M. Inguscio, W. Ketterle,\nand C. Salomon (IOS Press, Amsterdam, 2008).\n[15] S. Pilati, G. Bertaina, S. Giorgini, and M. Troyer,\narXiv:1004.1169; S.-Y . Chang, M. Randeria, N. Trivedi,\narXiv:1004.2680.\n[16] A. Lamacraft and F. M. Marchetti, Phys. Rev. B 77, 014511\n(2008).\n[17] N. V . Prokof’ev and B. V . Svistunov, Phys. Rev. B 77, 125101(2008).\n[18] A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of\nquantum field theory in statistical physics , (Dover Publications,\nNew York, 1975).\n[19] H. V . Lohneysen, A. Rosch, M. V ojta and P. Wolfle, Rev. Mod.\nPhys. 79, 1015 (2007)."    },    {        "title": "1502.00350v1.Resonant_magneto_tunneling_between_normal_and_ferromagnetic_electrodes_in_relation_to_the_three_terminal_spin_transport.pdf",        "content": "Resonant magneto-tunneling between normal and ferromagnetic electrodes in relation\nto the three-terminal spin transport\nZ. Yue and M. E. Raikh\nDepartment of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA\nThe recently suggested mechanism [Y. Song and H. Dery, Phys. Rev. Lett. 113, 047205 (2014)]\nof the three-terminal spin transport is based on the resonant tunneling of electrons between fer-\nromagnetic and normal electrodes via an impurity. The sensitivity of current to a weak external\nmagnetic \feld stems from a spin blockade, which, in turn, is enabled by strong on-site repulsion.\nWe demonstrate that this sensitivity exists even in the absence of repulsion when a single-particle\ndescription applies. Within this description, we calculate exactly the resonant-tunneling current\nbetween the electrodes. The mechanism of magnetoresistance, completely di\u000berent from the spin\nblocking, has its origin in the interference of virtual tunneling amplitudes. Spin imbalance in fer-\nromagnetic electrode is responsible for this interference and the resulting coupling of the Zeeman\nlevels. This coupling also a\u000bects the current in the correlated regime.\nPACS numbers: 72.15.Rn, 72.25.Dc, 75.40.Gb, 73.50.-h, 85.75.-d\nI. INTRODUCTION\nIn the past decade there was a remarkable progress in\nfabrication of lateral structures which combine ferromag-\nnetic and normal layers and exhibit spin transport. First\nexperimental evidence of spin injection from a ferromag-\nnet into a nonmagnetic material was obtained with the\nhelp of four-terminal (4T) technique. This technique was\ndeveloped in the pioneering papers Refs. 1, 2. It utilizes\ntwo ferromagnetic electrodes, injector and detector, cou-\npled to a normal channel. With detector circuit being\nopen, the charge current does not \row between the elec-\ntrodes. Instead, the current circulating in the injector\ncircuit leads to the voltage buildup between the detector\nand the normal channel. This nonlocal voltage is sup-\npressed by a weak magnetic \feld normal to the direction\nof magnetizations of the electrodes. Such a suppression,\ncalled the Hanle e\u000bect, re\rects the precession of the spin\nof carriers in course of di\u000busion between the electrodes.\nThus, the characteristic width of the Hanle curve is the\ninverse spin relaxation time.\nMore recently, experimental studies of spin injection\nwere carried out using the three-terminal3{13(3T) tech-\nnique. Unlike the 4T technique, in this technique the in-\njector and detector electrodes are combined. The signal\nmeasured is the contact voltage between the ferromagnet\nand the normal channel. Sensitivity of this signal to the\napplied magnetic \feld is simply the magnetoresistance.\nExperimental results reported in Refs. 3{13 consis-\ntently reveal two puzzling features of the 3T magnetore-\nsistance. Unlike the Hanle curves, the magnetoresistance\nshows up for bothorientations of the external \feld parallel\nand perpendicular to the magnetization of the injector.\nMoreover, the signs of magnetoresistance are opposite for\nthe two \feld orientations. In addition, the 3T magne-\ntoresistance curves are much broader than the inverse\nspin-relaxation times measured independently. In gen-\neral, the basic underlying physics of magnetoresistance in\ntransport between ferromagnetic and normal electrodes\neVFN\u0000z/2\u0000\u0000z/2\nI✓\nFIG. 1: [Color online] Schematic illustration of resonant\nmagnetotunneling between a normal electrode and a ferro-\nmagnet. External \feld, tilted by an angle \u0012from the direction\nof magnetization, causes a splitting, \u0001 z, of the the impurity\nlevel. For non-zero \u0012two Zeeman levels get coupled via a\ncontinuum of the states in a ferromagnet.\nconstitutes a puzzle. Indeed, since the normal electrode,\nacting as a detector, does not \\discriminate\" between dif-\nferent spin orientations, the current should not be sensi-\ntive to the spin precession.\nPossible resolution of these puzzles was proposed very\nrecently in the theoretical paper Ref. 14 and received\nsome experimental support in the subsequent publica-\ntions Refs. 15-17. The main idea of Ref. 14 is that the\npassage of current between the ferromagnet and the nor-\nmal electrode can be modeled as resonant tunneling via\nan impurity, see Fig. 1. On the qualitative level, the\nphysics uncovered in Ref. 14 can be explained as follows.\nWhen the current \rows from normal into ferromagnetic\nelectrodes, the spins of electrons arriving on the impu-\nrity do not have a preferential direction. Suppose that\nthe ferromagnet is fully polarized in \"direction. ThenarXiv:1502.00350v1  [cond-mat.mes-hall]  2 Feb 20152\nelectrons arriving with spin #will never tunnel into the\nferromagnet. External magnetic \feld induces precession\nof spins of the arriving electrons. Then the electrons,\nwhich were \\trapped\" on the impurity without magnetic\n\feld, get a chance to tunnel, unless the \feld is not paral-\nlel to the magnetization. As a result, the current, which\ndid not \row in a zero \feld, becomes \fnite. Characteristic\nvalue of magnetic \feld can be estimated by equating the\nperiod of precession to the waiting time for tunneling.\nThe mechanism is e\u000ecient if the spin relaxation rate is\nsmaller than the tunneling rate. Obviously, for the re-\nverse bias, when electrons \row from the ferromagnet this\nmechanism does not apply.\nThe key ingredient of the above scenario is a strong\nrepulsion,U, of\"and#electrons on the impurity. In-\ndeed, if the tunneling of the #electron is forbidden, then,\nwithout the repulsion, the current will be carried by \"\nelectrons, so that there will be no \\blockade\".\nIn the present paper we address a question: whether\nlargeUis indeed necessary to induce magnetoresistance.\nThe question is delicate, since, for U= 0, the current\ndoes not depend on the polarity of bias. Thus, if mag-\nnetoresistance is \fnite for tunneling into a ferromagnet,\nit should be the same for tunneling into a normal elec-\ntrode, which is highly non-obvious. On the other hand,\nforU= 0 the current can be calculated exactly. Indeed,\nresonant tunneling in external \feld can be viewed as a\ntwo-channel resonant tunneling18via the Zeeman-split\nlevels. Our main analytical result is that magnetoresis-\ntance is \fnite for U= 0, and its magnitude is about 50%.\nThe physical origin of the magnetoresistance is the inter-\nference of the two transport channels, or, in other words,\nthe coupling of Zeeman levels via a continuum of states in\nthe ferromagnet. We also trace how this coupling a\u000bects\nthe current in the regime of correlated transport14.\nThe paper is organized as follows. In Sect. II we derive\nand analyze the expression for non-interacting resonant\nconductance via two Zeeman levels and, subsequently, for\nthe net resonant current. In Sect. III we study how the\ncoupling of the Zeeman levels via a ferromagnet a\u000bects\nthe current in the presence of correlations. Concluding\nremarks are presented in Sect. IV.\nII. MAGNETORESISTANCE IN THE ABSENCE\nOF COULOMB CORRELATIONS\nA. General expression\nWithin a non-interacting picture we can view the tun-\nneling through a single impurity in a magnetic \feld\nas tunneling via two Zeeman-split levels. The non-\ninteracting current-voltage characteristics can be calcu-\nlated from the tunnel conductance, G(E), as follows\nI=Z\ndE\u0014\nf\u0010\nE\u0000V\n2\u0011\n\u0000f\u0010\nE+V\n2\u0011\u0015\nG(E);(1)\nwhereVis the bias, and f(E) is the Fermi distribution.\n-15-10-5510150.20.5\n-15-10-5510150.20.5\n-15-10-5510150.20.5\n-15-10-5510150.20.5(a)(b)\n(c)(d)G/e2⇡~G/e2⇡~\nG/e2⇡~G/e2⇡~E/\u0000N\nE/\u0000NE/\u0000N\nE/\u0000NFIG. 2: Di\u000berential conductance, G(E), in the units of e2=\u0019~\nis plotted from Eq. (12) for di\u000berent dimensionless magnetic\n\felds, in the units \u0001 z=\u0000N. (a)-(d) correspond to \u0001 z=\u0000N=\n2:5;4;6, and 10, respectively. All curves are plotted for \u0000 F=\n2\u0000Nand the orientation of magnetic \feld, \u0012= 15\u000e.\nIf the electrodes are normal, the tunneling via each\nZeeman level,\u0006\u0001z=2, proceeds independently, and G(E)\nis given by the Breit-Wigner formula\nG\u0006(E) =e2\n\u0019~h\u0000L\u0000R\n(E\u00061\n2\u0001z)2+1\n4(\u0000L+ \u0000R)2i\n;(2)\nwhere \u0000Land \u0000Rare the widths with respect to tunneling\ninto the left and right electrodes, respectively.\nTwo tunneling channels are independent because the\nnormal electrodes do not couple the Zeeman levels, since\nthe corresponding spinors are orthogonal to each other.\nBy contrast, a ferromagnetic electrode does introduce the\ncoupling between the levels for any orientation of mag-\nnetic \feld except for the \feld parallel to the magneti-\nzation. Indeed, if the angle between the magnetic \feld\nand magnetization is \u0012, the spinors corresponding to the\nZeeman levels are\n\u001f+= cos\u0012\n2\"+ sin\u0012\n2#; \u001f\u0000= sin\u0012\n2\"\u0000cos\u0012\n2#;(3)\nwhere\"and#are the spin states in the ferromagnet, and\nthe azimuthal angle is set to zero. Denote with \u0000\"\nLand \u0000#\nL\nthe widths of the Zeeman levels with respect to tunneling\ninto the ferromagnet for \u0012= 0. At \fnite \u0012, an electron\nin the state \u001f+can virtually tunnel into the \"-state\nof the ferromagnet. The amplitude of this tunneling is\ncos\u0012\n2. From the\"-state it can then virtually tunnel into\n\u001f\u0000with amplitude sin\u0012\n2. The electron can also proceed\nfrom\u001f+to\u001f\u0000via the#state of the ferromagnet. The\ncorresponding amplitude is \u0000sin\u0012\n2cos\u0012\n2, i.e. it has the\nopposite sign. As a result, the coupling matrix element\nbetween\u001f+and\u001f\u0000is equal to (\u0000\"\nL\u0000\u0000#\nL) sin\u0012\n2cos\u0012\n2. It\nis \fnite due to the di\u000berence in the densities of the inter-\nmediate states.3\n-10-5510-0.2-0.10.1-10-5510\n-0.02-0.010.005\n-10-5510-0.2-0.10.10.2-10-5510-0.02-0.01(a)(b)\n(c)(d)@2I@V2/e3⇡~\u0000N\n@2I@V2/e3⇡~\u0000N@2I@V2/e3⇡~\u0000N@2I@V2/e3⇡~\u0000N\u0000z/\u0000N\u0000z/\u0000N\nV/\u0000NV/\u0000N0.020.01\nFIG. 3: The second derivative,@2I\n@V2j\u0012=\u0019=2((a),(c)) and the\ndi\u000berence,@2I\n@V2j\u0012=\u0019=2\u0000@2I\n@V2j\u0012=0((b),(d)) is plotted in the\nunitse3=\u0019~\u0000Nfrom Eqs. (1), (12) versus dimensionless mag-\nnetic \feld, \u0001 z=\u0000N, (a) and (b), and versus dimensionless bias,\nV=\u0000N, (c) and (d). In (a) and (b) the bias is V= 2\u0000N, while\nin (c) and (d) the magnetic \feld is \u0001 z= 2\u0000N. In all plots\n\u0000F= 1:5\u0000N, polarization is p= 1=3, and temperature is\nT= 10\u0000N.\n-15-10-55101511.41.8\n✓=0✓=⇡2✓=⇡4\n\u0000z/\u0000NI/e\u0000N⇡~\nFIG. 4: Resonant current calculated numerically from Eqs.\n(1), (12) is plotted versus the dimensionless magnetic \feld,\n\u0001z=\u0000N, for di\u000berent \feld orientations. In all curves \u0000 F=\n2\u0000N, the bias is V= 10\u0000Nand the temperature is T= 10\u0000N.\nWith two Zeeman levels coupled, the tunneling into\nthe ferromagnet is described by a matrix\n^\u0000L=0\n@\u0000\"\nLcos2\u0012\n2+ \u0000#\nLsin2\u0012\n2(\u0000\"\nL\u0000\u0000#\nL) cos\u0012\n2sin\u0012\n2\n(\u0000\"\nL\u0000\u0000#\nL) cos\u0012\n2sin\u0012\n2\u0000\"\nLsin2\u0012\n2+ \u0000#\nLcos2\u0012\n21\nA:\n(4)This matrix enters into the calculation of the the di\u000ber-\nential conductance, which is given by the matrix gener-\nalization of the Landauer formula\nG(E) =e2\n\u0019~Tr(^\u0000L^S^\u0000R^Sy); (5)\nwhere the matrix ^\u0000R, describing the the tunneling into\nthe normal electrode, is diagonal\n^\u0000R= \u0000R\u0012\n1 0\n0 1\u0013\n: (6)\nThe matrix ^S, which is the Green function in the matrix\nform, is given by\n^S=\u0002\nE\u0000^E+i\n2(^\u0000L+^\u0000R)\u0003\u00001: (7)\nThe matrix ^Ein Eq. (7) encodes the energy level posi-\ntions\n^E=\u0012\n\u00001\n2\u0001z0\n01\n2\u0001z\u0013\n: (8)\nWe will present the result of the evaluating of the ma-\ntrix product Eq. (5) in the notations of Ref. 14, by\ndenoting with \u0000 N(instead of \u0000 R) the tunnel width for\nthe normal electrode and introducing the of polarization,\np, of the ferromagnetic electrode\np=\u0000\"\nL\u0000\u0000#\nL\n2\u0000F; (9)\nwhere \u0000F=1\n2(\u0000\"\nL+ \u0000#\nL) is the e\u000bective tunneling width\nfor the ferromagnetic electrode, so that\n\u0000\"\nL= \u0000F(1 +p);\u0000#\nL= \u0000F(1\u0000p): (10)\nWith the new notations, the matrix Eq. (4) assumes a\ncompact form\n^\u0000L= \u0000F\u0012\n1 +pcos\u0012 p sin\u0012\npsin\u00121\u0000pcos\u0012\u0013\n: (11)\nThe resulting expression for conductance, G(E), reads\nG(E) =2e2\n\u0019~\u0000N\u0000FE2+1\n4(\u00012\nz+ \u00002\nN)\u0000E\u0001zpcos\u0012+1\n4(1\u0000p2)\u0000F(2\u0000N+ \u0000F)\n\u0002\nE2\u00001\n4(\u00012z+ \u00002\nN+ 2\u0000F\u0000N+ (1\u0000p2)\u00002\nF)\u00032+\u0002\nE(\u0000N+ \u0000F)\u00001\n2\u0000F\u0001zpcos\u0012\u00032: (12)4\nB. Analysis\nNaturally, the dependence G(E) is an even function of\nEonly for the perpendicular orientation of magnetic \feld,\n\u0012=\u0019=2. The asymmetry of G(E) is maximal for the par-\nallel orientation. The asymmetry becomes progressively\npronounced with increasing magnetic \feld, as illustrated\nin Fig. 2.\nThe speci\fcs of tunneling from the ferromagnet, as\ncompared to the normal electrode, is that Eq. (12) de-\npends on the orientation of magnetic \feld. In Ref. 17\nthe tunneling from cobalt-iron electrode into silicon via\nan oxide was studied using the inelastic electron tun-\nneling spectroscopy. The curves@2I\n@V2exhibited di\u000berent\nbehavior for parallel and perpendicular orientations of\nmagnetic \feld. Motivated by this \fndings, in Fig. 3 we\nplot the@2I\n@V2calculated from Eq. (12) for \u0012=\u0019=2 as\na function of bias and magnetic \feld together with the\ndi\u000berence of@2I\n@V2for\u0012=\u0019=2 and\u0012= 0. The value at\n\u0012= 0 is \fnite due to \fnite polarization p= 1=3 of the fer-\nromagnet. All the plots correspond to high temperature\nT= 10\u0000N, so that only the magnitude, not the shape,\nof the curves is T-dependent. It is seen from Fig. 3 that\nthe relative di\u000berence of second derivatives is \u001810% and\nexhibits additional structure at small \u0001 zand at small\nbias. Still Fig. 3 cannot account for the results of Ref.\n17 where the observed anisotropy was really strong.\nAn interesting situation unfolds when the bias and\ntemperature are of the same order and are much big-\nger than the level width. Then the \u0001 z-dependence of\ncurrent, calculated numerically from Eq. (1), exhibits a\ngrowth for perpendicular orientation and a minimum for\nparallel orientation as it is seen in Fig. 4.\nC. The net current at large bias\nIn 3T experiments3{13the net current showed the de-\npendence on the magnitude and orientation of magnetic\n\feld. It is not obvious whether this dependence is cap-\ntured by Eqs. (1), (12). For tunneling between normal\nelectrodes, p= 0, there should be no magnetoresistance.\nIndeed, the expression Eq. (12) can be presented as a\nsum of two Lorentzians\nG(E) =e2\n\u0019~\u0000N\u0000F\n\u0002h1\n(E\u00001\n2\u0001z)2+(\u0000F+\u0000N)2\n4+1\n(E+1\n2\u0001z)2+(\u0000F+\u0000N)2\n4i\n;\n(13)\nso that the \u0001 z-dependence disappears upon integration\noverE. It turns out that magnetoresistance is nonzero for\na \fnitep. We will present the result for the net current\nassuming that ferromagnetic electrode is fully polarized,\n-10-5510678✓=⇡2✓=⇡4\n\u0000z/\u0000NI/e\u0000N⇡~\n✓=⇡8✓=0FIG. 5: Resonant current (in the units e\u0000N=\u0019~) in the ab-\nsence of correlations is plotted from Eq. (14) versus the di-\nmensionless magnetic \feld, \u0001 z=\u0000N, for di\u000berent \feld orien-\ntations. In all curves \u0000 F= 2\u0000N.\np= 1. Then the integration over Eyields\nI(\u0001z) =4e\n~\u0000F\u0000N\n\u0002(\u00012\nz+ \u00002\nN+ \u0000N\u0000F)(\u0000N+ \u0000F)\u0000\u0000F\u00012\nzcos2\u0012\n(\u00012z+ \u00002\nN+ 2\u0000N\u0000F)(\u0000N+ \u0000F)2\u0000\u0000F\u00012zcos2\u0012:\n(14)\nEq. (14) is our central result. Sensitivity of the net cur-\nrent to \u0001 zoriginates from the coupling of Zeeman levels\nvia the ferromagnetic electrode [nondiagonal element in\nmatrix Eq. (11)] and, thus, it is most pronounced for\n\u0000F\u001d\u0000N. In this limit Eq. (14) can be simpli\fed to\nI(\u0001z) =4e\n~\u0000N \n1\u0000\u0000N\u0000F\n\u00012zsin2\u0012+ 2\u0000N\u0000F!\n: (15)\nThe current Eq. (14) is a growing function of magnetic\n\feld for all orientations, \u0012, see Fig. 5. At large \u0001 zthe\ncurrent saturates at the value\nI1=4e\n~\u0000F\u0000N\u0000Fsin2\u0012+ \u0000N\n\u00002\nN+ 2\u0000F\u0000N+ \u00002\nFsin2\u0012: (16)\nThis saturation value can be understood from the follow-\ning argument. At large \u0001 zthe tunneling via upper and\nlower Zeeman levels get decoupled. The tunnel width of\nthe upper level is \u0000 Fcos2\u0012\n2+1\n2\u0000N, while the tunnel width\nof the lower level is \u0000 Fsin2\u0012\n2+1\n2\u0000N. The sum of the cur-\nrents corresponding to these widths yields Eq. (16). The\nsame saturation value can be obtained from purely clas-\nsical consideration, by introducing the probabilities of all\nfour variants of the occupation of the two Zeeman lev-\nels and solving the system of master equations for this\nprobabilities.\nIt is quite nontrivial that in Eq. (15) the characteristic\nscale of magnetic \feld, \u0001 z\u0018(\u0000F\u0000N)1=2, is much smaller5\n-4-2240.10.20.30.4I/2e\u0000F~\n\u0000z/\u0000F✓= 85\u0000✓= 55\u0000✓= 35\u0000\nFIG. 6: [Color online] The current from normal into ferro-\nmagnetic electrode (in the units 2 e\u0000F=~) in the correlated\nregime is plotted versus dimensionless magnetic \feld, \u0001 z=\u0000F,\nfor di\u000berent orientations, \u0012and \u0000N= 8\u0000F. Green curves are\nplotted from Eq. (31), while the purple curves are plotted\nfrom Eq. (32), Ref. 14.\nthan the level width \u0000 F=2. This suggests that, while the\ntunneling times for each of the Zeeman levels is \u0000\u00001\nF, i.e.\nshort, coupling of these levels via a ferromagnet modi-\n\fes them in such a way that one of the resulting levels\npossesses a long lifetime. Similarly to Refs. 18, 19, the\norigin of this long lifetime can be traced to the complex\npoles ofG(E) in Eq. (12). These poles correspond to the\ncondition: det ^S\u00001= 0, where the matrix ^Sis de\fned\nby Eq. (7). The secular equation reads\nh\nE\u00001\n2\u0001z+i\n2(\u0000N+ (1\u0000cos\u0012)\u0000F)i\n\u0002h\nE+1\n2\u0001z+i\n2(\u0000N+ (1 + cos\u0012)\u0000F)i\n=\u0010i\n2\u0000Fsin\u0012\u00112\n:\n(17)\nThe roots of Eq. (17) are\nE\u0006=1\n2h\ni(\u0000N+\u0000F)\u0006q\n\u00012z\u0000\u00002\nF\u00002i\u0001z\u0000Fcos\u0012i\n:(18)\nFor \u0000N\u001c\u0000Fand \u0001z\u001c\u0000Fthe imaginary parts of the\nroots are\nImE1=1\n2\u0000F; ImE2=\u0000N\n2+\u00012\nzsin2\u0012\n4\u0000F: (19)\nWe see that the time (Im E2)\u00001is long, and de\fnes the\nscale \u0001z\u0018(\u0000F\u0000N)1=2of magnetic \feld.\nIII. CORRELATED TUNNELING\nWith strong on-site repulsion, U, and the bias, V, ex-\nceeding the Kondo temperature, the mechanism of trans-\nport is sequential tunneling. The scenario of this sequen-\ntial tunneling is most simple for U\u001dV, when the dou-\nble occupancy of the impurity is forbidden. Then thepassage of current, say, from the ferromagnet ( F) into\nnormal electrode ( N) via impurity ( i) proceeds in simple\ncycles. At the \frst step, the electron tunnels from Fto\ni, and at the second step, from itoN. The current is in-\nversely proportional to the average duration of the cycle,\ni.e.\nIF!N=e\n\u001cF!i+\u001ci!N; (20)\nwhere\u001cF!iand\u001ci!Nare the average waiting times\nfor the corresponding tunneling processes. Similarly, the\ncurrent from NtoFis given by\nIN!F=e\n\u001cN!i+\u001ci!F: (21)\nFor a normal electrode, the time \u001ci!Nis related to\n\u001cN!ias21\n\u001ci!N= 2\u001cN!i; (22)\nre\recting the fact that tunneling from the electrode onto\nan empty impurity is possible for both spin directions,\nwhile the electron on impurity can tunnel only into the\nstates in the electrode having the same spin direction. If\nthe electrode Fwas unpolarized, the currents Eqs. (20)\nand (21) would be given by21\nIF!N=e\n2\u001ci!F+\u001ci!N; (23)\nIN!F=e\n\u001ci!F+ 2\u001ci!N: (24)\nFor a polarized electrode Fthe relation \u001ci!F= 2\u001cF!i\nis not valid. In calculating \u001ci!Fone should keep in mind\nthat electron can tunnel into Ffrom both Zeeman levels\ndescribed by spinors \u001f+,\u001f\u0000, Eq. (3), so that\n\u001ci!F=\u001ci!F\n++\u001ci!F\n\u0000\n2: (25)\nIn the same way, in calculating \u001cF!i, it should be taken\ninto account that the electron from Fcan tunnel into\nboth Zeeman levels. The net rate of tunneling is given\nby\n(\u001cF!i)\u00001= (\u001cF!i\n+)\u00001+ (\u001cF!i\n\u0000)\u00001: (26)\nUpon these modi\fcations, the times \u001ci!Fand\u001cF!ican\nbe very di\u000berent. Suppose that the polarization is full,\np= 1, and that the magnetic \feld is directed along\nthe direction of magnetization. Then for \u001ci!F\n+ we have\n\u001ci!F\n+ = (2\u0000F)\u00001, while\u001ci!F\n\u0000 =1, re\recting the fact14\nthat electron with spin #cannot tunnel into F, where all\nspins are\". For a \fnite angle, \u0012, between magnetization\nand magnetic \feld this blockade is lifted.\nIn calculating the tunneling times, it is very impor-\ntant that electron tunnels into Fnot from pure Zeeman\nlevels, but from the levels coupled via F. This coupling\nis described by the non-diagonal element of the matrix6\nEq. (11). Then the corresponding partial times are given\nby18\n\u001ci!F\n+ =\u001cF!i\n+ =~\n2ImE+; (27)\n\u001ci!F\n\u0000 =\u001cF!i\n\u0000 =~\n2ImE\u0000; (28)\nwhereE+andE\u0000are given by Eq. (18) with \u0000 N= 0.\nIt can be easily seen from Eq. (18) that the relation\n\u001cF!i\n++\u001cF!i\n\u0000 =~\n2\u0000F(29)\nholds. This, in turn, means that the current IF!Nis\nsimply equal to2e\n~\u0000F\u0000N=(2\u0000F+ \u0000N), i.e. it does not\nexhibit any magnetic-\feld dependence14. On the other\nhand, with times given by Eq. (27), the current IN!F\nacquires a non-trivial \u0001 z-dependence. Taking into ac-\ncount that\nImE\u0006=\n\u0000F\n2\u00061\n2h\n\u00001\n2(\u00012\nz\u0000\u00002\nF)+1\n2q\n(\u00012z\u0000\u00002\nF)2+ 4\u00002\nF\u00012zcos2\u0012i1\n2;\n(30)\nwe get\nIN!F=\u00102e\n~\u0011\n\u0002\u0000N\u0000\n\u00012\nz+ \u00002\nF\u0000p\n(\u00012z\u0000\u00002\nF)2+ 4\u00002\nF\u00012zcos2\u0012\u0001\n4\u0000N\u0000F+ \u00012z+ \u00002\nF\u0000p\n(\u00012z\u0000\u00002\nF)2+ 4\u00002\nF\u00012zcos2\u0012:\n(31)\nIt is instructive to compare the result Eq. (31) with\ncorresponding expression from Ref. 14 for p= 1, which\nreadsIN!F=\u00102e\n~\u0011\u0000F\u0000N\u00012\nzsin2\u0012\u0002\n(2\u0000N+ \u0000F)\u00012z+ 2\u00002\nF\u0000N\u0003\n\u0000\u0000F\u00012zcos2\u0012:\n(32)\nAt small\u0012we can expand the square root in Eq. (31) as\nq\n(\u00012z\u0000\u00002\nF)2+ 4\u00002\nF\u00012zcos2\u0012\u0019\u00012\nz+ \u00002\nF+2\u00012\nz\u00002\nF\u00122\n\u00012z+ \u00002\nF:\n(33)\nIt follows from Eq. (33) that the two results coincide at\nsmall\u0012. Otherwise, they are di\u000berent, see Fig. 6. The\ndi\u000berence is most pronounced for \u0000 F\u001c\u0000N, when the\ntunneling into Fdominates the current. For example,\nfor particular values \u0001 z= \u0000Fand\u0012=\u0019=2, the current\nEq. (31) is two times bigger than IN!Fgiven by Eq.\n(32). The origin of the discrepancy is the form of the\nHamiltonian, adopted in Ref. 14, where strong Coulomb\nrepulsion is ascribed to electrons in the states \"and#.\nThis is permissible only for \u0012= 0. For nonzero \u0012, the\nrepulsion takes place between the electrons occupying the\neigenstates \u001f+and\u001f\u0000, see Eq. (3). Comparison of Eqs.\n(31) and (32) is presented in Fig. 6.\nAt large \u0001 zthe current Eq. (31) saturates at the value\nIN!F\n1 =2e\n~\u0000F\u0000Nsin2\u0012\n\u0000Fsin2\u0012+ 2\u0000N: (34)\nIn this limit, the coupling between the Zeeman levels\nis negligible, so that the value IN!F\n1 follows from\nEq. (23), with \u001cN!i= 1=2\u0000Nand\u001ci!F= 1=\u0000Fsin2\u0012.\nNaturally, the large-\u0001 zlimit of Eq. (32), in which the\ncoupling of the Zeeman levels is completely neglected,\ncoincides with Eq. (34).\nIn closing of this Section we present the expression for\nthe current which generalizes Eq. (31) to the case of a\n\fnite polarization of the ferromagnetic electrode\nIN!F=\u00102e\n~\u0011\u0000N\u0000\n\u00012\nz+ (2\u0000p2)\u00002\nF\u0000p\n(\u00012z\u0000p2\u00002\nF)2+ 4p2\u00002\nF\u00012zcos2\u0012\u0001\n4\u0000N\u0000F+ \u00012z+ (2\u0000p2)\u00002\nF\u0000p\n(\u00012z\u0000p2\u00002\nF)2+ 4p2\u00002\nF\u00012zcos2\u0012: (35)\nIV. CONCLUDING REMARKS\n\u000fOur main physical message is that in resonant\nmagneto-tunneling between the normal electrode\nand the ferromagnet, the e\u000bect of coupling of Zee-\nman levels via a ferromagnetic electrode a\u000bects\nthe current both in correlated and non-correlated\nregimes. At this point we would like to draw a\nlink to the earlier studies, Refs. 19, 20, wherethe correlated resonant transport between the nor-\nmal electrodes via a two-level system, e.g. two\nquantum dots in parallel20, was studied. The au-\nthors realized that the current is strongly a\u000bected\nby the coupling between the levels via continuum\nof the states in the electrodes, and that the rate-\nequations-based description is invalid due to this\ncoupling. They demonstrated that this coupling\ngives rise to a strong dependence of current on the7\n-40-2020400.10.20.3\n-40-2020400.10.20.3\n-40-2020400.10.20.3\n-40-2020400.10.20.3(a)(b)\n(c)(d)I/2e\u0000F~I/2e\u0000F~\nI/2e\u0000F~I/2e\u0000F~\n\u0000z/\u0000F\u0000z/\u0000F\u0000z/\u0000F\u0000z/\u0000F\nFIG. 7: The e\u000bect of pseudomagnetic \feld on magnetotun-\nneling. The current (in the units 2 e\u0000F=~) in the correlated\nregime is plotted versus dimensionless magnetic \feld, \u0001 z=\u0000F,\nfor orientations \u0012= 70\u000e(a),\u0012= 55\u000e(b),\u0012= 25\u000e(c), and\n\u0012= 10\u000e(d). The plots correspond to \u0000 N= \u0000Fand pseudo-\nmagnetic \feld \u0001 0= 10\u0000F.\nenergy separation of the levels. In our situation,\nthis separation is simply the Zeeman splitting, \u0001 z.\nIn Refs. 19, 20, the ferromagnet was mimicked by\nthe asymmetry of coupling of the components of\nthe two-level system to the electrodes. In our sit-\nuation, the source of asymmetry is the angle, \u0012,\nbetween the magnetic \feld and the magnetization.\nThe e\u000bect analogous to \\magnetoresistance\" was\ncaptured in Refs. 19, 20 by numerically solving\nthe master equations. Our situation, when only\none electrode is ferromagnetic, is simpler, which al-\nlowed us to get the analytical result Eq. (31) for\nthe correlated current.\n\u000fIn the correlated regime, the magnetoresistance is\npresent only for one current direction N!F. Our\nresult Eq. (14) suggests that outside the block-\naded regime V >U , when the current is the same\nfor both voltage polarities, the magnetoresistance\nis still \fnite and strong. Probably, this prediction,\nequal mangetoresistances for both bias polarities,\nfor high enough bias can be tested in 3T spin-\ntransport experiments.\n\u000fExcept for the papers Refs. 23, 25, the bulk of\ntheoretical studies22{24,26{30of resonant transport\nbetween two ferromagnetic electrodes was focused\non the low-temperature Kondo regime. As it was\npointed in Ref. 23, outside the Kondo regime, in\naddition to blocking, there is another prominent\nmany-body e\u000bect, which results from the polariza-\ntion of the electrodes and might a\u000bect the trans-\nport. Namely, the on-site repulsion gives rise to apseudomagnetic \feld\n\u00010=\u0000F\n\u0019ZV=2\n\u0000V=2d\"\u00101\n\"\u0000U\u00001\n\"\u0011\n(36)\ndirected along the magnetization. The structure\nof Eq. (36) suggests that the underlying mech-\nanism is similar to cotunneling. Incorporating\nof this \feld into Eq. (31) is performed by re-\nplacing \u0001 zcos\u0012with \u0001zcos\u0012+ \u0001 0and \u00012\nzwith\n\u00012\nz+ \u00012\n0+ 2\u0001z\u00010cos\u0012. The e\u000bect of pseudo-\nmagnetic \feld on the shape of magnetoresistance\ncurves is illustrated in Fig. 7. We see that for\nlarge enough \u0001 0\u001810\u0000Fthe shapes can undergo\na dramatic transformation becoming asymmetric\nand even non-monotonic. Still these shapes do not\nexplain experimental observation that the current\ngrows with \u0001 zat\u0012=\u0019=2 and drops with \u0001 zat\n\u0012= 0. To account for this observation it was as-\nsumed in Ref. 14 that, in addition to external \feld,\na strong in-plane hyper\fne \feld is present.\n\u000fThroughout the paper we assumed that the impu-\nrity level position, E0, is zero. In fact, we required\nthatE0lies within the interval ( \u0000V\n2;V\n2), see Fig. 1.\nForE0lower than\u0000V\n2the resonant tunneling is for-\nbidden. It will be allowed again21whenE0falls into\nthe interval (\u0000V\n2\u0000U;V\n2\u0000U) (impurity of the \\type\nB\" in the language of Ref. 14). Then the interme-\ndiate state for the passage of current will be doubly\noccupied, and magnetoresistance will be present14\nforIF!N, but absent for IN!F. IfE0is lower than\n\u0000V\n2but aboveV\n2\u0000U, the mechanism of passage of\ncurrent is cotunneling, i.e. an elastic two-electron\nprocess in course of which one electron leaves the\nimpurity to Nand another arrives from F. The co-\ntunneling rate, \u001c\u00001\nc, is given by \u001c\u00001\nc\u0018\u0000F\u0000N=E0, so\nthat the magnitude of current is Ic=e=\u001cc. There is\na question whether or not the cotunneling current,\nIF!N, exhibits the magnetic \feld dependence. In\nour opinion it does. Indeed, without the magnetic\n\feld and for fully polarized Felectrode, the state\nof the impurity after a single cotunneling act is\n\". 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Tserkovnyak, arXiv:1412.4663."    },    {        "title": "0801.2922v1.Selective_Spin_Injection_Controlled_by_Electrical_way_in_Ferromagnet_Quantum_Dot_Semiconductor_system.pdf",        "content": "arXiv:0801.2922v1  [cond-mat.mes-hall]  18 Jan 2008Selective Spin Injection Controlled by Electrical way in\nFerromagnet/Quantum Dot/Semiconductor system\nZhen-Gang Zhu\nCenter for Advanced Study, Tsinghua University, Beijing 10 0084, China;\nInstitut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg,\nNanotechnikum-Weinberg, Heinrich-Damerow-St. 4, 06120 H alle, Germany;\nSilicon Nano Device Lab (SNDL), ECE Department, National Un iversity of Singapore;\nAbstract\nSelective and large polarization of current injected into s emiconductor (SC) is predicted in Fer-\nromagnet (FM)/Quantum Dot (QD)/SC system by varying the gat e voltage above the Kondo\ntemperature. In addition, spin-dependent Kondo effect is als o revealed below Kondo temperature.\nIt is found that Kondo resonances for up spin state is suppres sedwith increasing of the polarization\nPof the FM lead. While the down one is enhanced. The Kondo peak f or up spin is disappear at\nP= 1.\nPACS numbers: 72.25.-b, 73.40.-c, 73.21.La, 72.15.Qm\n1Effective spin injection into semiconductor is the central issue of sp in-related semicon-\nductor devices, such as the so-called spin-field-effect transistor (SFFT) proposed by Datta\nand Das [1] which may be the original starting point of spintronics [2]. Spin-valve effect\nwas predicted in it via controlling the gate voltage which controls the R ashba spin-orbit\ncoupling parameter [1]. Some experimental attempts were then per formed to realize it but\nonly small signal of spin injection had been observed. Schmidt et al.pointed out that the\nmismatch of conductance of FM and SC is the reason of the low efficien cy of spin injection\n[3]. However, Rashba proposed that a tunnel barrier can be insert ed between the FM and\nSC to overcome this problem [4]. Soon, many experiments were then r eported to confirm\nRashba’s idea [5, 6]. For example, hot electron current with a highspin polarizationof about\n98% can be obtained [5]. On the other hand, other methods for spin fi lter or spin injection\ninto semiconductor are also proposed, such as a FM tip of scanning t unnelling microscope is\nused to inject spin-polarized electrons into SC [7] and a triple tunne l barrier diode is utilized\nas spin source to enhance the spin-filtering efficiency even to 99 .9% [8].\nMore recently, new attempts to realize the devices where the spin c haracter of the in-\njected and detected electrons could be voltage selected [9][10], hav e been made. In these\ndevices, the source-drain voltage-controlled spin filter effect is inv estigated in a magnetic\nresonant tunnelling diode structure in which the central spacer is m ade of dilute magnetic\nSC ZnMnSe. Zhu and Su [11] proposed a magnetic filed dependence sp in filter effect based\non ZnSe/ZnMnSe/ZnSe/ZnMnSe/ZnSe structure in which resonan ces of different spin com-\nponents occur at different magnitude of magnetic field. These rese arches open new ways to\ncontrollable spin filter effect. However, these proposed structur es are involved in dilute mag-\nnetic SC whose Curie temperature is known blow room temperature, preventing its further\napplication in devices. In addition, for the difficulty of operating individ ual spin by external\nmagnetic field, new attempt called all electrical devices is proposed in which the controlling\nare all via electrical ways.\nIn this letter, such a selective spin injection into semiconductor is pr edicted in Ferromag-\nnet (FM)/Quantum Dot (QD)/SC system by varying the gate voltag e which controls the\nstates of the QD. A FM layer holding high Curie temperature (above r oom temperature)\nis used as a spin source and polarized electrons flowing out of it tunne l through a vertical\nQD (VQD) [12] into SC. Between the two tunnel barrier a quantum we ll is defined as a QD\nwith strong Coulomb interaction. The energy levels of QD can be tunn ed by a gate voltage\n2Vg. It is found the polarization of current is large and can be controlled by tunning Vgfrom\nnegative to positive (from down-spin filtering to up-spin filtering) be cause of the mixed roles\nof Coulomb interaction and the splitting of spin subbands of FM. It is w orth pointing out\nthat the splitting of energy levels of QD for different spins are large a nd corresponds to\nthe Curie temperature order. This large splitting guarantees the w ell-defined separation of\npolarized current with different spins and the spin filter effect.\n/s40/s99/s41/s48/s40/s97/s41/s83/s105/s100/s101/s32/s103/s97/s116/s101\n/s83/s67/s32/s72/s101/s116/s101/s114/s111/s115/s116/s114/s117/s99/s116/s117/s114/s101/s83/s67/s81/s68/s70/s77\n/s48/s40/s98/s41/s48/s43/s85\n/s48\n/s40/s99/s41/s48\n/s48\n/s40/s99/s41/s48\n/s48\n/s40/s100/s41/s48\n/s48\nFIG. 1: (color online). (a) The model configuration. (b) The f ormation of spin-dependent Kondo\nresonances for spin-polarized lead. (c) and (d) The develop ed spin-dependent resonant tunnelling.\nε0↑andε0↓have finite width for the imaginary of selfenergy.\nThe Hamiltonian is H=Hleads+Hdot+HT,Hleads=/summationtext\nkσεL\nkσa†\nkσakσ+/summationtext\nqσεqRb†\nqσbqσ,\nHdot=/summationtext\nσǫ0d†\nσdσ+Und↑nd↓,HT=/summationtext\nkσ[tσ\nkLa†\nkσdσ+h.c.] +/summationtext\nqσ[tσ\nqRb†\nqσdσ+h.c.], where\nεL\nkσ=εkL−µL−σM,M=gµBh/2,gis Land´ e factor, µBis Bohr magneton, his the\nmolecular field, εkLis the single-particle dispersion of the left FM, µL(R)is the Fermi level\nof the left (right) lead, ndσ=d†\nσdσ,εqR=/planckover2pi12q2/2m∗,m∗is effective mass of electrons in the\nright lead,tσ\nkL(qR)denotes the tunnelling amplitude through the left (right) barrier.\nThen following the standard equation of motion method, and assumin g that higher-order\nspin-correlations in the leads can be neglected [13], the Green funct ion/an}bracketle{t/an}bracketle{tdσ|d†\nσ′/an}bracketri}ht/an}bracketri}htrcan be\n3obtained\n/an}bracketle{t/an}bracketle{tdσ|d†\nσ′/an}bracketri}ht/an}bracketri}htr=(ε−/tildewideǫσ+U/an}bracketle{tnσ/an}bracketri}ht)δσσ′−U/angbracketleftBig\nd†\nσdσ/angbracketrightBig\nδσσ′\n(ε−/tildewideǫσ)(ε−ǫ0−Σ0σ)+UΣ1\nσ, (1)\nwhere/tildewideǫσ=ǫ0+U+ Σ0\nσ+ Σ3\nσ, ΣL(R)0\nσ=/integraltextdε′\n2πΓL(R)\nσ(ε′)\nε−ε′+iη, Σ2\nσ= Σ3\nσ−Σ1\nσ, ΣL(R)λ\nσ=\n/integraltextdε′\n2πΓL(R)\nσ(ε′)B̥(ε′), (B= 1,λ= 3;B=fL(R),λ= 1), where ̥(ε′) =1\nε−(2ǫ0+U)+ε′+iη+\n1\nε−ε′+iη, ΓL\nσ(ε′) = 2πρL(ε′+σM)|tσ\nL(ε′)|2, ΓR\n↑= ΓR\n↓= ΓR(ε′) = 2πρR(ε′)|tR(ε′)|2,ρL(R)is\ndensity of state (DOS) of the left (right) lead and Σγ\nσ= ΣLγ\nσ+ΣRγ\nσ(γ= 1,2,3). The retard\nselfenergy can be derived from Dyson equation Σr= (gr)−1−(Gr)−1, wheregris the retard\nGF of QD without coupling to the leads but with Coulomb interaction. To get/an}bracketle{tnσ/an}bracketri}ht, the\nselfconsistent calculation must be preformed [14]. And this procedu re needs lesser Green\nfunction which is subject to the Keldysh formula G<=GrΣ<Ga. The lesser self-energy is\ntaken the form as Σ<=1\n2[Σ<\n0(Σr\n0−Σa\n0)−1(Σr−Σa)+(Σr−Σa)(Σr\n0−Σa\n0)−1Σ<\n0] [15], where\nΣr(a,<)\n0aretheselfenergies ofthenoninteracting system while Σr(a,<)areselfenergies withfull\ninteraction. In fact, one method without solving G<and only with calculating the integral\n/integraltext\ndεG<(ε) exactly has been developed to round the calculation of the lesser G reen function\n[16]. However, the approximation used here to derive the lesser Gre en function can give a\nqualitatively correct results. We shall mention that/angbracketleftBig\nd†\nσdσ/angbracketrightBig\nin principle tends to zero with-\nout spin flip scattering. We keep it here to avoid any uncertainty whic h might be caused\nby self-consistent calculation procedure and its value can be given b y the self-consistent\ncalculation.\nNo losing generality, we shall do numerical calculations in the limit U→ ∞. We use ΓL\n0\nas the unit of energy, which is defined in terms of the unpolarized par abolic energy bands\nparameters, and J0=eΓL\n0//planckover2pi1as current unit. We set P=ρL\n↑−ρL\n↓\nρL\n↑+ρL\n↓,t↑\nqR(kL)=t↓\nqR(kL)=tR(L),\nthenχ↑= ΓL\n↑/ΓL\n0= (2\n1+χ2)1\n2,χ↓= ΓL\n↓/ΓL\n0=χχ↑, whereχ=1−P\n1+P. LetχR= ΓR/ΓL\n0=α.\nThe left lead is FM and the right lead is SC, and the ΓL(R)is in proportion to the DOS of\nthe left (right) lead. So we may estimate the αwill be between 10−4→10−3(for the right\nSC lead, we use 3D DOS rather than using 2D DOS to avoid the complexit y). But if the\ntunnelling matrix t(L)Rcan be tuned to be different, the parameter αmay be tuned till 1\n(this case is considered in Fig. 4). To get the retard Green function , selfenergy Σ1\nσwill be\ncalculated analytically as [17]\nΣL1\nσ(ΣR1) =ΓL\nσ(ΓR)\n2π{1\n2lnε1ε2\n(2πiT)2−ψ(z)−i\n2π}, (2)\n4whereε1=ε−µL(R)−D,ε2=ε−µL(R)+D,z=1\n2+ε−µL(R)\n2πiT,Dis the half bandwidth, and\nwe set it as 1500 in this letter, Tis temperature.\n/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s49/s50/s51/s52\n/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s50/s51/s40/s97/s41\n/s32/s32/s76/s68/s79/s83\n/s40/s98/s41\n/s32/s32/s82/s101/s40/s83 /s101/s108/s102/s101/s110/s101/s114/s103/s121/s49/s41\n/s32/s32/s76/s68/s79/s83\n/s69/s110/s101/s114/s103/s121/s69/s110/s101/s114/s103/s121 \n/s32/s32/s82/s101/s40/s83 /s101/s108/s102/s101/s110/s101/s114/s103/s121/s49/s41\n/s69/s110/s101/s114/s103/s121 \nFIG. 2: (color online). In the four graphs, dash-dot lines ar e generated at P= 0, and solid\n(dot) lines for ρ↑(ρ↓) orℜΣ1\n↑(ℜΣ1\n↓). Along the directions of the arrows, the lines are sequentl y\ncorresponding to P= 0.2,0.4,0.6,and 1. LDOS vs. energy in (a) and (b), ℜΣ1\nσin insets. α= 1\nin (a) and its inset; α= 0.001 in (b) and its inset. The other parameters are T= 0.01,ε0=−4,\nandV= 2 (ε0andVare the same in all figures).\nLocal density of state (LDOS) [18] ρσ(ε) andℜΣ1\nσin QD vs. energy for different polariza-\ntionPareshown inFig. 2. It canbefoundthat ρ↑(ε) andρ↓(ε) aresplit because ofFMlead.\nThe main peaks of LDOS exist at the resonant energy ε=ε0σ, whereε0σ=ε0+ℜΣ1\nσindi-\ncates that the original spin-independent energy level is modified as spin-dependent energy\nlevels because of Coulomb interaction on QD and the polarization P.ℜΣ1\nσis proportional\nto the tunnelling rate ΓL\nσwhich depends on P.ℜΣ1\n↑(���Σ1\n↓) increases (decreases) with Pand\nshow two resonances at −V/2 andV/2 except for ℜΣ1\n↓atP= 1 as shown in the inset of Fig.\n(2a). This increasing (decreasing) gives rise to the shift of the main peak ofρ↑(ε) (ρ↓(ε))\n5towards to the lower (higher) energy.\nKondo resonances (KRs) of ρ↑(ε) andρ↓(ε) existing about at µL=V/2 andµR=−V/2\nshown in Fig. (2a) are consequences of nonequilibrium effect [19]. i) Wit h increasing P,\nthe magnitude of KRs for down (up) spin component is higher (lower) . While the peak\nofℜΣ1\n↓disappears at V/2 andP= 1 in the inset of Fig. (2a), the corresponding KR of\nρ↑(ε) disappears also. But the other KRs of ρ↑(ε) bounded at −V/2 remain. The reason\nis there are no itinerant spin-down electrons in the FM layer for form ation of spin singlet\nwith the electrons on QD now. ii) The positions of spin-up KRs move to lo wer energy\nand the spin-down ones move to the opposite direction. Spin-depen dent Kondo effect was\nfirstly investigated in FM-QD-FM system in Ref. [20]. Then further th eoretical [21] and\nexperimental [22] investigations are evaluated to show the splitting of the Kondo resonances.\nHowever it has no influences on the spin filter effect which is mainly inves tigated in a\ntemperature scale much above the Kondo temperature.\nHigher order cotunneling processes [23] depicted in Fig. (1b) acco unt for the formation\nof these KRs. Initially, an up-spin electron occupies the QD, it can ju mp to the left (right)\nlead at a time scale /planckover2pi1/(µL(R)−ε0). Almost at the same time, a down-spin electron of the\nright (left) lead can jump into the QD. Then the final state is a spin-fl ip state. A large\nnumber of coherent superpositions of these events will give rise to KRs at Fermi levels. The\nconduction electrons tend to screen the nonzero spin on QD such t hat a many-body spin\nsinglet state forms. This process can transfer charges from one lead to the other and then\nthe KR may enhance the conductance or current [24]. Another co ntributing process is that a\nspin occupying the QD jumps into one lead and almost at the same time a n opposite spin in\nthe same lead tunnels into QD, which can be clearly seen from the Eq. ( 239) in Ref. [25] in\nwhichtheHamiltonianconsistingtheQDandleadsistransformedintoa Hamiltoniansimilar\nto the conventional Kondo Hamiltonian via Schrieffer-Wolff transfor mation [26]. There JLL\nandJRRdescribe the coupling of the local spin on QD and the spins of itinerant electrons\nin the left or right leads respectively. This process doesn’t transfe r charge from one lead to\nthe other.\nWhenαis very small (for example α= 0.001 in Fig. (2b) and its inset), ΣR1contributes\nlittle toℜΣ1\nσ. The channel of formation KRs between QD and the right lead is supp ressed.\nSo in Fig. (2b) and its inset, the peaks about at −V/2 all disappear, but the KRs about at\nV/2 are still present. The spin-splitting of ρ↑(ε) andρ↓(ε) remains, giving rise to the spin\n6filter effect described in the following.\nTo investigate the spin-filter effect, we shall calculate the current through this structure.\nBy using the nonequilibrium Green function technique [27], the stead y current with up\n(down) spin in unit J0is\nJ↑(↓)/J0= ∆1(2)/integraldisplay\ndε[fL(ε)−fR(ε)]ρd↑(d↓)(ε), (3)\nwhere ∆ 1=χ↑α\nχ↑+αand ∆ 2=χ↓α\nχ↓+α. When a gate voltage is applied, we set new energy level\non QD isε′\n0=ε0+Vg. The spin polarization of current is defined as Pout=J↑−J↓\nJ↑+J↓, which\nis not the polarization of the DOS of the left lead. When P= 0, i.e. the injector is spin\nindependent, ∆ 1= ∆2andPout= 0. When P= 1, i.e. the injector is fully polarized (for\nexample half-metal material), ∆ 2= 0 andPout= 1. When 0 <P <1,at the limit α→0,\n∆1≈∆2≈α, thenPoutjust depends on the difference of the LDOS of QD for different\nspins.\n/s45/s50 /s48 /s50 /s52 /s54 /s56/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s49/s50/s51/s52/s53/s54\n/s32/s32/s80\n/s111/s117/s116\n/s86\n/s103/s32/s32/s80/s61/s48/s46/s50\n/s32/s32/s80/s61/s48/s46/s52\n/s32/s32/s80/s61/s48/s46/s55\n/s40/s98/s41/s32/s80/s61/s48/s46/s50\n/s32/s80/s61/s48/s46/s50\n/s32/s80/s61/s48/s46/s52\n/s32/s80/s61/s48/s46/s52/s32\n/s32/s80/s61/s48/s46/s55\n/s32/s80/s61/s48/s46/s55/s74\n/s74\n/s32/s32/s99/s117/s114/s114/s101/s110/s116/s32/s40/s49/s48/s45/s52\n/s74\n/s48/s41\n/s40/s97/s41\nFIG. 3: (color online). The Vgdependenceof current in (a) and Poutin (b).α= 0.001 andT= 0.5.\nQuiteusually, devicesareoperatedinroomtemperaturewhichismuc hhigherthanKondo\ntemperature, and KRs disappear. The Vgdependence of J↑andJ↓are presented in Fig.\n7(3a). It is noted that currents have resonant peaks which are sp lit under nonzero Pbecause\nofε0↑/ne}ationslash=ε0↓. And the splitting becomes larger with increasing P. It can be understood\nthat whenε0↑(↓)is in the energy range [ −V/2,V/2] as shown in Fig. (1c) and (1d), resonant\ntunnelling occurs and a resonant peak of J↑(↓)present. When ε0↑(↓)is out of this range,\ncurrent is suppressed. We call this range as resonant window (RW) . Asε0↓> ε0↑,ε0↓first\nenters into the RW with increasing Vgas shown in Fig. (1c), now J↓is on-resonant and J↑\nis off-resonant. Increasing Vgfurther,ε0↑enters into the RW as shown in Fig. (1d), and\nthe case is opposite to the former. When Vg> Vg0(we setJ↑=J↓atVg0),Pout>0;Pout\nfirst increases and then decreases with Vg. WhenVg<Vg0,Pout<0;Poutalso first increases\nand then decreases with Vgas shown in Fig. (3b). Even αis very small, there is still a\nlargePout. For example, when P= 0.7, the peak magnitude of Poutis 0.58 (Vg= 5.83), and\nthe peak magnitude is enhanced by increasing P.Poutvaries from negative to positive with\nincreasing gate voltage, which means the spin filter effect can be con trolled by tuning the\ngate voltage.\n/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s45/s50 /s48 /s50 /s52 /s54 /s56/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s40/s97/s41\n/s32/s32/s76/s68/s79/s83\n/s69/s110/s101/s114/s103/s121\n/s40/s98/s41\n/s32/s32/s80\n/s111/s117/s116\n/s86\n/s103/s32/s32 /s61/s48/s46/s48/s48/s48/s53\n/s32/s32 /s61/s48/s46/s48/s48/s53\n/s32/s32 /s61/s48/s46/s48/s53\n/s32/s32 /s61/s48/s46/s49\n/s32/s32 /s61/s48/s46/s53\n/s32/s32 /s61/s49\nFIG. 4: (color online). LDOS vs. energy in (a) and Vgdependence of Poutin (b) for different α.\nIn (a) along the directions of arrows the lines correspond to α= 0.001,0.05, 0.1, 0.3, 0.5, 0.7, and\n1. The other parameters are P= 0.4, T= 0.5.\n8LDOS vs. energy in Fig. (4a) and Vgdependence of Poutin Fig. (4b) for different α.\nρ↑andρ↓become lower and fatter with increasing α. It means the local electrons on QD\ntend to be nonlocal and tunnel to the right lead. And the main peaks are shifted towards\nthe right direction shown in Fig. (4a). It is found that negative Poutis reduced and even\nbecomes positive with increasing αin Fig. (4b). On the other hand, the positive Poutwill\nbe enhanced with α. For example, when α= 1, there is no negative Pout. But the maximum\nofPoutis enhanced to give 0.464 for α= 1 andP= 0.4.\nThe effect predicted here is the consequence of well-defined spin-d ependent energy levels\nof QD. So spin relaxation in QD may reduce the effect and we may estima te its order. For\ncomparison, in Ref. [9], spin relaxation time (SRT) is shorter in ZnMnSe layer because of\nthe spin-dependent scatterings in it. However, SRT is much longer in our case. Firstly, it is\nbecause the QD is formed in the nonmagnetic semiconductor quantu m well, spin-dependent\nscatterings are sparse. Secondly, the zero dimensionality of elect ron states in QDs leads to\na significant suppression of the most effective 2D spin-flip mechanism s [28], and the electron\nspin states in QDs are expected to be very stable. Recent electrica l transport measurements\nof relaxation between spin triplet and singlet states confined in a VQD give relaxation time\n>200µsatT≤0.5 K [29]. Finally, we estimate the transit time. For a typical value\nΓ = 150µV[23] (Γ can be changed by changing the barrier thickness [30]), the estimated\ntransit time is about 5 ps. So it seems reasonable to assume that the spin relaxation on QD\nhas little effect in this model.\nIn Ref. [9], the spin dependent energy levels are induced by Zeeman s plitting under\nan external magnetic field in magnetic semiconductor ZnMnSe quant um well. While in\nthis letter the tunnelling rates for up and down spins are split becaus e of the splitting of\nDOS of FM. This splitting likes an effective magnetic field (EMF) but much stronger than\nconventional magnetic field, even reach 50 ∼70 T [31], leading to the well defined spin-\ndependent energy levels on QD. Further an upper limit on the local ma gnetic field (LMF)\nwhich isgenerated byFMleadinQDisestimated tobe0.6TforNi [32]. It seems reasonable\nto neglect this LMF.\nIn summary, selective and large polarization of current injected int o semiconductor is\npredicted inFerromagnet/QuantumDot/semiconductor system b y varying thegatevoltage\nabove the Kondo temperature. A FM layer is used as a spin source an d electrons tunnel\nthrough a QD into SC. Spin-dependent Kondo effect is revealed below Kondo temperature.\n9KRs for up spin state is suppressed with P. While the down one is enhanced. The KR for\nup spin is disappear at P= 1. With increasing the gate voltage, the polarization of current\nvaries from negative to positive, which means spin filter effect can be controlled by gate\nvoltage. A large efficient spin injection can be obtained.\nThis work was supported by the Natural Science Foundation of Chin a (Grant Nos.\n10574076, 10447118), and by the Program of Basic Research Dev elopment of China (Grant\nNo. 2006CB921500).\n[1] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n[2] S. A. Wolf et al., Science 294, 1488 (2001).\n[3] G. Schmidt et al., Phys. Rev. B 62, R16267 (2000).\n[4] E. I. Rashba, Phys. Rev. B 62, R4790 (2000).\n[5] X. Jiang et al., Phys. Rev. Lett. 90, 256603 (2003); S. van Dijken et al., ibid.90, 197203\n(2003); Phys. Rev. B 66, 094417 (2002); Appl. Phys. Lett. 80, 3364 (2002).\n[6] T. Manago and H. 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Ono et al.,\nScience297, 1313 (2002).\n[13] It means that correlation function which invovle unlik e spin-indices, invovle different operators\nof theleftandtherightleads orbothcreat operators(annih lation operators)equaltozero, and\nfactorize the correlation functions with like spin /an}bracketle{t/an}bracketle{ta†\nk′σak′′σdσ|d†\nσ′/an}bracketri}ht/an}bracketri}htr≈f(εL\nk′σ)δk′k′′/an}bracketle{t/an}bracketle{tdσ|d†\nσ′/an}bracketri}ht/an}bracketri}htr.\n10[14]/an}bracketle{tnσ/an}bracketri}ht=ℑ/integraltextdε\n2π/an}bracketle{t/an}bracketle{tdσ|d†\nσ/an}bracketri}ht/an}bracketri}ht<,/angbracketleftBig\nd†\nσdσ/angbracketrightBig\n=ℑ/integraltextdε\n2π/an}bracketle{t/an}bracketle{tdσ|d†\nσ/an}bracketri}ht/an}bracketri}ht<, where/an}bracketle{t/an}bracketle{ta|b/an}bracketri}ht/an}bracketri}ht<means lesser GF derived\nfromthereturdedGF byusingKeldyshformula.Given thesetw o initial values, thensubstitute\nthem into the retard Green function and Keldysh formula to ge t them for the second time.\nThis procedure continues to give a constant /an}bracketle{tnσ/an}bracketri}htand/angbracketleftBig\nd†\nσdσ/angbracketrightBig\n. 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B 50, 5528 (1994); H. Haugand A. -P. Jauho, Quantum Kinetics\nin Transport and Optics of Semiconductors (Springer-Verlag, Berlin, 1998).\n[28] A. V. Khaetskii and Y. N. Nazarov, Phys. Rev. B 61, 12639 (2000).\n[29] T. Fujisawa, et al., Nature (London) 419, 278 (2002).\n[30] P. Gu´ eret, et al., J. Appl. Phys. 66, 278 (1989).\n[31] A. N. Pasupathy, et al., Science 306, 86 (2004).\n[32] R. Hanson, et al., Phys. Rev. Lett. 91, 196802 (2003).\n11"    },    {        "title": "0812.4953v1.Temperature_dependent_dynamic_and_static_magnetic_response_in_magnetic_tunnel_junctions_with_Permalloy_layers.pdf",        "content": "1 \n Temperature dependent dynamic and static m agnetic response in magnetic tunnel \njunctions with Permalloy layers \nJ. F. Sierra, V.V.Pryadun and F.G.Aliev* \n \nDpto. Física de la Materia Condensada, C-III, Universidad Autónoma de Madrid, 28049, Madrid, \nSpain . \n \nS. E. Russek \n \nNational Institute of Standards and Technology, Boulder, CO 80305, USA. \n \nM. García-Hernández \nInstituto de Ciencia de Materiales Madrid, CSIC, Cantoblanco, 28049, Madrid, Spain \n \nE. Snoeck \nCEMES- CNRS, Toulouse, France \n \nV. V. Metlushko \nDept. Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, \nIllinois, USA \n \nREVISED VERSION: September 25-2008 \nAbstract \n \nFerromagnetic resonance and static magnetic properties of CoFe/Al 2O3/CoFe/Py and \nCoFe/Al 2O3/CoFeB/Py magnetic tunnel junctions and of 25nm thick single-layer Permalloy \n(Py) films have been studied as a function of temperature down to 2K. The temperature \ndependence of the ferromagnetic resonance exci ted in the Py layers in magnetic tunnel \njunctions shows “knee-like” enhancement of the resonance frequency accompanied by an \nanomaly in the magnetization near 60K. We attribute the anomalous static and dynamic \nmagnetic response at low temperatures to in terface stress induced magnetic reorientation \ntransition at the Py interface which could be influenced by dipolar soft-hard layer coupling \nthrough the Al 2O3 barrier. 2 \n Since the discovery of large room temper ature magnetoresistance in magnetic tunnel \njunctions (MTJs)1,2 there have been continuous efforts to improve the quality of MTJs and \noptimise their dynamic response, important for different applications.3-5 In order to \noptimize the low field performance, the MTJ sensors with amorphous Al 2O3 barriers \nemploy magnetically soft ferromagnetic (FM)  films typically made of Permalloy (Ni 80Fe20). \nKnowledge of the dynamic and static magnetic re sponse of Py in a wide temperature range \nis therefore crucial in a view of potential lo w temperature applications of these spintronic \ndevices. It was observed in the late 1960' s that the temperature dependence of the \nferromagnetic resonance (FMR) for in-plane ma gnetized single Py films shows anomalous \nvariation of the FMR linewidth6 which has been attributed to spin-impurity interaction \nenhanced damping. More recently, enhancement of the FMR frequency in single Py films with decreasing temperat ure has been observed\n7 and explained in terms of spin \nreorientation transition at the Py interface belo w 100K. This Letter presents an investigation \nof the temperature dependence (2K<T<300K ) of the dynamic and static magnetic \nproperties of about 25nm single-layer Py films a nd of Py films inserted as a free layer in \nMTJs with Al 2O3 tunneling barriers. Our results indica te a magnetic reorientation transition \n(RT) in Py films at low temperatures (T<T R). The static and dynamic response in MTJs \ncould be affected by soft-hard layer dipolar  coupling and interface stress in Py films. \nThe 25nm single-layer Py film (sample A)  was grown by electron-beam evaporation \non standard Si(100) wafer covered by a 2.5nm naturally-oxidized SiO 2 layer without an \nexternal magnetic field applied during deposit ion and without a top capping layer. In \naddition, two different MTJs optimised for low field sensors applications8 were grown in a \nhigh vacuum sputtering chamber. Both MTJs had the following structure: Pinned-3 \n FM/Al 2O3(1.8nm)/Free-FM  with Pinned-FM be ing exchanged biased pinned magnetic \nelectrode composed of Ir 20Mn 80(10nm)/Co 90Fe10(3nm). The free ferromagnetic electrodes \n(Free-FM) were Co 60Fe20B20(2nm)/Ni 80Fe20(23nm) for sample B and \nCo90Fe10(3nm)/Ni 80Fe20(28nm) for sample C. The entire wafers were covered with a \nTa(5nm)/Cu(5nm) layers to prevent oxidation an d were annealed at 250 ºC in the sputtering \nchamber during one hour in an in-plane applie d magnetic field of 20mT. For more details \non sample growth and characterization see references.8,9 \nThe low temperature FMR experiments were carried out with a commercial Agilent \nVector Network Analyzer (VNA) work ing up to 8.5GHz by employing VNA-FMR \ntechnique10 which uses a coplanar wave guide  to create the pumping field h RF. For \ncryogenic measurements (2K<T<300K) we used a variable temperature cryostat with an RF \ninsert. In order to determine the dynamic re sponse of the Free-FM layer, an in-plane \nexternal bias field H ap was applied along the easy axis with h RF being transverse to H ap. The \ndata analysis is described in more de tails in Ref.10. The resonance frequency f 0 and \nlinewidth ∆f0 were evaluated by fitting the resonance peaks to a lorentzian curve. A \nQuantum Design SQUID magnetometer was used to measure the field dependence of the \nin-plane magnetization M up to 0.5T at diffe rent fixed temperatur es. In addition, the \nstructures of the different layers and interf aces were investigated at room temperature by \ntransmission electron microscopy (TEM) in high resolution mode (HRTEM) using a FEI-\nF20 microscope with point  resolution of 0.12 nm. \nFigure 1 shows the loss profile (imaginary part of microwave permeability \nparameter U(f), see Ref.10) and compares the magnetic field dependence of the FMR \nfrequency measured at 150K and at 10K for the three samples investigated. In order to 4 \n describe in more detail the temperatur e dependence of FMR, Fig. 2 compares f 0(T) and \n∆f0(T) for samples A-C determined with field µ 0Hap=20mT. Both MTJ samples (B and C) \nshow independent of the magnetic  field history an increase of the FMR frequency below a \ntemperature T R of about 65K, with the increase being more pronounced for the MTJ-C (see \nalso Fig. 1). These changes in the FMR are acco mpanied by nearly step-like changes in the \nFMR linewidth below T R for both MTJs. The single Py film (sample A), however, shows a \nmore gradual enhancement of the FMR fre quency and the linewidth below roughly 100K, \nwhich is in a good agreement with previous studies.6,7  The presence of a weak step-like \nincrease in the FMR linewidth  just below 55K in sample A can also be observed in Fig. 2b.  \nFigure 3a illustrates quasi-static magnetiza tion curves for the MTJ-C measured for \nthree temperatures close to 60K. Here we sh all focus on the field intervals where both Free-\nFM and Pinned-FM layers are forced to the mo stly parallel alignment (P) and on the region \nof the antiparallel alignment (AP). By analyzi ng the low field hysteresi s loops of the Free-\nFM layer near P=>AP transition we determine the switching field of the Free-FM layer as a \nfunction of temperature (Fig. 3b). One clearly sees that in both type s of the MTJs studied \nthe switching field is substantially lower th an observed in the single Py film. This \ndifference may indicate the presence of dipolar fields i nduced interacti on between hard and \nsoft layers in magne tic tunnel junctions  which influences domain walls nucleation and \npropagation in the soft layer.  \nFor all samples the high field magnetizati on curves M(H) in the P state show \nmagnetization saturation above few 0.2T, except in the proximity to 55-70K where some \ndeviation from the saturation behavior with an anomalous peak and dip in the M(H) \ndependence below and above a critical temperat ure is observed. Data for the MTJ-C, where 5 \n this unusual behavior in M(T,H) in the P-st ate is mostly remarkable, with up to 20% \ndeviation in the high field magne tization, is shown in Fig. 3a. In order to compare M(H,T) \ndependences for samples A-C, Fig. 3c plots the temperature dependence of the high field \nmagnetization in the applied field of 0.5T (further M* S) normalized by the corresponding  \nM* S(5K) values. For the all samples studied M*\nS(T)/ M* S(5K) is close to one, except in \nproximity to the temperature interval ar ound 55-70K where an anomaly is observed. One \nclearly sees that this magnetizati on anomaly is strongest and shows qualitatively different \ntemperature dependence with peak and dip close to 60K  for the MTJ samples B,C in \ncomparison with the single Py film (A) where only weak maximum (about 3% deviation) is seen (see inset of Fig. 3c) which may reflect transition from perpendicularly oriented Py \ninterface spins at T«T\nR  to in-plane disordered Py interface spins at T»T R. \nWe believe that the high field magnetizat ion anomaly is evidence of a magnetic \nreorientation transition roughly below T R≈60K in the magnetically soft layer, supporting \npreviously described FMR vs. temperature resu lts. To show this more clearly, we marked \nwith arrows the positions of the corresponding anomalies in M*\nS(T) in Fig. 2a, showing \nFMR(T). The differences in the temperatur e dependence of FMR and magnetization in \nMTJ-B, C compared to single Py film (A), ma y indicate some fundamental changes in the \nRT occurring in the soft layer, most proba bly related to the presence or absence of \nmagnetically hard layer. \nIn order to understand these differences in the reorientat ion transition which include \nthe anomalous behavior in the M*\nS(T) close to T R accompanied by clear  “knee-like” \nvariation of the FMR frequency in MTJs B and C, we propose a simple model (see sketch \nin Fig. 4) which considers the mutual influence of the pinned (hard) la yer and the free (soft) 6 \n one, induced by dipolar antiferro magnetic coupling through the Al 2O3 barrier with some \nanticorrelated roughness.11 Figure 4a sketches magnetization in the regions of the soft and \nhard layers interfacing Al 2O3 barrier at temperatures T«T R , when the MTJ stack is situated \nin 0.5T field. These layers may have magnetic moments directed mostly out-of-plane due to \nthe surface anisotropy in the Py and dipolar  coupling. This means that both interface \nmagnetic moments tend to occupy a relative minimum of the energy corresponding to an \nout-of-plane magnetization (see sketch for the re lated energy profiles in Figs. 4a-d). Here \n“springs” indicate dipolar coupl ing between the hard and th e soft layers sketched by \n“balls”. Close to T R; at T≤TR (Fig. 4b) the soft layer starts  to undergo an orientational \ntransition from the out-to-plane to in-plane alignment enhancing the in-plane magnetization \nM*\nS(T), by moving the soft ferromagnetic system  (Py) toward its global energy minimum. \nFigure 4c schematically shows what may happen at T ≥TR when the hard layer, due to its \ncoupling to the soft layer and due to the st rong difference in the soft and hard layer \nmetastable energy profiles, is pushed towards in -plane (trending to be  antiparallel to the \nsoft layer) magnetization configuration, reducin g therefore the total in-plane magnetization \nM*\nS(T). Finally, at T»T R (Fig. 4d) both soft and hard layers turn to have in-plane \nmagnetization (both \"balls\" in the sketch in Fig. 4d occupy their absolute minima of \nenergy) in equilibrium conditions with an  antiparallel alignm ent showing a total \nmagnetization of the stack nearly the same as at T →0. Within our model, the quantitative \ndifferences between response observed in MTJs  samples B and C could be attributed to \ndifferent materials interfacing the Py layer (CoFe or CoFeB) which could determine the \nstress at the Py interface.  7 \n In Conclusion , temperature dependent dynamic and static magnetic response in \nmagnetic tunnel junctions with Permalloy laye rs shows a magnetic reorientation transition \nbelow 60K which is qualitatively different  from one reported for single Py films,7 most \nprobably due to dipolar soft-hard layer coupli ng. These findings could be important for low \ntemperature applications of de vices incorporating Permalloy.12 \n \nAcknowledgements \nWe thank Referee for suggesting soft-har d layer coupling mechanism to describe \ntemperature dependent hi gh field magnetization. Discussion with A.Levanuyk and \nR.Heindl is gratefully acknowledged. Authors acknowledge support by Spanish MEC \n(MAT2006-07196; MAT2006-28183-E; MAT-2005-06024- C02-01) and U.S. NSF (ECCS-\n0823813 (VM)).  8 \n REFERENCES \n1 J. Moodera, L. Kinder, R. Wong, and R. Merservey, Phys. Rev. Lett. 74, 3273 (1995). \n2 T. Miyazaki, and N. Tezuka, J. Magn. Mag. Mat. 139, L231 (1995). \n 3 S. S. P. Parkin, K. P. Roche, M. G. Samant , P. M. Rice, R. B. Beyers, R. E. Scheuerlein, \nE. J. O'Sullivan, S. L. Brown, J. Bucchigano, D. W. Abraham, Yu Lu, M. Rooks, P. L. \nTrouilloud, R. A. Wanner, and W. J. Gallagher, J. Appl. Phys. 85, 5828 (1999). \n4 C. Chappert, A. Fert, and  F. N. Van Dau, Nat. Mater. 6, 813 (2007). \n5 P. P.Freitas, R. Ferreira, S. Cardoso a nd F. Cardoso, J. Phys.: Condens. Matter. 19, \n165221 (2007).  \n6 C. E. Patton and C. E. Wilts, J. Appl. Phys. 38, 3537 (1967). \n7M. Díaz de Sihues, P.J. Silva,  and J.R. Fermin, Physica B 354, 361 (2004). \n 8 J. M. Shaw, R. Geiss and S. E. Russek, Appl. Phys.  Lett. 89, 212503 (2006). \n9 W. Xu, D. B. Watkins, L. E. DeLong, K. Ri vkin, J. B. Ketterson, and V. V. Metlushko, J. \nAppl. Phys. 95, 6645 (2004). \n10 S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva \nand J. P. Nibarger, J. Appl. Phys. 99, 093909 (2006). \n11 P. Vargas and D. Altbir, Phys. Rev. B 62, 6337, 2000. \n12 S. van Dijken and J. M. D. Coey, Appl. Phys. Lett. 87, 022504 (2005). \n  \n \n 9 \n  \nFIGURE CAPTIONS \n \nFIG. 1.(Color on-line): Contour plots of the FMR at different external bias fields. The color \nscale indicates the magnitude of the imaginary part of the susceptibility (see Ref. 10). The \ntop panel (a-c) shows resonant frequency f 0(H) measured at 150 K while the bottom panels \n(d-f) show f 0(H)  at T=10K.    \n FIG. 2. (Color on-line) (a) Temperature de pendence of the FMR frequency. The inset \nhighlights the temperature dependence of FMR for the sample B.  (b) FMR linewidth vs. \ntemperature for samples A-C.  All data are meas ured in the external field of 20mT. Arrows \nindicate the correspond ing anomalies observed in the M(T) curve. \n FIG. 3. (Color on-line) (a) Magnetic field de pendences of magnetization measured for the \nsample C in the proximity to 60K where an  anomaly in the high field magnetization is \nobserved. P and AP denote parallel and antip arallel alignment of free-pinned layers \nrespectively.(b) Temperature dependence of th e soft layer switching field for the three \nsamples studied. (c) Temperature dependence of the M\n*\nS (T) normalized by M* S (5K). The \ninset highlights dependence of M*\nS (T)/M*\nS (5K) for the single Py film. \n FIG. 4. (Color on-line) Top: TEM images of th e single Py film and one of the MTJs (B). \nThe dashed lines indicate the insulating barr ier profile which shows presence of regions \nwith anticorrelated roughness (marked with dot ted arrows). Bottom: Sketch explaining \nproposed magnetization configuration and the ener gy profiles of both soft and hard layers \nin the regions with anticorrelated roughness. Relative and absolute minima correspond to \nout-of-plane and in-plane magnetizations respec tively while spring indicates soft/hard layer \ncoupling for: (a) T«T\nR. (b) T ≤TR. (c) T ≥TR and (d) T»T R conditions. Dotted lines indicate \nstray fields. Dashed arrows show  non-equilibrium magnetization.    01 002 003 0040080012000\n1 002 003 0045670\n1002003004.134.204.27Δ\nf0 (MHz)T\n (K)(a) sample A \nsample B \nsample Cf0 (GHz)T\n (K)(b) f0(GHz)T\n(K)sample B02 004 00-200-10001002005\n01 001 500.81.01.24\n08 00.951.001.0501 002 003 000123(\nc)(b)µ0 H (mT) M (nA/m2) \n50K \n60K \n70K(a)P\nA P \nT(K)M*S\n/MS(5K) sample A \nsample B \nsample CSwitching Field (mT)T\n(K)Si\nPy 5nmPyCoFeBCoFe\nAl2O3Sample A Sample B\n10nm\n"    },    {        "title": "0804.3272v1.Local_electronic_structure_of_Cr_in_the_II_VI_diluted_ferromagnetic_semiconductor_Zn___1_x__Cr__x_Te.pdf",        "content": "arXiv:0804.3272v1  [cond-mat.mtrl-sci]  21 Apr 2008Local electronic structure of Cr in the II-VI diluted ferrom agnetic\nsemiconductor Zn 1−xCrxTe\nM. Kobayashi,1,∗Y. Ishida,1,†J. I. Hwang,1G. S. Song,1A. Fujimori,1C.-S. Yang,2\nL. Lee,2H.-J. Lin,2D. J. Huang,2C. T. Chen,2Y. Takeda,3K. Terai,3S.-I. Fujimori,3\nT. Okane,3Y. Saitoh,3H. Yamagami,3K. Kobayashi,4A. Tanaka,5H. Saito,6and K. Ando6\n1Department of Physics, University of Tokyo,\n7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\n2National Synchrotron Radiation Research Center, Hsinchu 3 0076, Taiwan\n3Synchrotron Radiation Research Unit,\nJapan Atomic Energy Agency, Sayo-gun, Hyogo 679-5148, Japa n\n4Japan Synchrotron Radiation Research Institute, 1-1-1 Kou to,\nMikaduki-cho, Sayou-gun, Hyogo 679-5148, Japan\n5Department of Quantum Matter, ADSM,\nHiroshima University, Hagashi-Hiroshima 739-8530, Japan\n6Nanoelectronics Research Institute, AIST, Tsukuba Centra l 2,\nUmezono 1-1-1, Tsukuba Ibaraki 305-8568, Japan\n(Dated: October 30, 2018)\n1Abstract\nThe electronic structure of the Cr ions in the diluted ferrom agnetic semiconductor Zn 1−xCrxTe\n(x= 0.03 and 0.15) thin films has been investigated using x-ray magn etic circular dichroism\n(XMCD) and photoemission spectroscopy (PES). Magnetic-fie ld (H) and temperature ( T) depen-\ndences of the Cr 2 pXMCD spectra well correspond to the magnetization measured by a SQUID\nmagnetometer. ThelineshapeoftheCr2 pXMCDspectraisindependentof H,T,andx,indicating\nthat the ferromagnetism is originated from the same electro nic states of the Cr ion. Cluster-model\nanalysis indicates that although there are two or more kinds of Cr ions in the Zn 1−xCrxTe samples,\nthe ferromagnetic XMCD signal is originated from Cr ions sub stituted for the Zn site. The Cr 3 d\npartial density of states extracted using Cr 2 p→3dresonant PES shows a broad feature near the\ntop of the valence band, suggesting strong s,p-dhybridization. No density of states is detected at\nthe Fermi level, consistent with their insulating behavior . Based on these findings, we conclude\nthat double exchange mechanism cannot explain the ferromag netism in Zn 1−xCrxTe.\nPACS numbers: 75.50.Pp, 75.30.Hx, 78.70.Dm, 79.60.-i\n2INTRODUCTION\nFerromagnetic diluted magnetic semiconductors (DMS’s) have open ed a way for the ma-\nnipulation of the spin degree of freedom of electrons through inter action between the local\nmoments of doped magnetic ions and the spins of charge carriers in t he host semiconductors.\nTherefore, ferromagnetic DMS’s have been considered to be key m aterials for semiconduc-\ntor spin electronics or spintronics [1, 2], which is intended to manipula te both the charge\nand spin degrees of freedom of electrons in semiconductors. If fe rromagnetism occurs as\na result of interaction between the local magnetic moments of dope d ions and the spins\nof charge carriers, the magnetism is called carrier-induced ferrom agnetism, and the III-V\nDMS’s Ga 1−xMnxAs and In 1−xMnxAs are prototypical systems of carrier-induced ferromag-\nnetism [3]. Using the III-V DMS’s, new functional devices such as circ ular polarized light\ndetectors [4], spin-related light-emitting diodes [5], and field-effect t ransistors controlling fer-\nromagnetism [6] have been fabricated. However, these devices on ly act at low temperatures\nsince the Curie temperatures ( TC’s) of the DMS’s are below room temperature. Therefore,\nferromagnetic DMS’s having TCabove room temperature are strongly desired for practical\napplications of spintronic devices. Ever since the theoretical pred iction of ferromagnetism\nhavingTCexceeding room temperature in wide-gap semiconductor-based DM S’s [7], there\nhave been many reports on room-temperature ferromagnetism o f wide-gap DMS’s such as\nGa1−xMnxNandZn 1−xCoxO[8]. Inorder to see whether the ferromagneticproperties arein-\ntrinsic or extrinsic, anomalous Hall effects [9], magnetic circular dichr oism (MCD) in visible-\nto-ultraviolet region [10], and carrier-doping dependence of the fe rromagnetism [11, 12] have\nbeen studied since these properties of DMS are derived from intera ction between the host\nsemiconductor and the doped magnetic ions.\nThe II-VI semiconductor ZnTe crystallizes in the zinc-blend struct ure as shown in\nFig. 1(a), has a band gap of ∼2.4 eV, and shows p-type electrical conductivity. Cr-doped\nZnTe crystal, in which the Cr concentration is below 1%, has been inve stigated before the\ndiscovery of ferromagnetism in heavily Cr-doped ZnTe thin films. Inf rared absorption [13]\nand electron spin resonance [14] studies of bulk Zn 1−xCrxTe have suggested that the Cr\nions are divalent and are subjected to tetragonal Jahn-Teller dist ortion. MCD measure-\nments in visible-to-ultraviolet region on bulk Zn 1−xCrxTe crystals have revealed a positive\np-dexchange constant Nβ, that is, the exchange interaction between the hole spin and the\n3Zn2+ Te2- Cr2+(a) (b)\nFIG. 1: Crystal structure of Zn 1−xCrxTe. (a) Zinc-blend structure. (b) [CrTe 4]6−cluster.\nlocal magnetic moment is ferromagnetic [15]. Recently, Saito et al. [16] have succeeded to\nprepare ZnTe thin films doped with high concentration of Cr atoms ( x∼20%) by the molec-\nular beam epitaxy (MBE) technique. The Zn 1−xCrxTe thin films showed ferromagnetism at\nroom temperature and their MCD signals observed at the absorptio n edge of ZnTe showed\nmagnetic-field ( H) and temperature ( T) dependences which follow these of magnetization\n(M), indicating that there is strong interaction between the spins of h osts,p-band elec-\ntrons and the magnetic moments of the doped Cr ions [16, 17]. There fore, Zn 1−xCrxTe has\nattracted much attention as an intrinsic DMS with strong s,p-dinteraction.\nThe ferromagnetic properties of Zn 1−xCrxTe thin films have been investigated so far [18].\nWhile thin films with high Cr concentrations have shown a clear hystere sis loop, those with\nlow Cr concentrations have exhibited unusual hysteresis: the loop of the hysteresis becomes\nthin near the zero field in the M-Hcurve, which can be explained by a superposition of\nferromagnetic and superparamagnetic components. In the M-Tcurves of ferromagnetic\nZn1−xCrxTe thin films, blocking phenomena, namely, difference in the magnetiza tion be-\ntween the zero-field-cool (ZFC) and field-cool (FC) have also bee n observed. The blocking\ntemperature ( TB), where the ZFC and FC curves start to separate, increased with increas-\ning Cr concentration: TB∼5 K forx= 0.01 and∼95 K for x= 0.17. These results\nimply that ferromagnetic Zn 1−xCrxTe is a magnetically random system with some antifer-\nromagnetic interaction between the Cr ions and that the magnetism is sensitive to the Cr\n4concentration.\nEffects of doping on the ferromagnetism of Zn 1−xCrxTe have been studied [19, 20, 21].\nIodine (I), which is expected to be an electron dopant, enhances t he ferromagnetism while\nnitrogen (N), which is expected to be a hole dopant, suppresses it [ 19, 20]. These effects have\nbeen explained based on the double-exchange mechanism [19, 22]. Ho wever, carrier-induced\nferromagnetism in Zn 1−xCrxTe is doubted because the Zn 1−xCrxTe films are highly insulat-\ning. Furthermore, the tendency that N doping increases hole carr ier concentration and sup-\npresses the ferromagnetism is opposite to the observation for Ga 1−xMnxAs [23], in which the\nferromagnetic property is enhanced by theincrease of hole conce ntration. Recently, spatially\ninhomogeneous distributions of the Cr ions have been pointed out to influence the magnetic\nproperties [24, 25, 26]. Spatially resolved energy-dispersive x-ray spectroscopy study has re-\ncently revealed that co-doping with I induces inhomogeneous forma tion of Cr-rich (Zn,Cr)Te\nnano-regions, whereas co-doping with N results in homogeneous Cr -ion distributions [21].\nX-ray magnetic circular dichroism (XMCD) and resonant photoemiss ion spectroscopy\n(RPES) are powerful tools to investigate the electronic structur e of DMS [27, 28, 29, 30, 31,\n32, 33, 34, 35, 36]. XMCDisdefined asthedifference between theco re-level x-rayabsorption\nspectroscopy (XAS) spectra taken with right-handled ( µ+) and left-handled ( µ−) circularly\npolarized x rays. Because XMCD is sensitive only to magnetically active species, it is\nvery efficient to extract information about the electronic and magn etic properties of doped\nmagnetic ions. In RPES, when the incident photon energy is adjuste d to the 2 p→3dcore\nexcitation energy, the photoemission intensity of the 3 dpartial density of states (PDOS) is\nresonantly enhanced [37, 38].\nOur previous Cr 2 pXMCD measurements on Zn 1−xCrxTe (x= 0.045) thin film [39] have\nrevealed that the orbital moment of the Cr 3 delectrons is largely quenched compared with\nthe value of the Cr2+ion in the tetrahedral crystal field. The XMCD intensity increased\nwith increasing Hup to 7 T, indicating the existence of paramagnetism and/or superp ara-\nmagnetism in Zn 1−xCrxTe, and that the magnetically active Cr ions had a single chemical\nenvironment although a small amount of magnetically inactive Cr ions e xisted. Atomic\nmultiplet theory analysis has suggested that the Cr ions are divalent and are subjected to\ntetragonal Jahn-Teller distortion, whose distortion axes are equ ally distributed in the X,Y\nandZdirections. The valence-band PES spectra showed suppressed sp ectral weight near\nthe Fermi level ( EF) [39]. Based on these observations, we have proposed that the sp ec-\n5tral suppression is originated from the Jahn-Teller distortion and/ or Coulomb interaction\nbetween Cr 3 delectrons.\nHowever, some points remain to be confirmed concerning the above suggestions. Since\nthe XMCD spectra were recorded only in the applied magnetic field par allel to the caxis,\nthe direction of Jahn-Teller distortion was not confirmed experimen tally. In addition, the\neffects of inhomogeneous Cr distribution in Zn 1−xCrxTe on the electronic structure of the Cr\nions should be elucidated. In order to address these issues, we hav e performed x,H,T, and\nincident-photon angular dependences of XAS and XMCD on ferroma gnetic Zn 1−xCrxTe. In\nthis paper, we repot on the results of XMCD and PES measurements combined with CI\ncluster-model analysis on Zn 1−xCrxTe thin films having different Cr concentrations. The x\nandHdependences ofXMCD spectra comparedwith Mmeasured by a SQUID magnetome-\nter provided experimental evidence that we measured the bulk mag netic properties. The T\nand angle dependences of XMCD, the former of which is sensitive to t he orbital degree of\nfreedom and the latter of which is sensitive to the direction of distor tion, revealed quenching\nof the orbital moment and isotropic electronic configuration of the Cr ions. In addition, the\nCI cluster-model analysis suggested that the Cr ions are subject ed to Jahn-Teller distortion\nwith isotropic distribution of the distortion axes and the ferromagn etism is caused by the Cr\nions substituted for the Zn sites. The p-dexchange coupling constant Nβestimated from\nthe electronic structure parameters was consistent with the MCD measurements. The Cr\n3dPDOS obtained for the 2 p→3dRPES was suppressed in the whole Cr concentrations,\nimplying that carrier-induced ferromagnetism is not effective in Zn 1−xCrxTe.\nEXPERIMENTAL\nZn1−xCrxTe(x= 0.03and0.15)thinfilmsweregrownoninsulatingGaAs(001)substrate s\nby the MBE technique. The total thickness of the Zn 1−xCrxTe thin films was 50 nm on a\n20 nm ZnTe buffer layer. During the deposition, the substrate was k ept at a temperature of\n∼250◦C. Details of the sample preparation were described in [16]. Since XMCD is a surface\nsensitive probe, the films were covered by an Al capping layer of 2 nm thickness so as to\navoid surface oxidization. Ferromagnetism was confirmed by magne tization measurements\nusing a SQUID magnetometer (Quantum Design, Co. Ltd). The Curie temperatures of the\nx= 0.03 and 0.15 films were estimated to be ∼150 and∼280 K, respectively.\n6XMCD measurements were performed at the Dragon Beamline BL11A of National Syn-\nchrotron Radiation Research Center in the total-electron-yield mo de. An XMCD hysteresis\nloopattheCr2 p3/2edgewas measured bythetotal-fluorescence-yield mode. The mon ochro-\nmator resolution was E/∆E>10000. The polarization of incident photons was fixed and\nXMCD ( µ+−µ−) spectra were obtained by switching the magnetic field. The backgr ound\nof the XAS spectra at the Cr 2 pedge was assumed to be a superposition of the spectrum\nof ZnTe in this region and hyperbolic tangent functions. The Cr 2 p→3dRPES mea-\nsurements were performed at BL23SU of SPring-8 using a Gammada ta Scienta SES-2002\nhemispherical analyzer operated in the transmission mode. The mon ochromator resolu-\ntion was E/∆E>10000. The spectra were taken at room temperature in a vacuum b elow\n5.0×10−8Pa. The total resolution of the spectrometer including temperatu re broadening\nwas∼200 meV.\nSOFT X-RAY MAGNETIC CIRCULAR DICHROISM\nAs described in the introduction, the ferromagnetism of Zn 1−xCrxTe is enhanced with Cr\nconcentration. From the magnetization measurements described below, the enhancement\nmay be related to the inhomogeneous Cr distribution. A question the n arises what happens\nin the electronic structure related to the ferromagnetism if a high c oncentration of Cr ions\nare doped into a ZnTe thin film. The xdependence of XMCD measurement can approach\nthis problem. In addition, we have measured H,T, and angle dependences of XMCD and\nXAS so as to obtain the information of the electronic structure of t he Cr ions.\nCr-concentration dependence\nIn order to investigate the Cr-concentration dependence of the ferromagnetism in\nZn1−xCrxTe, we compare in Fig. 2(a) the Cr 2 pXMCD spectra of the Zn 1−xCrxTe thin\nfilms ofx= 0.03 and 0.15 recorded under a fixed condition, H= 1.0 T andT= 25 K. Here,\nthe spectra of different x’s have been normalized to the XAS [( µ++µ−)/2] peak height at\n∼576 eV. Both the XAS and XMCD spectra of the x= 0.03 and 0.15 samples have nearly\nidentical line shapes except in the region around 578 eV, where the X AS spectrum of the\nx= 0.03 film is slightly stronger than that of the x= 0.15 one, and at H= 1.0 T the\n7100\n50\n0\n-20-100Intensity (arb. units)\n590 585 580 575 570\nPhoton Energy (eV)0 Normalized XMCD\n580 575 570\n-1.0 -0.5 0.0 0.5 1.0\nMagnetic Field (T)-20-1001020– XMCD (arb. units)\n-3-2-10123Magnetization ( /s109B/Cr)\n-404M ×10−5 (emu)\n-101Zn1-xCrxTe\nCr 2p\nXAS\nXMCDT=25 K(a)\nx=0.03\n 1.0 T\n \nx=0.15\n 0.1 T\n 0.7 T\n 1.0 TXMCD at\nh/s110=576.3 eV\n x=0.03\n x=0.15\n x=0.15 TFY\nMagnetization\n x=0.03\n x=0.15(b)\nT=25 K\nFIG. 2: Magnetic-field dependence of the Cr 2 pXMCD spectra of Zn 1−xCrxTe thin films. (a)\nXMCD spectra at various H. (b) XMCD intensity as a function of Hcompared with the mag-\nnetization curves. The diamagnetic component of the substr ate has been subtracted from the\nraw magnetization curves (inset). The XMCD hysteresis loop has been measured by the total-\nfluorescence-yield (TFY) mode.\nXMCD intensity increases with increasing x. Both the XAS and XMCD spectra are similar\nto those of a x= 0.45 sample previously reported [39]. These results suggest that alth ough\nthere may be two or more kinds of Cr ions having different electronic s tructures as observed\nby XAS, the ferromagnetic properties of these films reflected on t he XMCD spectra are orig-\ninated from a common magnetically active component. One can also se e in Fig. 2(a) that\nthe magnetically active component of the x= 0.15 film shows a stronger XMCD intensity\nthan that of the x= 0.03 film. Considering that XMCD is sensitive only to local electronic\nstates, the increase of the magnetization per Cr ion with Cr concen tration suggests that the\ndistance between the Cr ions affects the magnetism of (Zn,Cr)Te.\nMagnetic-field dependence\nXMCD measurements with varying Hare useful to investigate chemically and magneti-\ncally inhomogeneous samples [32, 33]. At low H, the XMCD spectra predominantly reflect\nthe ferromagnetic component of the Cr ions while at high H, the paramagnetic compo-\n8nent is superimposed. Figure 2(a) shows the Hdependence of the XMCD spectra of the\nx= 0.15 film measured at T= 25 K. The XMCD intensity increases with increasing H,\nwhile the spectra maintain the same line shape up to H= 1.0 T, as shown in the inset of\nFig. 2(a), where the XMCD spectra have been normalized to the neg ative peak at 576 eV.\nIn magnetic fields which are sufficient to saturate the magnetization of the ferromagnetic\ncomponent, the XMCD intensity hardly increases with increasing H, indicating that there\nare little paramagnetic magnetization in both the x= 0.03 andx= 0.15 thin films below\nH= 1.0 T, consistent with the previous XMCD measurements [39]. These re sults suggest\nthat the XMCD signal is dominated by the ferromagnetic one below H= 1.0 T, although\nat high fields the paramagnetic (or superparamagnetic) and antife rromagnetic XMCD sig-\nnals become detectable [39]. In order to see the consistency betwe en the macroscopic and\nmicroscopic magnetic measurements, comparison of the XMCD inten sity with the magne-\ntizationMis made in Fig. 2(b). The ratio of the XMCD intensity to Mat high magnetic\nfields almost coincide between the different films, providing experimen tal evidence that the\nXMCD intensity reflects the bulk magnetic properties of Zn 1−xCrxTe. The good agreement\nbetween the M-Hcurve and the XMCD intensity as a function of Hgives evidence for\nferromagnetism induced by the Cr ion. It is therefore likely that maj ority of the doped Cr\nions in Zn 1−xCrxTe magnetically interact with each other and give rise to the ferroma gnetic\nbehavior.\nTemperature dependence\nFor a 3dtransition-metal ion in a tetrahedral-symmetry ( Td) crystal field with an open t2\nshell, astrongtemperaturedependence oftheXASspectra isexp ected duetothedegeneracy\noftheorbitaldegreeoffreedominthegroundstate. FortheCr2+ionhaving the d4electronic\nconfigurationinthe Tdcrystal field, the t2statesarepartiallyoccupiedandtheorbitaldegree\nof freedom survives. However, the Cr L2,3XAS spectra of Zn 1−xCrxTe did not change with\ntemperature as shown in Fig. 3. This result implies that the orbital de generacy is lifted due\nto a Jahn-Teller distortion which splits the t2levels [40]. Alternatively, the 3 dshell may be\ncompletely filled, i.e., the Cr ion is in the Cr+state and has the d5electronic configuration.\n9100\n50\n0XAS\n590 585 580 575 570Photon Energy (eV)-20-100XMCD 25 K\n 100K\n 200KZn0.85Cr0.15Te\nH=1.0 TCr 2p\nXAS\nXMCD\nFIG. 3: Temperature dependence of the Cr 2 pXMCD spectra of the Zn 1−xCrxTe (x= 0.15) thin\nfilm atH= 1.0 T.\nXMCD sum rules\nBy applying the XMCD sum rules [41, 42], one can estimate the spin ( Mspin) and orbital\nmagnetic moments ( Morb) of the Cr ion separately by\nMorb=−2q\n3r(10−Nd), (1)\nMspin+7MT=−3p−2q\nr(10−Nd), (2)\nwherep,q, andrare the integrated intensities of the XAS and XMCD spectra as show n\nin Fig. 4(a), Ndis the number of 3 delectrons, and MTis the expectation value of the\nmagneticdipoleoperator, whichisnegligiblysmallwithrespect to Mspinbecauseofrelatively\nweak spin-orbital coupling in 3 delectrons [41]. The ratio Morb/Mspinwas estimated to be\n∼0.11±0.12, indicating that the orbital moment is significantly suppressed, c onsistent with\nthe previous XMCD measurements [39]. Thus, the candidate electro nic structures for the\nmagnetically active Cr ions are the Cr2+ion inD2dsymmetry or Cr+(d5) configuration, as\nshown in Fig. 4(b). Below, we shall discuss the electronic structure of the Cr ions under\nthese constraints.\nIncident photon angle dependence\nWhen the Cr2+ion is under the uniaxial distortion of tetragonal symmetry D2d, the\nelectronic state should have an anisotropy. Calculations using atom ic multiplet theory for\n10-20-100XMCD (arb. units)\n590585580575570\nPhoton Energy (eV)100\n50\n0XAS (arb. units)600\n500\n400\n300\n200\n100\n0XAS Integral\n-200XMCD Integral\n XMCD\n integ. XAS\n integ. r \n pqXAS\nXMCD(a)\nZn1-xCrxTe(b)\nt2\neCr2+ (d4) TdCr2+ (d4) D2dCr+ (d5)\nb2\ne\nb1\na1\nFIG. 4: Application of the XMCD sum rules to the Cr 2 pXAS and XMCD spectra of Zn 1−xCrxTe\nthin films. (a) XAS and XMCD spectra and their spectral integr als. (b) Candidates for the\nelectronic structure of the Cr ions. The orbital degree of fr eedom of the Cr2+ion in the Tdcrystal\nfield (left column) is quenched in the Cr2+ion in the D2dcrystal field (middle column) or in the\nCr+configuration (right column).\ntheTdandD2dsymmetries have been performed by varying the incident angle θof x ray as\nshown in Fig.s 5(a) and (b). Here, θ= 0◦is defined by the angle between the surface normal\nand the propagation vector of the incident x rays as shown in Fig. 5( c). Racah parameters\nhave been assumed to be the same as those of the Cr2+ion. Both the XAS and XMCD\nspectra for Tdare independent of θ, while the spectra for D2dshow systematic changes with\nθ. The analysis thus indicates that if the Cr ion is under the uniaxial dist ortion, the XAS\nand XMCD spectra would show θdependence.\nFigure 5(d) shows the XAS and XMCD spectra measured at several θ’s. There is little\nθdependence in the XAS and XMCD spectra of the x= 0.15 film, suggesting that the\nferromagnetic Cr ion consists of an almost isotropic electronic confi guration. Taking into\naccount the quenching of the orbital moment, the observation imp lies an equal distribution\nof the axis of the Jahn-Teller distortion in the a-,b-, andc-directions, or the d5electronic\nconfiguration of the Cr ion.\n11Intensity (arb. units) Intensity (arb. units)\n590585580575\nPhoton Energy (eV)590585580575\nPhoton Energy (eV)100\n50\n0\n-20-15-10-505Intensity (arb. units)\n590 585 580 575 570\nPhoton Energy (eV)/s113 =\n90\n80\n70\n60\n50\n40\n30\n20\n10\n090\n80\n70\n60\n50\n40\n2030\n10\n0\n90\n80\n70\n60\n50\n40\n30\n20\n10\n090\n80\n70\n60\n50\n40\n30\n20\n10\n0Cr2+ Td\nXASCr2+ D2d[100]\nXAS(a)\nXMCD XMCD/s113/s32h/s110/s32H \nsample\nZn1-xCrxTe\nx=0.15 XAS\nXMCD\n /s113 = 0    /s113 =30\n /s113 =60  /s113 =75(d)\nH=1.0 T, T=25 K(b) (c)\n[001]\nFIG. 5: Incident photon angle dependence of the Cr 2 pXAS and XMCD spectra of Zn 1−xCrxTe\nthin films. (a), (b) XAS and XMCD spectra calculated using ato mic multiplet theory for Cr2+Td\nandD2dwith varying θ. (c) Experimental setup for the angle-dependent measureme nts. (d) Cr 2 p\nXAS and XMCD spectra with various θ.\nCONFIGURATION-INTERACTION CLUSTER-MODEL CALCULATION\nThe substitutional magnetic ion in a II-VI DMS is tetrahedrally coord inated by anions as\nshown in Fig. 1(b), and therefore the electronic structure of the magnetic ion is influenced\nby the crystal field of the ligand ions and hybridization with ligand orbit als. In order to\ndetermine the electronic structure of the ferromagnetic compon ent, we have performed CI\ncluster-model analysisfortheCr2 pXASandXMCDspectra. Whileatomicmultiplet theory\ntreats the symmetry and strength of the crystal field through t he strength of the crystal-\nfield splitting of the energy levels of one-electron orbitals, hybridiza tion between ligand and\n3dorbitals is explicitly taken into account in CI cluster model. Here, CI me ans interac-\ntion between different charge-transfer electronic configuration s. Therefore, by applying CI\ncluster-model analysis, one can obtain more detailed information of electronic structure of\nthe Cr ions such as Nβ. CI cluster-model analysis is useful for describing the local electr onic\nstructure of 3 dions in DMS and enables us to estimate their electronic structure par ameters:\n12charge-transfer energy ∆, d-dCoulomb interaction energy Udd, and Slater-Koster parameter\n(pdσ).\nLetusfirst consider thepossibilityofthe d4electronic configuration. Accordingtoreports\non the dependence of the lattice constant on the Cr concentratio n in Zn 1−xCrxTe [18, 43], at\nlow Cr concentrations ( x/lessorsimilar0.04), thexdependence of the lattice constant obeys Vegard’s\nlaw, indicating that the Cr ions substitute for the cation sites. At hig her Cr concentrations\n(x/greaterorsimilar0.05), the lattice constant of Zn 1−xCrxTe becomes the same as that of ZnTe and\nZn1−xCrxTe shows cubic symmetry due to relaxation of the lattice constant. Therefore,\nif the Cr2+ions are accommodated in the Zn 1−xCrxTe lattice, it is likely that the Jahn-\nTeller axes along [001], [010], and [100] are distributed isotropically as a consequence of the\nrelaxation of lattice constant.\nNext, let us examine the possibility of the d5electronic configuration. Recently, a Ra-\nman electron paramagnetic resonance study has reported that C r+ions existed in bulk\nZn1−xCrxTe crystal ( x <0.02) as an acceptor although Cr+is a minority electronic configu-\nrationincontrasttothedominantCr2+one[44]. AccordingtoapreviousMCDmeasurement\non Zn 1−xCrxTe thin film [45], the p-dexchange constant Nβis positive (ferromagnetic)\nlike bulk Zn 1−xCrxTe. For the half-filled or more than half-filled 3 dorbitals as realized\nin Ga1−xMnxAs [46] and Zn 1−xCoxTe [45], the p-dexchange becomes negative (antiferro-\nmagnetic) because the up-spin states of the 3 dorbitals are fully occupied so that only a\ndown-spin pelectron can hop into the 3 dorbitals. It follows from these reports that the\nhalf-filled electronic configuration Cr+is minority in Zn 1−xCrxTe, if it exists.\nBased on these considerations, we have performed CI cluster-mo del calculations on Cr2+\nin aD2dcrystal field assuming the isotropic distribution of the Jahn-Teller a xes and on Cr2+\nwith negative ∆. In the letter case, the ground state electronic co nfiguration becomes 3 d5L\n(Ldenotes a hole in the host valence band), which is referred to “Cr+” hereafter. Figure 6\nshows comparison of the experimental spectra with calculated one s, where the calculated\nspectra are shifted to adapt the Cr 2 p3/2peak at 576 eV to the experimental one. The\ncalculated spectra for “Cr+” are narrower than the experimental ones, where the electronic\nparameters are chosen as ∆ = −1.0,Udd= 3.0, and (pdσ) =−0.75 eV. Even if one chooses\nothersetsofparameterswithnegative∆, thewidthofthecalculat edspectra doesnotchange\nconsiderably. Furthermore, the value of Morb/Mspinof the calculated XMCD spectrum is too\nsmall compared to that of the experimental one. Therefore, the analysis suggest that the\n130\n0Intensity (arb. units)\n590 585 580 575 570\nPhoton Energy (eV)c\nb\naCr 2p\nXASZn1-xCrxTe\n(x=0.15)\nXMCD\n Exp.\n Calc. Cr2+, Calc. “Cr+”\nMorb/Mspin\n   Experiment ~ −0.11±0.12\n   Calculation Cr2+: −0.121, \n   “Cr+”: −0.03(a) (b)\nFIG. 6: Configuration-interaction cluster-model analysis for the Cr 2 pXAS and XMCD spectra\nof Zn1−xCrxTe. (a) Comparison between theory and experiment. Arrows de note the minority\ncomponents. (b) Jahn-Teller distortions of the CrTe 4tetrahedron.\nd5Lelectronic configuration cannot explain the experimental spectra . On the other hand,\nthe calculated XMCD spectrum for Cr2+agrees well with the experimental one, although\nthere are some discrepancies between the Cr 2 pXAS spectrum and the calculated one.\nThe calculated Morb/Mspin=−0.121 is similar to the Morb/Mspinvalue estimated from the\nXMCD sum rules ∼ −0.11±0.12. The electronic structure parameters ∆, Udd, and (pdσ)\nare 4.0±0.5, 3.5±0.5, and−1.1±0.1 eV, respectively. Using these parameters, one can\nestimate the p-dexchange constant Nβ, which is given by [47]\nNβ=16\nS/parenleftbigg1\nδeff−1\nδeff+4j/parenrightbigg/parenleftBigg\n1\n3(pdσ)−2√\n3\n9(pdπ)/parenrightBigg2\n1\n3\n+16\nS/parenleftbigg\n−1\nδeff+6j−0.64\n−δeff+u′−j/parenrightbigg/parenleftBigg\n1\n3(pdσ)−2√\n3\n9(pdπ)/parenrightBigg2\n2\n3,(3)\nwhereu′andjare Kanamori parameters, δeff= ∆eff+WV/2, ∆effis the effective charge-\ntransfer energy, and WVis the width of the host valence band. In the distorted CrTe 4\ncluster, the first term of equation (3) becomes dominant [47]. The p-dexchange constant\n14estimated from the parameters is Nβ= 1.3±0.9 eV (>0: ferromagnetic), consistent with\nthe result of the MCD measurements [45]. The results indicate that t he ferromagnetism in\nZn1−xCrxTe is caused by a single kind of Cr ions most likely substituting the Zn site . Other\nkinds of Cr ions are detected only in XAS and not in XMCD, since they ar e magnetically\ninactive at least in the present low magnetic fields H≦1 T. These minority Cr ions may\nbe antiferromagnetically coupled with each other. In fact, the line s hape of the XMCD\nspectrum around the minority component peak position is slightly cha nged at high magnetic\nfields above H∼2 T [39]. The results indicate that the majority of Cr ions are divalent\nand are subjected to Jahn-Teller distortion with isotropic distribut ion of Jahn-Teller axes,\ngiving support to the previous XMCD and PES results [39].\nRESONANT PHOTOEMISSION SPECTROSCOPY\nAlthough the previous measurements on Zn 1−xCrxTe (x= 0.045) have shown a suppres-\nsion of DOS near EF[39], it is not obvious whether the Zn 1−xCrxTe with high Cr con-\ncentration has a finite intensity at EFor not. If double exchange interaction is dominant,\nthe peak of the Cr 3 dPDOS becomes broad with increasing Cr concentration. Therefore ,\nthe Cr-concentration dependence of the Cr 3 dPDOS provides useful information about the\norigin of the ferromagnetism.\nIn order to obtain the Cr 3 dPDOS in the valence-band region, RPES measurements\nhave been performed using photons at the Cr 2 p3/2absorption edge. Since photoemission\nspectroscopy is a surface sensitive technique, the spectra involv e signals from the Al capping\nlayer, too. However, since the Cr ions are expected to exist in the Z n1−xCrxTe layer and\nnot in the Al capping layer, the spectral intensity enhanced at the Cr 2pedge is probably\ncomposed of signals from the Cr ions in the Zn 1−xCrxTe layer. We have observed a clear\nCr 2p→3dresonance in the valence band as shown in Fig. 7(a). Here, the on- a nd off-\nresonance spectra have been taken at hν= 576 and 572 eV, respectively. The Cr 3 dPDOS\nobtained by subtracting the off-resonance spectra from the on- resonance one shows a peak\nat the top of the valence band, indicating that if hole carriers are do ped, the hole carriers\nshould have dcharacter. The line shape of the PDOS is broad and almost independe nt ofx,\nreflecting the strong hybridization between the host s,pbands and the localized dorbitals\nin Zn1−xCrxTe. The EFof thex= 0.15 sample is located closer to the top of the valence\n1515 10 5 0\nBinding Energy (eV)Intensity (arb. units)\nIntensity (arb. units)\n580 576 572\nPhoton Energy (eV)\n15 10 5 0\nBinding Energy (eV)Intensity (arb. units)Zn 3d\nCr 3dCr 2p3/2\nXASZn1-xCrxTe\n(x=0.15)\nCr 2p–3d\nRPES\n572574576577h/s110=\n580 eV(a)\nx=0.03(b)  On (h/s110=576 eV)\n Off (h/s110=572 eV)\nx=0.15Zn 3d\n Difference ×4\n(Cr 3d PDOS)\n Difference ×4\n(Cr 3d PDOS) On (h/s110=576 eV)\n Off (h/s110=572 eV)\nZn 3d\nFIG. 7: Cr 2 p→3dresonant photoemission spectra of Zn 1−xCrxTe. (a) Cr 2 p→3dresonant\nphotoemission series. The photon energies ( hν’s) are indicated on the Cr 2 p3/2XAS spectrum by\ntriangles in the inset. (b) On- ( hν= 576 eV) and off-resonance ( hν= 572 eV) spectra of the\nx= 0.03 andx= 0.15 thin films. The difference spectra show the Cr 3 dPDOS.\nband than that of the x= 0.03 sample as shown in Fig. 7(b). However, there is no density\nof states (DOS) at EFfor both the x= 0.03 and 0.15 samples, corresponding to their\ninsulating behaviors. Based on these findings, it is likely that the spec tral suppression near\nEFisoriginatedfromboththeJahn-TellerdistortionandCoulombintera ctionbetweenCr3 d\nelectrons as proposed in the previous measurements [39]. Since the ferromagnetism realized\nby the double-exchange mechanism requires a finite DOS at EF[22, 48], the observations\nindicate that the double-exchange mechanism appears difficult in Zn 1−xCrxTe. The present\nresults suggest that the ferromagnetic interaction between the spins of the s,pband electrons\nand local moments of the dorbitals is strong but the ferromagnetic interaction between the\ndorbitals is shortranged because the top of the valence band has dcharacter [49]. In\norder to obtain further understanding of the ferromagnetic inte raction, systematic XMCD\nmeasurements of “carrier-doped” (I and/or N doped) Zn 1−xCrxTe are highly desired.\n16CONCLUSION\nWe have performed XMCD and RPES measurements on the ferromag netic DMS\nZn1−xCrxTe (x= 0.03 and 0.15) thin films. The xandHdependences of the XMCD spectra\nprovide experimental consistency with the magnetization measure ments. The XMCD line\nshape is independent of x,H,T, andθ, indicating that the Cr ions responsible for the\nferromagnetism have a single chemical environment and a spatially iso tropic electronic con-\nfiguration. According to the XMCD sum-rule analysis and the Tdependence of XMCD, the\nground states of the ferromagnetic Cr ions should involve the elect ronic configuration with\nthe quenched orbital moment. The analysis using CI cluster-model calculation suggests that\nthe Cr ions responsible for the ferromagnetism substitute for the Zn site and are subjected\nto Jahn-Teller distortion with isotropic distribution of the distortion axes, and there are two\nor more kinds of Cr ions. The minority Cr ions may be antiferromagnet ically coupled with\neach other. The p-dexchange constant Nβhas been estimated from the electronic structure\nparameters and is consistent with MCD measurements. The Cr 3 dPDOS at EFis sup-\npressed even at x= 0.15, indicating that carrier-induced ferromagnetism appears difficu lt\nin Zn1−xCrxTe. The suppression of spectral weight at EFis probably caused by Jahn-Teller\ndistortion and d-dCoulomb interaction.\nFrom the report on the doping effects [21], both the hole-carrier co ncentration and the\ninhomogeneous Cr distribution affect the ferromagnetic propertie s of Zn 1−xCrxTe thin film.\nIn the present work, we have measured Zn 1−xCrxTe thin films having different Cr concentra-\ntion. In this case, it is likely that the effects of the Cr inhomogeneity o n the ferromagnetism\nare more dominant than these of the carrier concentration. Cons idering the fact that the\nferromagnetic XMCD signal of Zn 1−xCrxTe is independent of Cr concentration and that\nZn1−xCrxTe is highly insulating, it is likely that the carrier-induced ferromagnet ism is not\nimportant in Zn 1−xCrxTe and the inhomogeneous distribution of Cr atoms dominantly in-\nfluences the ferromagnetic properties of Zn 1−xCrxTe.\nAcknowledgment\nWe thank T. Mizokawa for fruitful discussions. 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Matter 19, 436227 (2007).\n21"    },    {        "title": "1410.3689v1.Spin_precession_mapping_at_ferromagnetic_resonance_via_nuclear_resonant_scattering.pdf",        "content": "Spin precession mapping at ferromagnetic resonance via nuclear resonant\nscattering\nLars Bocklage,1, 2,\u0003Christian Swoboda,3, 2Kai Schlage,1Hans-Christian Wille,1\nLiudmila Dzemiantsova,1, 2Saˇsa Bajt,1Guido Meier,4, 3, 2and Ralf R ¨ohlsberger1, 2\n1Deutsches Elektronen-Synchrotron, Notkestraße 85, 22607 Hamburg, Germany\n2The Hamburg Centre for Ultrafast Imaging,\nLuruper Chaussee 149, 22761 Hamburg, Germany\n3Institut f ¨ur Angewandte Physik und Zentrum f ¨ur Mikrostrukturforschung,\nUniversit ¨at Hamburg, Jungiusstrasse 11, 20355 Hamburg\n4Max-Planck Institute for the Structure and Dynamics of Matter,\nLuruper Chaussee 149, 22761 Hamburg, Germany\n(Dated: November 27, 2021)\nAbstract\nWe probe the spin dynamics in a thin magnetic film at ferromagnetic resonance by nuclear resonant\nscattering of synchrotron radiation at the 14.4 keV resonance of57Fe. The precession of the magnetization\nleads to an apparent reduction of the magnetic hyperfine field acting at the57Fe nuclei. The spin dynamics\nis described in a stochastic relaxation model adapted to the ferromagnetic resonance theory by Smit and\nBeljers to model the decay of the excited nuclear state. From the fits of the measured data the shape of the\nprecession cone of the spins is determined. Our results open a new perspective to determine magnetization\ndynamics in layered structures with very high depth resolution by employing ultrathin isotopic probe layers.\nPACS numbers: 76.50.+g, 76.80.+y, 75.25.-j, 78.70.Ck\n1arXiv:1410.3689v1  [cond-mat.mtrl-sci]  14 Oct 2014Spin waves, collective excitations of the magnetization, are key features of magnetic materials\nas they determine switching times and energy losses during magnetization reversal. They play a\ncrucial role for new concepts of logical operations, spin-torque oscillators, or signal processing,\nand constitute the basis of the emerging field of magnonics [1]. Spin waves are often probed\nby inelastic scattering techniques like Raman scattering, Brillouin light scattering, or inelastic\nneutron scattering that rely on analysis of the energy transfer to the scattered particles. We show\nthat resonant spin dynamics can be probed by coherent elastic scattering, namely nuclear resonant\nscattering of synchrotron radiation (NRS) [2, 3]. Since NRS is an x-ray technique, it opens the\npossibility to combine it with diffraction methods to obtain very high spatial resolution [4] down\nto atomic length scales [2]. Moreover, as NRS probes the nuclear decay of an excited M ¨ossbauer\nisotope it allows to employ ultrathin isotopic probe layers to achieve sub-nm depth resolution [5].\nThe application of conventional M ¨ossbauer spectroscopy as well as NRS has given indirect\naccess to thermally excited spin waves in ferromagnets via the temperature dependence of the\nmagnetic hyperfine field that exhibits the same temperature dependence as the magnetization in\nthese materials [6–8]. In contrast to a broad thermally excited spin wave spectrum we induce a\nsingle coherent mode by resonant excitation with radio frequency (rf) magnetic fields. This is one\napproach typically used in magnonic devices to obtain the desired functionality [9, 10].\nThe impact of rf magnetic fields on conventional M ¨ossbauer spectroscopy has so far been stud-\nied at MHz frequencies that are well below ferromagnetic resonances. It was shown that low-\nfrequency spin waves in paramagnetic media can induce magetic phase modulations of the nu-\nclear states [11]. A collapse of the magnetic hyperfine field arises from a fast periodic switching\nof the magnetization [12, 13]. The sideband effect originates from acoustic vibrations induced by\nmagnetostriction [13, 14]. Also Rabi oscillations have been observed where the rf field directly\ncouples to the nuclear transition [15]. Ferromagnetic or spin wave resonances at GHz frequencies\nhave not been investigated with M ¨ossbauer spectroscopy or NRS so far. This is the case to be\nstudied here.\nIn this letter we show that NRS measurements enable to extract the trajectory of the spins dur-\ning coherent precession at ferromagnetic resonance. The opening angle of the precession is an\nessential parameter for spintronics like, e.g., in spin pumping [16, 17] or in the spin dynamo [18].\nSo far the determination of the opening angle from experiments appeared to be challenging. Values\nfor opening angles averaged over time and space have been determined by anisotropic magnetore-\nsistance measurements for assumed circular trajectories [19, 20], an assumption that is justified\n2only in a few special cases like in a spherical particles without crystalline anisotropy.\nWe performed NRS measurements on ferromagnetic thin films excited at ferromagnetic reso-\nnance. The influence of spin dynamics on the NRS signal is analyzed in a stochastic relaxation\nmodel [21, 22] adapted to the ferromagnetic resonance theory of Smit and Beljers [23]. With this\nmethod the exact shape of the precession orbit is determined. This capability arises from the high\nsensitivity of NRS to the magnetization direction.\nNRS under grazing incidence [3] is performed at the Dynamics beamline P01 [24] of PE-\nTRA III at DESY (Hamburg, Germany) in 40 bunch mode with a bunch separation of 192 ns and a\nbunch duration of about 50 ps. The energy is tuned to the nuclear transition energy of 14.4125 keV\nof the M ¨ossbauer isotope57Fe that has a natural lifetime of 141.11 ns. The bandwidth of the syn-\nchrotron radiation is reduced to 1 meV by a high resolution monochromator. A Kirkpatrick-Baez\nmultilayer mirror system focuses the beam to a spot size of about 10 \u000210\u0016m2. The synchrotron\npulse excites all six allowed nuclear transitions simultaneously. The frequency differences man-\nifest as quantum beats in the temporal evolution of the nuclear decay. The recorded NRS time\nspectra are fingerprints for the magnetic spin structure of the sample.\nSamples are prepared on GaAs substrates. All layer geometries are defined by electron-beam\nlithography and lift-off processing. Electrical striplines are prepared by thermal evaporation of\n7 nm Cr, 118 nm Ag, and 20 nm Au. Hydrogen silsesquioxane (HSQ) with a thickness of 140 nm\nis used to electrically insulate the strip line from the ferromagnetic film and to provide a smooth\nsurface. The 800 \u0002800\u0016m2film is prepared by sputter deposition of 4 nm Cr, 18 nm Pd, 13 nm\nisotopically enriched permalloy (Ni 8157Fe19), and 2 nm Pd. A scheme of the layer system is shown\nin Fig. 1(a). The stoichiometry of permalloy is confirmed by energy-dispersive x-ray spectroscopy.\nThe sideband effect is negligible in permalloy due to the low magnetostriction and the high fre-\nquencies [15]. Because the ferromagnetic resonance of the film is only effectively excited right\nabove the 10 \u0016m wide stripline, any NRS signal from non-excited parts of the magnetic film is\nblocked by a highly x-ray absorbing bilayer of 7 nm Al and 30 nm Au on top of the magnetic film.\nThe sample is illuminated under grazing-incidence at an angle 'of 4.36 mrad, the critical angle\nfor total reflection where the nuclear signal reaches its maximum [25]. The 10 \u0002800\u0016m2part of\npermalloy film is completely illuminated by the microbeam under grazing incidence.\nExternal fields are applied in the plane of the film. The stripline is connected to a vector net-\nwork analyzer that serves as a signal source for the high frequency excitation as well as detector\nfor the transmitted signal. Figure 1(b) shows an electrical absorption spectrum of the film where\n3/s56\n/s52\n/s48/s102/s32/s40/s71/s72/s122/s41\n/s45/s56/s48/s45/s52/s48 /s48/s52/s48/s56/s48\n/s181/s48/s72/s101/s120/s116/s32/s40/s109/s84/s41\n/s40/s97/s41 /s40/s98/s41FIG. 1. (color online) (a) Schematic layer system of the sample. The angle of incidence of the photon wave\nvectork0is'. (b) Absorption spectrum of the permalloy film depending on external field and excitation\nfrequency. Black indicates high absorption. Dashed red line is the fit to the Kittel formula.\nhigh absorption (black) indicates the excitation of the ferromagnetic resonance (Kittel mode) of the\npermalloy film [26], a spin wave mode with zero wave vector that corresponds to a uniform pre-\ncession of the magnetic moments. A fit with the Kittel formula yields a saturation magnetization\nofMS=666 kA/m and a damping parameter of \u000b=0.012.\nThe precise evaluation of the NRS time spectra requires the exact knowledge of the hyperfine\nfield distribution in the sample. This can be obtained from the evaluation of NRS time spectra\nwithout rf magnetic field excitation at different in-plane angles \u001ebetween the incoming photon\nwave vector k0and an external field of 70 mT. Because of the low coercivity of the permalloy film\nof less than 1 mT [27] the magnetization mand the external field are parallel. The NRS time spec-\ntra are shown in Fig. 2 complemented with fits using the NRS evaluation package CONUSS [28].\nFrom the calculations the hyperfine field distribution of the permalloy film at about 27.6 T is de-\nduced as shown in the inset of Fig. 2. Isomer shift and quadrupole splitting are 0.134 mm/s and\n0.020 mm/s, respectively, similar to values found in previous studies on permalloy [29].\nIn the following the Kittel mode is excited at different static fields applied parallel to the in-\ncoming beam. Time spectra for a constant external field of 5 mT at different radio-frequency\nmagnetic field amplitudes hrfare shown in Fig. 3(a). The resonance frequency is 1.93 GHz as\ndetermined from the electrical absorption measurements. The overall shape of the time spectra al-\nters with increasing excitation field. A shift of the extrema to later times with increasing dynamic\nfield is visualized in Fig. 3(b). In addition a slightly faster decay of the time spectra is observed\nduring ferromagnetic resonance. The shift of the extrema indicates that the effective magnitude\nof the hyperfine field is lower. It results in a reduction of the magnetic hyperfine splitting and\na correspondingly larger period in the temporal beat pattern. The effect resembles the tempera-\nture dependence of the hyperfine field that originates from thermal excitation of spin waves [6–8].\n4/s48/s46/s50\n/s48/s46/s49\n/s48/s46/s48/s114/s101/s108/s46/s32/s119/s101/s105/s103/s104/s116 /s51/s48 /s50/s56 /s50/s54 /s50/s52\n/s104/s121/s112/s101/s114/s102/s105/s110/s101/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s108/s111/s103/s32/s40/s105/s110/s116/s101/s110/s115/s105/s116/s121/s41\n/s49/s54/s48 /s49/s52/s48 /s49/s50/s48 /s49/s48/s48 /s56/s48 /s54/s48 /s52/s48 /s50/s48\n/s116/s105/s109/s101/s32/s40/s110/s115/s41/s102/s32/s61/s32/s48/s176\n/s102/s32/s61/s32/s52/s53/s176\n/s102/s32/s61/s32/s57/s48/s176/s109/s107/s48/s102FIG. 2. (color online) NRS time spectra from the permalloy film at different in-plane angles \u001ebetween the\nphoton wave vector k0and the magnetization m. The red lines show fits calculated with the NRS evaluation\npackage CONUSS. Curves are offset for clarity. The deduced hyperfine field distribution of the permalloy\nfilm is given in the inset.\nHowever, here we excite only one coherent mode which allows to determine dynamic magnetic\nproperties under conditions of ferromagnetic resonances.\nThe hyperfine interaction energy Ehf= (mege\u0000mggg)Bhf=~!hfdetermines the time scale for\ndynamic effects on the NRS signal [30, 31]. Here mgandmeare the magnetic quantum numbers\nandggandgeare the g-factors of the ground and excited states, respectively. For hyperfine fields\nobserved in permalloy of 27.6 T the frequency !hf=2\u0019is in the order of a few ten MHz. For\nany change of the hyperfine field much faster than 1=!hfthe nucleus cannot follow the hyperfine\nfield and it experiences an effective hyperfine field resulting from temporal averaging over the\nrelatively long life time of the nuclear excited state. This situation corresponds to the fast relax-\nation regime. As a consequence a reduced effective hyperfine field is observed for a precession of\nthe magnetization around the equilibrium direction. Moreover, the high sensitivity of NRS to the\nmagnetization direction, as shown in Fig.2, yields the capability to determine the precession orbit\nof the magnetization because its dynamic in-plane and out-of-plane components influence the time\nspectra.\nFor a quantitative evaluation of the influence of the spin wave on the time spectra calculations\nwithin the stochastic relaxation model are performed [21, 22] that is implemented in CONUSS to\naccount for dynamics of the hyperfine field. The stochastic model assumes discrete hyperfine field\n5/s48/s46/s56\n/s48/s46/s52\n/s48/s46/s48/s109/s48/s104/s114/s102/s32/s40/s109/s84/s41\n/s49/s54/s48 /s49/s52/s48 /s49/s50/s48 /s49/s48/s48 /s56/s48 /s54/s48 /s52/s48 /s50/s48\n/s116/s105/s109/s101/s32/s40/s110/s115/s41/s108/s111/s103/s32/s40/s105/s110/s116/s101/s110/s115/s105/s116/s121/s41/s49/s54/s48 /s49/s52/s48 /s49/s50/s48 /s49/s48/s48 /s56/s48 /s54/s48 /s52/s48 /s50/s48/s116/s105/s109/s101/s32/s40/s110/s115/s41/s40/s97/s41\n/s40/s98/s41\n/s50/s46/s53 /s48/s46/s48/s108/s111/s103/s40/s105/s110/s116/s41/s48/s46/s57/s32/s109/s84\n/s48/s32/s109/s84FIG. 3. (color online) (a) Time spectra at various excitation field amplitudes. The lowest time spectrum\nis not excited. The excitation field \u00160hrfincreases up to 0.9 mT in steps of 0.1 mT. Curves are offset for\nclarity. Red lines are fits with CONUSS. (b) Logarithmic intensity map of the time spectra shown in (a)\nwith varying excitation field hrf.\ndirections with transition rates tnmbetween field directions nandm. We model the spin precession\nby eight points on the precession cone as shown in Fig. 4 (a). The transition matrix is chosen to\nallow transitions between neighboring points in a way that supports only one sense of rotation\nas realized in the experiment [25]. The transition rates equal the inverse period of the spin wave\nmeaning that every transition is performed once per cycle.\nThe field directions on the precession trajectory have to be modeled for the CONUSS calcu-\nlations. In the thin film the demagnetization field has to be considered that leads to an elliptical\nprecession. We deduce the precession trajectory of the magnetization from the Smit-Beljers for-\nmulation of the ferromagnetic resonance [23]. The Smit-Beljers formulation yields a set of coupled\ndifferential equations for the dynamic components \u000e'and\u000e#of the azimuthal angle 'and polar\nangle#of the magnetization (see Fig. 4) [32]\n\u0000\r\u00001MSsin(#0)\u000e_#=F'#\u000e#+F''\u000e'\n\r\u00001MSsin(#0)\u000e_'=F##\u000e#+F#'\u000e' (1)\n6/s100/s106/s100/s74 /s40/s97/s41\n/s40/s98/s41\n/s40/s100/s41/s110/s109\n/s116/s110/s109\n/s52/s49/s48/s50/s52/s49/s48/s48 /s32/s100/s106/s47/s100/s74\n/s48/s46/s49 /s49/s49/s48/s49/s48/s48\n/s181/s48/s72/s101/s120/s116/s32/s40/s109/s84/s41/s49/s46/s48\n/s48/s46/s57/s66/s40/s104/s114/s102/s41/s47/s66/s48\n/s48/s46/s56 /s48/s46/s52 /s48/s46/s48\n/s181/s48/s104/s114/s102/s32/s40/s109/s84/s41/s54/s48\n/s52/s48\n/s50/s48\n/s48/s100/s106/s32/s40/s100/s101/s103/s41\n/s54/s48\n/s52/s48\n/s50/s48\n/s48/s100/s106/s32/s40/s100/s101/s103/s41\n/s51/s48 /s50/s48 /s49/s48/s48\n/s109/s48/s32/s72/s101/s120/s116/s32/s40/s109/s84/s41/s54/s48\n/s52/s48\n/s50/s48\n/s48/s100/s106/s32/s40/s100/s101/s103/s41\n/s51 /s50 /s49\n/s102/s114/s102/s32/s40/s71/s72/s122/s41/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s32/s40/s99/s41\n/s40/s101/s41FIG. 4. (color online) (a) Scheme of the discretization of the spin wave precession cone for the stochastic re-\nlaxation model. Gray indicates the film plane and red the out-of-plane direction. \u000e'and\u000e\u0012are the dynamic\nin-plane and out-of-plane angles, respectively. The black dots represent the magnetization directions with\ntransition rates tnmused in the CONUSS calculations of the NRS time spectra. (b) Ratio of the dynamic\nangles for the permalloy film. (c)-(e) Maximum in-plane angle \u000e'of the spin precession cone in depen-\ndence of the excitation field amplitude at an external field of 5 mT with a resonance frequency of 1.93 GHz\n(c), of the excitation frequency at a field amplitude of 0.4 mT (d), and of the external field at the resonance\nfrequency with rf magnetic field amplitudes of 0.4 mT and 0.8 mT (smaller symbols)(e). Data deduced with\nCONUSS from the measurements (circles) and data from micromagnetic simulations (triangles) are shown.\nIn (c) the reduced hyperfine field is given in dependence of the rf magnetic field (rhombi). Lines are guides\nto the eye.\nwhere\ris the gyromagnetic ratio, #0is the equilibrium polar angle, and Fis the free energy\ndensity. The indices of Findicate partial derivatives at equilibrium positions. From Eq. (1) we\nderive the ratio of the dynamic angles at resonance\n\u000e'\n\u000e#=\u0000F'#+iq\nF''F##\u0000F2\n'#\nF'': (2)\nTo get the correct ratio for our evaluation the angular components have to be decoupled ( F'#=\nF#'= 0) meaning that the magnetization lies in the azimuthal plane. Then, Eq. (2) simplifies to\n7\u000e'=\u000e# =\u0000ip\nF##=F''.\nFor a thin film magnetized in the plane without crystalline anisotropy the free energy density\nisF=\u0000\u00160MSHextsin#cos'+1\n2\u00160M2\nScos2#, where'is the in-plane angle between magneti-\nzation and external field. At equilibrium the magnetization and external field are parallel and we\nget\n\f\f\f\f\u000e'\n\u000e#\f\f\f\f=r\nHext+MS\nHext: (3)\nThis ratio for the film, shown in Fig. 4(b), is inserted in the CONUSS calculations for the\nparametrization of the precession cone. For example, the experimental conditions presented\nin Fig. 3 with an external field of 5 mT yield a ratio of 13.0 (see Fig. 4(b)). The maximum dy-\nnamic in-plane angle \u000e'is the only free parameter in the fits of the time spectra under spin wave\nexcitation.\nFigure 4(c)-(e) show the in-plane opening angle \u000e'deduced from the time spectra. Fig. 4(c)\nand (d) display data at an external field of 5 mT where the resonance frequency is 1.93 GHz.\nFor increasing excitation fields the opening angle increases as well, while the gain is lower at\nhigher fields (Fig. 4(c)). With an excitation field of 0.4 mT the opening angles around resonance\nin Fig. 4(d) are obtained. The resonant behavior clearly shows that the hyperfine field does not\ndecrease due to Joule heating or eddy currents. Figure 4(e) shows the opening angle at resonance\nfor different static external fields. For increasing fields the opening angle is reduced.\nThe values of the extracted dynamic angles are compared to micromagnetic simulations. These\nsimulations have been carried out with the program package MicroMagnum [33] with a film thick-\nness of 12.9 nm using a cell size of 5 x 5 x 4.3 nm3, a saturation magnetization of 666 kA/m, a\nGilbert damping of 0.012, and periodic boundary conditions for the x and y direction. The exter-\nnal field has been slightly adjusted in the simulation to fit the experimentally observed resonance\nfrequency. The opening angles calculated from the simulations are also shown in Fig. 4(c)-(e).\nThe calculated angles assort well with the values obtained from the NRS spectra. The excellent\nagreement of the measured and simulated data for the thin film demonstrates how powerful NRS\nis to sense spin dynamics. Especially for nanoscaled samples, where theoretical modeling might\nget difficult, NRS could provide distinct advantages in determining dynamic properties.\nThe measurements indeed allow to distinguish between different assumed trajectories. The\nsimplest model is a circular precession cone. This model fits the data quite reasonably because the\naverage hyperfine field is the projection on the equilibrium direction of the magnetization. The\n8slightly faster decay of the NRS time spectra due to the dynamics is reproduced as well. However,\nthe dynamic in-plane and out-of plane components of the magnetization precession change for\nthe circular cone compared to the elliptical trajectory. This change generates differences in the\nfit quality and yields poorer fits with least squares up to 9% higher. However, we can simply\ndeduce the reduction of the effective hyperfine field from the projection of the circular cone on the\nhyperfine field axis. The in this way calculated reduced hyperfine field Bhf(hrf)=Bhf(0)is shown\nin Fig. 4(c). At an excitation field of 0.9 mT a reduction of 10% is obtained. For small opening\nangles the hyperfine field reduction becomes less significant. Because the effective hyperfine field\nscales with the average of the cosine of the dynamic angles the change of the hyperfine field is\ntiny. For the smallest deduced in-plane angle of 6\u000e, the out-of-plane angle is 0.46\u000eand we get\nhyperfine field reductions of 0.55 % and 0.003 %, respectively. However, these small reductions\nare sufficient to induce changes of the NRS time spectra compared to the static case.\nIn summary we have shown that the magnetic hyperfine field is significantly reduced at ferro-\nmagnetic resonance. From this reduction one can deduce the cone angle of the spin precession in\nthe thin fim. The measured hyperfine field reduction should be present for all kinds of spin dynam-\nics not only at ferromagnetic resonance. The technique is also applicable to non-zero wave vectors\nlike propagating or confined spin waves due to the nature of the scattering process. The overall\ngood agreement of the micromagnetic simulations and the evaluated data demonstrate the feasibil-\nity to study spin dynamics with high accuracy via NRS. The method’s isotopic-sensitivity can be\nemployed to study depth profiles and interface effects [5] related to spin dynamics. Combination\nof grazing-incidence diffraction and time-resolved NRS will enable three-dimensional mapping of\nspin waves confined in nanostructures.\nWe thank U. Merkt and T. Matsuyama for persistent encouragement and fruitful discussions,\nM. V olkmann and A. Berg for excellent technical support, A. Aquila for software code for KB\nmirror calculations, A. Rothkirch for help during the data processing, C. Adolff for support during\nlithography, J. Major for production of a sputter target and D. Schumacher as well as T. Guryeva\nfor sputter deposition. Financial support of the Deutsche Forschungsgemeinschaft via excellence\ncluster ‘The Hamburg Centre for Ultrafast Imaging - Structure, Dynamics and Control of Matter on\nthe Atomic Scale’, via Sonderforschungsbereich 668, and via Graduiertenkolleg 1286 is gratefully\nacknowledged.\n9\u0003lars.bocklage@desy.de\n[1] V . V . Kruglyak, S. O. Demokritov, and D. Grundler, Journal of Physics D: Applied Physics 43, 264001\n(2010).\n[2] E. 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V onsovskii (Pergamon, 1966)\npp. 12 – 77.\n[33] MicroMagnum Fast Physical Simulator for Computations on CPU and Graphics Processing Unit\nhttp://micromagnum.informatik.uni-hamburg.de/.\n11"    },    {        "title": "1808.01347v1.Ferromagnetic_resonance_in_thin_films___cross_validation_analysis_of_numerical_solutions_of_Smit_Beljers_equation__Application_to_GaMnAs.pdf",        "content": "Ferromagnetic resonance in thin \flms { cross-validation analysis of numerical\nsolutions of Smit-Beljers equation. Application to GaMnAs\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: August 7, 2018)\nThe new method of numerical analysis of experimental ferromagnetic resonance (FMR) spectra in\nthin \flms is developed and applied to (Ga,Mn)As thin \flms. Speci\fcally, it starts with the \fnding\nof numerical solutions of Smit-Beljers (SB) equation and continues with their subsequent statistical\nanalysis within the cross-validation (CV) approach taken from machine learning techniques. As a\nresult of this treatment we are able to reinterpret the available FMR experimental results in diluted\nferromagnetic semiconductor (Ga,Mn)As thin \flms with resulting determination of magnetocrys-\ntalline anisotropy constants. The outcome of CV analysis points that it is necessary to take into\naccount terms describing the bulk cubic anisotropy up to the fourth order to reproduce FMR exper-\nimental results for (Ga,Mn)As correctly. This \fnding contradicts the wide-spread conviction in the\nliterature that only \frst order cubic anisotropy term is important in this material. We also provide\nnumerical values of these higher order cubic anisotropy constants for (Ga,Mn)As thin \flms resulting\nfrom SB-CV approach.\nI. INTRODUCTION: THE EXPERIMENTAL\nDATA\nGallium manganese arsenide, (Ga,Mn)As, is probably\none of the most thoroughly studied diluted ferromagnetic\nsemiconductors. Simultaneous presence of magnetism\nand conductivity in this material makes it possible to\ncontrol of both the charge and the spin degrees of free-\ndom of the charge carriers. This creates potential spin-\ntronic applications. Another reason for the intense re-\nsearch on (Ga,Mn)As are its remarkable magnetic prop-\nerties between which magnetic anisotropy plays an im-\nportant role. It determines, among others, the orienta-\ntion of magnetization in the absence of an applied mag-\nnetic \feld1. Although its understanding is important for\nprospective applications such as e.g., memory devices, its\norigins are far from being fully explained. Magnetocrys-\nralline anisotropy in (Ga,Mn)As, usually described by\nthe single-domain model2,3, is being investigated by var-\nious experimental techniques, such as ferromagnetic reso-\nnance (FMR) and spin-wave resonance (SWR)4. Most of\nthese methods have been used to obtain information on\nanisotropy bulk properties of this material. Recently we\nhave proposed5that one can use the SWR to get infor-\nmation on such magnetic properties as surface anisotropy\nand surface pinning energy of (Ga,Mn)As thin \flms and\ntheir dependence on the orientation of magnetization in\nthe material.\nThe ferromagnetic resonance spectroscopy has long\nbeen a good tool for examining magnetocrysralline\nanisotropy, see e.g., recent review on FMR in (Ga,Mn)As\nthin \flms4. In FMR experiment, since the equilibriumposition of the total magnetic moment Mof the sample\ndoes not coincide with the direction of magnetic \feld H\ndue to the presence of magnetocrysralline anisotropy, M\nprecesses around its equilibrium position with a speci\fc\n(microwave) frequency !. By changing the magnetic \feld\nHone hits a resonance \feld Hr: the precession frequency\nofMis equal to the frequency of the spectrometer. The\nvalue of the resonance \feld Hrstrongly depends on its\norientation with respect to the examined sample, which\nis determined by angles \u0012Hand\u001eH, see Fig. 1, due to\nmagnetocrysralline anisotropy.\nFIG. 1: The coordinate system in which an orientation\nof the applied magnetic \feld His described with\nrespect to the sample in the FMR experiment. The \feld\ndirection is characterized by angles #Hand'H\nmeasured relative to the sample [001] and [100] axes.\nThe equilibrium direction of the sample\nmagnetization Mis represented by angles #and'.\nThis article presents the results of the analysis of bulk\nmagnetocrystalline anisotropy in (Ga,Mn)As based on\nexamination of the uniform mode in SWR resonance inarXiv:1808.01347v1  [cond-mat.mtrl-sci]  3 Aug 20182\nFIG. 2: Resonance \feld6Hrof the uniform SWR mode\nas a function of the magnetic \feld orientation for the\nout-of-plane con\fguration (plane Hr-#H) and for the\nin-plane con\fguration (plane Hr-'H).\n(Ga,Mn)As thin \flm6. The motivation to carry out this\nanalysis was two-fold: First { on the basis of examina-\ntion of surface mode in the same SWR experiment6we\nhave schown5that magnetocrystalline surface anisotropy\nin (Ga,Mn)As thin \flms contains cubic terms up to third\norder, which is not commonly found among ferromagnets.\nWe wonder if this is also true for bulk magnetocrystalline\nanisotropy. Second { it was originally shown6that in or-\nder to reproduce the experimental dependence of the res-\nonance \feld on the orientation of the magnetic \feld with\nrespect to the sample, only the \frst order term of cubic\nanisotropy should be taken into account, which, to some\nextent, is contradictory to the analysis carried out for the\nsurface5. In the meantime, numerical tools have been de-\nveloped that allow a thorough analysis of this problem.\nThat is why we have considered the old problem again.\nAt the begining let us recall the angular dependencies\nof resonance \feld for the uniform mode in ferromagnetic\n(Ga,Mn)As thin \flm6. They are shown in Ref. [6] in\nFig. 5 for the out-of-plane geometry and in Fig. 6 for the\nin-plane geometry, respectively. We show them again in\nFig. 2, to clearly emphasize the di\u000berence between reso-\nnance \feld resulting from the uniaxial anisotropy (plane\nHr-#H) and that resulting from the cubic anisotropy\n(planeHr-'H). We focus on the interpretation of this\nexperiment because the authors very carefully identi\fed\nresonance from uniform SWR modes and distinguished\nit from that for surface modes.\nII. PHENOMENOLOGICAL FREE ENERGY\nThe starting point for the interpretation of experimen-\ntal data of ferromagnetic resonance in (Ga,Mn)As is the\nphenomenological formula for the free energy of the in-\nvestigated sample. We assume that there exists a single\nhomogeneous magnetic domain within the sample and\nthat the free energy of unit volume of the sample con-\nsists of Zeeman term FZ, demagnetization term FD, and\nmagnetocrystalline anisotropy terms (cubic FCand uni-axiallFU):\nF=FZ+FD+FC+FU: (1)\nExpressing free energy in terms of \fctitious \felds one\nobtains\nf(#H;'H;#;' ) =F\nM=HZ+HD+HC+HU;(2)\nMstands here for the value of homogeneous magnetiza-\ntion. Dependence of \fctitious anisotropy \felds in Eq. (2)\non the direction in space is expressed by unit vectors\nalong the applied magnetic \feld Hand along the mag-\nnetization of the sample M. Their spatial orientation is\ndetermined by angles #H,'Hand#,'with respect to\nthe [001] and [100] axes, see Fig. 1. The unit vectors are\ngiven by\nH\nH= [nH\nx;nH\ny;nH\nz] = [cos'Hsin#H;sin'Hsin#H;cos#H];\n(3a)\nM\nM= [nx;ny;nz] = [cos'sin#;sin'sin#;cos#]:\n(3b)\nZeeman \feld is given by\nHZ(#H;'H;#;' ) =\u0000H(nxnH\nx+nynH\ny+nznH\nz):(4)\nDemagnetization \feld of a thin \flm may be approxi-\nmated by the expression describing demagnetization \feld\nof an in\fnite plane\nHD(#) = 2\u0019Mn2\nz: (5)\nThe \feldHC(#;') should be invariant under the cubic\nsymmetry transformations. It follows that it is possible\nto expand it into basis functions with the same symme-\ntry. Typically this expansion is limited to some low-order\nterms of systematically decreasing basis functions. The\nway of constructing such basis functions is presented, e.g.,\nin Ref. [7]: they are chosen from all terms of the expan-\nsion of the identity ( n2\nx+n2\ny+n2\nz)n= 1 (n= 2;3:::) which\nare invariant under permutation of the indices x,y, and\nz. Expansion up to n= 7 is used in what follows:\nHC(#;') =Hc1(n2\nxn2\ny+n2\nyn2\nz+n2\nzn2\nx)+\nHc2(n2\nxn2\nyn2\nz)+\nHc3(n4\nxn4\ny+n4\nyn4\nz+n4\nzn4\nx)+\nHc4(n4\nxn4\nyn2\nz+n4\nxn2\nyn4\nz+n2\nxn4\nyn4\nz)+\nHc5(n4\nxn4\nyn4\nz)+\nHc6(n6\nxn6\nyn2\nz+n6\nxn2\nyn6\nz+n2\nxn6\nyn6\nz):(6)\nAll terms included in the expansion of cubic \feld\nHC(#;') are shown in Fig. 3 (a)-(f). Let us emphasize\nthat every next term is smaller than the previous one -\nthe expansion (6) is convergent. Let us note, however,\nthat the set of six basis functions used in expansion (6)3\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 3: The basis functions up to 6thorder (a-f) used\nin the expansion (6) of the cubic magnetocrystalline\n\feld. Although each next function of a higher order is,\nfor given#;', smaller than the preceding one, they are\nshown here as being comparable in size.\n(a)\n (b)\n (c)\n (d)\nFIG. 4: Terms representing contributions to uniaxial\nmagnetocrystalline anisotropy of (Ga,Mn)As thin \flm\nin spherical coordinate system. Uniaxial anisotropy\nalongzaxis 1stto 3rdorder:n2\nz- (a),n4\nz- (b),n6\nz- (c).\nUniaxial anisotropy along [110] axis ( ny\u0000nx)2- (d).\nis neither orthogonal nor complete. Consequently, the\nexpansion may not be unique8. Nevertheless, it is widely\nused, at least in the cases of analyzing systems with lower\norder magnetocrystalline anisotropies. One can also ex-\npand the cubic magnetocrystalline \feld HC(#;') into\nother basis functions, such as, e.g., spherical harmonics,\nremembering however, to use only those with the appro-\npriate symmetry9.\nIt is recognized4,7that uniaxiall anisotropy \feld\nHU(#;') in (Ga,Mn)As consist of two terms H1[001]n2\nz\nandH[110](ny\u0000nx)2representing uniaxial anisotropy\nalongzaxis and uniaxial anisotropy along [110] axis, re-\nspectively. We add, however, two additional terms of 4th\nand 6thorder:\nHU(#;') =\u00001\n2H1[001]n2\nz\u00001\n2H2[001]n4\nz\u00001\n2H3[001]n6\nz\n\u00001\n2H[110](ny\u0000nx)2:\n(7)\nNote that two terms with cos2#Hare present in expan-\nsion 2: 2\u0019M and\u00001\n2H1[001] . Usually they are grouped\ntogether and referred to as e\u000bective anisotropy \feld:\nHeff\n[001]= 2\u0019M\u00001\n2H1[001] ). Four terms entering the ex-\npansion of uniaxial \feld HU(#;') are shown in Fig. 4.The right side of Eq. (2) depends on 15 variables:\nHr;#H;'H;#;';Hc1;:::;Hc6;Heff\n[001];H2[001];H3[001];\nH[110]. The \frst three will be considered as independent\nvariables that are measured in the experiment and are\nshown in Fig. 2. The pairs of angles ( #H;'H) and\n(#;') are not independent. The angles determining\nthe equilibrium orientation of M,#and', should\nminimize free energy. For the set of \fxed parameters\nHr;#H;'H;Hc1;:::;Hc6;Heff\n[001];H2[001];H3[001];H[110],\none \fnds them from the equilibrium condition\n@f\n@#= 0;@f\n@'= 0: (8)\nThus the right-hand side of Eq. (2) re-\nally depends on 10 \fctitious anisotropy \felds\nHc1;:::;Hc6;Heff\n[001];:::;H 3[001];H[110] which we col-\nlectively denote by the vector\nh\u0011(Hc1;:::;Hc6;Heff\n[001];H2[001];H3[001];H[110]):(9)\nThe questions should then be asked: how to \fnd\nanisotropy \felds and how many of them are necessary\nto reproduce the experimental dependence Hr(#H;'H)\nwell? These questions are answerd in the next section.\nIII. WHICH ANISOTROPY FIELDS ARE\nIMPORTANT FOR (GA,MN)AS?\nThe resonance condition (in the case of uniform mag-\nnetization) is given by10,11\n\u0012!\n\r\u00132\n=1\nsin2#(f##f\u001e\u001e\u0000f2\n#\u001e); (10)\nwheref#\u001e=@f\n@#@f\n@',\r=g\u0016B\n~withgbeing the spectro-\nscopic splitting factor, \u0016Bthe Bohr magneton and ~- the\nPlanck constant. The 9.46 GHz spectrometer was used\nin the considered experiment6, thus Eq. (10) reads\n45:6834 =g2\nsin2#(f##f\u001e\u001e\u0000f2\n#\u001e);[kOe2]: (11)\nAt resonance Eq. (11) should be met for any given\ndirection of Hr(#H;'H), i.e., in the case under consider-\nation, for all points shown in Fig. 2. Let us treat gand\ncomponents of the vector has not known parameters and\ndenote the right-hand side of the Eq. (11) calculated at\ni-th experimental point by Ri,\nRi(g;h) =g2\nsin2#(fi\n##fi\n\u001e\u001e\u0000(fi\n#\u001e)2): (12)\nNote that free energy derivatives (e.g., fi\n##) are calcu-\nlated ati-th the experimental point | for given values\nHr;#H;'H.\nOne should \fnd such values of unknown coe\u000ecients\ng;hthat the Eq. (11) is met as accurately as possible for4\nTABLE I: Models of cubic magnetocrystalline\nanisotropy in (Ga,Mn)As for which cross-validation was\ncarried out. In the second column the terms are given\nincluded in the expansion of the free energy (6) for\nmodels C1 - C6. Uniaxial anisotropy for C1 - C6 models\ndoes not change | only \felds Heff\n[001]andH[110]are\npresent.\nModel Cubic anisotropy Uniaxial anisotropy\nC1 Hc1 Heff\n[001]; H[110]\nC2 Hc1; Hc2 Heff\n[001]; H[110]\nC3 Hc1\u0000Hc3 Heff\n[001]; H[110]\nC4 Hc1\u0000Hc4 Heff\n[001]; H[110]\nC5 Hc1\u0000Hc5 Heff\n[001]; H[110]\nC6 Hc1\u0000Hc6 Heff\n[001]; H[110]\neach experimental point. So the following error function,\nbeing the positive square root of the sum of squares of\nresiduals,\nEN\nRMS(g;h) =s\n1\nNX\ni\u0000\nRi(g;h)\u000045:6834\u00012;(13)\nshould be minimized in 11-dimensional parameter space\n(g;h). The sum in Eq. (13) runs over all Nexperimen-\ntal points shown in Fig. 2. This least squares approach\nto \fnding the unknown parameters represents a speci\fc\ncase of maximum likelihood approach12,13. Actual values\nof the spectroscopic splitting factor gand magnetocrys-\nralline anisotropy \felds are those for which Eq. (13) has\na minimum close to zero.\nWe have included 10 magnetocrysralline anisotropy\n\felds into the formula for free energy. Now we will check\nwhich ones are really essential to describe the experimen-\ntal results well using a simple cross-validation scheme12.\nFor this purpose anisotropy models are de\fned in Tables\nI and II. For example, in the model C3 (third row of Ta-\nble I) the cubic anisotropy is expanded up to 3rdorder,\nthe uniaxial anisotropy along zaxis up to 1storder, the\nuniaxial anisotropy along [110] axis up to 1storder and\nsimilarly for other models.\nThe cross-validation, within leave-one-out technique ,\nruns as follows: We divide the N(= 55) element set\nof experimental data into two subsets: the training one\nand the test one. The \frst one contains N\u00001 elements,\nthe second one | 1 element. One can do it in Npos-\nsible ways. Subsequently the Nsubsets obtained in this\nway are used to train, i.e., to determine the values of\nthe unknown parameters ( g;h) by minimizing the error\nfunctionEN\u00001\nRMS(g;h), de\fned in Eq. (13) for each model\nunder consideration. Simultaneously the error function\nE1\nRMS(g;h) is calculated for one left test point for eachTABLE II: Models of uniaxial magnetocrystalline\nanisotropy in (Ga,Mn)As for which cross-validation was\ncarried out. In the second column the terms are given\nincluded in the expansion of the free energy (4) for\nmodels U1 - U3. Cubic anisotropy for U1 - U3 models\ndoes not change | only \felds Hc1\u0000Hc4are present.\nModel Uniaxial anisotropy Cubic anisotropy\nU1 Heff\n[001]; H[110] Hc1\u0000Hc4\nU2 Heff\n[001]; H2[001]; H[110] Hc1\u0000Hc4\nU3 Heff\n[001]; H2[001]; H3[001]H[110] Hc1\u0000Hc4\nFIG. 5: The values of error functions\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\nde\fned in Eq. (12) for all models de\fned in\nTables I and II. The values error functions were\ncalculated for the corresponding \feld values taken from\nthe Tables III and IV.\nmodel. Note that its value informs us how well we are\ndoing in predicting the values of anisotropy \felds for\na particular model. After Nminimizations one exam-\nines how averages\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\ndepend on the\nmodel, i.e., on the order od expansion (6) or (7). We use\nthe following criterion to assess the quality of the model:\nthe model describes magnetocrystalline anisotropy well if\ntaking into account higher order terms in the expansions\n(6) and (7) does not improve its predictive ability given\nby the average\nE1\nRMS\u000b\n.\nThe procedure described above allowed to \fnd the av-\nerage values of anisotropy \felds, g-factor and error func-\ntions\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\nfor each model after N mini-\nmizations for the sample in the experiment under consid-\neration. They are collected in Tables III and IV. The pre-\ndictive ability, measured by the error function\nE1\nRMS\u000b\n,\nfor all models de\fned in Tables I and II is shown in Fig.\n5.\nE1\nRMS\u000b\ndecreases for C1 - C4 models and remains5\nTABLE III: Anisotropy \felds [Oe] in bulk (Ga,Mn)As related to cubic and uniaxial symmetry and values of g-factor\ncalculated for models C1-C6 according to the procedure described in the text. In the last two columns error\nfunctions\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\nare shown.\nModel Hc1 Hc2 Hc3 Hc4 Hc5 Hc6 Heff\n[001]H[110] g\nEN\u00001\nRMS\u000b \nE1\nRMS\u000b\nC1 91.81 4765 65.53 1.978 0.95 0.69\nC2 92.21 -87.94 4774 70.72 1.979 0.88 0.68\nC3 77.06 -4.241 57.00 4776 61.70 1.982 0.75 0.61\nC4 78.07 -534.1 43.93 1405 4811 66.29 1.985 0.59 0.50\nC5 79.57 -583.4 41.68 1790 2080 4814 64.50 1.984 0.58 0.48\nC6 78.78 -419.0 41.82 436.1 2841 2700 4809 63.81 1.984 0.57 0.51\nLiu et. al6197a4588 77 1.98\naNote that de\fnition of cubic anisotropy in Ref. [6] is slightly di\u000berent. It takes into account tetragonal distortion in (GaMn)As thin\n\flms. The value of cubic anisotropy \feld calculated from this de\fnition should be approximately equal to 2 Hc1.\nTABLE IV: Anisotropy \felds [Oe] in bulk (Ga,Mn)As related to cubic and uniaxial symmetry and values of g-factor\ncalculated for models U1-U3 according to the procedure described in the text. In the last two columns error\nfunctions\nEN\u00001\nRMS\u000b\nand\nE1\nRMS\u000b\nare shown.\nModel Heff\n[001]H2[001] H3[001] H[110] Hc1 Hc2 Hc3 Hc4 g\nEN\u00001\nRMS\u000b \nE1\nRMS\u000b\nU1 4811 66.28 78.07 -534.1 43.93 1405 1.985 0.59 0.50\nU2 4779 21.83 66.08 80.12 -449.3 40.76 1222 1.987 0.59 0.51\nU3 4778 21.99 6310 66.13 80.15 -449.3 40.59 1221 1.987 0.59 0.50\nroughly constant for the C4 - C6 models. It means that\nhigher order terms used in free energy expansion (2) for\nC5 and C6 models do not improve their ability to pre-\ndict cubic magnetocrystalline anisotropy on the basis of\nexperimental data from Fig. 2. Similarly we see that tak-\ning into account higher order terms in uniaxial anisotropy\nexpansion for models U2 and U3 does not improve their\npredictive ability. It follows that the correct description\nof magnetocrystalline anisotropy in (GaMn)As requires\nthat the free energy expansion should include terms de-\nscribing cubic anisotropy up to fourth order and that is\nenough to take into account uniaxial anisotropies up to\n\frst order.\nOne can also see the result of minimization in Fig. 6\nin the form of a collapse: for the real minimum of error\nfunction at g\u0003;h\u0003its valuesRi(g\u0003;h\u0003) for all experimen-\ntal points fall onto a line 45.6834. By comparing the\nscattering of points for C1 model (a) and C4 model (b)\nwe note the important thing: the addition of the higher\norders of cubic anisotropy \felds improves \ftting not only\nforin-plane experimental points but also for out-of-plane\nexperimental points. This is due to the occurrence of par-\ntial mixed derivatives of the free energy in the determi-\nnant from Hessian (representing a local curvature of free\nenergy in#H;'Hspace) in Smit-Beljers equation (10)\nFIG. 6: The values of function Ri(g;h) de\fned in\nEq. (12) for model C1 - (a) and for model C4 - (b) for\nall experimental points from Fig. 2. Squares { in-plane\ngeometry:#H= 90\u000e,'Hchanges; circles { out-of-plane\ngeometry:#Hchanges,'H=\u000045\u000e. The values of\nRi(g;h) were calculated for the corresponding \feld\nvalues taken from the Table III.\nwhich we solve numerically14for all experimntal points\nsimultaneously treating them on equal footing. There-\nfore, to get as accurate as possible anisotropy \feld values\nit is important to measure resonance \felds in di\u000berent\ngeometries.6\nFul\flling condition (8) while solving Eq. (13) leads to\n\fnding dependences #(#H;'H) and'(#H;'H). for all\nmodels from Tables III and IV. They are shown for C4\nmodel in Figs. 7 and 8. Function #(#H;'H) for a given\n'Hit always is a concave function. For angle \u001eH=\n45\u000eand 135\u000ewe see a ripple, which is the result of the\npresence of cubic symmetry. Note also, that function\n'(#H;'H) for a given #His for all#Ha linear function\n'/'H(does not depend on #H).\nFIG. 7: Equilibrium magnetization angle #versus the\nexternal \feld angles #H;'Hdetermined from the\ncondition (8) for the (Ga,Mn)As thin \flm studied in\nRef. 6.\nFIG. 8: Equilibrium magnetization angle 'versus the\nexternal \feld angles #H;'Hdetermined from the\ncondition (8) for the (Ga,Mn)As thin \flm studied in\nRef. 6.\nLet us summarize this section by stating that for the\ncorrect description of magnetocrystalline anisotropy in\n(GaMn)As, the free energy expansion should include\nterms describing cubic anisotropy up to fourth order and\nthat is enough to take into account uniaxial anisotropies\nup to \frst order.IV. BACK TO THE EXPERIMENT: WHAT IS\nTHE EFFECT OF INCORPORATING OF\nANISOTROPY FIELDS OF HIGHER ORDERS?\nLet us now examine the dependence of the resonance\n\feldHron#Hand'H. The problem can be stated in\nthe following way: Given the values of Hron the bound-\nary of the box presented in Fig. 2, determine the reso-\nnance \feld inside the box. One \fnds a solution in two\nstages. First, one determines the anisotropy \felds, for\nwhich Eq. (11) is satis\fed on the boundary of the box.\nThis stage has been described in the Section III. Second,\nto get the resonance \feld for each #Hand each'Hone\nshould solve Eq. 11 numerically for the anisotropy \felds\ndetermined in the \frst stage (collected in Table III) with\ncondition (8) met for each tentative point obtained dur-\ning the numerical solving procedure. In Figs. 9 and 10\nFIG. 9: Comparison of the Hr-values calculated\nnumerically from Eq. 10 with experimental data for\nout-of-plane geometry for C1-C4 models. The resonance\n\feld for in-plane geometry measurement is visible as\na small rectangle on the vertical axis for #H= 90\u000e.\none can see dependencies Hr(#H;\u000045\u000e), i.e, for out-of-\nplane geometry and Hr(90\u000e;'H) { for in-plane geome-\ntry, respectively. Uniaxial anisotropy is the most visible\nfor out-of-plane geometry (Fig. 9) and although the use\nof higher order cubic terms does improve the agreement\nof the calculated Hr-values with the experimental data,\nthis improvement is not particularly visible in the scale\nof Fig. 9 because the cubic anisotropy is much smaller\nthan the uniaxial one. The improvement, however, can\nbe seen for in-plane geometry: the use of higher order\nterms in of cubic anisotropy expansion given by Eq. (6)\nbecomes necessary to describe dependence Hr(90\u000e;'H)\nmore precisely.\nSpatial dependence of the resonance \feld on angles\n#Hand'His shown in Fig. 11. For small angle #H\nwe see resonance \feld whose source is mainly uniaxial\n[001] anisotropy (with two-fold symmetry), whereas for\nangle#H\u001990\u000ethe resonance \feld with four fold sym-7\nFIG. 10: Comparison of the Hr-values calculated\nnumerically from Eq. 10 with experimental data for\nin-plane geometry for C1-C4 models. Taking into\naccount higher order terms of cubic anisotropy one\nimproves the agreement of calculated Hr-values with\nexperimental data.\nFIG. 11: Spatial angular dependence of resonance \feld\nHr(#H;'H) resulting from our theory. Experimental\ndata (squares) in the plane ( Hr;#H) correspond to\nout-of-plane geometry while those in the ( Hr;'H) plane\n{ to in-plane geometry. The surface in the \fgure is the\nnumerical solution of Eq. (11) with condition (8).\nmetry becomes more noticeable. It is the result of cubic\nanisotropy, although Hr(90\u000e;45\u000e)< Hr(90\u000e;135\u000e) due\nto small uniaxial [110] anisotropy, see also Fig. 10.\nThe spatial dependence of the cubic anisotropy \feld is\nshown in the Fig. 12. We see that really the magnetic\n\feld determines hard/easy directions, not hard/easy\naxes: The largest cubic anisotropy \feld is for #H= 23:7\u000e\nand'H= 45\u000ewhereas the position of hard axis direction\nis given by #H= 54:7\u000eand'H= 45\u000e.\nThe values of \fctitious anisotropy \felds are impor-\ntant in that they allow to reproduce the spatial de-\npendence of the energy of the sample in a magnetic\nFIG. 12: Cubic anisotropy \feld for C4 model versus #H\nand'H.\n\feld due to magnetocrystalline anisotropy and to \fnd\neasy and hard axes. To \fnd this spatial dependence\none needs to know the saturation magnetization. Then\nthe magnetic anisotropy constants can be easily ex-\npressed by corresponding anisotropy \felds, see e.g., very\nclearly written Ref. [15]. We have found the sat-\nuration magnetization16of the considered sample: it\namountsMs= 30:5 emu/cm3, which is a typical value\nfor (Ga,Mn)As containing a few percent of Mn atoms.\nReturning to the Eq. (6) and multiplying it by Msone\nobtains the spatial distribution of energy FC(#;') stored\nin bulk (Ga,Mn)As and related to its cubic magnetocrys-\ntalline anisotropy for the C4 model\nFC(#;') =MsHc1(n2\nxn2\ny+n2\nyn2\nz+n2\nzn2\nx)+\nMsHc2(n2\nxn2\nyn2\nz)+\nMsHc3(n4\nxn4\ny+n4\nyn4\nz+n4\nzn4\nx)+\nMsHc4(n4\nxn4\nyn2\nz+n4\nxn2\nyn4\nz+n2\nxn4\nyn4\nz)\u0011\nKc1(n2\nxn2\ny+n2\nyn2\nz+n2\nzn2\nx)+\nKc2(n2\nxn2\nyn2\nz)+\nKc3(n4\nxn4\ny+n4\nyn4\nz+n4\nzn4\nx)+\nKc4(n4\nxn4\nyn2\nz+n4\nxn2\nyn4\nz+n2\nxn4\nyn4\nz);(14)\nand similarly for the C1 model. Kc1-Kc4stand in\nEq. (14) for cubic anisotropy constants. Taking the nu-\nmerical values of anisotropy \felds Hc1-Hc4from Ta-\nble III one obtains the numerical values of anisitropy\nconstants for C1 and C4 models { they are collected\nin Table V. Let us note that the values of \frst order\ncubic anisitropy for (Ga,Mn)As are several dozen to sev-\neral hundred times smaller than the corresponding values\nfor such ferromagnets as Ni or Fe. Perhaps this is why\nanisotropies of higher orders become visible in resonance\nexperiments only for weak ferromagnets.\nTo assess the accuracy of the present method we used\nthe bootstrap method to evaluate errors for C1 and C4\nmodels. To do this we assumed that the error probabil-\nity distribution of experimental results was normal, and8\n(a)\n (b)\nFIG. 13: Spatial dependence of cubic\nmagnetocrystalline energy in spherical coordinate\nsystems for C1 (a) and C4 (b) models.\nFIG. 14: Spatial dependence of the di\u000berence of cubic\nmagnetocrystalline energy between C1 and C4 models\nin spherical coordinate systems.\nconsequently the errors of solution of Smit-Beljers equa-\ntion had also a normal distribution. Then we could de-\ntermine the approximated errors of obtained anisotropy\nconstants. They are listed in the Table V.\nFinally, let us show, how taking into account the higher\norders of anisotropy \felds changes the cubic anisotropy\nenergy surface. It might seem that correction will be\nof little importance. However, this is not the case: the\nSmit-Beljers equation (10) describing the curvature of\nenergy surface is nonlinear with respect to the second\nderivatives. This leads to signi\fcant corrections in en-\nergy values. Figure 13 shows the spatial dependence of\nenergy from cubic anisotropy on the same scale for mod-\nels C1 and C4. Axes [100], [010] and [001] are easy axes\nwith respect to cubic anisotropy and axis [111] is a hard\none. For example, for hard axis we have FC1([111]) =933\u00066,FC4([111]) = 769\u00064 [erg/cm3]. The C1 model\nthus overestimates the anisotropy energy along hard axis\nby about 20%. Figure 14 shows the spatial dependence\nof energy di\u000berence between C1 and C4 models.\nV. SUMMARY AND OUTLOOKS\nThe article presents how to determine bulk magne-\ntocrysralline anisotopy in (Ga,Mn)As thin \flm by nu-\nmerical solution of the Smit-Beljers equation for all data\ncollected in one FMR experiment, i.e., for di\u000berent spa-\ntial orientations of the magnetic \feld with respect to the\nsample, on equal footing. To avoid essential drawbacks\nof \ftting procedures (lack of information which \ftted\nconstants are relevant and possibility of over\fting) by\n\fnding anisotropy constants we cross-validated the nu-\nmerical solutions of Smit-Beljers equation for six models\n(C1-C6). The results of this cross-validation, i.e., the val-\nues of the function\nE1\nRMS\u000b\ndisplaying predictive ability\nfor models C1-C6 point that it is necessary to expand\nbulk cubic anisotropy up to the fourth order to repro-\nduce spatial dependence of the resonance \feld correctly\n| that is increasing the order of expansion of anisptropy\ndoes not change predictive ability of the model under\nconsideration. Such cubic anisotropy (up to fourth or-\nder) is visible in the resonant experiment. It means that\nthe models of \frst order cubic anisitropy applied so far\nto (Ga,Mn)As overstimated the value of this anisotropy.\nLet us stress that this description of the bulk anisotropy\nis consistent with the presented earlier description5of the\nsurface anisotropy (both descriptions require higher or-\nder expansion of cubic anisotropy). We also have shown\nthat FMR data allow one to \fnd the spectroscopic split-\nting factor with high accuracy. We intend to con\frm the\nusfullness of this new approach by applying it to other\navailable resonance experiments in the near future.\nAcknowledgment The authors would like to thank\nMarcin Tomczak for stimulating discussions. This study\nis a part of a project \fnanced by Narodowe Centrum\nNauki (National Science Centre of Poland), grant no.\nDEC-2013/08/M/ST3/00967. Numerical calculations\nwere performed at Pozna\u0013 n Supercomputing and Net-\nworking Center under Grant no. 284.\n\u0003Corresponding author: ptomczak@amu.edu.pl\n1U. Welp, V. K. Vlasko-Vlasov, X. Liu, J. K. Furdyna, and\nT. Wojtowicz, Phys. Rev. Lett. 90, 167206 (2003).\n2X. Liu, Y. Sasaki, and J. K. Furdyna, Phys. Rev. B 67,\n205204 (2003).\n3G. Nieuwenhuys, T. Prokscha, A. Suter, E. Morenzoni,\nD. Chiba, Y. Nishitani, T. Tanikawa, F. Matsukura,\nH. Ohno, J. Ohe, S. Maekawa, and Y. J. Uemura, Na-\nture Materials 9, 299 (2010).\n4X. Liu and J. K. Furdyna, Journal of Physics: Condensed\nMatter 18, R245 (2006).5H. Puszkarski and P. Tomczak, Surface Science Reports\n72, 351 (2017).\n6X. Liu, Y. Y. Zhou, and J. K. Furdyna, Phys. Rev. B 75,\n195220 (2007).\n7J. Zemen, J. Ku\u0014 cera, K. Olejn\u0013 \u0010k, and T. Jungwirth, Phys.\nRev. B 80, 155203 (2009).\n8E. Callen and H. Callen, Journal of Physics and Chemistry\nof Solids 16, 310 (1960).\n9H. Puszkarski, Progress in Surface Science 9, 191 (1979).\n10J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955).\n11L. Baselgia, M. Warden, F. Waldner, S. L. Hutton, J. E.9\nTABLE V: Cubic anisotropy \felds ( Hc) in Gaussian units [Oe] and SI units [kA/m] and cubic anisotropy constants\n(Kc) in [erg/cm3] and [J/m3] for bulk (Ga,Mn)As calculated for models C1 and C4.\nHc1 Hc2 Hc3 Hc4 Kc1 Kc2 Kc3 Kc4\nModel C1\nGaussian 91.81 \u0006.55 2800 \u000617\nSI 7.306 \u00060.044 280.0 \u00061.7\nModel C4\nGaussian 78.07 \u0006.40 -534 \u000628 43.9 \u00061.2 1410 \u000670 2381 \u000613 -16280 \u0006860 1330 \u000637 42900 \u0006220\nSI 6.213 \u00060.032 -42.5 \u00062.3 3.493 \u00060.096 112.2 \u00065.6 238.1 \u00061.3 -1628 \u000686 133.0 \u00063.7 4290 \u000622\nDrumheller, Y. Q. He, P. E. Wigen, and M. Mary\u0014 sko,\nPhys. Rev. B 38, 2237 (1988).\n12C. M. Bishop, Pattern Recognition and Machine Learning\n(Springer, 2006).\n13C.-H. W. A. G. D. C. R. C. K. F. D. J. S. Pankaj Mehta,Marin Bukov, arXiv:1803.08823v1.\n14We use Python packages scipy.optimize and Numdi\u000btools .\n15M. W. Gutowski, arXiv:1312.7130v1.\n16Details of this calculation will be published in a separate\npaper."    },    {        "title": "1204.2423v1.Magnetostatic_spin_waves_and_magnetic_wave_chaos_in_ferromagnetic_films__II__Numerical_simulations_of_non_linear_waves.pdf",        "content": "arXiv:1204.2423v1  [cond-mat.other]  11 Apr 2012Magnetostatic Spin Waves and Magnetic-Wave Chaos in\nFerromagnetic Films.\nII. Numerical Simulations of Non-Linear Waves\nYu.E. Kuzovlev, Yu.V. Medvedev, and N.I. Mezin∗\nA.A.Galkin Physics and Technology Institute of NASU,\nul. R.Luxemburg 72, 83114 Donetsk, Ukraine\nAbstract\nA method and some results of numeric simulations of magnetos tatic spin waves in ferromagnetic\nfilms are exponded, in comparison with the theory earlier pre sented in arXiv preprint 1204.0200.\nIn particular, roles of films finiteness (edges) and defects i n formation of linear and non-linear\nmagnetostatic wave patterns, excitation and evolution of t wo-dimensional solitons, and chaotic\nnon-linear ferromagnetic resonance are considered.\nPACS numbers: 75.30.Ds, 75.40.Mg, 76.50.+g\n∗Electronic address: kuzovlev@fti.dn.ua\n1Introduction\nThis preprint represents first, preparatory, stage of numerica l investigation of magneto-\nstatic spin waves’ chaos in ferrite films realized between 2001 and 20 03 under particular\nsupport of Multimanetic Solutions Ltd. Its main contents will be subm itted in separate\npreprint, as Part III of the manuscript whose Part I was devoted to theory of magnetostatic\nwaves and already presented by our arXiv preprint 1204.0200.\nOur primary purposes here, in Part II, and in the next Part III wer e: (i) to test numeric\nalgorithmsbasedonspatial discretization offilm’s volume; (ii)to estim ateanextent towhat\nthe theory developed for idealized infinite films is applicable to real finit e-size (and may be\ndefective) films; (iii) to examine theoretical concepts, - e.g. “quas i-local magnetic energy\ndensity” (see below), - which have no unambiguous theoretical defi nition (because of strong\nnon-locality od dipole interactions or by other reasons) but can be u seful for description\nof non-linear magnetic-wave phenomena; (iv) to visually indicate mec hanisms and forms of\nmagnetostatic-wave chaos, (v) to see what of them are most app ropriate for practical use,\nand numerically investigare possibilities of control and synchronizat ion of this chaos.\n6. NUMERICAL METHODS AND NUMERICAL SIMULATIONS OF EXTER-\nNALLY DRIVEN FILMS\nThe modern theory of nonlinear wave processes in ferromagnets a nd ferrites is not devel-\noped to an extent sufficient for fruitful applications to so complex p henomenon as magnetic\nchaos. By this reason, numerical simulations must be in anyway usef ul, since they\n(i) serve as powerful “microscope” to watch for magnetization dy namics at its natural\ntemporal scales, from the one tenth part of nanosecond up to te ns microseconds, and in this\nway allow to\n(ii) verify existing theoretical models, find prompts for improving an alytical theory, and\n(iii) obtain concrete practically acceptable estimates, conclusions a nd predictions.\nAlthoughmany aspects onnonlinear wave dynamics may be modelled in t erms of popular\nNSE (nonlinear Shr¨ odinger equation, see Sec.5), it does not appro ach for above purposes,\nbecause it contains no natural amplitude limit for waves and solitons. Indeed, in Eq.11\nitself and in its solitonic solutions (5.25) and (5.26) the amplitudes, |Ψ|andA, may be\n2arbitrary large, while physically, according to relations (5.9), they c an not exceed a level\nof order of unit. This means that at certain critical time moments of chaotic dynamics\nother higher-order non-linearities play significant role. Besides, NS E neglects odd powers of\nnon-linearity and related parametric processes.\nTherefore, numerical simulationalgorithmsshouldbebasedoncomp leteLandau-Lifshitz-\nGilbert equation (2.1). At this approach, trivial normalization of spin length to unit ensures\ntaking into account all orders of non-linearity.\n6.1. NUMERICAL ALGORITHM.\nTo solve Eq.2.1, the classical third-order Runge-Kutta algorithmwa s used, in its adaptive\nversion with time step being automatically chosen as large as possible a t fixed precision.\nNevertheless, running of Eq.2.1 appears not a fast procedure. Th e matter is that non-local\ndipole interaction should be calculated by means of Fourier transfor m, which even in its\nmost fast form takes much greater time than, say, calculation of g radients and Laplasians.\nMoreover, if the transform was located within the sample volume the n the non-local\ninteraction would be represented by a function of two spatial point s,r1andr2instead\nof only their difference r1−r2, and Fourier transform could not be applied. Therefore,\nwhen discretizing the sample volume into Nx×Ny×Nzcells (“spins”), it is necessary\nto consider at least N′\nx×N′\ny×N′\nzcells, - with N′\nx,y,z≥2Nx,y,z−1 , - in order to correctly\nrepresent thedipole interaction kernel inthe ( r1−r2)-space(fortunately, N′\nx,y,z= 2Nx,y,z−\n1 orN′\nx,y,z= 2Nx,y,zare always sufficient numbers). The award for this complication is\nthat all the finite-size and boundary effects are taken into accoun t.\nThe algorithm was tested by modeling magnetization of small ferroma gnetic slabs, with\nsizes of order of tens r0(exchange interaction radius). It was possible to observe (i) form a-\ntion of magnetic vortices and domains at sufficiently weak field, (ii) the ir death at moderate\nfields, withresidual demagnetizationatformerdomainboundaries, (iii)almostuniformmag-\nnetization at strong fields with significant demagnetization at sample boundaries only, (iv)\nhysteretic effects, characteristic hysteresis curves on H−B-plane, re-magnetization chaos\nand noise at periodically varying field (chaotic hysteresis).\n6.2. CONCRETIZATION OF NUMERICAL TASKS.\nWe will be most interested in comparatively large-scale phenomena in Y IG films whose\ntypical size is about 7 mm ×2 mm ×10 micron. Besides, we want to apply such\nkind of film magnetization which ensures maximum MW frequencies at min imum value of\n3magnetizing external field H0. This requirement is satisfied if (i) film is tangentially\nmagnetized and (ii) the surface magnetostatic waves (MSW) are ex plored which propagate\nperpendicularly to static magnetization (see Secs.3-4). Practically suitable wave length, λ\n, of these MSW lies in the interval 0 .1÷1 mm , that is λ/D/greaterorsimilar10 (Dis film thickness),\nwhile suitable cross sizes of MSW inductors (antennas) are compara ble with D.\n6.3. ASSUMPTIONS AND SIMPLIFICATIONS.\nIfwe tookthe r0(exchange radius ∼5·10−6cm) bethescale ofspatial discretization,\nthen (at above mentioned film sizes) Nx×Ny×Nzwould be ∼1012. Hence, literal\nsimulation of real samples is impossible.\nThe natural possibility to simplify numerical problems arises from the fact that surface\nMSW with λ/D/greaterorsimilar10 are almost uniform across film’s thickness (i.e. in z-direction).\nTherefore, we can use the averaging over thickness. Then the lat ter becomes basic spatial\nscale, and we come to two-dimensional discretization lattice, with Nz= 1 and Nx×Ny\ndetermined by the film length to thickness and wide to thickness ratio s.\nIn this approach, we inevitably neglect bulk MSW modes with N >0 which are\nessentially non-uniform in respect to thickness (see Part I, Sec.4) . These non-uniform bulk\nwaves are not directly excitable by thick antennae, but they can be generated from surface\nMSW by means of nonlinear fourth-order G-P-processes or third- order P-processes (Sec.5)\nand thus influence their dynamics. Nevertheless, our formally cruc ial simplification has\nphysical grounds as follow.\nIn fact, there are similar fourth-order interactions between sur face MSW themselves. We\ncan suppose that just these interactions are dominating in surfac e MSW dynamics because\nrealize in resonant way, while interaction with most of non-uniform bu lk modes is far from\nresonances. Indeed, their frequencies lie below uniform precessio n frequency, ωu, while\nfrequencies of surface MSW are higher than ωu. If comparing the dispersion law (3.38)\nfor DE-waves and Eqs.4.28 and 4.29 for dispersion of bulk modes, one can verify that their\nfrequencies are close at\nqN±≈πN\nD∼2π\nr0/radicalBigg\n2πD\n(H0+2π)λ∼1.5·105cm−1, N∼40, (1)\nonly. In reality, so short-wave modes must be damped certainly str onger than long surface\nwaves. At the same time, the latter effectively influence one upon an other around (in k-\n4space) equi-frequency curves (shown in Fig.4a, see Sec.4). By the se reasons, we expect that\nfourth-order interaction with non-uniform modes is relatively insign ificant. The same can\nbe argued in respect to third-order parametric by surface waves (all the more, at ωu/2<\nω1, i.e. at H0/greaterorsimilar4π/3 , this process is forbidden at all).\nAt present this is (i.e. was ten years ago) unavoidable simplification, s ince atNz/greaterorsimilar2N\n, that is at Nz/greaterorsimilar80 , again numerical simulation would be unrealistic even with Pentium-\nIV in our order. Rigorous analysis of role of non-uniform (multi-layer ed) MW in nonlinear\nlong-wave dynamics is the interesting task for future. Now, we yet are forced to deal with\nnumerical model rather than with literal numerical simulation.\n6.4. NUMERICAL STRATEGY.\nAfter the simplification, film thickness takes the role of length unit. M oreover, the size of\ndiscretization lattice, in units of D, can be chosen n×n×1 , which allows to simulate\nthe film area nNxD×nNyD. The permissible additional roughening, n, depends on\ncharacteristic minimum wave length in a situation under analysis. To sp eed up calculations,\nwe may choose a greater nbut then lower it if necessary.\nIn fact, the values 1 ≤n≤8 , 20 ≤Ny=≤170 , and 70 ≤Nx≤256 were\nused [ten years ago, while now it is possible to take greater Nx, Nyand simultaneously\nNz>1 , i.e. fractional n]. Of course, the dipole interaction of discrete cells (effective\nspins) was calculated with taking into account their shape as determ ined by n. A choice\nof definite “magic” NxandNynumbers ensured most fast FFT on a N′\nx×N′\nylattice\n(withN′\nx,y≥2Nx,y−1). Nevertheless, typically from 1 to 5 real time seconds were elaps ed\nper one period of spin precession, ∼0.3 ns, since several FFT’s and inverse FFT’s should\nbe performed at each time step.\nAt given uniform magnetizing field H0, firstly static magnetization pattern was cal-\nculated and conserved in memory, then serving as initial ground sta te for wave and soliton\nstructures caused by time-varying currents.\nThe anisotropy is what can be simply taken into account with no numer ical problems.\nBut it evolves several parameters at once. At present stage, we want to obtain numerical\n“reference point data” with minimum amount of free parameters an d therefore intentionally\nomit anisotropy.\n6.5. NUMERICAL FRICTION.\nItiswell knownthatthetimediscretizationwhennumericallysolvingdiff erential dynamic\n5equations inevitably results in more or less effective friction (or may b e negative one). In our\ncase this artifact also takes place leading to energy relaxation even if the friction coefficient\nis put on be zero, γ= 0 . Interestingly, this excess numerical relaxation excellently obe ys\nexponential law and therefore works as increase of γ,γ→γeff=γ+γnum.\nThe value of γnumdepends on mean time step. Typically, the latter was between one\nthirtieth and one twentieth part of the precession period resulting inγnum≈0.0004 . This\nfriction is just suitable to simulate good but not best samples. Howev er, it could be made\nlower than 0 .0001 if decrease mean time step to about one fortieth part of the p eriod.\nBelow, the designation γwill stand for γeff.\n6.6. MSW EXCITATION BY WIRES AND LOOPS.\nFirst of all, excitation of weak magnetic waves in small-area film by wire a nd loop in-\nductors was numerically watched for. At present, we confined our selves by inductors with\nround cross-section and radius greater than D, oriented along y-axis in parallel to mag-\nnetizing field. Corresponding current induced field, h(r,t) ={hx(x,y,t),0, hz(x,y,t)}\n, was calculated from usual magnetostatics formulas. For relation s between physical and\ndimensionless time and frequency units, see Sec.3.\nExamples of such the simulations are illustrated by Figs.6a-c. Two rat her obvious con-\nclusions do follow from these pictures.\n(i) At given microwave frequency of linear wire or loop current, not a single plane wave is\ninduced (as it would be in infinite-size film), but a spectrum of waves wit h different length,\nincluding ones with non-zero y-component of wave vector (notice that at H0= 3 the\nuniform precession frequency ωu≈6.83 ). In fact, in Fig.6a and Fig.6b we observe\neigen-modes (or compositions of nearly degenerated eigen-modes ) of small (finite-size) film.\nComparisonofthesefiguresshowsthat,naturally,loopinductore nsuresbetterwaveselection\nand simpler magnetization pattern.\n(ii) At the same time, the spatial Fourier spectrum of excited patte rn can be intelligently\ninterpreted in terms of infinite film theory (Sec.4), as illustrated by c ontour plot in Fig.6b.\nIn this plot, a number of lines surrounding some point of k-plane indicates its contribution\nto summary picture. Clearly, spectrum maxima well agree with equi-f requency curves in\nk-plane of infinite film (Fig.4a in Sec.4), in spite of not large film length to wa velength\nratio (≈5 ). Hence, so visible characteristic rhombic structures in Figs.6a-b directly reflect\ncharacteristic slopes of equi-frequency curve responding to the excitation frequency.\n6The Fig.6c justifies that particular eigen-modes and eigenfrequenc ies of even small-size\nfilm may be quantitatively close to waves in infinite film. One particular mo de is selected\nby equating pump frequency to that of plane surface (Damon-Esh bach) wave with kx=\n±2π/2landky= 0 which would be generated by the same loop in infinite film. We see\nthat result is almost plane wave too, i.e. indeed resonance takes plac e. Nevertheless, this\nmode contains some contribution from plane waves with kx≈ ±3·2π/2landky/ne}ationslash= 0\nwhich possess the same eigenfrequency and occur resonantly exc itable by the same loop.\n6.7. ROLE OF FILM EDGES.\nIn the top of Fig.7, static distributions of the internal field, W0, and magnetization in\nsmall-area film are shown, at moderate value of external field, H0= 3Ms. Clearly, in most\npart of the film practically uniform magnetization realizes, with W0≈H0. Substantial\ndemagnetization takes place at narrow strips only which adjoin film ed ges perpendicular to\nexternal field and have width ∼D(as it was stated in Sec.3).\nSince the internal field is lowered at these demagnetized strips, a loc al spin precession\nfrequency there also is lowered. Therefore, usually these strips t ake almost none part in\nshaping and propagation of waves, as if spins were partially pinned th ere. These statement\ncan be illustrated by Figs.6a-c.\nIn principle, specific edge waves can be excited in the demagnetized r egions. However,\nthis is rather exotic phenomenon, and it was not a case in practically a ll of our numerical\nsimulations.\n6.8. A SOLITON FED UP BY WEAK CONSTANT PARAMETRIC PUMP.\nOne more exotic phenomenon is illustrated at bottom of Fig.7, concre tely, very small-\namplitude soliton (spatially localized wave packet) created and then s upported by extremely\nweak parametric pump. The loop current parallel to external field ( i.e. toy-axis) induces\nfield with amplitude of its x-component ∼10−4(i.e∼0.01 Oe in real units) and\nfrequency ωe= 14.5 which is far out off total MSW frequency band (with upper bound\n=H0+ 2π≈9.3 , see Sec.4). As the result, long weak envelope soliton is formed\nwhose carrying frequency equals to half of ωe, with magnitude /bardblS⊥x/bardbl ∼10−5and\nwidth∼150D(at film length = 432 D). Interestingly, the two latter quantities\napproximately satisfy the relation between amplitude and width of br ight solitons which\nfollows from Eqs.5.15 and 5.25. Although bright solitons moving perpen dicular to external\nfield are formally forbidden in infinite film, this simulation shows that simila r objects are\n7permitted for finite-size film. Besides, we detect that parametric e xcitation allows to create\nsmall-amplitude soliton avoiding the restriction (5.23).\nFurther behavior of this object is even more intriguing. It oscillates between film ends\n(see Fig.7) undergoing some decay after reflection from the distan t end but more or less am-\nplification while reflecting from the edge where inductor takes place. The dot line separates\nperiod of strong amplification of the soliton due to occasionally “good ” relation between\nphases of its carrier and pump. Dimensionless time and frequency in F ig.7 are real ones as\nexpressed in units of τ0andf0, respectively (see Sec.2 and 3).\nFrom the point of view of Eq.5.1 the parametric process under discus sion must be de-\nscribed by quadratic term /an}bracketle{tS0,h/an}bracketri}ht[S0,S⊥] insecond row. Interestingly, in view of hy= 0\nwe should conclude that pump, /an}bracketle{tS0,h/an}bracketri}ht, mostly acts at the demagnetized edges. Among\nour numerical collection, this is exclusive example when film edges play a key role.\nBelow, we will deal with non-parametric pump whose frequency belon gs to the MSW\nband. Like here, in all forthcoming examples inductors are parallel t o external magnetizing\nfield.\n6.9. CREATION OF SOLITONS BY NON-PARAMETRIC PULSE PUMP.\nThe Fig.8a shows the consequence of intensive radio-impulse of curr ent passed through\nwire inductor at one of ends of relatively large-area film. The impulse d uration was about\ntwenty periods of the carry frequency, ωe= 7.5 . At a distance from the inductor, the in-\nduced magnetization precession impulse is deformed. If its initial amp litude was sufficiently\nlarge then further it breaks into a chain of pulses with almost zero dip s between highest of\nthem. The picture in Fig.8a can be qualified as formation of gray soliton s inside finite-length\nwave packet.\nThecriticalbreakinglevelofamplitudeisjustitsmaximumafterbrea king,inthisexample\n/bardblS⊥x/bardbl ≈0.15 . This observation is in reasonable agreement with the estimate of this level\nwhich follows from Eqs.5.9, 5.10 and 5.23,\n/bardblS⊥x/bardbl ≈Amin√p=/radicalbig\nΓp/|κ| ≈0.1 (2)\n(at friction γ= 0.0005 what tookplace). In all the below discussed numerical simulatio ns,\nclosethresholdvalues, 0 .1/lessorsimilar/bardblS⊥x/bardbl/lessorsimilar0.2 , marktransitiontobrightlyexpressednon-linear\neffects and to magnetic chaos.\n6.10. CHAOS UNDER UNIFORM RESONANT MICROWAVE FIELD.\n8In most of previously reported experiments on magnetic chaos, th e latter was excited by\nnearly uniform microwave magnetic field, h(r,t) , either using parallel parametrical pump\nwhenh/bardblS0and excitation frequency ωe∼2ωuor through perpendicular ferromagnetic\nresonance (FMR) when h⊥S0andωe∼ωu. Consider the second variant since it is\nmore close to chaotic auto-generation of MSW to be under our inter est.\nThe Fig.8b demonstrates modeling of nonlinear FMR in moderate-area film atH0= 3\nandγ= 0.0007 , under uniform microwave field parallel to x-axis. Due to finite-size\neffects (edge demagnetization), factual (numerically found) unif orm precession frequency,\nωu≈6.785 (at H0= 3 ), is slightly lower then theoretical value for infinite film,\nωu∞≈6.83 . Taking ωesufficiently close to ωu, it is possible to obtain strong response\nto weak perpendicular field h∼0.001Ms∼0.1 Oe. Naturally, the response is indicated\nby amplitude of uniform component of S⊥, i.e./an}bracketle{tS⊥/an}bracketri}ht ≡/integraltext\nVS⊥dr/VwhereVstands\nfor film volume.\nAfter sharp two times increase of pump we observe increase of /an}bracketle{tS⊥/an}bracketri}ht’ amplitude which\nmonotonically tends to nearly two times larger value. However, next increases of hby\nthe same step result in smaller and non-monotone response. At h= 0.007 , the response\ntransforms into periodic oscillations. At last, when h= 0.008 , these oscillations turn into\nchaotic one, at time moment marked as “burst” on Fig.8b. More care ful repetition of this\nprocess (with smaller step) allows to notice at least one or two period -doubling bifurcations\nof regular oscillations (as well know in theory of dissipative chaos [1]).\nInterestingly, to reach the chaos, well satisfied resonance cond itionωe≈ωuis quite\nnecessary. For instance, at ωe=ωu∞(i.e. at frequency deviation less than 1% )\nthe only result of even very intensive pump, h∼1 , is strongly nonlinear but regularly\noscillating long-wave structure.\nOther characteristic observation is the hysteresis of chaos: if his lowered from 0 .008\nthan chaotic regime remains at least down to h= 0.005 .\n6.11. SHORT-WAVE EXPLOSION AND TRANSITION TO CHAOS.\nUsually, experimental magnetic chaos is analyzed in terms of particu lar wave modes, that\nis in momentum space (see, for instance, [2,3]). In our simulation, we c an view also how it\nlooks in real space, and watch for spatial-temporal picture of tra nsition from regular motion\nto chaos. Characteristic scenario of this transition is illustrated in F ig.9.\nWhen FMR is still regular, rather smooth magnetization pattern tak es place with one or\n9three maximums of oscillations and precise mirror symmetry (with res pect to middle lines\nof rectangular film area). However, the closer is the transition the higher and narrower is\nthe central maximum. This means that spectrum of excited MSW bec omes more and more\nwide, but still coherent, in the sense that all the waves are mutually connected by some\nrigid phase relations. At critical “burst” time moment the central m aximium collapses into\npeak very narrow in x-direction and rather flat (elonged) in y-direction (i.e. along static\nmagnetization). Then this peak blows up giving freedom to to the sho rt waves. The latter\nincoherently scatter in all the directions in their turn giving rise to co mplicated (chaotic)\nmagnetization pattern. Beginning of this unstable explosive stage is shown at bottom of\nFiq.9. Notice that corresponding magnitude of S⊥xconfirms estimate (2).\nImportant sign of transition to chaos is violation of the mirror symme try. The symmetry\nwith respect to 180orotation only remains after the explosion. Clearly, this symmetry is\ninvoked by that of the static S0pattern (see Fig.7).\nAt later time, most short of the explosively induced waves decay. In further stationary\nchaotic regime, magnetization picture more or less restores both s moothness and mirror\nsymmetry. But, naturally, increase of pump rises both short-wav e contents and asymmetry.\nMoreover, under sufficiently intensive pump new similar explosions (bu rsts) are repeated\nfrom time to time, serving as “discharges” of excessive energy acc umulated by long waves.\nThis may be called strong chaos.\n6.12. EXCESS ENERGY AND POWER ABSORPTION.\nThe top of Fig.9 demonstrates typical chaotic behavior of excess fi lm energy, E, and\nof power absorption by film, P, both related to unit volume (discretization cell). Here\nand below, “excess energy” (or simply “energy”) will term increase of magnetic energy due\nto the excitation, S⊥, but excluding direct S⊥’s interaction with pump field (i.e.\nexcept−/integraltext\n/an}bracketle{th,S⊥/an}bracketri}htdr/V, see Sec.2). The power absorption, P, describes energy flow\ninto film which is spent for both Eand dissipation in the film interior. Hence, in general\ndE\ndt=P−Pdis, P=/angbracketleftbigg/integraldisplay/angbracketleftbigg\nh(r,t),dS⊥\ndt/angbracketrightbiggdr\nV/angbracketrightbigg\nT, (3)\nwithPdisbeing dissipated power per unit volume. Symbol /an}bracketle{t/an}bracketri}htTdesignates time\naveraging with respect to spin precession. These equations direct ly follow from the basic\nEq.2.1.\nIt is natural to expect that\n10Pdis≈2Γ(E−E0), (4)\nat some E0, and Γ being previously considered dissipation rate of magnetization . At\nγ= 0.0007 first of Eqs.5.10 gives 2Γ ≈0.013 , then energy relaxation time is ∼1/2Γ≈\n80 . Indeed, approximately such the time scale can be viewed at plot ( A) in Fig.9. Further,\nlet us put on /an}bracketle{tS⊥x/an}bracketri}ht=/bardblSx/bardblsin[ωet−φ] where /bardblSx/bardblandφdesignate amplitude and\nphase, respectively, of the uniform component of spin precession . Then\nP=h/angbracketleftbigg\nsin(ωet)d/an}bracketle{tS⊥x/an}bracketri}ht\ndt/angbracketrightbigg\nT≈1\n2ωeh/bardblSx/bardblsinφ (5)\nAt/bardblSx/bardbl ≈0.1 (as prompted by Fig.8b), h= 0.008 and φ≈π/2 (which means good\nresonance) this relation results in P∼2.5·10−3, in agreement with plot (B) in Fig.9.\nHence, rough average characteristics of chaotic variables are ea sy explainable.\nMuch harder task is to explain chaotic deviations from average value s. For instance, in\nplots (A) and (B) energy and power nearly follow one another. It is c lear: the energy comes\nfrom power absorption, but the latter depends on the phase φwhich in its turn must be\nsensitive to energy, because of non-isochronity of spin precessio n (see Sec.5). But without\nan adequate dynamical model for this connection we can not estima te details of chaotic time\nseries.\n6.13. INSTANT FREQUENCY.\nDirectly, our numerical algorithm produces fast oscillating time serie s. To extract from\nthem relatively slow time-varying (“instant”) amplitudes and phases (like/bardblSx/bardblandφ\nabove), and corresponding instant frequencies (e.g. ωe−dφ/dt ), there are two ways.\nOne is to build analytical signals by means of (discrete) Gilbert transf ormation. Another\nway is to indicate maximums, minimums and zero-crossings of oscillating variable. If tn,\nn=...,−1,0,1,..., are estimates of time moments when zero-crossing does occur, t hen\ninstant frequency at t≈tncan be determined as ωin= 2kπ/(tn+k−tn−k) , where\nk≥1 .\nIf disposing this quantity,the phase can be restored by discrete nu merical integration.\nThis method is more comfortable and fast than Hilbert transformat ion, but ensures not\nworse (usually better) accuracy what was confirmed by special te sts.\n6.14. QUASI-LOCAL ENERGY DENSITY.\n11Because of relations (2.9) and (4.1), the excess energy (per unit v olume) can be repre-\nsented in the form\nE=Eloc+Enonloc, E loc=/integraldisplay\nelocdr\nV, eloc≡H0(1−Sy)+2πS2\nz(6)\nHereEnonlocis contribution from non-local (non-singular) part of dipole interac tion in\nplate geometry (Sec.4), while Elocconsists of its local (singular) part and also local first\nterm of Eq.2.9.\nFor uniform precession (even let large-amplitude and strongly non- linear), Enonloc\nvanishes, hence, the sum eloc−/an}bracketle{th(t),S/an}bracketri}htplays the role of Hamiltonian of the average spin.\nIn particular, in autonomous regime of uniform precession, at h= 0 , elocbecomes\nintegral of motion.\nImportantly, Enonlocmay be negligible as compared with Elocin non-uniform chaotic\ncase too. Indeed, according to equations (4.1-2), (4.6) and (6), the non-local contribution\ncan be estimated as\nEnonloc≈πD/integraldisplay/parenleftbiggk2\nx\n|k|/vextendsingle/vextendsingle/vextendsingle/tildewideSx(k)/vextendsingle/vextendsingle/vextendsingle2\n−|k|/vextendsingle/vextendsingle/vextendsingle/tildewideSz(k)/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\ndk , (7)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleEnonloc\nEloc/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarπD/an}bracketle{t|k|/an}bracketri}ht\nH0,/an}bracketle{t|k|/an}bracketri}ht ≡/integraldisplay\n|k|/vextendsingle/vextendsingle/vextendsingle/tildewideSx(k)/vextendsingle/vextendsingle/vextendsingle2\ndk/parenleftbigg/integraldisplay/vextendsingle/vextendsingle/vextendsingle/tildewideSx(k)/vextendsingle/vextendsingle/vextendsingle2\ndk/parenrightbigg−1\n,\nwith/tildewideSx,z(k) denoting spatial Fourier transform of magnetization. If long MSW (with\nD|k| ≪1 ) are dominating in magnetization pattern, then the excess energ y is well char-\nacterizable by Eloc. As the consequence, the quantity elocbehaves like local integral of\nmotion and thus can be termed quasi-local energy density.\n6.15. WEAKLY CHAOTIC FMR.\nSeemingly weak chaos realizes at h= 0.005 . The Fig.10 shows evidence for (i) good\ncorrelation between the energy and dissipation (with E0= 0 and Γ well related to\nactual friction coefficient γ), and, at plot (D), (ii) rigid correlation between energy and\naverage longitudinal component of magnetization,/angbracketleftbig\nS/bardbl/angbracketrightbig\n. Here/an}bracketle{t../an}bracketri}htdenotes space-time\naveraging. Notice that in most part of film area S/bardbl=/an}bracketle{tS0,S/an}bracketri}ht ≈Sy.\nThe latter correlation just gives the evidence that non-local ener gy contribution is rela-\ntively small. Indeed, because of relations\n12/bardblSz/bardbl2≈ /bardblSx/bardbl2/p2,/angbracketleftbig\nS2\nx/angbracketrightbig\n+/angbracketleftbig\nS2\nz/angbracketrightbig\n≈2(1−/angbracketleftbig\nS/bardbl/angbracketrightbig\n),\nwherepis characteristic eccentricity, Eloccan be expressed as\nEloc≈/parenleftbigg\nH0+4π\np2+1/parenrightbigg\n(1−/angbracketleftbig\nS/bardbl/angbracketrightbig\n) (8)\nHence, at Enonloc≪Eloctotal excess energy also well reduces to/angbracketleftbig\nS/bardbl/angbracketrightbig\n. If we equated\npto small-amplitude uniform precession eccentricity given by Eq.5.10, t hen atH0= 3\nthe Eq.8 would yield dE/d/angbracketleftbig\nS/bardbl/angbracketrightbig\n≈dEloc/d/angbracketleftbig\nS/bardbl/angbracketrightbig\n≈ −5 . This value is lower by 10%\nthan the slope ( ≈ −5.6 ) at Fig.10.D. The difference can be explained if take into account\nthat increase in spatial non-uniformity of precession result in decr ease of its eccentricity.\nPlot (G) in Fig.10 presents spectrum of power absorption (absolute value of Fourier\ntransform of P(t) ). At Ms≈140 Oe (as for YIG), the dimensionless frequency unit\ncorresponds to ≈390 MHz, hence, the dominating frequency in P(t) ’s spectrum is\n≈20 MHz. Chaos in instant frequency of precession, as shown in plot ( B), and thus in\nits phase is characterized by significantly wider frequency band (in p art contained by plot\n(C) in Fig.9). Nevertheless, plot (F) in Fig10 visualizes anti-correlatio n between instant\nfrequency and energy which corresponds to negative sign of non- isochronity (see Sec.5).\nFor fractal dimension (Sec.5.9) of these data it was found that dcor<3 (see below).\nThis implies that chaos is governed by three relevant variables only, a nd therefore chaotic\nattractor could be represented in 3-dimensional space. It seems doubtless that E,P\nand/bardblSx/bardblare relevant variables (due to Eq.5, Pand/bardblSx/bardbldetermine also the phase\nφ). Plot (C) in Fig.10 gives show of the attractor in these coordinates .\n6.16. ANALYSIS OF CORRELATION DIMENSION.\nLet usdiscuss practical calculationof thecorrelationdimension, dcor(Sec.5.9). At given\ncorrelation sum, one may estimate dcorby two ways:\ndcor=dlnσ(R)\ndlnRordcor=ln[σ(R)/Ω]\nlnR(9)\nThe first of them may be called differential dimension while the second in tegral. Under\nformal limit N→ ∞ these quantities are expected to coincide one with another. But\nreal calculation needs in more than N2operations, therefore Ncan not be as large as\n13wanted. Under realistic N, the differential estimate satisfactorily works at moderate\nvalues of Ronly,Rmin≪R≪Rmax(whereRminis minimum of Rij).In\nopposite, the integral estimate better works at lower end of this in terval.\nNaturally, theintegralestimateisless sensitive tofiniteness of N, butinsteaditrequires\nto know the coefficient Ω . It is rather obvious that the best genera l assumption about Ω\nis that it equals to volume of dcor-dimensional unit-radius sphere, i.e. Ω = πdcor/2/Γ(1+\ndcor/2) , where Γ means gamma-function. By special tests we verified th at this recipe\nindeed constantly improves precision of dcorestimates.\nNotice that wide class of tests is presented by the Kaplan-Yorke ch aotic system [1]. It is\ndescribed by the set of difference (discrete time) equations:\nXj(t+1) =F(Xj(t)), j= 1...n , Y (t+1) =αY(t)+n/summationdisplay\nj=1fj(Xj(t)),|α|<1,(10)\nwhereF(X) is some chaotic one-dimensional map (for instance, tent map, F(X) = 1−\n|2X−1|), andfj(X) are any smooth functions. Dependently on nandα, fractal\ndimension of chaotic sequence Y(t) can be equated to arbitrary number (for example,\nfor the tent map n= 2 and α= 1/16 lead to dcor= 2.5 while n= 3 and α= 1/64\ntodcor= 3.5).\n6.17. FRACTAL DIMENSION OF CHAOTIC FMR.\nTypical example of evaluation of dcoris shown in Fiq.11a (left plot). Here “log of Cell\nSize” in horizontal axis means ln( Rmax/R) while vertical axis presents both the estimates\n(9). The dependencies of differential and integral dimensions on ln( Rmax/R) are drawn\nby thin and fat lines, respectively.\nWe see that plateau in the first of them well coincides with the upper v alue of the\nsecond. From above mentioned tests, it is known that just this valu e should be taken as\nbest estimate of dcor, and that the coincidence signify good reliability of this estimate.\nIn more complicated case when the plateau differs from upper (most right-hand) value of\nintegral dimension, the latter must be preferred. But such the sit uation testifies that either\nthe data (finite chaotic series) are not enough representative or they possess essential multi-\nfractality.\nIn case under consideration, fractal dimension of the power abso rption time series under\nweakly chaotic FMR can be estimated as dcor≈2.4 .\n14To form better representative data, the time separation, τ, ind-dimensional embed-\nding points {x(t0+nτ), x(t0+nτ+τ), ...,x(t0+nτ+dτ)}should be a few times shorter\nthan characteristic correlation time of x(t) , while number of points must be sufficient for\nminimum “filling” of all of ddimensions, at least N≥2d.\n6.18. CHAOTIC FMR IN DEFECTIVE FILM.\nReal films always have more or less amount of defects. The right-ha nd plot in Fig.11a\ndemonstrates static magnetization by tangential field in film with per iodic lattice of defects\n(punctures) which touch about 10% of film area. Naturally, mean int ernal field, W0, and\nthuscharacteristicprecession frequencies arelowered bydefec ts. Butwefoundnoqualitative\ndifference between chaotic FMR in defective film and “good” film.\nThe top and left bottom plots in Fig.11b illustrate how the uniform comp onent of spin\nprecession behaves at small time scale (of order of 1 ns) and at moderate time scale (of\norder of 0 .1µs). On right hand, contour plot of spatial Fourier transform of S⊥xshows\nthat two groups of MSW modes form magnetization pattern, both b elonging to the same\nfrequency rangearound ωu, withtheshort-wavegroupinducedbydefect lattice. Thelevels\nfor this plot were chosen specially to highlight short-wave modes. In fact, their contribution\nto energy is of order of a few percents.\n6.19. SYNCHRONIZATION OF CHAOTIC FMR.\nIn the work [4] synchronization of magnetic chaos under non-linear FMR in normally\nmagnetized YIG film was experimentally realized. First, the signal, PM(t) , related to\npower absorption was recorded into a memory. Characteristic fre quencies of this signal were\nbetween 0.5 MHz and 10 MHz. The similar actual “slave”signal, PS(t) , was compared\nwiththerecorded“master”signal, andthedifferencewasdirected , withsomeproportionality\nconstant, K, to perturb the external magnetizing field, H0→H0+K(PM−PS) .\nAt suitable choice of K, after a transient time ∼10µs, excellent coincidence between\nPS(t) andPM(t) was observed.\nThe Fig.12 illustrates the attempt to numerically reproduce such the experiment but\nwith tangentially magnetized film. The above mentioned defective film m odel is explored,\natH0= 3 and h= 0.005 . The power absorption per unit volume, P(t) , is taken\nto serve as the control signal. The master signal to be addressed to the feedback, PM(t) ,\neither equals to P(t) or formed from it by slight time-smoothing (over 5 ÷30 periods of\nprecession). Typical magnitude of chaotic P(t) ’s variations is ∼5·10−5. The feedback\n15coefficient, K, is changed in the range between −30 and −600 .\nUnfortunately, 10 µs is rather large time for our numerical simulations. Total duration\nof numerical runs was just about 10 µs. Best signs of synchronization were observed at\nK∼200 . Hence, magnitude of the bias field modulation was ∼0.01 , i.e. ∼1.5 Oe in\nreal units. For comparison, in [4] essentially smaller values ∼0.1 Oe were in action.\nAccording to Fig.12, with no doubts synchronization takes place, bu t its quality is far\nfrom so excellent as reported in [4]. Possibly, this is due to wider frequ ency range of the\ncontrol signal in our system, up to ∼50 MHz (see plot of P(t)’s spectra in Fig.12) and\nto not long enough duration of the numeric experiment.\nREFERENCES\n1. A.J.Lichtenberg andM.A.Lieberman. Regularandstochasticmotion . Springer-Verlag,\n1988.\n2. S.M.Rezende and F.M.de Aguiar. Proc. IEEE, 78 (1990) 893.\n3. J.Beeker, F.Rodelsperger,Th.Weyrauch, H.Benner, W.Just and A.C enys. Phys.Rev.\nE59 (1999) 1622.\n4. D.W.Peterman, M.Ye and P.E.Wigen. Phys.Rev.Lett. 74 (1995) 1740.\nConclusion\nThe above expounded material leads to conclusions as follow:\n(i) Concepts, formulas and observations of linear theory of magne tostatic spin waves\n(MSW) in infinite films (Sections 2-4 of Part I, see arXiv preprint 1204 .0200), as well as\nthat of quasi-linear MSW theory (Sec.5 in the Part I), appear quite a dequately useful for\ninterpretation of numeric simulations of even non-linear and chaotic MSW even in small-size\nfilms;\n(ii) Most effective way to MSW chaos is via parametric resonance and p arametric non-\nlinear transformations of MSW;\n(iii) MSW chaos typically is determined by a few relevant variables only, - i.e. character-\nized by rather low fractal dimension , - and therefore seems allowing its control and more\nor less satisfactory synchronization.\nThe two last features will be further investigated in next Part III o f this manuscript,\n16concentrating on not externally driven but auto-generated MSW c haos.\n1710203040506070805101520\n10203040506070805101520\n−1 −0.5 0 0.5 1−1−0.500.51\nkxDkyD  Fig.6a.  Instant S⊥ z−pattern in small−area in−plane magnetized film under weak                \nnon−resonant excitation by wire positioned at x/D=9. Both external field and wire are                   \nparallel to y−axis. Parameters: H0=3, γ=0.003, exciting frequency ωe=7 .                    \n  Fig.6b.  (Top) The same as in Fig.6a but inductor is loop taking lines x/D=9 and x/D=17.                   \nBoth external field and loop are parallel to y−axis. Parameters: H0=3, γ=0.003, ωe=7.3.         \n   (Bottom) Contour plot of spatial spectrum of S⊥z−pattern. The curve marked by arrows is                   \ntheoretical equi−frequency line ω=ωe=7.3 for surface waves (see Fig.4a).                                          1810203040506070805101520\nx/Dy/D\n10203040506070805101520\nx/D, S⊥xy/D, S⊥z\n−1.5 −1 −0.5 0 0.5 1 1.5−1−0.500.51\nkxDkyD\n  Fig.6c.  (Top)  Instant S⊥x pattern under nearly resonant excitation by loop current with \nfrequency ωe=8.26 ≈ωDE(2πD/2l), l=8D. Other parameters as in Fig.6b.  \n   (Middle) Corresponding instant distribution of the vector S⊥. \n   (Bottom) Contour plot of spatial spectrum of S⊥z. The curve marked by arrows is \nequi−frequency line corresponding to ω=ωe (see Fig.4a).\n190123W0(x,y)\n0100200300400−1−0.500.511.5x 10−5\nt (time)S⊥x envelope at x=200D\n4 6 802468x 10−5\nω (frequency)S⊥x spectrum\nωe=14.5\n Fig.7. (Top & Middle) Internal field, W0, and x−component of static magnetization, S0x, in small−area \nfilm under external field H0=3 parallel to y−axis. (Bottom) Parametric subharmonics generation under \nexcitation by weak loop induced field hx=±0.0001sin( ωet) applied to left edge, at H0=3, γ=0.0005, \nωe=14.5. Left: magnetization disturbation in the middle of film. Right: its spectrum occurs near ωe/2.0\n20\n40\n60\n8005101520−0.500.5S0x(x,y)x/D y/D \nωe/2 \n2030 40 50 60 70 80 90−0.2−0.100.10.2\nt (time, in τ0 units)S⊥x(x=200D,t)\n0 500 1000 1500 2000 250000.020.040.060.080.10.120.140.16\nt (in τ0 units)amplitude of <S⊥x>\n2000 2500 3000 3500 40000.10.15\nt (in τ0 units)<S⊥x>  Fig.8a. Beginning of soliton formation under strong wire−induced field with hx∼0.2sin(ωet) \napplied to left edge, at H0=3, γ=0.0005, ωe=7.5. The envelope of S⊥x(x=200D,t) is shown.\n  Fig.8b.  Nearly resonant excitation of in−plane magnetized film by uniform radio−frequency field,                 \nhx=h⋅sin(ωet) , at H0=3, γ=0.0007, ωe=6.785. (Top) Transition from stable FMR regime to chaos\nas the field, h, grows from 0.001 to 0.008. (Bottom) Chaotic oscillations of uniform                                \nmagnetization component <S⊥x>=∫VS⊥x(r,t)dr/V.  In both plots its amplitude is shown.              h=0.001 h=0.002 h=0.003 h=0.005h=0.008 \nchaos  burst \n210.040.050.060.070.08Energy\n20002500300035004000450011.522.53\ntimePower⋅103\n6.7 6.8 6.900.511.522.5\nω (frequency)spectrum of <S⊥x>\n  Fig.9.  Chaotic ferromagnetic resonance of in−plane magnetized film excited by uniform                                     \nradio−frequency field, hx=0.008⋅sin(ωet), at H0=3, γ=0.0007, ωe=6.785. (A) Excess energy              \nper unit volume. (B) Power absorbtion per unit volume. (C) Spectrum of particular chaotic                                    \nrealization of <S⊥x>. (D) Instant S⊥x pattern before transition to chaos. (E) Characteristic S⊥x        \npattern at the moment of transition to chaos (marked \"burst\" in Fig.8b). The explosive birth                                 \nof short magnetic waves and mirror symmetry violation are clearly seen.                                                      (A) \n(B) (C) \n(D) \n(E) \n220.060.070.080.09E (energy)\n40005000600070006.756.86.85\ntimeFrequency\n0.98 0.985 0.990.060.080.1\n<S>E (energy)\n0.05 0.16.766.776.786.796.86.816.82\nE (energy)Zero Crossing Frequency\n0 0.05 0.100.511.5\nEnergyPdis⋅1030.06\n0.07\n0.08\n0.09 0.050.10.15\n00.511.52\n||S⊥x||\nE (energy)Power⋅103\n00.10.200.511.522.533.544.55\nω (frequency)P(t)′s spectrum, a.u.dE/d<S>≈−5.6 \nPdis≈2ΓE\nΓ≈0.007       (A) \n(B) (C) \n(D) \n(F) \n(E) (G) \n  Fig.10. Chaotic FMR in uniform field, hx=0.005⋅sin(ωet), at H0=3, γ=0.0007, ωe=6.785.\n(A) Excess energy. (B) Instant frequency of <S⊥x(t)> determined by its zero crossings.                 \n(C) Fragment of chaotic attractor. (D) Correlation between energy, E, and <S>.                     \n(E) Correlation between energy, E, and instant frequency. (F) Correlation between                             \ndissipated power, Pdis, and energy. (G) Spectrum of power absorption.                                      232 4 60.511.522.533.5\nlog of Cell SizeFractal Dimensiondifferential\nintegral\n−0.100.1S0x(x,y)\n1.158 1.16 1.162 1.164 1.166 1.168−0.100.1<S⊥x>\n11.21.41.60.050.10.15\ntime, µs<S⊥x> envelope\n−0.4−0.200.20.40.6−0.8−0.6−0.4−0.200.20.40.6\nkxDkyD  Fig.11a. Left: Fractal (correlation) dimension of absorbed power signal, P(t), obtained       \nby its embedding into 10−D space. The result is dcor≈2.4 . Right: S0x−component of  \nstatic magnetization in periodically punctured film.                                            \n Fig.11b. (Top) Small piece of uniform component of magnetization excitation, <S⊥x> (dotted                 \nlines mark periods of exciting field). (Bottom left) Fragment of <S⊥x>′s envelope in chaotic FMR       \nregime of in−plane magnetized film. (Bottom right) Contour plot of typical S⊥x′s pattern.              \nDotted lines are theoretical equi−frequency lines for ω=ωu, i.e. |ky|/|kx|=(4π/H0)1/2, at H0=3.245 5.2 5.4 5.6−0.100.10.20.30.40.50.6Power, r.u.\n6.9 7 7.1 7.2 7.30.050.10.150.20.250.30.350.4Power, r.u.master\nslave\n020406000.511.5\nFrequency, MHzPower Spectrum\n7.4 7.6 7.8 88.2 8.4 8.6 8.8 900.10.20.30.40.50.6\ntime, µsPower, r.u.\n Fig.12. Synchronization of chaotic power absorption signal, Pslave(t), at one film undergoing FMR \nby power signal, Pmaster(t), from another film also subject to chaotic FMR. The films are identical, \nboth periodically punctured, tangentially magnetized by field H0=3Ms and excited by the same  \nuniform field, hx=0.005Mssin(ωet), whose frequency nearly equals to the uniform precession \nfrequency, ωu≈5.65. For concreteness, in plots the value Ms=140 Oe is substituted.master\nslave\n25"    },    {        "title": "1306.6268v2.Actuation__propagation__and_detection_of_transverse_magnetoelastic_waves_in_ferromagnets.pdf",        "content": "arXiv:1306.6268v2  [cond-mat.mes-hall]  1 Nov 2013Actuation, propagation, and detection of transverse magne toelastic waves in\nferromagnets\nAkashdeep Kamraa, Gerrit E. W. Bauera,b\naKavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nbInstitute for Materials Research and WPI-AIMR, Tohoku Univ ersity, Sendai 980-8577, Japan\nAbstract\nWe study propagation of ultrasonic waves through a ferromagnet ic medium with special attention to the boundary\nconditions at the interface with an ultrasonic actuator. In analogy to charge and spin transport in conductors, we\nformulate the energy transport through the system as a scatte ring problem. We find that the magneto-elastic coupling\nleads to a non-vanishing magnetic (elastic) energy accompanying th e acoustic (spin) waves with a resonantly enhanced\neffect around the anti-crossing in the dispersion relations. We demo nstrate the physics of excitatinig of magnetization\ndynamics via acoustic waves injected around the ferromagnetic re sonance frequency.\nKeywords: A. ferromagnets; D. ultrasound; D. magnetoelastic coupling; D. s pin pumping\nPACS:75.50.Dd; 43.35.+d; 75.80.+q; 72.25.Pn\n1. Introduction\nWhile the exchange interaction is the largest energy\nscale offerromagnets, explaining Curie temperatures ofup\nto 1000 K, the static equilibrium and dynamic properties\nof the magnetization field in ferromagnetic materials are\ngoverned by the dipolar and crystal anisotropy fields [1].\nSince the total angularmomentum of an isolated system is\nconserved,anychangeofthemagnetizationexertsatorque\non the underlying lattice, as measured by Einstein and de\nHaas [2]. Vice versa, a rotating lattice can magnetize a de-\nmagnetized ferromagnet[3]. The coupled equations ofmo-\ntion of lattice and magnetization fields have been treated\nin a seminal paper reported by Kittel [4]. The magne-\ntoelastic coupling parameters are material constants well\nknown for many ferromagnets [5].\nInterest in magnetoelastic coupling has recently been\nrevived in the context of the “spin mechanics” concept\ncovered by the present special issue [Editorial SSC]. Here\nwe are interested in the magnetization dynamics acous-\ntically induced by injecting ultrasound into ferromagnets\nby piezoelectric actuators as bulk [6] or surface [7] plane\nacoustic waves. The magnetization dynamics in these ex-\nperiments is conveniently detected by spin pumping [8]\ninto a normal metal that generates a voltage signal via the\ninverse spin Hall effect [9].\nSome of the consequences of magnetoelastic coupling\nhave alreadybeen investigated theoretically [4, 10, 11] and\nexperimentally [12, 13] in the literature. While the cou-\npled magnetoelastic dynamics has been well understood\ndecades ago [4, 10, 14], much less attention has been de-\nvoted to the interfaces that are essential in order to under-\nstand modern experiments on nanostructures and ultra-thin films. The Landauer −B¨ uttikerelectrontransport for-\nmalism based on scattering theory is well suited to handle\nthese issues thereby helping to understand many problems\nin mesoscopic quantum transport and spintronics [15, 16].\nHereweformulatescatteringtheoryoflatticeandmagneti-\nzationwavesin ferromagnetswith significantmagnetoelas-\ntic coupling. Rather than attempting to describe concrete\nexperiments, we wish to illustrate here the usefulness of\nthis formalism for angular momentum and energy trans-\nport.\nWe consider magnetization dynamics actuated by ul-\ntrasound for the simplest possible configuration in which\nthe magnetization direction is parallel to the wave vector\nof sound with transverse polarization (shear waves). The\ncorresponding bulk propagation of magnetoelastic waves\nwas treated long ago by Kittel [4] who demonstrated that\nthe axial symmetry reduces the problem to a quadratic\nequation. The injected acoustic energy is partially trans-\nformed into magnetic energy by the magnetoelastic cou-\npling that can be detected by spin pumping into a thin\nPt layer. For this symmetric configuration and to leading\norder a purely AC voltage is induced by the inverse spin\nHall effect (ISHE) [17] that might be easier to observe by\nacoustically induced rather than rf radiation induced spin\npumping, since in the former the Pt layer is not directly\nsubjected to electromagnetic radiation. The configuration\nconsidered by Uchida et al. [6], in which pressure waves\ngenerate a DC ISHE voltage by a magnetization parallel\nto the interfaces, will be discussed elsewhere.\nPreprint submitted to Solid State Communications November 7, 20182. Kittel’s equations\nWe consider a ferromagnet with magnetization texture\nM(r,t) with constant saturation magnetization |M|=\nM0. Inthe followingweconsidersmallfluctuations around\nthe equilibrium magnetization M0z.The classical Hamil-\ntonian can be written as the sum of different energies\nH=HZ+Hex+Hme+Hp (1)\nThe magnetic Zeeman energy reads\nHZ=ω0\n2γM0/parenleftbig\nM2\nx+M2\ny/parenrightbig\n(2)\nwhereγ=|γ|is the gyromagnetic ratio, ω0=µ0γHis\nthe magnetic resonance frequency for an effective mag-\nnetic fieldHzandµ0the permeability of free space .The\nexchange energy cost of the fluctuations\nHex=A\nM2\n0/bracketleftBig\n(∇Mx)2+(∇My)2/bracketrightBig\n(3)\nwhereAis the exchange constant. The magnetoelastic en-\nergyforcubiccrystalsandmagnetizationinthe z-direction\ncan be parameterized by the magnetoelastic coupling con-\nstantb2:\nHme=b2\nM0/parenleftbigg\nMx∂Rx\n∂z+My∂Ry\n∂z/parenrightbigg\n(4)\nwhereR= (Rx,Ry,0) is the displacement vector of a\ntransverse lattice wave propagating in the z-direction.\nHmecan be interpreted as a Zeeman energy associated\nwith a dynamic transverse magnetic field b2∂zR. The cor-\nresponding elastic energy reads\nHp=ρ\n2˙R2+α\n2/bracketleftBigg/parenleftbigg∂Rx\n∂z/parenrightbigg2\n+/parenleftbigg∂Ry\n∂z/parenrightbigg2/bracketrightBigg\n(5)\nin terms of the mass density ρand shear elastic constant\nα.\nThe total Hamiltonian Hdefines the equations of mo-\ntion ofthe coupled RandMfields.The resultsin momen-\ntum and frequency space X(t) =x(k,ω)ei(kx−ωt)can be\nsimplified by introducing circularly polarized phonon and\nmagnonwaves m±=mx+iσmy,r±=rx+iσry(σ=±1),\nleading to [4]\n/parenleftbiggi(ω−σωm)σγb2k\nib2k\nM0ω2ρ−k2α/parenrightbigg/parenleftbiggmσ\nrσ/parenrightbigg\n= 0 (6)\nwhereωm=ω0+Dk2andD= 2Aγ/M 0is the spin wave\nstiffness. This secular equation is quadratic in k2with 4\nroots (s=±1) :\n(kσ\ns)2=ρω2\n2α−ω0−σω\n2D+γb2\n2\n2αM0+s√\n∆σ(7)with discriminants\n∆σ=/parenleftbiggρω2\n2α−ω0−σω\n2D+γb2\n2\n2αDM0/parenrightbigg2\n+ω2ρ\nαD(ω0−σω).\n(8)\nThe corresponding eigenstates are given by the spinor\nψσ\ns(ω) =/parenleftbiggmσ\ns\nrσ\ns/parenrightbigg\n=Nσ\ns/parenleftBiggM0\nib2kσ\ns//parenleftBig\nω2ρ−(kσ\ns)2α/parenrightBig/parenrightBigg\n,\n(9)\nwhereN±\nsis a dimensionless normalization factor.\nThe dispersion is plotted in Fig. 1 for the parameters\nappropriate for Yttrium Iron Garnet (YIG): M0= 1.4×\n105A/m,b2= 5.5×105J/m3,H= 8×104A/m,D= 8.2×\n10−6m2/s,γ= 2.8×1010HzT−1,ρ= 5170kg/m3,α=\n7.4×1010Pa [18, 19, 20] and µ0= 4π×10−7NA−2. In\nFig. 1(a) we plot the solutions for waves rotating with\nthe magnetization that appear to be completely phonon\n(small dispersion) or magnon like (large dispersion). The\nlatter are evanescent ( k2<0) below the spin wave gap ω0.\nThe low-frequency anticrossing is better seen in Fig. 1(c)\nin which the momentum is plotted on an expanded scale.\nSpin waves precessing against the magnetization, σ=−1,\nare always evanescent and there is no (anti)crossing with\nthe propagating phonons. When the magnetoelastic cou-\npling is switched off ( b2→0) andα >4ρDω0the pure\nlattice wave ωp=/radicalbig\nα/ρkand spin wave ωm=ω0+Dk2\ndispersions may cross twice\nωc=α\n2ρD/parenleftBigg\n1±/radicalbigg\n1−4ρDω0\nα/parenrightBigg\n(10)\n4ρDω0≪α=/braceleftbiggω0\nα\nρD=2.8GHz\n1.7THz(11)\nwhere in the second step we take the limit of small\nD. Around these degeneracy points, of which only the\nlow frequency one is relevant here, the effects of the\nmagnon−phonon coupling are most pronounced. In the\nzero frequency limit the solution with ( kσ\ns)2→0 repre-\nsents a phonon mode with zero wave number. ( kσ\ns)±=\nγb2\n2/(αM0)−ω0/Disapurelyevanescentmagnonforsmall\nb2that in principle may become a real excitation when the\ncoupling of the lattice is strong enough to overcome the\nspin wave gap.\n3. Energy flux\nEnergy conservation implies ∇·/vectorF=−∂H/∂t,where\nthe energy flux /vectorF=Fˆ zconsists of phonon and magnon\ncontributions. In time and position space [21]\nF(z,t) =−/integraldisplay\ndz∂H\n∂t=−2A\nM2\n0∂Mx\n∂z∂Mx\n∂t\n−/parenleftbigg\nα∂Rx\n∂z+b2\nM0Mx/parenrightbigg∂Rx\n∂t+(x←→y).(12)\n2For a plane wave oscillating with frequency ω\nX(z,t) =x(z,ω)e−iωt+x∗(z,ω)eiωt(13)\n=x(ω)ei(kx−ωt)+x∗(ω)e−i(kx−ωt)(14)\nthe time-averaged energy flux reads\n¯F(z)x=−2ωIm/bracketleftbiggD\nγM0mx∂zm∗\nx+αrx∂zr∗\nx+b2\nM0rxm∗\nx/bracketrightbigg\n.\n(15)\nIn the absence of magnetoelastic coupling, pure phonon\nand magnon waves\nψ(m)\ns=N(m)\nsM0/parenleftbigg1\n0/parenrightbigg\nei(ksz−ωt)(16)\nψ(p)\n0=Np/parenleftbigg\n0\nk−1/parenrightbigg\nei(kz−ωt)(17)\ncarry, respectively, the energy fluxes ¯F(p)and¯F(m)\ns:\n¯F(p)=N2\np2αω/k=N2\np2α/radicalbig\nα/ρ (18)\n¯F(m)\ns=/parenleftBig\nN(m)\ns/parenrightBig22DM0\nγωks (19)\n=/parenleftBig\nN(m)\ns/parenrightBig2/braceleftbigg2D\nγM0ω/radicalbig\n|sω−ω0|/D\n0forsω>ω 0\nsω<ω 0.\n(20)\nOne of the magnon states is always evanescent, while\nthe other becomes propagating for frequencies above the\nmagnon gap ω0.It is then convenient to define\n(Np)2=1\n2α√αρ;/parenleftBig\nN(m)\ns/parenrightBig2\n=γ\n2ωM0Θ(sω−ω0)/radicalbig\nD|sω−ω0|.\n(21)\nsuch that each state carries a unit of flux. The flux carried\nby propagating (Im kσ\ns= 0) mixed states reads\n¯Fσ\ns= 2ωkσ\ns(Nσ\ns)2\nα/parenleftBigg\nb2k±\ns\nω2ρ−/parenleftbig\nk±s/parenrightbig2α/parenrightBigg2\n−b2\n2\nω2ρ−/parenleftbig\nk±s/parenrightbig2α+DM0\nγ/bracketrightBigg\n. (22)\nwhile the time-averaged flux for evanescent waves with\nImkσ\ns/negationslash= 0 and Re kσ\ns= 0 can be shown to vanish identi-\ncally. By setting ¯Fσ\nsto unit flux in Eq. (22) we define the\ndimensionless flux normalization factor Nσ\nsfor the mixed\nstate. Here and in the following σis the chirality and s\nthe root of an eigenstate. Note that this normalization\nis rather arbitrary. We could have also used angular mo-\nmentum flux normalization, or fix the amplitude of one\ncomponent to unity. We believe, however, that for more\ngeneral situations with reduced symmetry and many wave\nvectors, the present choice is most convenient.4. Interface boundary conditions\nWe consider a weakly damped ferromagnetic structure\nactuated by a piezoelectric layer (Fig. 2) that is excited\nat a given resonance frequency ω.We assume that any re-\nflection vanishes at the end of the ferromagnet, e.g. by\nattaching an acoustic absorber [6]. Both actuator and fer-\nromagnetarethusconsideredtobe reservoirsadiabatically\nconnected to the scattering region. In and outgoing waves\nare then all propagating. We then may disregard negative\nwave numbers in the ferromagnet as well as the scattering\ncoefficients r′andt′, the reflection and transmission coeffi-\ncients of wavesfrom the magnetic side. On the left side we\nhave incoming circularly polarized phonons with chirality\nσthat is conserved when reflected at a flat interface to\na ferromagnet with magnetization along the propagation\ndirection\nχσ\nL(z,t) =NL\np\nkL/bracketleftbigg/parenleftbigg0\n1/parenrightbigg\neikLz+/parenleftbigg0\nrσ\nL/parenrightbigg\ne−ikLz/bracketrightbigg\neiωt,\n(23)\nwhereNL\np=/parenleftbig\n4α3\nLρL/parenrightbig−1/4, kL=vLω=/radicalbig\nαL/ρLωin\nterms of the acoustic parameters of the actuator and rσ\nL\nis the reflection coefficient determined below. This state\ncarriesan energy flux of Fσ=Fσ\n0/parenleftBig\n1−|rσ\nL|2/parenrightBig\n,whereFσ\n0is\nthe actuator power density. On the right side we can scat-\nter into the two mixed eigenstates at the same frequency\nwith transmission coefficients t.The axial symmetry pre-\nvents mixing between states with different polarizations\nand\nχσ\nR(z,t) =eiωt/summationdisplay\nstσ\ns(ω)/parenleftbigg\nmσ\ns\nrσ\ns/parenrightbigg\neikσ\nsz(24)\nwhere the magnetic and lattice components are flux-\nnormalized as described above. Energy conservation dic-\ntates thatF\nF0= 1−|rσ\nL|2=/summationdisplay\ns|tσ\ns|2(25)\nwhich reflects the unitarity of the scattering matrix com-\nposed by randt(as well as by r′andt′).\nAt the interface z= 0 we demand continuity of the\nlatticeRσ(0−) =Rσ(0+),which leads to\nNL\np\nkL(1+rσ\nL) =/summationdisplay\nstσ\nsrσ\ns. (26)\nContinuity of the stress or energy current at the interface\nz= 0,¯F(0−) =¯F(0+),leads to the boundary condition\nb2\nM0/summationdisplay\nstσ\nsmσ\ns=iαLNL\np\nkL(1−rkL)−iαR/summationdisplay\nstσ\nsrσ\nskσ\ns(27)\nIntegrating the equation of motion over the interface\nleads to free boundary condition for the magnetization\n∂mσ/∂x(0+) = 0,whichimpliesthattheenergyandangu-\nlar momentum carried by spin waves vanish at the bound-\nary, leading to a relation for the transmission coefficients\n/summationdisplay\nstσ\nsmσ\nskσ\ns= 0 (28)\n3We have now three linear equations with three unknown\nvariables for a choice of chirality, viz.tσ\n1,tσ\n2andrσ\nL.We\ncan easily derive the coefficients as analytic functions of\nωand constituting parameters, but the expressions are\nlengthy, hence, not given here. We used parameters of\ngadolinium gallium garnet (GGG) for the actuator: ρL=\n7085kg/m3,αL= 9.0×1010Pa, which implies small but\nfinite acoustic mismatch. In Fig. 3, the flux components\nat the interface are plotted as a function of frequency for\nthe parameters used in the dispersion relations (Fig. 1).\nFar from the (anti) crossing we observe weak reflection of\nthe incoming sound wave, which is a consequence of the\ngood acoustic impedance matching assumed here. Trans-\nmission into the phonon-like root sthat changes signs be-\ntween low and high frequencies dominates. The transmis-\nsion and reflection probabilities look complicated because\nthe spin wave dispersion is so flat; the anticrossing over-\nlaps with the FMR frequency ω0for spin wave excitation,\nindicated by the arrow on the abscissa. This is so because\nthe exchange energy at the anticrossing is insignificant as\ncompared to the Zeeman energy. In the neighborhood of\nthe anticrossing we observe the typical mode conversion\nplus an additional contribution to the back reflection of\nacoustic energy into the actuator.\n5. Detection of magnetization dynamics by spin\npumping\nUchidaet al. [6] and Weiler et al.[7] detected the\nacoustically induced magnetization dynamics by the spin\ncurrentpumped [8] intoathin layerofPtwith asignificant\nspin Hall angle θSH[22]. In terms of the spin mixing con-\nductanceg↑↓\nrthe magnitude and polarization of the spin\ncurrent reads [8]\nIpump\ns=/planckover2pi1\n4πg↑↓\nr\nM2\n0M×dM\ndt(29)\n=/planckover2pi1\n4πg↑↓\nr\nM2\n0/parenleftBigg\nM0˙Mxˆ y−M0˙Myˆ x\n+/parenleftBig\nMx˙My−My˙Mx/parenrightBig\nˆ z/parenrightBigg\n.(30)\nForM2\nx+M2\ny≪M2\n0the average cone angle\nΘ =/radicalBig/angbracketleftbig\nM2x+M2y/angbracketrightbig\nz,t/M0 (31)\n= 2/radicalBigg/summationdisplay\ns/parenleftBig/vextendsingle/vextendsinglet+s/vextendsingle/vextendsingle2/vextendsingle/vextendsinglem+s/vextendsingle/vextendsingle2+/vextendsingle/vextendsinglet−s/vextendsingle/vextendsingle2/vextendsingle/vextendsinglem−s/vextendsingle/vextendsingle2/parenrightBig\n/M0(32)\nisaconvenientmetricfortheexcitationofthemagneticde-\ngree of freedom per unit acoustic energy flux. The angular\nbrackets indicate a time and position average over inter-\nference fringes that are an artifact of the one-dimensional\nand monochromatic approximations. Θ( ω) is plotted in\nFig. 4. The qualitative features can be understood in\nterms of competition between magnetic character and ex-\ncitation efficiency of the eigenmodes. As the frequency ap-\nproaches the anticrossing from below, the increasing mag-\nnetic character of the propagating normal mode leads toan increasing Θ. Just below the FMR frequency, the nor-\nmal mode becomes flat, corresponding to a small group\nvelocity which reduces mode excitation efficiency (see Fig.\n3). Just above the FMR frequency, a new propagating\nnormal mode becomes available which restores a relatively\nefficient mode excitation. The highest Θ is achieved close\nto the crossing at which the acoustic and magnetic modes\nare fully mixed. Finite dissipation, finite quality factor of\nthe actuator, and disorder disregarded here will broaden\nthe sharp features and reduce the magnetization ampli-\ntude, but the effects are believed to be minor for high\nquality YIG and proper actuator design.\nWhen the Hall contacts of the Pt layer are short cir-\ncuited, the spin current induces a Hall charge current in\nthe Pt layer with direction and magnitude [22]\nIc=2e\n/planckover2pi1θSHˆ z×Ipump\ns (33)\nwith\n(Ic)x,y=−eg↑↓\nr\n2πθSH˙Mx,y\nM0(34)\n/radicalBigg/angbracketleftbigg\n|(Ic)x|2+/vextendsingle/vextendsingle/vextendsingle(Ic)y/vextendsingle/vextendsingle/vextendsingle2/angbracketrightbigg\n=eg↑↓\nrθSH\n2πM0/radicalbigg/angbracketleftBig\n(˙Mx)2/angbracketrightBig\n+/angbracketleftBig\n(˙My)2/angbracketrightBig\n(35)\n=eg↑↓\nrθSHωΘ\n2π. (36)\nWe see that the DC spin current component does not gen-\nerate an inverse spin Hall effect in this configuration, but\na pure AC inverse spin Hall effect is expected [17]. The\ncurrents in the xandy-direction oscillate at the frequency\nof the actuating ultrasound with amplitude ∼Θ,in con-\ntrast to the DC inverse spin Hall effect that scales with\nΘ2and is orders of magnitude smaller for small acoustic\nenergy fluxes. The FMR generated AC inverse spin Hall\neffect is difficult to observe since the Pt layer is directly\nsubjected to rf radiation, which causes strong electrody-\nnamic artifacts at the resonance frequency. The acousti-\ncally generated AC inverse spin Hall effect does not suffer\nfrom this disadvantage when the piezoelectric material is\nspatially separated from the magnetic material.\n6. Conclusions\nWe computed the acoustically stimulated magnetiza-\ntion dynamics and the associated spin-pumping induced\nac inverse spin Hall effect for a symmetric configuration\nof transverse acoustic waves polarized normal to the mag-\nnetization direction. The theory can be extended to in-\nclude longitudinal phonons (pressure waves) and arbitrary\nmagnetization direction, finite quality factor of the actu-\nator, finite magnetizations damping, multilayered struc-\ntures, diffuse scattering by disorder, etc., if necessary. In\nprinciple this formalism can be extended as well to ob-\ntain spin and heat transport through arbitrary structures\n4under a temperature difference between the reservoirs by\nincluding incoming and outgoing waves with all wave vec-\ntors.\nAcknowledgement\nTheauthorsthankPengYanandSebastianG¨ onnenwein\nfor useful discussions. This work was supported by the\nFOM Foundation, Marie Curie ITN Spinicur, Reimei pro-\ngram of the Japan Atomic Energy Agency, EU-ICT-7\n“MACALO”, the ICC-IMR, DFG Priority Programme\n1538 “Spin-Caloric Transport”, and Grand-in-Aid for Sci-\nentific Research A (Kakenhi) 25247056.\nReferences\n[1] C. Kittel, Introduction to Solid State Physics, (John Wiley,\nHoboken, 2005).\n[2] A. Einstein and W. J. de Haas, Deutsche Physikalische\nGesellschaft, Verhandlungen 17, 152 (1915).\n[3] S. J. Barnett, Phys. Rev. 6, 239 (1915); S. J. Barnett, Rev.\nMod. Phys. 7, 129 (1935).\n[4] C. Kittel, Phys. Rev. 110, 836 (1958).\n[5] S. Chikazumi, Physics of Ferromagnetism, (Oxford University\nPress, Oxford, 1997).\n[6] K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrand s,\nS. Maekawa, and E. Saitoh, Nat. Mater. 10, 737 (2011).\n[7] M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka, R. Gross, and\nS. T. B. Goennenwein, Phys. Rev. Lett. 108, 176601 (2012).\n[8] Y. Tserkovnyak, A. Brataas, and G. E.W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n[9] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys .\nLett.88, 182509 (2006).\n[10] T.Kobayashi, R.C.Barker, J.L. Bleustein, and A. Yelon , Phys.\nRev. B7, 3273 (1973).\n[11] L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gros s, M.\nS. Brandt, and S. T. B. Goennenwein, Phys. Rev. B 86, 134415\n(2012).\n[12] H. Boemmel and K. Dransfeld, Phys. Rev. Lett. 3, 83 (1959).\n[13] I. Feng, M. Tachiki, C. Krischer, and M. Levy, J. Appl. Ph ys.\n53, 177 (1982).\n[14] H. F. Tiersten, J. Math. Phys. 5, 1298 (1964).\n[15] S. Datta, Electronic Transport in Mesoscopic Systems , (Cam-\nbridge University Press, Cambridge, 2003).\n[16] Y.V.Nazarovand Y.M.Blanter, Quantum Transport: Introduc-\ntion to Nanoscience , (Cambridge University Press, Cambridge,\n2009).\n[17] H.Jiaoand G.E.W.Bauer, Phys.Rev.Lett. 110,217602 (2013).\n[18] F. G. Eggers and W. Strauss, J. Appl. Phys. 34, 1180 (1963).\n[19] P. Hansen, Phys. Rev. B 8, 246 (1973).\n[20] YIG Crystal Specification Sheet, Deltronic Crystal Ind ustries\n(Inc.), http://deltroniccrystalindustries.com .\n[21] A. Akhiezer, V. Bar’yakhtar, and S. Peletminskii, Spin Waves\n(North Holland Publishing Company, Amsterdam 1968).\n[22] A. Hoffmann, IEEE Trans. Magn. 49, 5172 (2013).10−1100101−4−20246810x 1014\nω (GHz)k2 (m−2)\n  \n(k2)1+\n(k2)2+\n(a)\n10−1100101−16−14−12−10−8−6−4−202x 1014\nω (GHz)k2 (m−2)\n  \n(k2)1−\n(k2)2−\n(b)\n2.7 2.75 2.8 2.85 2.900.511.522.533.5x 106\nω (GHz)k (m−1)\n  \nk1+\nk2+\n(c)\n2.7 2.75 2.8 2.85 2.900.511.522.533.5x 106\nω (GHz)k (m−1)\n  \nk1−\nk2−\n(d)\nFigure 1: Dispersion relations of magnetoelastic waves in\nYttrium Iron Garnets according to Eq. (7). (c-d) [(a-b)]\ndepict the [squared] momenta of the eigenstates for polar-\nization along and against the magnetic order parameter as\na function of frequency. Imaginary momenta appear on\nthe abscissa corresponding to zero real part. The FMR\nresonance frequency ω0is indicated by the arrow on the\nabscissa.51\nrt\nFigure 2: Schematic scattering problem of a phonon re-\nflected and transmitted at the interface of a ferromagnet.\nThe red arrow is the magnetization ( z-) direction.\n2.78 2.79 2.82.81 2.82 2.83 2.84 2.8500.20.40.60.81\nω  (GHz)Flux normalised to incoming flux\n  \nReflected wave\nTransmitted wave 2\nTransmitted wave 1\nTotal flux\nFigure 3: Normalized flux components |rσ\nL|2,|tσ\n1|2,and\n|tσ\n2|2as a function of frequency close to the anticrossing\nbetween magnon and phonon modes. The labels of the\ntransmitted waves are consistent with the labeling of the\nwave numbers in Fig. 1 (a) and (c). Flux conservation\n|rσ\nL|2+|tσ\n1|2+|tσ\n2|2= 1 is demonstrated. The FMR reso-\nnance frequency ω0is indicated by the arrow on the ab-\nscissa.2.782.792.82.812.822.832.842.8500.0050.010.0150.020.0250.030.035\nω  (GHz)Θ\nFigure 4: Θ (Eq. (31)) vs. frequency of the injected sound\nwave for unit injected acoustic energy F0= 1J//parenleftbig\nm2s/parenrightbig\n(Θ∼√F0) . Θ is the average precession cone angle and a\nmetric for the efficiency of acoustic excitation of the mag-\nnetic degree of freedom. The FMR resonance frequency\nω0is indicated by the arrow on the abscissa.\n6"    },    {        "title": "0712.0404v2.Finite_size_effects_on_spin_torque_driven_ferromagnetic_resonance_in_spin_valves_with_a_Co_Ni_synthetic_free_layer.pdf",        "content": "arXiv:0712.0404v2  [cond-mat.mes-hall]  14 Feb 2008Finite size effects on spin-torque driven ferromagnetic res onance in spin-valves with a\nCo/Ni synthetic free layer\nW. Chen, G. de Loubens, J-M. L. Beaujour, A. D. Kent\nDepartment of Physics, New York University, New York, NY 100 03\nJ. Z. Sun\nIBM T. J. Watson Research Center, Yorktown Heights, NY 10598\n(Dated: October 18th, 2007)\nSpin-torque driven ferromagnetic resonance (ST-FMR) is us ed to study magnetic excitations\nin Co/Ni synthetic layers confined in nanojunctions. Field s wept ST-FMR measurements were\nconducted with a magnetic field applied perpendicular to the layer surface. The resonance lines\nwere measured under low amplitude excitation in a linear res ponse regime. The resulting resonance\nfields were compared with those obtained using conventional rf field driven FMR on extended films\nwith the same Co/Ni layer structure. A lower resonance field i s found in confined structures. The\neffect of both dipolar fields acting on the Co/Ni layer emanati ng from other magnetic layers in the\ndevice and finite size effects on the spin wave spectrum are dis cussed.\nOne approach to study ferromagnetic resonance\n(FMR) of a magnetic layer in a confined structure is\nto use the spin transfer interaction [1, 2] in a current-\nperpendicular (CPP) nanojunction. An rf current is ap-\npliedtoamagnetictunnel junction[3]orspinvalve[4], to\ndrive FMR, in a method known as the spin-torque-driven\nferromagnetic resonance (ST-FMR). This new technique\nenables quantitative studies of magnetic properties of\nmaterials in nanopillars, such as their magnetic excita-\ntions, anisotropy and damping.\nSpin-transfer devices that incorporate materials with\nperpendicular magnetic anisotropy are of great interest.\nThis is because of their potential to lead to faster ST-\ndevices, with lower power dissipation [5] and critical cur-\nrent [6]. Recently, Mangin et al.studied perpendicular\nspin valves with a Co/Ni multilayer free layer, where a\nlarge magnetoresistance value and a high spin torque ef-\nficiency were observed [7].\nIn this work, we present ST-FMR studies on bilayer\nspin valves, where the thin (free) layer is composed of\na Co/Ni synthetic layer and the thick (fixed) layer is\npure Co. By comparing the ST-FMR resonance fields\nwith those of conventional rf field driven FMR of ex-\ntended Co/Ni films with the same layer structure, we il-\nlustrate interactions important in ST-FMR of nanojunc-\ntions. Specifically, we discuss both dipolar interactions\nbetween the Co/Nilayerand other magneticlayersin the\ndevice, and finite size effects on the magnetic excitation\nspectrum.\nPillar junctions with submicron lateral dimensions and\nrectangular shape, shown in Fig. 1a, were patterned on\na silicon wafer using a nanostencil process [8]. Junctions\nwere deposited by evaporation, and have the layer struc-\nture/bardbl1.5 nm Cr|100 nm Cu |20 nm Pt |10 nm Cu |[0.4\nnmCo|0.8nmNi] ×3|10nmCu |12nmCo |200nmCu /bardbl.\nThe ST-FMR measurement setup is shown in Fig. 1(a).\nAnrfcurrentgeneratedbyahighfrequencysourceiscou-\npled with a dc current through a bias-T (the dashed-line\nbox in Fig. 1(a)) into the spin valve. At resonance, the rfcurrentandspin valveresistanceoscillateatthe samefre-\nquency. This results in a dc voltage ( V=< I(t)R(t)>)\n[3, 4]. Assuming a small angle circular precession of the\nfree layer on resonance,\nV=1\n4(RAP−RP)Irfsinβsinθ (1)\nwhereβis the angle between the free and fixed layers\n(before applying the rf current) and θis the precession\nangle.Irfrepresentsthe rf current amplitude in the junc-\ntion, and RAP(RP) is the static junction resistance when\nfreelayerandfixedlayerareantiparallel(parallel)toeach\nother. With a perpendicular magnetic field greater than\nthe free layer’s easy-plane anisotropy field, the free layer\nmagnetization is normal to the surface, while the fixed\nlayer, which has a larger easy-plane anisotropy field, is\nstill mainly magnetizedin the film plane. In this way, the\nsignal is maximized, according to Eq. 1. To improve the\nsignal (typically in the sub- µV range) to noise ratio, we\nmodulate the rf current on and off at 800 Hz and use a\nlock-in amplifier to detect the voltage at this frequency.\nExtended films with the same stack as the free layer\nwere deposited on an oxidized Si wafer using the same\ndeposition technique. It has the same Co/Ni synthetic\nlayer sandwiched between 10 nm of Cu on each side. As\nis shownin Fig. 1(b), it wasmeasuredby placingthe film\nonto a 50-Ohm-matched coplanar waveguide [9]. An rf\ncurrent wassent through the waveguideand generatesan\nrf field that drives the magnetic film into resonance. The\ntransmission of the rf signal was measured using a net-\nwork analyzer, as a function of rf frequency and external\nmagnetic field.\nThe magnetoresistance(MR) of the nanojunctions was\nmeasured using a four-point geometry with the magnetic\nfield applied in the film plane. A typical MR hysteresis\nloop of a 50 ×150 nm2junction is shown in the inset of\nFig. 2(a). MR= ( RAP−RP)/RP, is≃2.3±0.2% for a\ntotal of 10 junctions studied.\nST-FMR measurements were conducted with the ex-\nternal magnetic field Happappliednearlyperpendicular2\nFIG. 1: (a): Spin valve layer structure and ST-FMR circuit.\n(b): Field-driven FMR on same-stack extended films using\nthe flip-chip method.\nto the film plane (2◦off the normal direction as shown in\nFig. 1(a), where the small in-plane component is along\nthe easy-axis of the junction, in order to avoid vortex\nstates in the free layer). This was measured using a two-\npoint geometry. Fig. 2(a) shows a typical field-swept\nresonance line at a fixed rf frequency of 18 GHz and zero\ndc current. It was measured on the same junction on\nwhich the data in the inset was taken. The resonance is\nfit by a Lorentzian, indicated by the solid line. From the\npeak height Vpeak, we estimate the precession angle to be\n1.9◦using Eq. 1. We verified that this data was taken\nin a linear response regime with Vpeak/I2\nrfindependent of\nIrf.\nA series of ST-FMR resonance lines at different rf fre-\nquencies fwere measured within the low amplitude lin-\near regime. Those with 7 different rf frequencies (4 ∼16\nGHz in 2 GHz steps) are plotted in Fig. 2(b), with each\nadjacent curve offset by 0 .2µV. The resonance field Hres\n(/trianglesolidin Fig. 2(b)) increases linearly with fgreater than\n4 GHz. At f <4 GHz, the perpendicular magnetic field\nat resonance is lower than the easy-plane anisotropy field\nof the free layer. Therefore, the free layer magnetization\ntends to tilt into the plane, leading to a lower resonance\nfield. Similar dispersion relationships have been found\non junctions with other lateral dimensions.\nThe resonance field for both extended films and nano-\njunctions as the function of fare plotted in Fig. 3(a).\nBlack dots are for a similar spin valve with the same lat-\neraldimensionand pinkdots forthe same-stackextended\nfilm. Further details on the conventional FMR experi-\nments can be found in Ref. [10]. Red and blue solid\nlines are their corresponding linear fits above 4 GHz. By\ncomparing these two sets of data, we find a slight dif-\nference in slope and a small shift in the zero frequency\nintercept. The relationship between fandHresin the\nextended film is given as:h\nµBf= g(Hres−4πMeff) [11]\nin the case where the magnetization is normal to the film\nsurface. Here g is the Land´ e g factor of the film, and\nthe easy-plane anisotropy is 4 πMeff= 4πMs−HP, where\nMsandHPrepresent the saturation magnetization, and\nthe perpendicular anisotropy field. A direct linear fit of\neach data set gives a slope g=2.17, field-axis intercept\n4πMeff= 2.58 kOe for the extended film, and a slightly\nlarger slope (2.28) and a smaller field-axis intercept (1.92FIG. 2: (a): ST-FMR voltage signal ( /squaresolidpoints) as a func-\ntion of applied perpendicular magnetic field together with a\nLorentzian fit (solid line). The measurement was done on a\n50×150 nm2junction with an rf amplitude of Irf=560µA at a\nfrequency of 18 GHz. Inset: MR hysteresis loop on the same\njunction with the magnetic field applied in-plane. (b): Zero\ndc current lock-in voltage signal as the function of applied\nmagnetic field at different frequencies from 4 GHz up to 16\nGHz in 2 GHz steps.\nkOe) for the confined structure in the spin valves. Such a\nconsistency confirms that the main peak of the ST-FMR\nsignal comes from the Co/Ni synthetic free layer rather\nthan the fixed Co layer [12].\nWe now estimate the effect of dipolar fields Hdipfrom\nothermagneticlayersin spinvalves, which comefromthe\nfixed Co layer and the junction level magnetic residuals\noutside the stencil holes [8]. Hdipfrom the normal com-\nponent of the fixed Co layer is not negligible. At f= 10\nGHz, the Co layer is tilted ∼16◦out-of-plane at Hres,\nand the component of Hdipnormal to the film surface\nis 500 Oe. At higher frequencies, the normal component\nofHdipis larger since the fixed Co layer magnetization\nis more tilted out-of-plane for larger Hres.Hdipof the\njunction level Co residual is estimated to be ∼15 times\nsmaller, while that of the junction level Co/Ni residual\nhas a constant -120 Oe field contribution normal to the\nsurface. The in-plane component of the dipolar fields\n(/lessorsimilar260 Oe) shifts Hresdown by ∼30 Oe at 4 GHz, and\nthis shift is significantly reduced at higher frequencies.\nTherefore, the dipolar fields from the normal component\nof the fixed Co layer and from the junction level Co/Ni\nresidual areimportant in biasing Hresof free layerin spin\nvalves. The dashed line in Fig. 3(a) is the linear fit of\nHreswith the dipolar field correction. In the new linear\nfit, the slope becomes 2.18, which is almost the same as\nthe g factor of the extended film, and the field-axis inter-\ncept becomes 2.13 kOe. A 450 Oe shift between them at\nall frequencies still remains.\nSimilar results have been found in other spin valve\njunctions. ST-FMR results taking into account the dipo-\nlar fields from other magnetic layers in the device show\na similar g-factor (2.18 ±0.03) with a 370 ∼570 Oe lower\nresonance field than that of the extended film. Numeri-3\nFIG. 3: (a): Comparison of resonance field as the function\nof frequency between the spin valve junction (black dots) an d\nsame-stack extendedfilm (pink dots). The spin valve junctio n\nis a different one from that in Fig. 2, but has the same lateral\ndimension. Solid lines are linear fits. Dashed line: linear\nfit of resonance field with estimated dipolar fields corrected .\n(b): The dispersion of the lowest four spin wave modes on a\n50×150×3.6 nm3Co/Ni synthetic structure using OOMMF\nsimulation (dots) and the analytical model discussed in the\ntext (dashed lines). Solid line is the linear fit of Hresof the\nextended film. Corresponding mode profiles are shown in the\nlower-right corner, in the order of ( nx,ny)=(1, 1) (1, 2) (1, 3)\n(2, 1) from bottom to top.\ncal calculation on the normal spin wave (SW) modes in\nelliptical Py disks was presented in Ref. [13], and the\nresonance shift of different SW modes due to the finite\nsize effect was discussed. In order to estimate the reso-\nnance shift in our rectangular Co/Ni synthetic nanoele-\nment, wehavealsodonemicromagneticsimulationsusing\nOOMMF [14], which includes the Zeeman, exchange and\nmagnetostatic contributions to the energy (but not the\nspin transfer interaction). The modes of a 50 ×150 nm2\nelement are shown in lower-right of Fig. 3(b). The mode\nresonanceis shifted towardslowerfield asthe orderofthe\nSW modes becomes higher (dots in Fig. 3(b)). The low-\nest order SW mode is shifted by ∼1 kOe from the thin\nfilm result (solid line). This shift is the same order of\nmagnitude as found in our experiments. It is larger than\nthat observed possibly because the lateral dimensions of\nthe pillar are larger than the nominal dimensions. The2nd mode is close to the lowest one, while 3rd and 4th\nones are further apart. This shows the effect of finite size\nand mode structure on the resonance field. Furthermore,\nmultiple peaks found at severalfrequencies (see Fig. 2(b)\natf= 6 GHz, for example), may be due to the excitation\nof higher order SW modes.\nAn analytical estimation of the energy of the SW\nmodes in a rectangular element enables a better under-\nstanding of the physics. The idea is to assume a sinu-\nsoidal profile of the eigenmode in the finite rectangular\nelement [15], and to use the dipole-exchange dispersion\nof the SW modes in a perpendicularly magnetized film\n[16]:\nω2\nk=γ2(Hin+2Ak2\nMs)[Hin+2Ak2\nMs+4π(1−1−e−kt\nkt)Ms](2)\nwhereγis the gyromagnetic ratio, and the internal field\nHin=Happ−4πNzMs+HP, whereNzis the demag-\nnetization factor in the normal direction.2Ak2\nMsis the\nexchange term, with A and k representing exchange con-\nstant and in-plane wave vector. 4 π(1−1−e−kt\nkt)Msis the\ndipolar term in which t is the thickness of the disk, and\nit describes the SW dynamic dipole-dipole interaction.\nMoreover, as discussed in Ref. [17], the oscillating mag-\nnetizationisonlypartiallypinnedattheboundary. Thus,\nalargereffectivelateraldimensionneedstobeintroduced\nin the analyticalcalculationtomimic the partialpinning.\nThe 1 kOe shift, like that in the OOMMF simulation,\ncan be found using Eq. 2 by introducing an effective lat-\neral dimension Lx×Ly∼70×170 nm2. Using these di-\nmensions, the higher order modes denoted as ( nx,ny) are\nalsoquantitativelyreproducedwith kx, y=nx, yπ/Lx, y,\nnx, y∈N∗(see dashed lines in Fig. 3(b)).\nIn summary, we used the ST-FMR technique to mea-\nsure the resonance fields of Co/Ni synthetic layers in a\nconfined spin valve structure and compared it to that\nof the same-stack extended film. The effects of dipolar\nfields and finite element size produce changes in the reso-\nnance field that are the same orderof magnitude as those\nobserved in the experiment.\nThis research is supported by NSF-DMR-0706322,\nARO-W911NF-07-1-0643 and an NYU-Research Chal-\nlenge Fund award.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub-\nota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira,\nN. Watanabe, and S. Yuasa, Nature 438, 339 (2005).\n[4] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N.\nKrivorotov, R.A. Buhrman, andD. C. Ralph, Phys. Rev.\nLett.96, 227601 (2006).\n[5] A. D. Kent, B. ¨Ozyilmaz, and E. del Barco, Appl. Phys.Lett.84, 3897 (2004).\n[6] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[7] S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey,\nB. D. Terris, and E. E. Fullerton, Nature Materials 5,\n210 (2006).\n[8] J. Z. Sun, Appl. Phys. Lett. 81, 2202 (2002).\n[9] J.-M. L. Beaujour, J. H. Lee, A. D. Kent, K. Krycka, and\nC.-C. Kao, Phys. Rev. B 74, 214405 (2006).\n[10] J.-M. Beaujour, W. Chen, K. Krycka, C.-C. Kao, J. Z.\nSun, and A. D. Kent, Eur. Phys. J. B 59, 475 (2007).4\n[11] C. Kittel, Introduction to Solid State Physics (John Wi-\nley & Sons, Inc., 1996).\n[12] FMR measurement on 10 nm thick extended pure Co\nlayers shows that the resonance field is much larger (a\nfew Tesla) for f >3 GHz, because of its high easy-plane\nanisotropy (17 kOe). At lower frequencies, the resonance\nfield decreases rapidly as the layer magnetization tilts\ninto the plane, with a slope far from the g ≃2 seen in the\nST-FMR studies.\n[13] R. D. McMichael and M. D. Stiles, J. Appl. Phys. 97,10J901 (2005).\n[14] M. J. Donahue and D. G. Porter, OOMMF User’s Guide\n(National Institute of Standards and Technology, 1999).\n[15] M. Bailleul, R. H¨ ollinger, and C. Fermon, Phys. Rev. B\n73, 104424 (2006).\n[16] B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013\n(1986).\n[17] K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and\nA. N. Slavin, Phys. Rev. B 66, 132402 (2002)."    },    {        "title": "0806.3805v1.Spin_dynamics_in_point_contacts_to_single_ferromagnetic_films.pdf",        "content": "arXiv:0806.3805v1  [cond-mat.mtrl-sci]  24 Jun 2008\n/CB/D4/CX/D2 /CS/DD/D2/CP/D1/CX\r/D7 /CX/D2 /D4 /D3/CX/D2 /D8 \r/D3/D2 /D8/CP\r/D8/D7 /D8/D3 /D7/CX/D2/CV/D0/CT /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /AS/D0/D1/D7/C7/BA /C8 /BA /BU/CP/D0/CZ /CP/D7/CW/CX/D21/B8 /CE/BA /CE/BA /BY/CX/D7/D9/D21,2/B8 /C1/BA /C3/BA /CH /CP/D2/D7/D3/D21/B8 /C4/BA/CH /D9/BA /CC /D6/CX/D4/D9/D8/CT/D21/B8 /BT/BA /C3 /D3/D2/D3 /DA /CP/D0/CT/D2/CZ /D32/B8 /CP/D2/CS /CE/BA /C3 /D3/D6/CT/D2/CX/DA/D7/CZ/CX2\n1/BU/BA /CE /CT/D6/CZ/CX/D2 /C1/D2/D7/D8/CX/D8/D9/D8/CT /CU/D3/D6 /C4 /D3/DB /CC /CT/D1/D4 /CT/D6 /CP/D8/D9/D6 /CT /C8/CW/DD/D7/CX\r/D7 /CP/D2/CS /BX/D2/CV/CX/D2/CT /CT/D6/CX/D2/CV/B8/C6/CP/D8/CX/D3/D2/CP/D0 /BT \r /CP/CS/CT/D1/DD /D3/CU /CB\r/CX/CT/D2\r /CT/D7 /D3/CU /CD/CZ/D6 /CP/CX/D2/CT/B8 /BI/BD/BD/BC/BF /C3/CW/CP/D6/CZ/CX/DA/B8 /CD/CZ/D6 /CP/CX/D2/CT/BN 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/D3 /DB /CT/D6 /D3/CU /CX/D6/D6/CP/CS/CX/CP/D8/CX/D3/D2 /CP/D8 /BE /BZ/C0/DE/B8 /CU/D3/D6/CA/BY /D4 /D3 /DB /CT/D6P /BP/BC/B8 /BE/BA/BG/B8 /CP/D2/CS /BF/BA/BIµ /CF /B4\r/D9/D6/DA /CT/D7 /BD/B8 /BE/B8 /CP/D2/CS /BF/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD/B5/BA /CC/CW/CT \r/D9/D6/DA /CT/D7 /CP/D6/CT /D7/CW/CX/CU/D8/CT/CS /DA /CT/D6/D8/CX\r/CP/D0/D0/DD/CQ /DD /BDΩ /CU/D3/D6 \r/D0/CP/D6/CX/D8 /DD /BA /CC/CW/CT \r/D3/D2 /D8/CP\r/D8 /CX/D7 /BV/D9/B9/BV/D3/B4/BD/BC/BC /D2/D1/B5/B8R0= 5.04 Ω /B8H⊥= 2.47 /CC/BA /CC/CW/CT /CX/D2/D7/CT/D8 /D7/CW/D3 /DB/D7 /D8/CW/CT/CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D8/CW/CT /CX/D2/CS/D9\r/CT/CS /D4 /CT/CP/CZ /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /CA/BY /D4 /D3 /DB /CT/D6/BA/BD/BE /BZ/C0/DE/B5 /CP/D2/CS /BD/BC /CS/BU/D1 /B4/BJ/B9/BD/BE /BZ/C0/DE/B5/B8 /D1/CT/CP/D7/D9/D6/CT/CS /CP/D8 /D8/CW/CT /D3/D9/D8/D4/D9/D8 /D3/CU /D8/CW/CT /CV/CT/D2/CT/D6/CP/D8/D3/D6/BA /CC/CW/CT /CW/CX/CV/CW/CT/D6/D4 /D3 /DB /CT/D6 /CP/D0/D0/D3 /DB /CT/CS /D8/D3 /CT/D2/CW/CP/D2\r/CT 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/C2/BA /BU/CP/D7/D7/B8 /CF/BA /BV/BA /BV/CW/CX/CP/D2/CV/B8 /CE/BA /CC /D7/D3/CX/B8 /CP/D2/CS /C8 /BA /CF /DD/CS/CT/D6/B8 /C6/CP/D8/D9/D6/CT/B8 /BG/BC/BI/B8 /BG/BI/B4/BE/BC/BC/BC/B5/BA/BD/BG"    },    {        "title": "2101.02411v1.Microwave_directional_dichroism_resonant_with_spin_excitations_in_the_polar_ferromagnet_GaV__4_S__8_.pdf",        "content": "arXiv:2101.02411v1  [cond-mat.str-el]  7 Jan 2021Microwave directional dichroism resonant with spin excita tions in the polar\nferromagnet GaV 4S8\nY. Okamura1, S. Seki2, S. Bord´ acs3,4,´A. Butykai3, V. Tsurkan5, I. K´ ezsm´ arki3, and Y. Tokura1,2\n1Department of Applied Physics and Quantum Phase Electronic s Center,\nUniversity of Tokyo, Tokyo 113-8656, Japan\n2RIKEN Center for Emergent Matter Science (CEMS),\nWako 351-0198, Japan\n3Department of Physics,\nBudapest University of Technology and Economics and MTA-BM E\nLendulet Magneto-optical Spectroscopy Research Group,\n1111 Budapest, Hungary\n4Hungarian Academy of Sciences,\nPremium Postdoctor Program, 1051 Budapest, Hungary\n5Experimental Physics V,\nCenter for Electronic Correlations and Magnetism,\nUniversity of Augsburg, 86159 Augsburg, Germany\nWe have investigated the directional dichroism of magnetic resonance spectra in the polar ferro-\nmagnet GaV 4S8. While four types of structural domains are energetically d egenerated under zero\nfield, the magnetic resonance for each domain is well separat ed by applying magnetic fields due to\nuniaxial magnetic anisotropy. Consequently, the directio nal dichroism as large as 20 % is clearly\nobserved without domain cancellation. The present observa tion therefore demonstrates that not\nonly magnetoelectric mono-domain crystals but also magnet oelectric multi-domain specimens can\nbe used to realize microwave (optical) diodes owing to the la ck of inversion domains.\nPACS numbers:\nThe light-matter interaction plays a fundamental role\nin various fields of physics, chemistry and engineering.\nThe optical response is highly sensitive to the material\nproperties even beyond the static ones and often help to\nrealize versatile functionalities. Multiferroics, materials\nwith simultaneouslybrokenspatialinversionand time re-\nversal symmetry, provide a unique arena to study novel\noptical phenomena that cannot show up in conventional\nferromagnets or ferroelectrics [1–4]. Such unconventional\noptical response, called the optical magnetoelectric (ME)\neffect, occurs when coexisting magnetic and ferroelectric\norders simultaneously interact with the oscillating elec-\ntric field and magnetic field of light. The discovery of\nthis phenomenon was first reported for a noncentrosym-\nmetric antiferromagnetCr 2O3[5], where the polarization\nrotationoflightisreverseddependingonthe propagation\ndirection of the light, although the effect is rather small.\nAnother type of the optical ME effect, termed as non-\nreciprocal directional dichroism (DD), was demonstrated\nsubsequently, where the light beams travelling in oppo-\nsitedirectionsareabsorbeddifferently[6–21]. Ithasbeen\nrevealed that the DD can become large and even show a\ncloakingfunction, thusbeingparticularlyimportantfrom\nthe view point of future applications [13, 14].\nThe DD is observed for various types of excitations\nfrom X-ray to microwave-frequency regions in multifer-\nroic materials [6–21]. Among them, the DD can often\nbe large for the collective spin excitations that are both\nelectric- and magnetic-dipole active owing to the strongME coupling. These novel elementary excitations, re-\nferred to as the magnetoelectric resonances, are accom-\npanied by the resonant motion of the magnetization ( M)\nas well as that of the electric polarization ( P), which en-\nhances the ME coupling and leads to large DD. Recently,\nas a new guiding principle towards large DD, magneto-\nelectric resonances in the type-I multiferroics, in which\nmagnetic orders develop within a pre-existing ferroelec-\ntric or pyroelectric phase [22], is attracting growing in-\nterests because this class of materials often exhibits large\nspin-induced changes of the electric polarization [15–17].\nThe lacunar spinel GaV 4S8is a new member of the type-\nI multiferroics, which is an excellent candidate material\nto show large DD since it is a rare polar ferromagnet\n[23, 24].\nIn this Letter, we have studied the microwave DD at\nthe ferromagnetic resonance in the polar magnetic semi-\nconductor GaV 4S8. While the sample consists of multi-\nple polar rhombohedral structural domains in which the\nmagnetic states aredegeneratein zeromagnetic field, the\ndegeneracy of the magnetic excitations on the different\ntypes of domains is split in external magnetic fields ow-\ning to the axial magnetic anisotropy. As the result, the\nDD as large as 20 % is clearly observed free from can-\ncelation among the domains. The magnitude critically\ndepends on each domain, indicating the vital role of the\nmicroscopicME coupling besides macroscopiclifted sym-\nmetry.\nThe lacunar spinel structure of GaV 4S8, which has the2\nP1\nHH\nm’\nsample\nHH = 0\nk(f) (g) (e) (b) (a) \nV\nS\nGa \n(d) (c) \ndomain A domain B domain C \n[112][111]\n[110] H/g90 E/g90\nCPWCPWP4P3P2\nP4P1[001] \n[010] [100] [001] \n[010] [100] \nP2\nH\nm’H\nm’P3\nFIG. 1: (color online) (a) Crystal structure of GaV 4S8at\nroom temperature. (b) The schematic illustration of V 4clus-\nters. The V ion is displaced along four equivalent cubic /angbracketleft111/angbracketright\naxes below the structural phase transition temperature. (c -\ne) Three types of structural domains classified based on the\nrelationship with respect to H/bardbl[11¯2].m′represents a mir-\nror plane combined with time reversal operation as indicate d\nby the blue shaded region. (f) Schematic illustration of the\ncoplanar wave guide (CPW). (g) Cross sectional view of the\nCPW; the spatial distribution of microwave electric and mag -\nnetic fields are illustrated.\nspace group symmetry F ¯43m at room temperature, is\nshown in Fig. 1(a) [26, 27]. It consists of a network of\n(V4S4)5+clusters that form a face centered cubic lattice.\nBelow 42 K, due to cooperative Jahn-Teller distortion,\nthe triply degenerate molecular orbitals of V 4clusters\nare lifted through the elongation of the lattice along any\nofthe four cubic /angbracketleft111/angbracketrightaxes[Fig. 1(b)]. The crystalstruc-\nturethenbecomespolar(R3m) witharelativelylargepy-\nroelectric polarization, P≃1µC/cm2[24]. It should be\nnoted that the opposite-polarization domains may exist\ndue to the non-centrosymmetric nature of the room tem-\nperature cubic F ¯43m phase. However, the previous PFM\nand static pyro/magnetocurrent measurements indicate\nthat such inversion domains do not exist in these sin-\ngle crystals [24]. Therefore, we analyze the observed DD\nwithout consideringthe coexistenceofinversiondomains,\nwhich would lead to partial cancellation of the DD, i.e.\nits magnitude characteristic to a single inversion domain\nwould be larger than the observed one. The magnetic\ntransition occurs at Tc∼12.7 K, well below the struc-\ntural transition temperature. The cycloidal spin state is\nstabilized at zero field between 6 −12.7 K and turns into\na collinear field-polarized ferromagnetic state in moder-ate magnetic fields ( H) [28]. The N´ eel-type skyrmion\nlattice is observed in a specific temperature and mag-\nnetic field region [23, 28]. The stability of each magnetic\nphase critically depends on the magnetic-field direction\ndue to the easy-axisanisotropywith respect to the rhom-\nbohedral axis, as discussed later [Figs. 1(c-e)].\nSingle crystals of GaV 4S8were grown with chemical\nvapor transport and the detail of the growth procedure\nis described elsewhere [23]. We performed broadband\nmicrowave spectroscopy to measure the transmission co-\nefficient of the sample mounted on a coplanar waveg-\nuide (CPW) as shown in Fig. 1(f). The signal line\nwas designed to be 20 µm in width so that it is much\nsmaller than the sample width; the sample dimension\nis typically ∼2×2×1 mm3. The directions of oscillat-\ning magnetic and electric field of the microwave, de-\nnoted as HωandEωrespectively, depend on the po-\nsition in space: the Hωjust above the center of sig-\nnal line is parallel to the plane of the CPW, while the\nHωbetween the signal line and the ground is perpen-\ndicular to the plane [Fig. 1(g)]. The transmission coef-\nficient for a microwave propagating along kωand−kω\ndirections is denoted as S12andS21, respectively, and\nwas recorded with a vector network analyzer (Agilent\nTechnology, E8363C). The absorption spectrum associ-\nated with the magnetic excitations, denoted as ∆ S12(or\n∆S21) for the + kω(or−kω) microwave, was obtained by\ncalculating ∆ S12(21)=−S12(21)+S12(21)(T > T c), where\nS12(21)(T > T c) taken at T > T cdoes not contain the\nmagnetic signal (0.01 −35 GHz) (for more details, see\nSupplemental Material [29]).\nTo observe the DD in this material, we focus on\nthe magnetic resonance when kω/bardbl[1¯10] andH/bardbl[11¯2], as\nsketched in Fig. 1(g), because the DD can emerge for ev-\nery types of structural domains from the symmetry point\nof view, as shown in Figs. 1(c-e). For this choice of the\nmagnetic field, the single mirror plane remains for do-\nmainAandC.Incontrast,twokindsofdomainBsarein-\nterchanged by this mirror reflection combined with time\nreversal operation. This unique mirror plane combined\nwith the time reversal operation allows the emergence\nof DD for light beams propagating perpendicular to it\nbut not for beams travelling parallel to the plane [30].\nThis symmetry is also compatible with a phenomenolog-\nical toroidalmoment T=P×Mpointing perpendicular\nto the mirror plane [31]. In all the three kinds of mag-\nnetic domains classified in terms of the angle between\ntheir/angbracketleft111/angbracketright-type easy axes and the direction of the mag-\nnetic field (90 deg, 61.9 deg and 19.5 deg), Tis finite,\nalthough it has different magnitudes in the different do-\nmains [Figs. 1(c-e) and Figs. 4(b,c)].\nAs revealed in the previous magnetic-resonance study,\nthe resonance frequency critically depends on the mag-\nnetic domains, or equivalently, the magnetic-field direc-\ntion due to the uniaxial anisotropy [32, 33]. Thus, we\nfirst investigated the evolution of the magnetic resonance3\n/g84ʪ111ʫ\n/g73\n(c) (b) \ndomain A \ndomain A domain B domain C domain B \ndomain C (a) 2.0\n1.5\n1.0\n0.5\n0.0/g39S12  (dB)\n30 20 10 0\nFrequency (GHz)0 mT100 mT 200 mT 300 mT 400 mT 500 mT 600 mT 700 mT H||[112] \nk||[110] T = 8 K \n0102030Frequency (GHz)\n0 200 400 600M\n090 \n/g84\nMagnetic Field (mT)(degree) \nFIG. 2: (color online) (a) The magnetic resonance spectra fo r\neach magnetic field when kω/bardbl[1¯10] andH/bardbl[11¯2]. The spectra\nare shifted vertically for clarity. (b) Calculated angle of θ,\ni.e., the angle between the magnetization direction and pol ar\n/angbracketleft111/angbracketrightaxis, numerically calculated based on Eq. (1). (c) The\nmagnetic-field dependence of the resonance frequency (cir-\ncles). The calculated resonance frequency of ferromagneti c\nstates (solid lines).\nwith field forthe different magnetic domains. Figure 2(a)\nshowsthe magneticresonancespectraat8Kfor kω/bardbl[1¯10]\ninvariousmagneticfields H/bardbl[11¯2]. Inthecycloidalphase\nat zero field, three resonance peaks are observed at 5,\n18 and 27 GHz, in accord with a previous study [32].\nBy applying the magnetic field of 100 mT, the inten-\nsity of resonance modes at 5 and 18 GHz are weakened\nand enhanced, respectively, while the resonancepeak dis-\ncerned at 27 GHz immediately disappears. The absorp-\ntion peak observed at 27 GHz, indicated as the purple\ndots in Fig. 2(c), is therefore the resonance mode charac-\nteristic of the cycloidal spin structure. With further in-\ncreasing the field, the lower-lying mode slightly shifts to-\nwardshigher-frequency regionand the higher-lyingmode\nsplits into two modes. In domains B and C the cycloidal\nphase is suppressed and the field-polarized ferromagnetic\nstate is reached by fields below ∼100 mT. In contrast,\nin domain A the transverse conical state formed in finite\nfields is robust to ∼500 mT, around which a spin-flop\ntransition occurs to the ferromagnetic state. In the fol-\nlowingwefocus onthe magneticresonancesandtheir DD\nin the ferromagnetic state, where a simple phenomeno-\nlogical model can be used to describe the observed DD.\nIn general, the resonance frequency of the fer-\nromagnetic state can be calculated on the ba-\nsis of the Smith-Suhl formula [34],/parenleftBig\nω\nγ/parenrightBig2\n=+700 mT \n-700 mT/g39S12 \n/g39S21 \n/g39S12 \n/g39S21 (a)(c) \n(b)\n(d)\ndomain C \ndomain B \ndomain A \n0 200 400 60001020\nMagnetic Field (mT)Magnitude of DD (%)2.0\n1.5\n1.0\n0.5\n0.0Absorption (dB)\n30 20 10 0\nFrequency (GHz)/g39S12 \n/g39S21 \n0 mT100 mT 200 mT 300 mT 400 mT 500 mT 600 mT 700 mT 0.5\n0.4\n0.3\n0.2\n0.1\n0.0Absorption (dB)\n0.5\n0.4\n0.3\n0.2\n0.1\n0.0Absorption (dB)\n30 20 10 0\nFrequency (GHz)H||[112] \nk||[110] T = 8 K \nFIG. 3: (color online) The magnetic resonance spectra for\n+kωand−kωdirections at +700 mT (a) and -700 mT (b).\n(c) The + kωand−kωspectra for each magnetic field. The\nspectra are shifted vertically for clarity. (d) The magneti c-\nfield dependence of the magnitude of the directional dichro-\nism. The red, green and blue circles correspond to the lower- ,\nintermediate-, and higher-frequency modes, respectively .\n1\nM2sin2θ/parenleftbigg\n∂2E\n∂θ2∂2E\n∂φ2−/parenleftBig\n∂2E\n∂θ∂φ/parenrightBig2/parenrightbigg\n, where γ,MandE\nare the gyromagnetic ratio, magnetization and free\nenergy, respectively. θandφare the polar angle and\nazimuthal angle of the M, respectively, in a spherical\ncoordinate system [Fig. 2(b), inset]. In the present\nsystem, the free energy is given by,\nE=−M·H−K(M·z/M)2, (1)\nwhereKandzrepresent the uniaxial anisotropy and\nunit vector along [111] axis, respectively. The resonance\nfrequency is thus calculated as,\nω\nγ=/radicalbigg\nH2sin2θH+H2cosθsin2θH\n2sinθ+2KHsinθHcos2θ\nMsinθ,\n(2)\nwhereθHis the polar angle of the Hfromzaxis. On\nthe basis of Eq. (1), θcan be numerically calculated for\neach domain so as to minimize E, as shown in Fig. 2(b).\nBy substituting this result into Eq. (2), it is found that\nthe theory well reproduces the field dependence of the\nresonances in the ferromagnetic state with K/M= 260\nmT [Fig. 2(c)]. Here we used g= 1.82 estimated in4\nRef. [33] and the deduced K/Mvalue almost coincides\nwith the value determined in Ref. [33]. Thus, the low-,\nintermediate- and high-frequency modes observed at 700\nmT are attributed to resonance modes in the domain A,\nB and C, respectively. Note that the discrepancy be-\ntween the observed and calculated results for the domain\nC in lower field region is due to the subsisting transverse\nconical state up to 500 mT, not the ferromagnetic state\nassumed in the calculation.\nFigures 3(a) and 3(b) show the absorption spectra\nwhen the microwave propagates along + kωand−kωdi-\nrections at the field of ±700 mT, where the magnetic\nstate is ferromagnetic for all the domains. We found\na clear signature of the DD for the intermediate- and\nhigh-frequency modes: The resonance peaks are higher\nin the ∆S21spectrum than that in the ∆ S12spectrum at\n+700mT forboth modes. Thisrelationshipisreversedin\n−700 mT, further verifying the nonreciprocal nature of\nthe transmission. The DD spectra change systematically\nby applying magnetic fields, as shown in Fig. 3(c). The\nmagnitude of the DD, which is defined as∆S12−∆S21\n(∆S12+∆S21)/2,\nincreases monotonically for the higher-lying mode and\nis basically unchanged for the intermediate mode as in-\ncreasing the fields [Fig. 3(d)]. The magnitude of the DD\namountstoapproximately20%at+750mT,whichisthe\nlargest value among those reported for multiferroics in\nthe microwave frequency range to the best of our knowl-\nedge [18–21].\nNotably, although kω,PandMare perpendicular\nto each other in the domain A satisfying the necessary\ncondition to observe the DD [6–10, 12, 16–18, 21], the\nDD in the domain A is negligibly small. On the other\nhand, the domain C exhibits a strong DD, although for\nthis type of domain PandMare nearly antiparallel,\ni.e. toroidal moment T=P×Mis smaller as com-\npared to the domain A [Figs. 4(b-c)]. Moreover, the do-\nmains A and C are related with each other by rotating\n70.5 degrees around the kω/bardbl[1¯10]. These facts highlight\nthe crucial role the direction of the Min the ME cou-\npling. It should be noted that, although the Damon-\nEshbach mode or magnetostatic surface wave (at non-\nzero wavenumber) may show the nonreciprocal propaga-\ntion, itdoesnotleadtothenonreciprocalabsorption[34].\nMoreover, the Damon-Eshbach mode cannot be clearly\ndiscerned in the present experiment, thus allowing us to\nexclude its contribution to the presently observed DD.\nTounderstandtheroleoftheMEcoupling, weconsider\nthe instantaneous responses induced by HωandEωin\nthe same manner as adopted in Ref. [19]. From the view\npoint of the symmetry, the Pcan be described by,\nPx=aMxMz−bMxMy,\nPy=−bM2\nx+bM2\ny+aMyMz,\nPz=cM2\nz+dM2,(3)\nwherex,yandzrepresent [11 ¯2], [¯110] and [111] axes,2.0\n1.0\n0.0\n-1.0PH /MH\n/g90 /g90 (a.u.) \n600 400 200 0\nMagnetic Field (mT)\nP1\nP1111\n4\n4\n14\n4P1H||[112]\nMM\nMMM M\nz\nx y(b) (a) (f) \n(e) (d) \nP4\ndestructive=\nequivalentdomain A\nconstructivek||[1-10]k||[1T\n--\n10]k||[1-10]\nE/g90\nPH/g90k||[1-10]\nE/g90-kH/g90\nH/g90H/g90\nH/g90-kE/g90E/g90(c) \n= 0 PH/g90= 0\nz\nx ydomain C\nP4P4\nPH/g90PH/g90M\nP\nTT\nk -k M\nP\nT\nH||[112]-domain A x0.4 domain B domain C \nz\nxyz\nx\ny\nFIG. 4: (color online) (a) Schematic illustration of the DD\nand toroidal moment. (b,c) The experimental configurations\nfor domain A (b) and domain C (c). (d,e) Instantaneous re-\nsponse of PandMfor theP1domain (d) and P4domain\n(e). (f) The magnetic-field dependence of the dynamical po-\nlarization induced by the microwave magnetic field in domain\nA, B and C.\nrespectively. The terms that contain Hsuch asPx=\na′HxHzare also allowed and have the same form as\nMbut we omit them here for clarity. Given that the\n+kω(/bardbl−y) microwave in the CPW contains Hω/bardblzand\nEω/bardblx[Fig. 1(g)] and that the Mpoints to the xdirec-\ntion in the domain A with P1, the magnetic resonance is\ninduced by Hω/bardblzand produces the dynamical magneti-\nzation,Mω\nH, alongthe zdirection[Fig. 4(d)]. Mω\nHsimul-\ntaneously induces dynamical polarization, Pω\nH, along the\nxdirection through the ME coupling given by Eq. (3);\nPω\nH,x=aMxMω\nH,z. ThisPω\nH,xmay interfere with the dy-\nnamical polarization ( Pω\nE) induced by Eω(/bardblx) and lead\nto the DD. In the present case, however, the DD is ex-\nperimentally not observed in the domain A and therein\nthe coefficient ashould be negligibly small.\nIn the domain C with P4, theMis slanted from both\nxandzaxes and therefore MandMω\nHhave both finite\nxandzcomponents [Fig. 4(e)]. As the result, Pω\nHis\ngenerated along zdirection, which can be described as\n2cMzMω\nH,z. The resulting Pω\nH,zis antiparallel to Pω\nE,zfor5\n+kωmicrowave whereas they are parallel for −kωmi-\ncrowave, as schematically illustrated in Fig. 4(d); here,\nwe note that HωandEωalso have both xandzcom-\nponents [Fig. 1(d)]. Therefore, the interference between\nPω\nH,zandPω\nE,zoccurs destructively for + kωmicrowave\nwhile constructively for −kωmicrowave. This difference\nmay lead to the DD, which is indeed observed in Fig. 3.\nHence, the DD observed here originates from the form\nof the ME coupling, Pz=cM2\nz, which is in accord with\nthe nature of the static ME effect reported in Ref. [24].\nThe microscopic origin of the ME coupling is therefore\nattributed either to anisotropic exchange mechanism or\nto single-site ME effect.\nThe ME coupling discussed above reproduces the\nmagnetic-field dependence of the DD in the ferromag-\nnetic state as well: Because the Hωcirculates around\nthe signal line and is perpendicular to the [110] axis in\nthe present experimental setup, we calculated Pω\nH,z=\n2cMzMω\nH,z, whichisproportionaltothemagnitudeofthe\nDD, for each domain by averaging all the contributions\ninduced by Hωlying in the (110) plane. The calculation\nqualitatively reproduces the experiment except for the\nlow-field region for domain A, where the transverse con-\nical state, not the ferromagnetic state presumed in the\ncalculation, is formed: The DD for domain A is negligi-\nbly small, for domain B shows a broad maximum and for\ndomain C keeps increasing with the magnetic field (see\nFigs. 3(d) and 4(f)).\nIt has been believed that the DD should be maximized\nwhenkω,PandMare perpendicular to each other,\nwhereas the present observation unambiguously demon-\nstrates the crucial role of the microscopic ME coupling\nas well as the macroscopic symmetry: The macroscopic\nsymmetry defines the necessary conditions for the ex-\nistence of the DD, as illustrated in Fig. 4(a), but leaves\nfreedom for different ME mechanisms to governthe mag-\nnitude of the DD; as clear from the comparison of the\nmagnetization and polarization dynamics for domain A\nand C, the DD can emerge even when kω,PandMare\nnot totally perpendicular to each other.\nInsummary,wehaveinvestigatedthemicrowaveDDin\nthe polar ferromagnetic state of GaV 4S8. The ferromag-\nnetic resonance is separated for the structural domains\nwith different directions of the electric polarization due\nto uniaxial anisotropyand we clearly observed the DD as\nlarge as 20 % for a specific domain without cancelation\namong the multi-domain states. Our findings widen the\nclass of materials that can potentially show the large DD\neven in the presence of the multi-domain states.\nThe authors thank M. Mochizuki, K. Penc and T.\nKurumaji for enlightening discussions. 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Sanada 3) \n \n1) Department of Physics, Keio University, Hiyoshi 3-14-1, Yokohama 223-8522, Japan \n2)  PRESTO, JST, Sanbancho 5, Chiyoda, Tokyo 102-0075, Japan \n3) Department of Electronics and Electrical E ngineering, Keio University, Hiyoshi 3-14-1, \nYokohama 223-8522, Japan \n \n* Corresponding authors’ E-mail: yamaguch@phys.keio.ac.jp (A. Y .), kmotoi@phys.keio.ac.jp \n(K.M.), and miyajima@phys.keio.ac.jp (M. H.)  2/33 [Abstract] \nThe broadband ferromagnetic resonance meas urement using the rectifying effect of \nNi81Fe19 wire has been investigated. One wire is deposited on the center strip line of the \ncoplanar waveguide (CPW) and the other one deposited between two strip lines of CPW. The \nmethod is based on the detection of the magnetoresistance oscillation due to the magnetization dynamics induced by the radio frequency field. The magnetic field dependences of the \nresonance frequency and the rectification spectrum are presented and analytically interpreted \non the standpoint of a uniform ma gnetization precession model. \n \n \n \n   \n \n   \n \n   \n \n   \n \n \n   3/33 I. Introduction \nFerromagnetic properties in the GHz region have been actively explored for \napplications in radio-frequency (RF)/microwave  devices by using many kinds of procedure, \nsuch as ferromagnetic resonance (FMR), Brillouin light scattering 1 – 3, the time-resolved \nmagneto-optical Kerr effect 4 - 6, and so on. These procedures, however, have their own \ndistinguishing advantages and disadvantages for the characterization of the high-frequency \nmagnetization dynamics. The development of new methods as well as measuring techniques is \na crucial issue, not only for the basic scien tific field but also technical applications. \nRecently, several new electrical measurements 7 – 13 were proposed, which are \nextremely sensitive and suitable for investigati ng the magnetization dyna mics in sub-micron \nscale magnets. One of them is pulse inductive measuring used to determine the time domain 9 \nof dynamical properties of component films and multilayer spin-valve stacks. It is especially \nuseful in the characterization of the intrinsic dynamical properties with recourse to expensive \nstorage-oscilloscopes for high-speed sampling  and wideband amplifiers. The broadband \nspectrometer is preferable since it allows one to study FMR in constant magnetic fields. The \nnovel inductive technique of FMR using vector network analyzer gives an insight into the \nmodal spectrum with respect to the frequency and effective damping of the various modes 7, 8.  \nCurrently, the investigation of spin-polarized  current has progressed, with applications   4/33 to spintronics devices such as magneto-res istive random-access memories and microwave \ngenerators. The spin-polarized current flowing through ferromagnetic multilayers is known to \ngenerate spin wave excitation 14, 15 together with magnetization reversal. The physical \nmechanism, as revealed by the experimental results, is a consequence of the spin angular \nmomentum transfer, which occurs due to the interaction between the spin-polarized current and \nthe ferromagnetic moment. One of the interesting properties is the rectification of RF current \noccurring in magnetic tunnel junction (MTJ) 10, in spin-valve structure 11 and in single-layered \nferromagnetic wire 12. This rectification is explained in terms of the by the magnetoresistance \noscillation attributable to spin-transfer torque. It should be noted that direct-current (DC) \nvoltage is produced whenever the resonant RF current flows through these systems. \nIn this study, we present the broadband FMR induced by RF current flowing in the \ncoplanar waveguide (CPW) and show the experimental results on sub-micron single-layered \nferromagnetic Ni 81Fe19 wires. We also discuss the FMR spectrum, excited not only by the \nmagnetic field but also by the difference in spin -transfer torque. In order to investigate the spin \ndynamics induced by the in-plane and out-of-plane RF field components, two kinds of devices \nwere prepared; one is Ni 81Fe19 wire, prepared on the center strip line of the CPW, and the other \nis prepared between the center strip lin e and the ground strip line of the CPW.   5/33 II. Sample fabrication a nd magnetic field mode \nThe 22 and 30-nm-thick polycrystalline Ni 81Fe19 wires are patterned by electron \nbeam lithography and lift-off processing (Fig. 1). Two kinds of wire are prepared; One is \nprepared on the top of the CPW comprising a Cr (5 nm)/Au (38 nm) conductive strip (Fig. \n1(a)), and the other is prepared in the aperture between the conductive strip lines of the CPW \ncomprising Cr (5 nm)/Au (80 nm) on MgO (100) substrates (Fig. 1(b)). The width of the \n22nm-thick Ni 81Fe19 wire prepared on the CPW (Fig. 1(a)) is 2 μm, and that of the CPW strip \nlines is 3 μm. On the other hand, the width of the 30nm-thick wire prepared between the strip \nlines (Fig. 1(b)) is 2 μm, and that of the CPW line is 5 μm. \nThe ground-signal-ground (GSG) type microwave probe is connected to the CPW, \nand the DC voltage difference induced by the RF current flowing through the system 12 is \ndetected by using a bias-tee circuit and a voltmeter, as illustrated in Fig. 1(c). The sinusoidal \nvoltage output of the signal generator (SG), with frequency range from 10 MHz to 15 GHz, \nproduces an approximately elliptical field pattern hRF around the central line. In the \narrangement in Fig. 1(a), the RF current flow ing through the center strip gives rise to the \ntransverse RF field, while in Fig. 1(b), the RF field acting on the wire is perpendicular to the \nplane due to the application of the RF current across the CPW. The external static magnetic \nfield extH  is applied in the substrate pl ane as a function of tilting angle  θ from the   6/33 longitudinal axis of the wire and the measurements are performed at room temperature.   \nFigures 2(a) and 2(b) show the RF field di stributions obtained by the High-Frequency \nStructure Simulator (HFSS) 17 for the cases of Figs. 1(a) and 1(b), respectively. The geometry \nand dimensions of the calculation are the same as those of the present sample except for the \nmagnitude of the applied microwave power. The RF field distribution whithin the real space, \nvisually shown in Figs. 2(a) and 2(b), is consis tent with the analytical model and the analysis \nfor the experimental results  as latterly described. \nAs shown in Fig. 2(a), the in-plane Oersted field along the y direction, produced by \nthe RF current I flowing at a distance z is Hy (I, z) in the configuration of Fig. 1(a), where \nthe y axis is transverse to the direction of the current. At a position close to the center \nconductor of the CPW but far from an edge, the cent er conductor appears as an infinite sheet of \ncurrent, which produces the field xyh= I 2 w  9, where w is the width of the center \nconductor of the CPW. The arrangement of Fig.  1(b) also provides an out-of-plane field \n()zz,2== HI y h I y  along the wire axis, as shown in Fig. 2(b), in which y is the \ndistance between the center of the wire and the CPW center strip line. \n \nⅢ Analytical model for the spin dynamics  \nThe dynamics of the magnetic moment in RF field are analytically described by the   7/33 Landau-Lifshits-Gilbert (LLG) equa tion. The LLG equation in the coordinate system shown in \nFig. 1(e) is expressed by \n() ( )0e f f R F(t) (t)(t) tttγα∂∂=− × + + ×∂∂mmmH h m ,                    ( 1 )  \nwhere ()mt denotes the unit vector along the local magnetization, (S =mM M and \n1=m ), 0γ the gyromagnetic ratio, effH  the effective magnetic field, including the \nexchange and demagnetizing fields, RFh the RF field produced by the RF current flowing \nthrough the center strip of the CPW, and α the Gilbert damping constant.  \nAs schematically shown in Fig. 1(e), we define the (),,xyz  coordinate system, in \nwhich each component corresponds to  the vertical wire axis, the longitudinal wire axis, and the \nnormal to the substrate plane, respectively. The external magnetic field is directed at angle θ \nfrom the x-coordinate axis. Subsequenctly, we also define the coordinate system (),,abc  \nwhere the a direction corresponds to the equilibrium direction of 0m along the effective \nmagnetic field eff ext A=+HHH  including the external field extH  and the shape magnetic \nanisotropy field AH. Now, we assume the uniform precession to be dominant and the \nexchange field to have disappeared. \nThe unit vector 0m inclines at an angle ψ from the x-coordinate axis. The \nmagnetization precession around effH  results in a small time-dependent component of the \nmagnetization perpendicular to 0m, which is almost parallel to the direction of effH .   8/33 Subsequently, we can decompose the unit vector ()mt as \n() () () () () () 0a b c ,, δ =+ =mm mtt m t m t m t , (2) \nwhere () 1t=m  and so ()2\nacm= 1 -m+ m2\nb . The external field extH , the anisotropy \nfield AH, and the effective field effH in the (),,abc  coordinate system are given by \n    ()\n()ext\next extcos\nsin\n0θψ\nθψ− ⎛⎞\n⎜⎟=−⎜⎟\n⎜⎟⎝⎠HH\nH ,   (3) \n    aa\nAS S b b\ncc⎛⎞\n⎜⎟=− ⋅ =−⎜⎟⎜⎟⎝⎠HN m \u0004Nm\nM MN m\nNm,  (4) \nand \n    ()\n()ae x t S a a\neff ext A b ext S b b\ncS c ccos\nsinHH M N m\nHH M N m\nHM N mθψ\nθψ−− ⎛⎞⎛⎞\n⎜⎟⎜⎟=+ = = − −⎜⎟⎜⎟⎜⎟⎜⎟ − ⎝⎠⎝⎠HHH ,                    ( 5 )  \nwhere N\u0004 is the demagnetization coefficient in the (),,abc  coordinate system, given by \n    ax x y\nby x y\ncz zcos sin 0 cos sin\nsin cos 0 sin cos\n00 1NN N N\nNN N N\nNN Nψψ ψ ψ\nψψ ψ ψ−− ⎛⎞ ⎛ ⎞ ⎛⎞ ⎛ ⎞\n⎜⎟ ⎜ ⎟ ⎜⎟ ⎜ ⎟== = +⎜⎟ ⎜ ⎟ ⎜⎟ ⎜ ⎟⎜⎟ ⎜ ⎟ ⎜⎟ ⎜ ⎟⎝⎠ ⎝ ⎠ ⎝⎠ ⎝ ⎠N\u0004 .             ( 6 )  \nIn the case of the wire prepared on the conductor strip line, the RF field RFh& in the plane is \n    () RF xy xy es i n, ec o s, 0it ithhωωψψ =h& .            ( 7 )  \nWhile, in the case of the wires prepared between the strip lines, the RF field perpendicular to \nthe plane is written by   9/33  () RF z 0,0, eithω\n⊥=h .              ( 8 )  \nHere, the aspect ratio of the wire elongated along the xaxis is large, and consequently Nx is \nalmost zero. In the small precession angle limit around the effective field effH  \n(aadd 0 , 1mt m =\u0011), the LLG equation can be linearized. In response to each driving field \nRFh& and RF⊥h  with angular frequency ω, we can solve Eq. (1), considering a small \nvariation, \n    () () () () bc bc0, , 0, e , eit itmtmt m mωωδ≈≈m  and ()01, 0, 0≈m .                    ( 9 )  \nThe linearized LLG equation for the magnetization is reduced to a partial differential equation \nfor the dynamic magnetization ()bb eitmt mω=  and ()cc eitmt mω= . Inserting them into Eq. \n(1) and taking only the linear terms in bm and cm into account, one obtains the following \ntwo cases: \n \nA. The driving field RFh& in the plane of the wire  \nAfter the substitution of Eqs. (5), (7), and (9 ) into Eq. (1) and the rearrangement of the \nterms, the following equation is obtained, neglecting the very small quadratic terms xy chm  \nand xy bhm : \n()\n()() ( )\n() ( )()\n()ca S c b c\n0\next a S b b xy c b sin [ ] cositmt H M N mt mt\nHH M N m t h e mt mtttω γαθψ ψ+ ⎛⎞ − ⎛⎞ ⎛ ⎞∂∂≈− +⎜⎟ ⎜⎟ ⎜ ⎟⎜⎟ −− + + ∂∂⎝⎠ ⎝ ⎠ ⎝⎠.         ( 1 0 )  \nRearranging Eq. (10), we obtain   10/33 [ ]\n[] ()b0 c c\n0 b b c 0 ext xy0\nsin e cositim H i m\nHi mi m H hωωγ ω α\nγ ωα ω γ θψψ−′ ⎧ ++ =⎪⎨′ ⎡ ⎤ −+ + = − − +⎪ ⎣ ⎦ ⎩,        (11) \nwhere the components of the static field are replaced by \n    b aS b\ncaS cHH M N\nHHM N′=+⎧\n⎨′=+⎩.             ( 1 2 )  \nForming a queue form Eq.(11), we obtain \n   [ ]\n[] ()b 0c\nc0 e x t x y 0b0\nsin e cositm iH i\nmH h Hi iωωγ ω α\nγθ ψ ψ γω α ω−⎛⎞ ′ ⎛⎞ +⎛⎞=⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ′ ⎡ ⎤ −− + −+ ⎝⎠ ⎝⎠ ⎣ ⎦ ⎝⎠,    (13) \nIn general, in ferromagnetic conductor is very small, and then 20 α≈ in the calculation. The \nsolution of Eq. (13) is given by \n()\n()() ()\n()22 22\n0c k k 0 c 0 ext xy b\n222 2 2 2 22 2ck ksin e cositHi H Hh m\nm iωγω ωω α ω ω γ γθ ψ ψ\nωω ω α ωα ω ω ω− ⎛⎞ ⎡ ⎤ ′′ −+ −− Δ ⎡⎤ −+ ⎛⎞ ⎣ ⎦ ⎣⎦ ⎜⎟⎜⎟⎜⎟⎝⎠ −+Δ −Δ − −⎝⎠\u0011 ,(14) \nwhere the FMR frequency kω and Δ are given by the following relations: \n    22\nk0 c b HH ωγ ′′ = ,                                             ( 1 5 )  \n ()0b cHHγ ′′ Δ= + .                                                     ( 1 6 )  \nNow, the RF field is given by () () 2 ht It w =  with () cos It I t ω = . The \ndynamic component bmδ  induces the AMR with a magnetization precession angle of ()tφ  \naround effH, namely, \n()bsin tφδ ≈⎡⎤⎣⎦mm  and ()bsin 2 2 tφδ⎡ ⎤ ≈⎣ ⎦mm ,                    ( 1 7 )  \nand the inductive voltage is bddVm t δ∝ . In the present case, a microwave current is   11/33 flowing, induced in the wire located betwee n the strip lines. The detection circuit has \ncapacitive and inductive coupling to the CPW st ructure (Fig. 2), as well as the source of \nrectification between the AMR and induced microwave currents. By considering these \nconditions, the product of the AMR and the current yields the DC voltage ()0Vt . \n  () () ()\n() ()\n() () ()0\n2\n22 2cos cos\n1cos cos 1 2cos sin sin 2 sin 22Vt I t R t\nIt R t\nIRt t tωψ φ\nωψ ψ φ ψ φ=⋅\n=⋅ Δ +\n⎡⎤=⋅ Δ + − −⎢⎥⎣⎦. (18) \nThe rectifying frequency spectrum of ()0Vt  is evaluated by the Fourier transformation of Eq. \n(18); \n    () ()() ()\n()22\n0S\n2\n2\nS\nb b1s i n2 s i n2 1 2c o s s i n2\n1sin 2 2 1 2cos2VR I t t\nRIωψ φ ψ φ\nδδψψ⎧⎫≈Δ ⋅ ⋅ − + − ⎨⎬⎩⎭\n⎧⎫ ⎡⎤⎡ ⎤ ⎪⎪≈Δ ⋅ ⋅ − ⋅ + − ⎢⎥⎢ ⎥ ⎨⎬⎢⎥⎢ ⎥ ⎪⎪ ⎣⎦⎣ ⎦ ⎩⎭mm\nmm,                   ( 1 9 )  \nwhere SI and CI are the current flowing through the wire and that flowing through the \nconductive strip line of the CPW, respectively.  The second harmonic term cannot produce \nrectification and the DC voltage spectrum ()0Vω is expressed as \n    () () () () ( )2 SC\n0 ext1sin 2 cos 2 1 2cos cos sin22RI IVA A Hwωω ψψ ψ ψ ωθ ψΔ⋅ ⋅ ⎡⎤≈⋅ − + − ⋅⋅ −⎢⎥⎣⎦,(20) \nwhere ()Aω is given by \n()()\n()22 2\n0c k\n222 2 2 2\nkH\nAγω ω\nω\nωω ω α′ −\n≈\n−+Δ.                       ( 2 1 )  \nWhen the external field extH  exceeds the anisotropy field AH , the \nmagnetization aligns almost parallel to extH, namely, ψθ≈. Then, ()0Vω is proportional   12/33 tosin 2 cosθθ . On the other hand, when extH  is smaller than AH, the magnetic moment \ndirects along the longitudinal wire axis, namely, 0 ψ≈D, and ()0Vt  is almost proportional to \nextsinH θ. Consequently, the field and angle de pendences of the induced DC voltage ()ωV  \nare summarized as follows: \n(i) ()0 sin 2 cos Vωθ θ∝  when ext A>>HH         ( 2 2 )  \n(ii) ()0e x t sin VHωθ∝   when ext A<<HH       ( 2 3 )  \nThese results are consistent with the previous works. Here, we must consider the \neffect of the inductive voltage signal (IVS),  which is closely related to the transverse \nmagnetization through Faraday’s la w. The IVS is proportional to ybdd c o s ddmt mt ψ =⋅  9. \nIn this case, the θ dependence of the IVS is given by 2cosθ and sinθ for \next A>>H H and ext A<<H H, respectively. Therefore, 0V is produced by the AMR \noscillation due to the RF field in the former case (i). The IVS, however, may contribute to the \nDC signal in the latter case (ii). It is important to note the type of ()0Vωdue to the RF field \nalways takes the dispersion one, while ()0Vω caused by the RF spin-polarized current \ndirectly flowing through the wire is given by the superposition of the Lorentzian and \ndispersion types 12. \nB. The driving field RF⊥h perpendicular to the plane of the wire \nWhere the wire is located between the center strip line and the ground strip line of the \nCPW, the RF field RFh given by Eq. (8) is perpendicular to the plane on the wire position.   13/33 Likewise, after the substitution of Eqs. (5), (8), and (9) into Eq. (1) and the rearrangement of \nthe terms we obtain, neglecting the very small quadratic terms zchm  and zbhm , \n   ()\n()()\n() ( )()\n()b c cc z\n0\nc b ext b be\nsinitmt mt mt H h\nmt mt Hm t H ttω\nγαθψ′ ⎛⎞ ⎛⎞ ⎛ −⎞ − ∂∂≈− +⎜⎟ ⎜⎟ ⎜ ⎟⎜⎟ ′ −− ∂∂⎝⎠ ⎝ ⎠ ⎝⎠,             ( 2 4 )  \nand rearranging, we obtain \n   () ( )0c z b\n0\n0b e x t c sin eitiH i h m\nHi i H mωωγ ω αγγω α ω θ ψ−′+− ⎛⎞ ⎛ ⎞ ⎛⎞≈− ⎜⎟ ⎜ ⎟ ⎜⎟′ −+ −⎝⎠ ⎝⎠ ⎝ ⎠.            ( 2 5 )  \nThe real part of the solution of Eq. (25) can be presented by \n   (){ }\n()() ( )\n() ( )22\n0kb z0 c e x t\n222 2 2 2c 0b z e x tksin e\nsin eit\niti m ih H i H\nm Hi h i Hω\nωγωω ω α ωγ ω α θ ψ\nγω α ω θ ψ ωω ω α−\n−−−Δ ′ ⎛⎞ ++ − ⎛⎞≈× ⎜⎟ ⎜⎟ ⎜⎟ ′+− − ⎝⎠ −+Δ ⎝⎠.        ( 2 6 )  \nThere is the source of rectification betw een the AMR and induced microwave currents. \nMultiplying them yields a DC voltage. The frequency spectrum of the induced voltage \n()0Vωis also expressed by the Fourier transformation of Eq. (26); \n() ()() ()\n()\n()\n() ()() () {} () { }22\n0S\n2\n2\nS\nbb\n2\n22 2 2 2\nk0 c k e x t1s i n2 s i n2 1 2c o s s i n2\n1sin 2 2 1 2cos2\n2\nsin 2 2 1 cos 2 sinCSVR I t t\nRI\nIIBRy\nBH Hωψ φ ψ φ\nδδψψ\nωω α\nψψ ωωω γ ωω θ ψ⎧⎫≈Δ ⋅ ⋅ − + − ⎨⎬⎩⎭\n⎧⎫ ⎡⎤ ⎡⎤ ⎪⎪≈Δ ⋅ ⋅ − ⋅ + − ⎢⎥ ⎢⎥ ⎨⎬⎢⎥ ⎢⎥ ⎪⎪ ⎣⎦ ⎣⎦ ⎩⎭\n⋅≈⋅ ⋅ Δ ⋅\n′ ×− Δ + − ⋅ ⋅ − ⋅ Δ − − −mm\nmm    ( 2 7 )  \nwhere SI and CI are the current flowing through the wire and that flowing through the \nconductive strip line of the CPW, respectively, and ()Bω is given by \n   ()\n()0\n222 2 2 2\nkBγω\nωω ω α=\n−+Δ.                ( 2 8 )    14/33 When extH  is much larger than AH , the magnetization directs along the applied field, \nthen ψθ≈and the rectifying voltage can be expressed as \n   () ()2\nSC\n0 sin 22II RVByωαωω θ⋅⋅ Δ ⋅ ⋅ ⋅ Δ=− ⋅ ⋅ .       (29) \nThis condition shows that ()0Vω is proportional to sin2θ, which is consistent \nwith the result of Costache 13. Conversely, extH  is smaller than AH, the magnetic moment \nis almost parallel to the direction of effH. Then, 0c o n s t .ψ θ <= <D, and the ()0Vω is \nalmost proportional to sinθ. \nIII. Experimental Results \nA. DC voltage induced by the in-plane RF field  \nIn the rectification of RF current in the wire, the magnetization dynamics induced by \nthe RF spin-polarized current result in the resi stance oscillation that originated from the AMR \neffect 12, which generates the DC voltage 0V combining with the RF current. Conversely, in \nthe system comprising wires prepared on the CPW (Fig. 1(a)), the majority of the RF current \nflows in the CPW line and the wire. As s hown in Fig. 1(d), the total resistance R of the \nsystem can be calculated by \n() () () () SC S C / RtR t RR tR=+ ,                ( 3 0 )  \nwhere ()SRt and CR are the resistance of the Ni 81Fe19 wire and that of the CPW line, \nrespectively. In the system, 0V is given by the multiple of ()It  and ()Rt. Here, as the   15/33 resistance of the CPW line is almost constant, 0V is reduced to \n   () () () () ()0S SVt I t R t It Rt =⋅ ≈ ⋅ ,                           ( 3 1 )  \nwhere CSII I=+ , ()CS C SII R R R=+  and ()SC C SII RRR=+ . \nIn the present frequency region, the CPW magnetic field distribution is similar to that \nof the quasi-TEM mode 16 as shown in Figs. 2(a) and 2(b) , with the microwave magnetic-field \npattern forming an ellipse about the central strip conductor. The microwave power injected into \nthe CPW is +5dBm. The DC resistance of the CPW line is RC = 55 Ω while that of Ni 81Fe19 \nwire is estimated to be RS = 600 Ω, assuming the resistivity of Ni 81Fe19 wire to be equal to that \nof bulk Ni 81Fe19. Due to the impedance mismatch among cables, the bias tee, the GSG typed \nmicrowave probe, and the CPW with the sample, the RF attenuation depends on the frequency. \nTo estimate the losses due to the impedance mi smatch, we measure the total impedance of the \nsystem as a function of frequency using a vector network analyzer. For a wire of with 2 μm \nprepared on the CPW, the impedance increases wi th frequency, while the change in impedance \ncan cause an increase of up to 10 percent incr ease, virtually independently of frequency within \nthe operational frequency range below 15 GHz (see Fig. 3). Therefore, we estimate that each \nRF current density flowing through the center coplanar strip line and the wire as \n()106.0 6.2 10 ×∼  A/m2 and ()97.0 7.2 10 ×∼  A/m2, respectively, the value of which are \nrelatively low for the RF Joule heating at input power of +5dBm within the range 0 to 15 GHz,   16/33 The amplitude of the RF field on the center strip line is estimated to be 15 Oe 7 - 9. \nThe magnetic field is applied to the substrate plane every 5D from the wire-axis. \nFigure 4(a) shows the field dependen ce of the DC voltage difference ()0Vω Δ  ( 30θ=D) \ngenerated in the 2 μm width wire prepared on the CPW, which is shown in Fig. 4(a). As seen in \nthe figure, the peak (dip) position of the reso nance spectrum shifts toward a higher frequency \nregion with an increasing magnetic field. Figure  4(b) shows the frequency variations of the \ncenter between the peak and the dip for differen t frequencies of the RF field. The (red) solid \ncircle is the experimental data, while the solid line fits with the Kittel formula for the uniform \nprecession mode, including the shape magnetic anisotropy 18; \n() ( )1\nB0 2\nAA S ()gHH H H H Mμμω =⋅ + ⋅ + + ⎡⎤⎣⎦=,               ( 3 2 )  \nwhere g is the Lande factor, and AH the effective anisotropy field, including \ndemagnetizing and exchange fields. The magnetic  field dependence of the resonance frequency \nis in positive agreement with that of calculati on using the Kittel’s Eq. (32), which is consistent \nwith Eq. (15). As the RF field induced by th e current precesses the magnetic moment in the \nwire, the AMR resistance also oscillates at the re sonance frequency. This resistance oscillation \ngenerates 0V combining with the RF current. \nThe rectifying spectra for the magnetic field of 900 Oe in the substrate plane at \n0 ,35 ,85 ,95 ,145θ=DDDD D, and D180  are shown in Fig. 4(c). As seen in the figure, the   17/33 spectrum strongly depends on θ, indicating that the uniform precession is affected by θ and \nthat the sign of 0V reverses when the sense of the field is reversed ( θ to 180θ+D). \nFigure 5 shows the θ dependence of 0VΔ in the static magnetic fields of (a) \next 900= H  Oe and (b) ext 30= H  Oe. The solid curve represents (a) sin 2 cosθθ  and \n(b) sinθ fitting curves which are obtained from the analytical calculation result of case (i) \nand (ii) of Eq. (20), respectively. These f itting curves are in good agreement with the \nexperimental results. \nIt is noted that ()0Vω is generated in conjunction with FMR, as shown in Fig. 6. \nThe (red) solid line shows ()0Vω induced by the in-plane RF field in the 22 nm-thick and 2 \nμm width wire located on the CPW, while the (b lack) dashed line show s the spectrum induced \nby the RF spin-polarized current in the 30 nm-thick wire of width 2.2 μm. The type of ()0Vω \ndue to the RF field is of the dispersion type, as described in Eq. (18), while that of ()0Vω \ndue to the RF spin-polarized current is of the Lorentzian resonant type. These results can be \nascribed to the difference in torque direction; with the torque direction caused by the in-plane \nRF field perpendicular to the plane but that  of the spin-transfer torque in plane. \n \nB. DC voltage induced by the out-of-plane RF field \nThe typical ()0Vω spectra for 150 θ=D for various Hext are shown in Fig. 7(a). \nBoth the resonance frequency and 0VΔ increase with the magnetic field. Figure 7(c) shows   18/33 ()0Vω Δ  in Hext=600 Oe for 0 180θ=DD∼ . The resonance frequency takes a minimum at \n90θ=Dand a maximum at 0θ=D and 180θ=D, while ()0Vω Δ  has a minimum at \n0, 9 0θ=DD and 180D. \nThe additional and weak peaks observed in th e trace of Figs. 7 (a) and 7(c) seem to \nbe the additional spin wave modes. We note, ho wever, only the dominant mode in order to \nproceed with the analytical calculation and the interpretation for the experimental data, since \nthe additional modes are very hard to solve using the calculations. Figure 7(b) shows the static \nfield dependence of the resonance frequency. The (red) solid circle is the experimental data, \nwhile the (black) solid line show s the fitting with the Kittel formula for the uniform precession \nmode given by Eq. (32) 18. Effective fitting demonstrates how the magnetization vector of the \nwire is driven in the uniform precession mode wi th respect to the dominant mode. The inset of \nFig. 7(b) shows the variation of ()0Vω Δ  with magnetic field, which the (blue) open square \nis the experimental data. The ()0Vω Δ  at the FMR frequency increases with increasing \nmagnetic field. According to the result of Eq. (29), ()0Vω Δ  at the FMR frequency kω is \nevaluated as follows: \n()23\n0k 0 S c b bcVM H H H H ωγ ′′ ′ ′ Δ∝ Δ = + .           ( 3 3 )  \nWhen the demagnetization field is much larger than the applied field, we obtain cSH M′≈  and \n0bVH ′ Δ∝ . These results (Fig. 7) indicate that the dominant mode is the uniform precessional   19/33 mode, consistent with the model described by Costache 13. \nThe θ dependence of 0VΔ in (a) Hext=600 Oe and (b) Hext=50 Oe is shown in Figs. \n8(a) and 8(b), respectively. The (red) solid circle is the experimental data. The (black) solid \nline in Fig. 8(a) shows a sin 2θ −  curve, which fits to the data well. This indicates that the \nuniform precession mode is dominated in the magnetic field to a sufficient extent to direct the magnetization parallel to the field. Conversely , the (black) solid line in Fig. 8(b) shows a \nsin 2 sinθθ − curve, which fits the data relatively well. The angle dependence, however, is \ndifficult to explain using solely the uniform precession mode generated by the RF field \nperpendicular to the plane. The origin of the θ dependence of ()0kVω Δ  is contributable to \nthe higher-order spin wave exc itation within the confined stru cture of the shape-controlled \nferromagnets 1 – 8. \nIn conclusion, we demonstrated the rectifying broadband FMR spectroscopy for the \nmicron-scale ferromagnetic wires induced by the in-plane or out-of-plane components of RF \nfield using the CPW method. We showed that  the experimental DC voltage spectrum ()0Vω \nproduced in the wires can be well explained by  the present phenomenological and analytical \nmodel; based on the coupling between the RF cu rrent and the AMR oscillation derived from \nthe uniform magnetization precession. \nWe find that the characteristic asymmetr ic rectifying spectrum produced by the   20/33 in-plane RF field differs in comparison with th e rectifying effect due to the spin polarized \ncurrent directly flowing through the wire, while the θ dependence of ()0Vω Δ is almost the \nsame. \nThe broadband rectifying effect generated by the in-plane or out-of-plane \ncomponents of the RF field using the CPW will provide a highly-sensitive measurement and \nanalysis of spin dynamics in nano-scale magnets. As the effect can be easily tuned to control \nthe resonance frequency, based on the shape of the wire or the magnetic field, this will \nfacilitate fresh developmen t for spintronics devices. \n \nAcknowledgements  \nThe present work was partly supported by MEXT Grants-in-Aid for Scientific \nResearch in Priority Areas, JSPS Grants-in- Aid for Scientific Research, and the Keio \nLeading-edge Laboratory of Science and Technology project 2007. \n \n   \n \n   \n \n   21/33 Reference \n1B. Hillebrands and K. E. Ounadjela, Spin Dynamics in Confined Magnetic Structures \n(Springer, Berlin, 2003), Vols. 1 - 3. \n2C. Bayer, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Appl. Phys. Lett. 82, 607 \n(2003).: C. Bayer, J. Jorzick, B. Hillebrands, S.  O. Demokritov, R. Kouba, R. Bozinoski, A. N. \nSlavin, K. Y . Guslienko, D. V . Berkov, N. L. Gorn, and M. P. Kostylev, Phys. Rev. B 72, \n064427 (2005).: S. O. Demokritov, B. Hillrbrands, and A. N. Slavin, Phys. Rep. 348, 441 \n(2001). \n3A. Ercole, A. O. Adeyeye, J. A. C. Bland, and D. G. Hasko, Phys. Rev. B 58, 345 (1998). \n4A. Barman, V . V . Kruglyak, R. J. Hicken, J. M. Rowe, A. Kundrotaite, J. Scott, and M. \nRahman, Phys. Rev. B 69, 174426 (2004). \n5M. Belov, Z. Liu, R. D. Sydora, and M. R. Freeman, Phys. Rev. B 69, 094414 (2004). \n6G. Gubbiotti, P. Candeloro, L. Businaro, E. Di Fabrizio, A. Gerardino, R. Zivieri, M. Conti, \nand G. Carlotti, J. Appl. Phys. 93, 7595 (2003). \n7F. Giesen, J. Podbielski, T. Korn, M. Steiner, A. Van Staa, and D. Grundler, Appl. Phys. Lett. \n86, 112510 (2005). \n8M. Bailleul, R. Höllinger, an d C. Fermon, Phys. Rev. B 73, 104424 (2006). \n9T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, J. Appl. Phys. 85, 7849 (1999). \n10A. A. Tulapurkar, Y . Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D.   22/33 Djayaprawira, N. Watanabe, and S. Yuasa, Nature 438, 339 (2005). \n11J. C. Sankey, P. M. Brahanca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. \nRalph, Phys. Rev. Lett. 96, 227601 (2006). \n12A. Yamaguchi, H. Miyajima, T. Ono, Y . Suzuki, and S. Yuasa, ., Appl. Phys. Lett. 90, 182507 \n(2007).: A. Yamaguchi, H. Miyajima, S. Kasai, and T. Ono, Appl. Phys. Lett. 91, 132509 \n(2007). \n13M. V . Costache, M. Sladkov, C. H. van der Wal, and B. J. van Wees, Appl. Phys. Lett. 89, \n192506 (2006).: M. V . Costache, S. M. Watts, M. Sladkov, C. H. van der Wal, and B. J. van \nWees, Appl. Phys. Lett. 89, 232115 (2006). \n14J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). \n15L. Berger, Phys. Rev. B 54 , 9353 (1996). \n16C. P. Wen, IEEE trans. Microwave theory and technique 17, 1087 (1969). \n173D Full-wave electromagnetic filed simulation, produced by Ansoft Corporation. \n18C. Kittel, Introduction to Solid State Physics Ch. 16 (John Wiley, New York, 1986), 6th ed. \n \n \n \n \n \n   23/33 Figure captions \n \nFigure 1  \nSchematic diagram and optical microscope image of the device in (a) in-plane and (b) \nout-of-plane magnetic field on the wire position. Th e external static magnetic field is applied in \nthe substrate plane at an angle of θ from the longitudinal axis of the wire. A schematic \ndiagram of electrical measurement circuit (c ) and equivalent circuit (d) for the device \nillustrated in Fig. 1(a). (e) The schematic co ordinate system adopted in this study. \n \nFigure 2 \nThe magnetic field distribution on the wire positi on on the center conductive strip line of CPW \nat the phase (a) 0D and (b) 180D, calculated by HFSS. The magnetic field distribution on the \nwire position between the center strip line and the ground line of CPW at the phase (c) 0D \nand (d) 180D. The phases 0D and 180D correspond to the direction of the current and are \nrespectively the same and opposite as/to the positive x direction, which is parallel to the \nlongitudinal axis of the wire. \n \nFigure 3 \nThe impedance ()Zω of the system comprising the wire on the center conductive strip line \nof the CPW as a function of the RF frequency 2ωπ . \n \n \n \n    24/33  \nFigure 4 \n(a) DC voltage spectra ()0Vω generated by the 22 nm-thick Ni 81Fe19 wire on the center strip \nline of the CPW in response to the RF current in the magnetic field of \next 100, 200, 400,600=H  and 800 Oe for 30θ=D. (b) The frequency of the center \nposition between the peak and the di p of the spectra as a function of the applied magnetic field. \nThe line is a fit of Eq. (25) to the data. (c) ()0Vω for different angles θ between the \ndirection of the applied magnetic field and the longitudinal axis of the wire. Each resonant \nresponse is vertically  shifted for clarity. \n \nFigure 5 \nThe (red) solid circles show the angle θ dependence of the DC voltage difference ()0kVω Δ  \nat the FMR frequency in the magnetic field of (a) ext 900=H  Oe and (b) ext 30=H  Oe. \nThe (black) solid lines represent th e corresponding fits from case (i) and (ii) of Eq. (18) (see \nText) \n \nFigure 6 \nThe rectifying spectra ()0Vω generated by the in-plane RF field (the red solid line) and that \nby the RF spin-polarized current (the black dash ed line) as a function of the RF frequency.   25/33 Figure 7 \n(a) DC voltage spectra ()0Vω generated in the 30 nm-thick Ni 81Fe19 wire prepared between \nthe center strip line and the ground line of the CPW (Fig. 1(b)) in response to the RF current in \nthe magnetic fieldsext 100, 200, 400,600=H  and 800  Oe for 150θ=D. (b) The \nfrequency of the center position between the peak and the dip of the spectra as function of the \nmagnetic field. The line is a fit of Eq. (25) to the data. ()0Vω as a function of the magnetic \nfield shown in the inset of Fig. 7(b). (c) ()0Vω for different angles θ between the direction \nof the applied magnetic field and the longitudinal  axis of the wire. Each resonant response is \nvertically shifted for clarity. \n \nFigure 8 \nThe (red) solid circles show the angle θ dependence of the DC voltage difference ()0kVω Δ  \nat the FMR frequency in the magnetic field of (a) ext 600=H  Oe and (b) ext 50=H  Oe. \nThe (black) solid line in Fig. 8(a) represents the corresponding fits fr om Eq. (29). The (black) \nsolid line in Fig. 8(b) represents a sin 2 sinθθ −  curve. The fitting line is qualitatively in \npositive agreement with the data. \n \n   26/33 \nSG\nG22 nm- thick Ni81Fe19 wire\nHext\nθxy\nZhRF\n100 μm\nSG\n30 nm- thick Ni\nGV100 μm81Fe19 wire\nhRF(a)\n(b)Bias tee\nSample V SG(c)\nRS\nRCR =RS+RCRSRC(d)\nψm0ferromagnetic wire\nzy\nxb\nacHext\nθhRF//hRF(e)\n \nFig. 1 \n \n \n  \n \n  \n \n \n \n   27/33 \nG GG G\nG G G GS S\nS SNi81Fe19 wirez\nzz\nzy y\ny yhRFhRF\nhRF hRF\nNi81Fe19 wire(a) (b)\n(c) (d)\n \nFig. 2 \n \n  \n \n \n   28/33 60\n55\n50 Z (ω) (Ω)\n15 10 5 0\n ω/2π (GHz) \nFig. 3 \n  \n \n \n \n  \n \n \n \n \n  \n \n   29/33 10\n5\n0ω/2π (GHz)\n1000 500\nHext (Oe)θ = 30(b)-20-100V0 (ω) (μV)\n15 10 5 0\nω/2π (GHz)800 Oe\n600400200100\nθ = 30(a)\nV0 (ω)  (μV)\n15 10 5 0xyH\nθext\nω/2π (GHz)20μV900 Oe\nw = 2 μm180\n85\n35\n0145\n95(c)\nhRFΔV0\n \nFig. 4 \n    \n \n  \n \n  \n \n   30/33 -10010\n360 270 180 90 0\n θ (deg.)Hext = 30 Oe\nsinθ(b)-20020ΔV0 (μV)\nxy H\nθext\nhRFHext = 900 Oe\nsin2θcosθ(a)ΔV0 (μV)\n \nFig. 5 \n \n    31/33 4\n3\n2\n1\n0V0 (ω) (μV) \n10 5\n ω/2π (GHz)8\n4\n0\n-4100 Oe\nRF currentRF field\nV0 (ω) (μV) \n \n \nFig. 6 \n \n \n  \n \n  \n \n  \n \n \n \n    32/33 -202V0 (ω) (μV)\n15 10 5 0\nω/2π (GHz)800\n600\n400\n200\n100 Oe ΔV0(a)\nθ = 150\n+10 dBm\n10\n5\n0ω/2π  (GHz)\n1000 500 0\nHext (Oe)(b)\nθ = 15010\n5\n0ΔV0 (μV)\n1000 500 0\nHext (Oe)\n15 10 5 0\nω/2π (GHz)y\nxH\nθhRFext\n0306090120150180\nΔV03 μV600 Oe\n+10 dBm(c)V0 (ω) (μV)\n \n \nFig. 7 \n \n  \n \n  \n \n \n   33/33  \n-6-3036ΔV0 (μV)\n360 270 180 90 0y\nxHext\nθ(a)\n600 Oe\n-sin2 θ\n-1.0-0.500.51.0\n360 270 180 90 0\n θ (deg.)(b)\n50 Oe\n-sin2 θsinθΔV0 (μV)\n \n \nFig. 8 \n "    },    {        "title": "2204.11057v2.Theory_of_inertial_spin_dynamics_in_anisotropic_ferromagnets.pdf",        "content": "Theory of inertial spin dynamics in anisotropic ferromagnets\nMikhail Cherkasskii,1,\u0003Igor Barsukov,2,yRitwik Mondal,3, 4,zMichael Farle,1and Anna Semisalova1,x\n1Faculty of Physics and Center of Nanointegration (CENIDE),\nUniversity of Duisburg-Essen, Duisburg 47057, Germany\n2Department of Physics and Astronomy, University of California, Riverside, California 92521, USA\n3Department of Spintronics and Nanoelectronics, Institute of Physics ASCR,\nv.v.i., Cukrovarnick\u0013 a 10, Prague 6, 162 53, Czech Republic\n4Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden\nRecent experimental observation of inertial spin dynamics calls upon holistic reevaluation of the\ntheoretical framework of magnetic resonance in ferromagnets. Here, we derive the secular equation\nof an inertial spin system in analogy to the ubiquitous Smit-Beljers formalism. We \fnd that the\nfrequency of precessional ferromagnetic resonances is decreased as compared to the noninertial case.\nWe also \fnd that the frequency of nutational resonances is generally increased due to the presence of\nmagnetic anisotropy and applied magnetic \feld. We obtain exact solutions of the secular equation\nand approximations that employ the terminology of non-inertial theory and thus allow for convenient\nestimates of the inertial e\u000bects.\nI. INTRODUCTION\nInertial e\u000bects of spin dynamics have mostly been ne-\nglected in the analysis of experimental spin resonance and\nspin transport results, since they mainly manifest them-\nselves at hardly accessible terahertz frequencies in fer-\nromagnets. With recent advances in spectroscopic tech-\nniques [1, 2], this fundamental phenomenon has been ob-\nserved in permalloy, cobalt and cobalt-iron-boron ferro-\nmagnetic \flms [3, 4]. Inertial spin dynamics o\u000bers novel\navenues for ultrahigh-frequency spintronic applications\nusing well-established ferromagnetic materials and is thus\nbecoming a prominent \feld of research.\nThe mathematical concept of inertia of magnetization\nwas introduced in the context of magnetoelastic coupling\nin ferromagnets by Suhl [5]. It was followed by an exten-\nsion of the breathing Fermi surface model which demon-\nstrated the emergence of a damping contribution linked\nto inertia [6, 7]. It was noticed that the Landau-Lifshitz-\nGilbert equation, which describes magnetization motion\nin analogy with a spinning top, required an inertial ten-\nsor of a rigid body. A revision [8] of this analogy within\na macroscopic Lagrangian approach suggested that iner-\ntia originates from generalization of gyromagnetic ratio\n| the magnetic moment is non-collinear to the angular\nmomentum. The Landau-Lifshitz-Gilbert equation was\nextended by including the second-order time derivative\nof magnetization, which now resembles Newton's equa-\ntion of motion for a massive point particle.\nThis inertial Landau-Lifshitz-Gilbert (ILLG) equation\n\u0003macherkasskii@hotmail.com\nyigorb@ucr.edu\nzPresent address: Department of Physics, Indian Institute of\nTechnology (ISM) Dhanbad, IN-826004, Dhanbad, India\nxanna.semisalova@uni-due.dereads\n@tM=\u0000j\rj\u00160M\u0002He\u000b+\u000b\nM0M\u0002@tM\n+\u0011\nM0M\u0002@ttM;(1)\nwhere\r=g\u0016B=~is the gyromagnetic ratio, \u0016Bis the\nBohr magneton, gis the g-factor, \u00160is the permeability\nof free space, Mis the magnetization with the magni-\ntudeM0,He\u000bis the e\u000bective magnetic \feld, \u000bis the\nGilbert damping, and \u0011is the inertial parameter. The\ninertial parameter has been discussed to be correlated\nto the Gilbert damping within the breathing Fermi sur-\nface model [6, 7] and torque-torque correlation model\n[9], whereas inertia has been considered independent of\ndamping within the classical Lagrangian approach [8].\nThe ILLG equation was derived within the framework of\nmesoscopic nonequilibrium thermodynamics [10] and the\nDirac-Kohn-Sham theory [11]. In an atomistic method,\nexchange interaction, damping, and inertia were calcu-\nlated from \frst principles [12]. The microscopic origin\nof inertia has been asserted in the relativistic spin{orbit\ncoupling [9, 13, 14].\nInertia leads mainly to nutation { a terahertz-\nfrequency motion of magnetization superimposed on the\nregular gigahertz-frequency precession [15] (Fig. 1). Nu-\ntational resonances have been discussed in ferromagnets\n[16] and antiferromagnets [17{20]. Moreover, traveling\nnutational spin waves [21{23] have been proposed. Be-\nsides nutational motion, inertia has been found to result\nin a frequency shift of the uniform magnetization pre-\ncession [15, 24] and spin waves [22] at gigahertz frequen-\ncies. Previous studies of inertial spin dynamics treated\nvarious ferromagnetic systems including nanoparticles\nand nanostructures [25{27]; however, a general approach\nbased on ILLG for a ferromagnet with an arbitrary mag-\nnetic anisotropy has not yet been proposed.\nHere, we develop a holistic theoretical framework for\na ferromagnet with an arbitrary magnetic anisotropy en-\nergy landscape in analogy to the Smit-Beljers (SB) ap-arXiv:2204.11057v2  [cond-mat.mes-hall]  29 Aug 20222\nFigure 1. The magnetization Mprecesses around the e\u000bec-\ntive magnetic \feld and He\u000bdue to the precessional torque\n(blue). The inertial torque (red) causes the magnetization to\nundergo a concurrent nutational motion. The total magneti-\nzation motion is subject to the damping torque (green).\nproach. For the last six decades [28], the Smit-Beljers\nformalism has been an indispensable tool for routinely\npredicting and analyzing macrospin ferromagnetic reso-\nnance and, with extensions, spin-wave resonances. Now,\nour theoretical framework allows for deriving the fre-\nquencies of the nutational and precessional resonances of\nanisotropic ferromagnets in the presence of inertia. We\nformulate the secular equation of inertial spin dynamics,\nprovide its exact and approximate solutions, and discuss\nthe interplay of inertia and magnetic anisotropy.\nII. PRECESSIONAL AND NUTATIONAL\nRESONANCE FREQUENCIES\nThe Smit-Beljers approach was initially developed via\nlinearizing the Landau-Lifshitz equation in spherical co-\nordinates and using small-angle approximation around\nthe equilibrium direction of magnetization [29, 30]. Em-\nploying the periodic solution ansatz, the ferromagnetic\nresonance (FMR) frequency was derived as\n!2\nSB=j\rj2\u0000\n1 +\u000b2\u0001\nM2\n0sin2\u00120\u0010\n@\u0012\u0012F@\u001e\u001eF\u0000(@\u0012\u001eF)2\u0011\n;(2)\nwhere\u00120is the equilibrium polar angle of magnetization.\nThe equation gives a closed-form relation between the\nFMR frequency and magnetic anisotropy energy Fand\nallows for e\u000ecient and convenient numerical analysis of\nexperimental data [31].\nIn our approach, we similarly write the ILLG equation\nin spherical coordinates which now contains \frst and sec-\nond derivatives of the angles. As detailed in Appendix A,\nthe relative magnitude of the terms in this resulting equa-\ntion can be analyzed by their prefactors expressed in the\norders of the Gilbert damping parameter \u000b. Since the lat-\nter is typically 10\u00003\u000010\u00002, we omit higher-order termsand arrive at\n8\n>>>>><\n>>>>>:@tt\u0012=j\rj\u00160H\u0012\n\u0011\u0000\u000bj\rj\u00160H'\n\u0011+@t'sin\u0012\n\u0011\n+(@t')2sin\u0012cos\u0012;\n@tt'sin\u0012=j\rj\u00160H'\n\u0011+\u000bj\rj\u00160H\u0012\n\u0011\u0000@t\u0012\n\u0011\n\u00002@t'@t\u0012cos\u0012:(3)\nUsing the small-angle approximation, we develop the\nequations around the equilibrium direction of magneti-\nzation which introduces second-order derivatives of the\nenergyF. The system of equations can be further lin-\nearized employing a Jacobian matrix of the angles (Ap-\npendix A). Using the periodic solution ansatz, we arrive\nat a fourth-order characteristic polynomial constituting\nthe secular equation of the inertial spin system:\n\"\n!2\nj\rj2\u0000\u0000\n1 +\u000b2\u0001\nM2\n0sin2\u00120\u0010\n@\u0012\u0012F@\u001e\u001eF\u0000(@\u0012\u001eF)2\u0011#\n\u0000\u00112!2\"\n!2\nj\rj2\u00001\n\u0011j\rjM0\u0012\n@\u0012\u0012F+@''F\nsin2\u00120\u0013#\n\u0000i!\u000b\nj\rjM0\u0012\n@\u0012\u0012F+@''F\nsin2\u00120\u0013\n= 0:(4)\nThe \frst group of terms corresponds to Eq. 2. The sec-\nond group of terms introduces inertia of magnetization.\nThe third group of terms corresponds to the frequency-\ndomain linewidth of the ferromagnetic resonance\n\u0001!SB=j\rj\u000b\nM0\u0012\n@\u0012\u0012F+@''F\nsin2\u00120\u0013\n(5)\nas it does in the noninertial case [28, 32]. The pre-\nsented approach has the advantage to converge to the\nSmit-Beljers secular equation when the inertial parame-\nter vanishes and can be written as\n\u0000\n!2\u0000!2\nSB\u0001\n\u0000\u00112!2\u0012\n!2\u00001\n\u0011\u000b\u0001!SB\u0013\n\u0000i!\u0001!SB= 0 (6)\nEquation 6 has two physical solutions: precessional res-\nonance!pat lower frequency and nutational resonance\n!nat higher frequency. In Appendix B, we calculate the\nexplicit (exact but complex) solutions, shown in Fig. 2,\nas a benchmark for the consecutive approximations.\nFirst, similarly to the original Smit-Beljers formalism,\nwe can omit the imaginary term in the secular Equation 6\n{ an approximation that we mark with 'a' { and derive\nan analytical form of the resonance frequencies:\n!(a)\np=\u0010\np\u0000p\np2\u0000q\u00111=2\n; (7)\n!(a)\nn=\u0010\np+p\np2\u0000q\u00111=2\n; (8)3\nwhere\np=1\n2\u00112+\u0001!SB\n2\u000b\u0011;\nq=!2\nSB\n\u00112:(9)\nThe analytical form can be used conveniently to calculate\nthe resonance frequencies via !SBand \u0001!SB, thus adding\njust a few extra steps compared to the Smit-Beljers for-\nmalism. However, the analytical form is still too complex\nand the e\u000bect of magnetic anisotropy on precessional and\nnutational behavior is not immediately clear.\nWe thus implement another approximation { marked\nwith 'b' { by expanding the analytical form into a Taylor\nseries (Appendix B) and neglecting higher-order terms in\n\u0001!SB\u0011\u001c1:\n!(b)\np=!SBr\n1\u0000\u0011\u0001!SB\n\u000b; (10)\n!(b)\nn=1\n\u0011+\u0001!SB\n2\u000b: (11)\nHere, we immediately see a systematic red-shift of the\nprecessional frequency as compared to the non-inertial\ncase of!SB. As shown in Fig. 1, the inertial torque vec-\ntor has a notable component that is antiparallel to the\nprecessional torque, thus e\u000bectively reducing the latter.\nThe nutational frequency !(b)\nn, on the other hand,\nshows a substantial frequency increase (a blueshift) as\ncompared to an earlier estimation !n\u00181=\u0011for an\nisotropic ferromagnet [3, 4]. Another approximation, for\ninstance employed in Ref. [15], accounts for a frequency\nshift due to an applied magnetic \feld\n\u0016!n=p\n1 +\u0011j\rj\u00160H0\n\u0011=1\n\u0011+j\rj\u00160H0\n2+::: (12)\nbut neglects the e\u000bects of magnetic anisotropy. While\nthe nutational frequency obtained in our model converges\nto the estimate \u0016 !nin the case of vanishing anisotropy,\nit demonstrates that magnetic anisotropy (both, shape\nand magnetocrystalline) shifts the nutation resonance\nfrequency as compared with \u0016 !n, and must be accounted\nfor according to the characteristic polynomial (Eq. 6) and\nits solutions.\nIII. EFFECT OF MAGNETIC ANISOTROPY\nON INERTIAL SPIN DYNAMICS\nWe calculate the e\u000bect of magnetic anisotropy on pre-\ncessional and nutational resonances for four concrete ex-\namples of magnetic samples that have been and may\nlikely be used in an experiment probing inertial spin dy-\nnamics. We consider single-crystal ferromagnetic thin\n\flms with cubic magnetocrystalline anisotropy (iron) anduniaxial magnetocrystalline anisotropy (hexagonal-close-\npacked cobalt) with magnetic parameters obtained from\nexperimental data of Refs. [3, 4]. The explicit (exact)\nsolutions of the characteristic polynomial of Eq. (6) are\nplotted in Fig. 2 for various con\fgurations of applied\nmagnetic \feld with respect to the \flm surface and crystal\nsymmetry axes. We use free energy density and equilib-\nrium angles de\fned in Appendices C and D. As shown in\nFig. 2, the e\u000bect of inertia is consistent in all calculated\nscenarios. The precessional frequency experiences a red-\nshift due to inertia as compared to resonance frequency\n!SBfor noninertial Smit-Beljers case. Both aligned pre-\ncessional modes (above the saturation \feld) and non-\naligned precessional modes (below the saturation \feld in\nhard-axes con\fgurations) [31] experience a redshift which\nincreases with increasing precessional frequency. For the\naligned modes, the redshift thus becomes stronger with\nincreasing magnetic \feld. For nonaligned modes, on the\nother hand, decreasing magnetic \feld can result in in-\ncreasing redshift.\nThe nutational frequency experiences a blue shift due\nto magnetic anisotropy and magnetic \feld. The blueshift\ntypically increases with increasing magnetic \feld. How-\never, below the saturation \feld in hard-axes con\fgura-\ntions, the blue-shift of the nutational frequency can be-\ncome stronger with decreasing magnetic \feld (see the red\nline in Fig. 2(a-c)). The exact behavior is dominated by\nthe \u0001!SBterm in Eqs. (10) and (11). Both shifts can be\nsubstantial (reaching up to 12% in the magnetic \feld of\n10T) for all calculated scenarios.\nWhile we use the explicit solutions of the secular\nequation in Fig. 2 to visualize the e\u000bects of magnetic\nanisotropy in inertial spin systems, the observed fre-\nquency shifts are in qualitative agreement with the ap-\nproximations. To assess the quantitative validity of our\napproximations 'a' and 'b', we compare them with the\nbenchmark of the explicit solutions. We \fnd that the an-\nalytical form 'a' (Eqs. (7-8)) accrues less than 0.5% error\nfor aligned modes when compared to the exact solution of\nEq. 4; however, it should be stressed that the character-\nistic polynomial Eq. 4 itself has been derived for inertial\nparameters \u0011\u001c1=\u0001!SB. The next-step approximation\nby the Taylor series 'b' (Eqs. (10-11)) introduces an error\nless than 6% for aligned modes with \u0011 <100 fs\u0001rad\u00001,\nwhile at higher values of the inertial parameter, the Tay-\nlor series causes a substantial error of the frequencies.\nWhile the approximation 'b' in Eqs. (10-11) should thus\nbe treated with caution, for comparison, we calculate\nthe explicit forms of precessional and nutational frequen-\ncies for aligned modes in the con\fgurations displayed in\nFig. 2:\n(a) The out-of-plane cubic case:\n!(b)\np2=j\rj2\u0000\n1 +\u000b2\u0001\u0012\n\u0000\u00160M0+\u00160H0+2Kcub1\nM\u00132\n\u0002\u0014\n1\u0000\u0011j\rj\u0012\n\u00002\u00160M0+ 2\u00160H0+4Kcub1\nM\u0013\u0015\n;\n(13)4\nFigure 2. Frequency-\feld relation of ferromagnetic resonance and nutational resonances. Explicit solutions of Eq. 6 for the\nprecessional resonance (orange) show a red-shift compared to the non-inertial Smit-Beljers case (dashed blue). Explicit solutions\nof Eq. 6 for the nutational resonance (red) show a blue-shift compared to the zeroth-order approximation 1 =\u0011. (a) The calculation\nparameters for a thin \flm with cubic magnetocrystalline anisotropy are: g= 2:09,\u00160M0= 2:1 T,\u000b= 0:002,\u0011= 75 fs \u0001rad\u00001,\nKcub1= 4:9\u0002104J\u0001m\u00003, Ref. [3, 4, 33]. (b)-(d) The following parameters for a thin \flm with uniaxial magnetocrystalline\nanisotropy have been used: g= 2:17,\u00160M0= 1:8 T,\u000b= 0:10,\u0011= 75 fs \u0001rad\u00001,Ku1= 4:1\u0002105J\u0001m\u00003, Ref. [3, 4, 34].\n!(b)\nn=1\n\u0011+j\rj\u0012\n\u0000\u00160M0+\u00160H0+2Kcub1\nM0\u0013\n(14)\n(b) The out-of-plane uniaxial case:\n!(b)\np2=\u0000\n1 +\u000b2\u0001\nj\rj2\u0012\n\u0000\u00160M0+\u00160H0+2Ku1\nM0\u00132\n\u0002\n\u0012\n1\u0000\u0011j\rj\u0012\n\u00002\u00160M0+ 2\u00160H0+4Ku1\nM0\u0013\u0013\n;\n(15)\n!(b)\nn=1\n\u0011+j\rj\u0012\n\u0000\u00160M0+\u00160H0+2Ku1\nM0\u0013\n: (16)\n(c) The in-plane perpendicular uniaxial case:\n!(b)\np2=\u0000\n1 +\u000b2\u0001\nj\rj2(\u00160M0+\u00160H0)\u0012\n\u00160H0\u00002Ku1\nM0\u0013\n\u0002\n\u0012\n1\u0000\u0011j\rj\u0012\n\u00160M0+ 2\u00160H0\u00002Ku1\nM0\u0013\u0013\n;\n(17)\n!(b)\nn=1\n\u0011+j\rj\u0012\u00160M0\n2+\u00160H0\u0000Ku1\nM0\u0013\n: (18)\n(d) The in-plane parallel uniaxial case:!(b)\np2=\u0000\n1 +\u000b2\u0001\nj\rj2\u0012\n\u00160H0+2Ku1\nM0\u0013\n\u0002\n\u0012\n\u00160M0+\u00160H0+2Ku1\nM0\u0013\n\u0002\n\u0012\n1\u0000\u0011j\rj\u0012\n\u00160M0+ 2\u00160H0+4Ku1\nM0\u0013\u0013\n;(19)\n!(b)\nn=1\n\u0011+j\rj\u0012\u00160M0\n2+\u00160H0+2Ku1\nM0\u0013\n: (20)\nIt is a common practice to analyze experimentally de-\ntermined dependences of ferromagnetic resonance fre-\nquency on applied magnetic \feld for evaluating magnetic\nparameters such as magnetic anisotropy and g-factor\n[31, 35{40]. Our work demonstrates that such evalu-\nation needs to be adjusted by taking into account the\ninertial redshift. In particular, measurements at higher\n\felds/frequencies have been considered to result in more\naccurate determination of magnetic parameters [41{45].\nOur model, however, shows that especially at high mag-\nnetic \felds, inertial redshift is strong and needs to be\ntaken into account.\nIt should be noted that in the framework of the ex-\ntended breathing Fermi surface model [6, 7], the inertial\nterm with negative sign was derived. Such negative in-\nertial term would formally result in a blueshift of the\nprecessional frequencies. However, since the origin of in-\nertia is still under discussion, we consider here only the\ne\u000bects of the positive inertial term suggested in Ref. [8].5\nIV. SUMMARY\nIn summary, we derived the secular equation for an in-\nertial spin system with an arbitrary magnetic anisotropy\nenergy in analogy with the Smit-Beljers approach. We\n\fnd that ferromagnetic resonance experiences a substan-\ntial redshift due to the inertia, while nutational resonance\nexperiences a blueshift due to magnetic anisotropy and\n\feld. For an accurate evaluation of magnetic parameters\nfrom magnetic resonance measurements, inertia needs to\nbe taken into account. Our model (Eq. 6) allows for con-\nvenient calculation of precessional and nutational reso-\nnances of an inertial spin system using parameters ( !SB\nand \u0001!SB) obtained from noninertial models.\nACKNOWLEDGMENTS\nM.Ch., M.F. and A.S. acknowledge funding by\nDeutsche Forschungsgemeinschaft (DFG, German Re-\nsearch Foundation) { Project No. 392402498 (SE 2853/1-\n1) and Project No. 405553726 CRC/TRR 270. R.M. ac-\nknowledges the funding from the Swedish Research Coun-\ncil via VR 2019-06313. I.B. acknowledges funding by the\nNational Science Foundation through Grant No. ECCS-\n1810541.\nAppendix A: SMIT-BELJERS APPROACH WITH\nINERTIA\nFirst, we transform the ILLG equation into a spher-\nical coordinate system and \fnd equilibrium angles of\nmagnetization. Second, we linearize the system of equa-\ntions describing magnetization dynamics at the equilib-\nrium point. Finally, we derive the eigenfrequencies cor-\nresponding to the resonances.\nIn a spherical coordinate system one writes M=M0er;\nHe\u000b=Hrer+H\u0012e\u0012+H'e';where the magnitude of the\nmagnetization vector persists over time. Using\n@tM=M0(@t\u0012e\u0012+ sin\u0012@t'e');\n@ttM=M0nh\n\u0000(@t\u0012)2\u0000(@t')2sin2\u0012i\ner\n+h\n@tt\u0012\u0000(@t')2sin\u0012cos\u0012i\ne\u0012\n+ [@tt'sin\u0012+ 2@t'@t\u0012cos\u0012]e'g;(A1)\none transforms the ILLG equation into\n8\n>>>>>>><\n>>>>>>>:@tt\u0012=j\rj\u00160H\u0012\n\u0011\u0000\u000b@t\u0012\n\u0011+@t'sin\u0012\n\u0011\n+(@t')2sin\u0012cos\u0012;\n@tt'sin\u0012=j\rj\u00160H'\n\u0011\u0000\u000b@t'sin\u0012\n\u0011\u0000@t\u0012\n\u0011\n\u00002@t'@t\u0012cos\u0012:(A2)\nHere, we introduce the \frst approximation, i.e. we re-\nplace the \"red\" terms in the system (A2) with the \"blue\"ones in the system (A5) as follows. Based on the ILLG\nequation, we write\n8\n>>>>>>><\n>>>>>>>:\u000b@t'sin\u0012\n\u0011=\u0000\u000bj\rj\u00160H\u0012\n\u0011+\u000b2@t\u0012\n\u0011\n\u0000\u000b(@t')2sin\u0012cos\u0012+\u000b@tt\u0012;\n\u000b@t\u0012\n\u0011=\u000bj\rj\u00160H'\n\u0011\u0000\u000b2@t'sin\u0012\n\u0011\n\u00002\u000b@t'@t\u0012cos\u0012\u0000\u000b@tt'sin\u0012(A3)\nand substitute these equations in (A2) instead of the cor-\nresponding \"color\" terms. We obtain\n8\n>>>>>>><\n>>>>>>>:@tt\u0012=j\rj\u00160H\u0012\n\u0011\u0000\u000bj\rj\u00160H'\n\u0011+@t'sin\u0012\n\u0011\u0000\n1 +\u000b2\u0001\n+(@t')2sin\u0012cos\u0012+ 2\u000b@t'@t\u0012cos\u0012+\u000b@tt'sin\u0012;\n@tt'sin\u0012=j\rj\u00160H'\n\u0011+\u000bj\rj\u00160H\u0012\n\u0011\u0000@t\u0012\n\u0011\u0000\n1 +\u000b2\u0001\n\u00002@t'@t\u0012cos\u0012+\u000b(@t')2sin\u0012cos\u0012\u0000\u000b@tt\u0012:\n(A4)\nThe last two terms in both equations are negligible, since\nthey are multiplied by \u000b<1, while \u0001!SB\u0011\u001c1. The\u000b2\nterms are much less than 1, hence the ILLG equation is\nconverted to\n8\n>>>>>>><\n>>>>>>>:@tt\u0012=j\rj\u00160H\u0012\n\u0011\u0000\u000bj\rj\u00160H'\n\u0011+@t'sin\u0012\n\u0011\n+(@t')2sin\u0012cos\u0012;\n@tt'sin\u0012=j\rj\u00160H'\n\u0011+\u000bj\rj\u00160H\u0012\n\u0011\u0000@t\u0012\n\u0011\n\u00002@t'@t\u0012cos\u0012:(A5)\nThis transformation is commonly adopted and was per-\nformed in Ref. [32]. The advantage of the \frst approx-\nimation is that the \fnal result, which is to be shown\nbelow, converges to the SB equation for \u0011= 0. Note\nthat the e\u000bective magnetic \feld He\u000b=\u0000\u0016\u00001\n0@MFin the\nspherical coordinate system is given by\nH\u0012=\u00001\n\u00160M0@\u0012F; H'=\u00001\n\u00160M0sin\u0012@'F: (A6)\nIn order to \fnd the eigenfrequencies from the nonlin-\near system of equations (A5), it is necessary to linearize\nit and to determine the equilibrium orientation of mag-\nnetization. The equilibrium given by the angles \u00120and\n'0is found from the extremum conditions\n@\u0012F= 0; @'F= 0 (A7)\nlimited by the conditions for the minimum, namely, the\ndeterminant of a Hessian matrix has to be positive\n@\u0012\u0012F@''F\u0000@\u0012'F@'\u0012F > 0 (A8)\nand one of the second derivative has to be positive as well\n@\u0012\u0012F > 0: (A9)6\nIn the excited state, magnetization is de\rected from the\nequilibrium orientation by the e\u000bective magnetic \feld\nchanges over time. Here, we introduce the second ap-\nproximation, which corresponds to the standard SB ap-\nproach, that is the de\rection from equilibrium is small\n\u0001\u0012(t) =\u0012(t)\u0000\u00120;\u0001'(t) ='(t)\u0000'0 (A10)\nand it is su\u000ecient to limit the expansion of free energy\nto the linear terms\n@\u0012F=@\u0012\u0012F\u0001\u0012+@\u0012'F\u0001';\n@'F=@\u0012'F\u0001\u0012+@''F\u0001';(A11)\nwhere the second derivatives are evaluated at the equilib-\nrium. Using the small-angle approximation, one obtains\nfrom expressions (A5)-(A11)\n@tt\u0001\u0012=\u0012\n\u0000j\rj\n\u0011M0@\u0012\u0012F+\u000bj\rj\n\u0011M0sin\u00120@\u0012'F\u0013\n\u0001\u0012\n+\u0012\n\u0000j\rj\n\u0011M0@\u0012'F+\u000bj\rj\n\u0011M0sin\u00120@''F\u0013\n\u0001'\n+@t\u0001'sin\u00120\n\u0011+ (@t\u0001')2sin\u00120cos\u00120;\n@tt\u0001'sin\u00120=\u0012\n\u0000j\rj\n\u0011M0sin\u00120@\u0012'F\u0000\u000bj\rj\n\u0011M0@\u0012\u0012F\u0013\n\u0001\u0012\n+\u0012\n\u0000j\rj\n\u0011M0sin\u00120@''F\u0000\u000bj\rj\n\u0011M0@\u0012'F\u0013\n\u0001'\n\u0000@t\u0001\u0012\n\u0011\u00002@t\u0001'@t\u0001\u0012cos\u00120:\n(A12)\nIn order to linearize this system of equations, the fol-\nlowing notations are introduced\na41=\u0000j\rj@\u0012'F\n\u0011M0sin2\u00120\u0000\u000bj\rj@\u0012\u0012F\n\u0011M0sin\u00120;\na42=\u00001\n\u0011sin\u00120;\na43=\u0000j\rj@''F\n\u0011M0sin2\u00120\u0000\u000bj\rj@\u0012'F\n\u0011M0sin\u00120;\n\u00174=\u00002 cot\u00120;\na21=\u000bj\rj@\u0012'F\n\u0011M0sin\u00120\u0000j\rj@\u0012\u0012F\n\u0011M0;\na23=\u000bj\rj@''F\n\u0011M0sin\u00120\u0000j\rj@\u0012'F\n\u0011M0;\na24=sin\u00120\n\u0011;\n\u00172= sin\u00120cos\u00120;\nx1= \u0001\u0012;\nx2=@t\u0001\u0012;\nx3= \u0001';\nx4=@t\u0001':(A13)Employing (A13), we rewrite the system (A12) as\n@tx1=x2;\n@tx2=a21x1+a23x3+a24x4+\u00172x2\n4;\n@tx3=x4;\n@tx4=a41x1+a42x2+a43x3+\u00174x2x4:(A14)\nAt the \fxed point x\u0003= (x\u0003\n1;x\u0003\n2;x\u0003\n3;x\u0003\n4) the dynamics of\nthe nonlinear system (A14) are qualitatively similar to\nthe dynamics of a linear system (A15) associated with\nthe Jacobian matrix J(x\u0003) [46], i.e.,\n0\nB@@tx1\n@tx2\n@tx3\n@tx41\nCA=\n0\nB@@x1f1(x\u0003)::: @x4f1(x\u0003)\n.........\n@x1f4(x\u0003)::: @x4f4(x\u0003)1\nCA0\nB@x1\nx2\nx3\nx41\nCA;(A15)\nwhere the right-hand sides of Eqs. (A14) are denoted\nasfi. The \fxed point is determined by equating the\nderivatives of the nonlinear system (A14) to zero, which\ngives the following equations\nx2= 0;\na21x1+a23x3= 0;\nx4= 0;\na41x1+a43x3= 0;(A16)\nwith the solution x\u0003\n1=x\u0003\n2=x\u0003\n3=x\u0003\n4= 0. The Jacobian\nmatrix of Eqs. (A14) is\nJ=2\n640 1 0 0\na21 0a23a24+ 2\u00172x4\n0 0 0 1\na41a42+\u00174x4a43\u00174x23\n75 (A17)\nand at the point x\u0003\n1=x\u0003\n2=x\u0003\n3=x\u0003\n4= 0 it provides the\nlinear system of equations\n@tx1=x2;\n@tx2=a21x1+a23x3+a24x4;\n@tx3=x4;\n@tx4=a41x1+a42x2+a43x3:(A18)\nThis third approximation goes beyond the SB approach\nand it is the linearization of the system (A14). The\neigenvalues of these equations give resonance frequencies,\nwhich are calculated from the characteristic polynomial\n!4+ (a21+a24a42+a43)!2\n\u0000i(a24a41+a23a42)!\u0000a23a41+a21a43= 0:(A19)\nRestoring the original variable notations, one \fnds the\nequation describing eigenfrequencies of a ferromagnet7\nwith inertia\n\"\n!2\nj\rj2\u0000\u0000\n1 +\u000b2\u0001\nM2\n0sin2\u00120\u0010\n@\u0012\u0012F@\u001e\u001eF\u0000(@\u0012\u001eF)2\u0011#\n\u0000\u00112!2\"\n!2\nj\rj2\u00001\n\u0011j\rjM0\u0012\n@\u0012\u0012F+@''F\nsin2\u00120\u0013#\n\u0000i!\u000b\nj\rjM0\u0012\n@\u0012\u0012F+@''F\nsin2\u00120\u0013\n= 0:(A20)\nNote that this equation can be converted to SB formula\n(2) if the inertial parameter vanishes.\nAppendix B: EXACT AND APPROXIMATE\nEXPRESSIONS OF RESONANCE FREQUENCIES\nThe quartic equation (A20) results in two pairs of\nroots. The \frst pair is precessional frequency modi\fed\nby inertia, one root of the pair is positive, the second is\nnegative. The same applies to the other pair correspond-\ning to the nutational frequency. Here we consider only\npositive roots. Let us use the Ferrari's solution for this\nquartic equation to write exact expressions of resonance\nfrequencies, and introduce the notations:\nAr=M2\n0\u00112;\nCr=\u0000M2\n0\u0000M0\u0011j\rj\u0012\n@\u0012\u0012F+@''F\nsin2\u00120\u0013\n;\nDr=iM0\u000bj\rj\u0012\n@\u0012\u0012F+@''F\nsin2\u00120\u0013\nEr=j\rj2\u0000\n1 +\u000b2\u0001\nsin2\u00120\u0010\n@\u0012\u0012F@\u001e\u001eF\u0000(@\u0012\u001eF)2\u0011\n;\nar=Cr\nAr; br=Dr\nAr; cr=Er\nAr:(B1)\nIn Ferrari's method, one determines a root of the nested\ndepressed cubic equation. In our case, the root is written\nyr=\u00005ar\n6+Ur+Vr; (B2)\nwhere\nUr=3s\n\u0000r\nP3\nr\n27+Q2\nr\n4\u0000Qr\n2;\nVr=\u0000Pr\n3Ur;\nPr=\u0000a2\nr\n12\u0000cr;\nQr=1\n3arcr\u0000a3\nr\n108\u0000b2\nr\n8:(B3)Thus, the exact precessional angular frequency modi\fed\nby inertia is given by\n!p=par+ 2yr\n2\n\u00001\n2s\n\u00003ar\u00002yr\u00002brpar+ 2yr;(B4)\nThe exact nutational angular frequency can be written\nas\n!n=par+ 2yr\n2\n+1\n2s\n\u00003ar\u00002yr\u00002brpar+ 2yr:(B5)\nNext, we write a few approximations allowing one to\nelucidate the physics behind Eqs. (B4)-(B4). The ap-\nproximation \"a\" of resonance frequencies is derived by\ntaking into account the real part of the quartic equa-\ntion (A20), which transforms this equation into a bi-\nquadratic one. Thus, the approximation \"a\" of preces-\nsional frequency reads\n!(a)\np=\u0010\np\u0000p\np2\u0000q\u00111=2\n: (B6)\nThe approximate nutational frequency is given by\n!(a)\nn=\u0010\np+p\np2\u0000q\u00111=2\n; (B7)\nwhere\np=1\n2\u00112+\u0001!SB\n2\u000b\u0011;\nq=!2\nSB\n\u00112;(B8)\nThis approximation introduces an additional error, which\ndoes not exceed 0.5% for the aligned modes for the pa-\nrameters employed in the main part of the paper. We\nthus \fnd that the solution 'a' can be considered su\u000e-\nciently accurate in the context of this work.\nExpressions (B6) and (B7) can be further simpli\fed by\nemploying Taylor series expansion and assumption that\n\u0001!SB\u0011\u001c1. The approximation 'b' of precessional fre-\nquency is\n!(b)\np=!SBr\n1\u0000\u0011\u0001!SB\n\u000b: (B9)\nFrom Eq. (B9), one can see that this expression of pre-\ncessional frequency modi\fed by inertia converges to the\nconventional expression of FMR at \u0011= 0. The approxi-\nmation \"b\" of the nutational frequency reads\n!(b)\nn=1\n\u0011+\u0001!SB\n2\u000b: (B10)\nThe series expansion leads to a further error of about\n6% for the parameters used for numerical calculations\npresented in the main part of the paper.8\nFigure 3. The orientation of coordinate frames in the case of\nmagnetic \feld applied in-plane (a) and out-of-plane (b).\nAppendix C: FREE ENERGY DENSITY\nWe consider two geometrical con\fgurations are of in-\nterest for the study of resonances in magnetic materials.\nIn the \frst con\fguration, a magnetic \feld rotates tangen-\ntially through the plane of the \flm surface the x0yplane\n(Fig. 3(a)). In the second con\fguration, the magnetic\n\feld goes from the tangential direction to the normal di-\nrection in the x0yplane (Fig. 3(b)). In the geometries\nselected here, the \flms are located di\u000berently relative to\nthe axes, thus the aforementioned planes do not coin-\ncide. Such choice of the axes allows one to avoid the\ndivision by zero in the out-of-plane applied \feld con\fgu-\nration (Fig. 3(b)). Otherwise, if one directs the magnetic\n\feld along the normal to the \flm in the con\fguration of\naxes shown in Fig. 3(a), one obtains singularity ( \u00120= 0)\nin Eqs. (4) and (2). Note that an alternative approach\nwas derived in the past by Baselgia et al. [47].\nIn order to write expressions of the energy contribu-\ntions, let us de\fne coordinate frames. In the general\ncase, the axes of magnetocrystalline anisotropy (cubic,\nuniaxial, etc) may not coincide with the demagnetiza-\ntion axes, therefore, one needs to make a transition from\none axis to another to calculate the energy. We indicate\nangles of magnetization vector as \u0012and'respectively\nto Cartesian coordinate system xyz, de\fning the demag-\nnetizing energy. The polar and azimuthal angles of the\nvector of the magnetic \feld are denoted by \u0012Hand'H\nwith respect to the same axes. The axes specifying the\nenergy of magnetocrystalline anisotropy are indicated by\nxayaza. For instance, we focus on uniaxial anisotropy in\nthe rotated coordinate system such that the c-axis ( za)\nis aligned with the y-axis. Thus, the free energy density\nis given by\nF=FZ+Fdm+Fa; (C1)\nwhereFZis the Zeeman energy density, Fdmis the de-\nmagnetizing energy density and Fais related to magne-\ntocrystalline anisotropy. Using representation of vectors\nFigure 4. The sequence of z-y-z rotations to the ~ \u000b;~\fand ~\r\nangles respectively.\nin a spherical coordinate system, one writes the Zeeman\nenergy as\nFZ=\u0000\u00160MH\n=\u0000\u00160M0H( sin\u0012sin\u0012Hcos ('\u0000'H)\n+ cos\u0012cos\u0012H)(C2)\nand the demagnetizing energy as\nFdm=1\n2\u00160M2\n0(Nxsin2\u0012cos2'+Nysin2\u0012sin2'\n+Nzcos2\u0012);(C3)\nwhereNx; Ny; Nzare demagnetizing factors. The mag-\nnetocrystalline anisotropy energy of a ferromagnet with\ncubic symmetry is given by\nFcub=Kcub1\u0000\n\u000b2\n1\u000b2\n2+\u000b2\n2\u000b2\n3+\u000b2\n1\u000b2\n3\u0001\n+Kcub2\u000b2\n1\u000b2\n2\u000b2\n3+:::\n=Kcub1cos2'asin2\u0012a\u0002\ncos2'a+\u0000\n1 + sin2\u0012a\u0001\nsin2'a\u0003\n+Kcub2sin4\u0012asin2'acos4'a+:::;\n(C4)\nwhere\u0012aand'aare polar and azimuthal angles of the\nmagnetization vector in the xayazaframe, and\u000b1;\u000b2and\n\u000b3are directional cosines with respect to the same frame.\nFinally, the uniaxial anisotropy energy can be written as\nFuni=Ku1sin2\u0012a+Ku2sin4\u0012a+Ku3sin6\u0012a\n+Ku4sin6\u0012acos 6'a+:::;(C5)\nwhere constants of anisotropy are denoted with Ki.9\nThe magnetization vector can be speci\fed in two\nequivalent ways\nM=M2\n4\u000b1\n\u000b2\n\u000b33\n5=M2\n4sin\u0012acos'a\nsin\u0012asin'a\ncos\u0012a3\n5; (C6)\ntherefore, one can write\n\u0012a= arccos\u000b3;\n'a= arctan\u000b2\n\u000b1:(C7)\nOn the other hand, one can match the vector components\nin thexayazaframe with the xyzframe using the Euler\nrotation matrix in the form\n2\n4\u000b1\n\u000b2\n\u000b33\n5=Eu\u0010\n~\u000b;~\f;~\r\u0011T2\n4sin\u0012cos'\nsin\u0012sin'\ncos\u00123\n5; (C8)\nEu\u0010\n~\u000b;~\f;~\r\u0011\n=2\n4c~\u000bc~\fc~\r\u0000s~\u000bs~\r\u0000c~\rs~\u000b\u0000c~\u000bc~\fs~\rc~\u000bs~\f\nc~\fc~\rs~\u000b+c~\u000bs~\rc~\u000bc~\r\u0000c~\fs~\u000bs~\rs~\u000bs~\f\n\u0000c~\rs~\fs~\fs~\rc~\f3\n5;(C9)\nthereby one rotates the xyz axes to the xayazaaxes.\nNote that the coordinate system rotates, not the vec-\ntor, therefore the Euler matrix is transposed. Here we\nintroduce the short notations for trigonometric functions\nc~\u000b= cos ~\u000b; s ~\u000b= sin ~\u000band so on. The given Euler ro-\ntation matrix describes a sequence of rotations to the\nangles ~\u000b;~\fand ~\raround the z, y and z local axes (Fig.\n4). Thus, one can express directional cosines \u000b1; \u000b2and\n\u000b3through the predetermined rotation angles ~ \u000b;~\f;~\rand\nangles of magnetization \u0012and'in thexyzframe; then\none can use formulas (C7) and the corresponding expres-\nsion of energy density to calculate anisotropy energy in\nthe thexyzcoordinate system.\nFor example, we focus on ferromagnets with uniax-\nial symmetry and cases shown in Fig. 3, then the con-\nsistency between the magnetization angles in xyz and\nxayazaframes is given by\nEu\u0010\u0019\n2;\u0019\n2;0\u0011T\n=2\n40 0\u00001\n\u00001 0 0\n0 1 03\n5; (C10)2\n4\u000b1\n\u000b2\n\u000b33\n5=2\n4\u0000cos\u0012\n\u0000sin\u0012cos'\nsin\u0012sin'3\n5; (C11)\n\u0012a= arccos (sin \u0012sin');\n'a= arctan (tan \u0012cos'):(C12)\nThe energy density is de\fned as\nF=\u0000\u00160M0H( sin\u0012sin\u0012Hcos ('\u0000'H)\n+ cos\u0012cos\u0012H)\n+Ku1\u0000\n1\u0000sin2\u0012sin2'\u0001\n+Fdm;(C13)\nwhere we neglect the high-order anisotropy terms. 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Mary\u0014 sko,\nDerivation of the resonance frequency from the free en-\nergy of ferromagnets, Physical Review B 38, 2237{2242\n(1988)."    },    {        "title": "1808.01296v1.Definition_of_the_interlayer_interaction_type_in_magnetic_multilayers_analyzing_the_shape_of_the_ferromagnetic_resonance_peaks.pdf",        "content": "arXiv:1808.01296v1  [cond-mat.mes-hall]  3 Aug 2018Definition of the interlayer interaction type in magnetic mu ltilayers analyzing the\nshape of the ferromagnetic resonance peaks\nO.G. Udalov,1,2,a)A.A. Fraerman,2and E.S. Demidov3\n1)California State University, Northridge, CA, USA\n2)Institute for Physics of Microstructures RAS, Nizhny Novgo rod,\nRussia\n3)Lobachevsky State University of Nizhni Novgorod, Nizhny No vgorod,\nRussia\n(Dated: 7 August 2018)\nWe present theoretical study of ferromagnetic resonance in a sy stem of two coupled\nmagnetic layers. We show that an interaction between the layers lea ds to the oc-\ncurrence of the so-called Fano resonance. The Fano resonance c hanges the shape of\nthe ferromagnetic resonance peak. It introduces a peak asymme try. The asymmetry\ntype is defined by the sign of the interaction between the magnetic la yers. Therefore,\nstudying the shape of the ferromagnetic resonance peaks one ca n define the type of\nthe interlayer coupling (ferromagnetic or antiferromagnetic). We show that using\nnumerical simulations one can estimate a magnitude of the interactio n by fitting the\nasymmetric resonance peaks.\na)oleg.udalov@csun.edu\n1I. INTRODUCTION\nFerromagnetic resonance (FMR) is a powerful tool for studying o f magnetic multilayer\nstructures [1–10]. The FRM method allows to obtain the information o n the magnetization\nmagnitude and magnetic anisotropy of each layer. It can be also use d for studying of the\ninterlayer coupling. Alotof effortswere spent oninvestigation ofth e coupling in thesystems\nwith magnetic layers separated by a metallic non-magnetic spacer [1, 8, 11–15]. In this case\nthe interlayer coupling is strong enough. This makes it relatively easy to define the coupling\nsign and magnitude studying shifts of FMR peaks.\nThesituationisdifferent formagneticmultilayers whereferromagne ticfilmsareseparated\nby an insulating spacer leading to a much weaker interlayer coupling [16 and 17]. Measuring\nthe coupling in this case is a tricky issue. The mutual shift of FMR peak s corresponding to\ndifferent layers is small comparing to the peaks width [3 and 4]. The situ ation becomes even\nmore complicated when resonant fields (frequencies) of the peaks are close to each other. In\nthis case a completely different approach is needed.\nIn the present work we propose to define the interlayer interactio n sign and magnitude\nby studying the FMR peaks shape rather than the shift. We will show that the interaction\ninduces an FMR peaks asymmetry. Such an asymmetry can be consid ered as the Fano\nresonance[18]inamagneticmultilayer. Studying theshapeofthisas ymmetryonecandefine\nthe interaction sign and magnitude. Such a method is particularly use ful when resonance\nfrequencies of two interacting layers are close to each other.\nStudying of the interaction sign and magnitude with the conventiona l method based on\nthe FMR peaks shift requires a reference sample without the interla yer interaction. This\nallows to measure the peak shift. The approach based on the peak s hape does not have such\na disadvantage. One can define the interaction sign and magnitude u sing a single sample.\nThe paper is organized as follows. In the Sec. II we analyse a simplified model in which\ntwo magnetic moments are placed into a strong magnetic field. Such a model allows an-\nalytical consideration providing the insight into the physics behind th e FMR peak shape\n(asymmetry). In Sec. III we study numerically magnetic bilayer sys tem (NiFe/Co) with an\narbitrary orientation of the external magnetic field.\n2/s122\n/s121/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s100/s105/s114/s101/s99/s116/s105/s111/s110/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121/s32/s100/s105/s114/s101/s99/s116/s105/s111/s110/s77\n/s49/s77\n/s50/s72\n/s101/s120/s116\n/s104\n/s120\nFIG. 1. A model system. Two magnetic moments placed in an exte rnal magnetic field Hext. An\nalternating magnetic field his applied perpendicular to Hext.M1,2show equilibrium orientation\nof the magnetic moments.\nII. SIMPLIFIED MODEL\nIn this section we consider a simplified model of two coupled magnetic m oments. We\ncalculate dissipation (FMR signal) inthis system anddemonstrate how theasymmetric peak\nof absorption appears. Consider two ferromagnetic (FM) films with uniform magnetizations\nM1,2(see Fig. 1). For simplicity we assume that the magnetic moments of b oth layers are\nthe same |M1,2|=M0. There is a uniaxial anisotropy in each film along the z-axis. It can be\ninduced by a demagnetizing field or by an internal anisotropy. The an isotropy constants are\nλ1,2. An external magnetic field Hext=H0z0is applied to the system. There is also a weak\nhigh-frequency alternating field along the x-axis h(t) =h(t)x0. Magnetic films interact with\neach other. The interaction energy is given by the expression\nEint=−˜J(M1M2). (1)\nWe linearize the Landau-Lifshitz-Gilbert (LLG) equations for both m agnetic moments\nM1,2in the vicinity of equilibrium positions M1,2=M0z0. The equations take the form\n\n\n˙m1x=−H1m1y−J(m1y−m2y)−˜α1˙m1y,\n˙m1y=H1m1x−J(m2x−m1x)+ ˜α1˙m1x−h,\n˙m2x=−H2m2y+J(m1y−m2y)−˜α2˙m2y,\n˙m2y=H2m2x+J(m2x−m1x)+ ˜α2˙m2x−h.(2)\n3Herem1,2are the corrections to the equilibrium magnetizations normalized by M0, the\nmagnitude of the effective field acting on the layers are H1,2=γ(H0+2λ1,2M0),J=γ˜Jis\nthe interaction constant multiplied by the gyromagnetic ratio γ. The renormalized damping\nconstants are ˜ α1,2. The system Eq. (2) can be transformed into two second order eq uations\nof the form\n\n\n¨m1x+α1˙m1x+ω2\n1m1x=A1m2x+D1˙m2x+h1,\n¨m2x+α2˙m2x+ω2\n2m2x=A2m1x+D2˙m1x+h2,(3)\nwhere we introduced the following notations\nα1,2=2˜α1,2(H1,2+J)\n1+ ˜α2\n1,2≈2˜α1,2(H1,2+J),\nω2\n1,2=(H1,2+J)2+J2\n1+ ˜α2\n1,2≈H2\n1,2+2JH1,2,\nA1,2=(H1+H2)J+2J2\n1+ ˜α2\n1,2≈(H1+H2)J,\nD1,2=J(˜α1+ ˜α2)\n1+ ˜α2\n1,2≈0,\nh1,2=H1,2h+ ˜α1,2˙h\n1+ ˜α2\n1,2≈H1,2h.(4)\nEquations (3) describe the system of two coupled oscillators with th e resonant frequencies\nω1,2. There are two types of coupling between the oscillators. We assum e that the damping\nis weak (˜ α1,2≪1) which is often the case for ferromagnets. In this limit one can neg lect\nthe dissipative coupling terms D1,2˙m1,2x. Also the retarded external excitation ˜ α1,2˙hcan\nbe omitted. For our purposes we can also neglect ˜ α2\n1,2in denominators in Eqs. (4). We\nassume that the coupling between the films Jis weak comparing to the effective fields H1,2.\nTherefore, we keep only the terms linear in J.\nA response of the system to a periodic external field h1,2=h(0)\n1,2eiωtcan be represented as\nm1,2x(t) =m1,2eiωt. The complex amplitudes m1,2are given by\n\n\nm1=(ω2\n2−ω2+iα2ω)h(0)\n1+A1h(0)\n2\n(ω2\n2−ω2+iα2ω)(ω2\n1−ω2+iα1ω)−A1A2,\nm2=(ω2\n1−ω2+iα1ω)h(0)\n2+A2h(0)\n1\n(ω2\n2−ω2+iα2ω)(ω2\n1−ω2+iα1ω)−A1A2.(5)\n4A. Layers with essentially different damping, but the same re sonant\nfrequencies\nLets now further simplify our consideration assuming that α2= 0 and ω1=ω2. This\nmeans that H1=H2,h(0)\n1=h(0)\n2, andA1=A2=A. Next we assume that the interaction is\nweak comparing to the damping ( α1≫A/ω1). In this case the oscillation amplitude of the\nfirst layer magnetization is given by\n|m1|2=((ω2\n2−ω2)+A)2(h(0)\n2)2\n(ω2\n2−ω2+A)2(ω2\n2−ω2−A)2+ω2α2\n1(ω2\n2−ω2)2. (6)\nIn the case of no interaction ( A= 0) we have an ordinarily resonance peak with the\nfrequency ω2−α2\n1/(4ω2). Introduction of the finite interaction Aleads to additional shift of\nthepeak, butwecanneglectitwhen α1≫A/ω1. Thefiniteinteractionisalsoresponsiblefor\nthe appearance of two peculiar points at ω=ω2±A/(2ω2). At the point ω=ω2−A/(2ω2)\nthe amplitude reaches its maximum. Oppositely, the oscillation amplitud e goes to zero at\nthe frequency ω=ω2+A/(2ω2). Such a reduction of the oscillation amplitude is called the\ndynamical damping and is very well known in the oscillation theory. Two periodic forces\nact on the the magnetic moment m1. The first one is due to the external field and the\nsecond one is due to the interaction with the second magnetic layer. Phases of the forces\ndepend on frequency. When the phase difference is πthe forces cancel each other. Such\na cancellation appears at ω=ω2+A/(2ω2) and therefore, m1does not oscillate at this\nfrequency. At ω=ω2−A/(2ω2) these two forces are in phase leading to enhancement of\noscillations. Finally, the shape of the resonance peak is distorted an d the peak asymmetry\nappears. Such a peculiarity in the frequency dependence of the os cillation amplitude is well\nknown as the Fano resonance [18].\nWhenwetakefinite α2intoaccountthereisnofulldampingandtheamplitudeisnotzero,\nbut one still has the minimum at ω=ω2+A/(2ω2) and the maximum at ω=ω2−A/(2ω2).\nImportant feature here is that if one changes the interaction sign the minimum and\nmaximum switch their positions. For A <0 (antiferromagnetic (AFM) interaction) the\ndynamical damping appears below ω2. ForA >0 (FM interaction) the dynamical damping\nappears above ω2. This feature can be used for defining the interaction sign.\nFigure 2 demonstrates behavior of |m1|2as a function of frequency for ω2= 100 a.u. and\n5/s57/s57/s46/s48 /s57/s57/s46/s51 /s57/s57/s46/s54 /s57/s57/s46/s57 /s49/s48/s48/s46/s50 /s49/s48/s48/s46/s53 /s49/s48/s48/s46/s56/s48/s46/s56/s53/s48/s46/s57/s48/s48/s46/s57/s53/s49/s46/s48/s48/s49/s46/s48/s53/s49/s46/s49/s48/s49/s46/s49/s53\n/s65 /s61/s48\n/s65 /s62/s48/s50/s124/s109\n/s49/s124/s50\n/s32/s91/s97/s46/s117/s46/s93\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s44/s32/s91/s97/s46/s117/s46/s93/s50/s50/s50/s65\n/s50\n/s65/s60 /s48\nFIG. 2. Amplitude of the magnetization of the first layer |m1|2as a function of frequency ω. The\nred line is for the zero interlayer coupling ( J= 0). The blue dashed line is for the finite AFM\ninteraction ( J <0). The green dash-dotted line corresponds to J >0. The black line shows the\namplitude of the second layer oscillation |m2|2(reduced 10 times to make it comparable to |m1|2).\nα1= 10 a.u. The solid red curve shows the case of zero interaction, A= 0. In this case there\nare no peculiarities in the amplitude behavior. Blue dashed curve in Fig. 2 shows|m1|2for\nfinite AFM interaction A=−3 a.u. One can easily see the asymmetry of the resonant peak.\nAccording to our consideration the dynamical damping occurs in this case below ω2= 100\na.u. Note that the curve is plotted for finite α2and therefore instead of zero amplitude\natω=ω2−A/(2ω2) we have finite oscillations. The dynamical enhancement appears at\nω=ω2+A/(2ω2). Dash-dotted green line shows |m1|2for positive FM interaction A= 3\na.u. One can see that the Fano resonance (asymmetry) is reflecte d with respect to ω=ω2\nin this case. So, the shape of the peak is clearly different for differen t sign of the interlayer\ninteraction.\nClosing this section we have to mention that the Fano resonance disa ppears if the dissi-\npation is the same in both layers.\nB. Layers with essentially different resonant frequencies\nSimilarbehavior occurswhentheresonant frequencies oftwo layer s arenotthesame. The\nFano resonance appears around the resonant frequency of the layer with lower dissipation.\nAgain, the sign of the interlayer interaction defines the shape (“dir ection”) of the Fano\n6/s57/s56 /s57/s57 /s49/s48/s48 /s49/s48/s49 /s49/s48/s50/s49/s46/s56/s50/s46/s48/s50/s46/s50/s50/s46/s52/s50/s46/s54/s50/s46/s56/s51/s46/s48/s51/s46/s50/s51/s46/s52\n/s65/s60/s48/s124/s109\n/s49/s124/s50\n/s32/s91/s97/s46/s117/s46/s93\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s44/s32/s91/s97/s46/s117/s46/s93/s50\n/s65/s62/s48\n/s49/s65/s61/s48\nFIG. 3. Amplitude of the magnetization of the first layer |m1|2as a function of frequency ω. The\ncase when the resonant frequencies of the layers are different . Red line is for the zero interlayer\ncoupling ( J= 0). Blue dashed line is for the finite AFM interaction ( J <0). Green dash-dotted\nline corresponds to the finite FM interaction ( J >0).\nresonance. Fig. 3 shows the amplitude |m1|2as a function of frequency for ω1= 100 a.u.\nandω2= 101 a.u., α1= 3 a.u. and α2= 0.1 a.u.,A= 0,±3 a.u.\nImportant to note that the Fano peculiarity disappears as the res onance frequencies\nbecome far from each other and there is no overlap between the FM R peaks.\nC. Absorption.\nIn the FMR experiment the measured quantity Wis the absorption or imaginary part of\nthe system response\nW/ω=M0h(0)Im(m1x+m2x)∼α1|m1x|2+α2|m2x|2. (7)\nFigure 4 shows the absorption as a function of frequency for two in teracting magnetic\nmoments. Resonance frequencies are ω1,2= 100 a.u., α1= 0.1 a.u.,α2= 2 a.u., A= 0,±50\na.u. One can see that at zero interaction the absorption peak is sym metric, while for finite\ninteraction the peak asymmetry appears. At that the asymmetry is defined by the interlayer\ninteraction sign.\n7/s57/s57 /s49/s48/s48 /s49/s48/s49/s48/s49/s50/s51/s52/s53/s54\n/s32/s32/s68/s121/s110/s97/s109/s105/s99/s97/s108/s32\n/s101/s110/s104/s97/s110/s99/s101/s109/s101/s110/s116/s65/s61/s48\n/s65/s62/s48/s87 /s32/s91/s97/s46/s117/s46/s93\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s44/s32/s91/s97/s46/s117/s46/s93/s65/s60/s48/s68/s121/s110/s97/s109/s105/s99/s97/s108/s32\n/s32/s32/s100/s97/s109/s112/s105/s110/s103\nFIG.4. Absorption W(Eq.(7))asafunctionoffrequency ω. Thecaseofequalresonantfrequencies\nofthemagneticlayers isshown. Redlineisforthezerointer layer interaction ( J= 0). Bluesolidline\nis for finite FM interaction ( J >0). Green dash-dotted line corresponds to finite AFM interac tion\n(J <0).\nIII. NUMERICAL SIMULATIONS\nIn the previous section on the base of the simplified model it was show n that the FMR\npeak asymmetry arises due to a weak interaction of the magnetic lay ers. The frequency\ndependencies of FMR signal were studied which was relevant for com parison of magnetic\nmultilayer systems withothersystems showing theFanoresonance s. IntheFMRexperiment\nthe field dependence is ordinarily measured at a fixed frequency of a lternating field.\nBesides, in the model a limit of strong field was considered in which magn etizations M1,2\nwere co-directed with each other and with the external field. In a r eal FMR experiment the\nmagnitude of the external field is limited. Therefore, the coincidenc e of resonance fields of\nthe magnetic layers ( Hr1≈Hr2) may appear in the situation when the external magnetic\nfield and the equilibrium magnetic moments of the layers are not co-dir ected. Analytical\nsolution of the problem in this situation is not feasible. Therefore, he re we present numerical\ndemonstration of the FMR peak asymmetry in a realistic situation.\nWeuseawell knownnumerical algorithmtosolve theLLGequations fo rmagneticfilms [1\nand 19]. The system energy is given by\nE=EZ+ED+EA+Eint, (8)\n8/s40/s48/s41\n/s67/s111\n/s122\n/s121/s77\n/s49\n/s77\n/s50/s72\n/s101/s120/s116\n/s104\n/s120/s83/s112/s97/s99/s101/s114/s78/s105/s70/s101\n/s100\n/s50/s100\n/s49/s49\n/s50/s72/s40/s48/s41\nFIG. 5. System geometry used in our numerical modeling. Two m agnetic layers (NiFe and Co)\nwith thicknesses d1,2are placed in an external magnetic field Hext. The field makes angle θHwith\nthe layers normal. The alternating magnetic field his applied perpendicular to Hext. Equilibrium\nmagnetic moments M1,2make angles θ(0)\n1,2with the normal.\n/s48 /s51 /s54 /s57 /s49/s50/s50/s48/s52/s48/s54/s48/s56/s48\n/s72/s32/s32/s32/s32 /s44/s32/s91/s107/s79/s101/s93\n/s101/s120/s116/s91/s100/s101/s103/s93/s67/s111\n/s78/s105/s70/s101\nFIG. 6. Equilibrium angles θ(0)for Co and NiFe layers as a function of external field magnitud e.\nThe external field is applied by the angle θH= 5.8 deg with respect to the sample normal.\nwhere the Zeeman energy is\nEZ=−/summationdisplay\ni=1,2di(MiHext), (9)\nthe magneto-dipole shape anisotropy is\nED=/summationdisplay\ni=1,22πdiM2\nicos2(θi), (10)\n9the uniaxial anisotropy is\nEA=/summationdisplay\ni=1,2diKicos2(θi). (11)\nHereθ1,2arethepolaranglesofmagnetizations(seeFig.5). Theexternal m agneticfield Hext\nis inclined by an angle θHwith respect to the sample normal. Kis the anisotropy constant.\nEquilibrium angles of magnetizations θ(0)\n1,2are defined by minimization of the system energy\nEq. (8). We use the parameters approximately corresponding to t he NiFe/I/Co magnetic\nbilayer. The thickness of NiFe and Co is d= 1 nm, g-factors are g1,2= 2, the frequency\nof the alternating field is ω= 9.5 GHz, the saturation magnetizations are M1= 325 Gs,\nM2= 1420 Gs, the uniaxial anisotropy constants are K1=−7.5·105Gs·Oe andK2= 4·106\nGs·Oe, the damping parameters are α1= 0.006 andα2= 0.04.\nFigure6showsbehaviourofequilibriummagnetizationanglesasafunc tionoftheexternal\nfield magnitude at θH= 5.8 deg. The field magnitude and angle are chosen in the region\nwhere we will observe the FMR peak asymmetry. One can easily see th at the equilibrium\nmagnetic moments are not co-directed with each other and with the magnetic field.\nFigure 7 shows the dependence of the FRM signal as a function of th e external magnetic\nfield magnitude ( W(Hext)) at a fixed frequency of the alternating field. The upper and\nlower panels correspond to different sign of the exchange interact ion˜J=±0.001 J/m2.\nEach figure shows several plots for different angle θHof the applied field. When the angle\nθH>6.5 deg and θH<5.5 deg, one sees two separate peaks corresponding to NiFe and Co\nlayers. The NiFe peak is the narrow one and the Co peak is the wide one . Changing the\nangle of the applied field one shifts the resonance field of NiFe and Co fi lmsHr1,2. Since the\nmagnetic anisotropy of these films is quite different Hr1,2(θH) the dependencies are not the\nsame and intersect each other at a certain angle θH. One can see that peaks overlap at the\nangleθH≈5.9 deg.\nThere is no peak asymmetry when NiFe and Co peaks are far from eac h other. This is\nin agreement with our analytical model. The asymmetry appears whe n the peaks overlap.\nComparing upper an lower panel one can see that the peak asymmet ry is different for FM\nand AFM interaction. Therefore, one can define the interaction sig n by measuring FMR\nspectrum at conditions of intersection of peaks. If the slope of th e narrow peak is higher\non the left side the interaction is FM. If the slope is higher on the right side the interaction\nis AFM. Fitting experimental data one can even define the magnitude of the interlayer\n10/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s56 /s49/s48 /s49/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s72/s32/s32/s32/s32/s32 /s44/s32/s91/s107/s79/s101/s93\n/s101/s120/s116/s61/s32/s55/s32/s100/s101/s103\n/s72/s61/s32/s54/s46/s53/s32/s100/s101/s103\n/s72/s61/s32/s54/s32/s100/s101/s103\n/s72/s61/s32/s53/s46/s57/s32/s100/s101/s103\n/s72/s61/s32/s53/s46/s56/s32/s100/s101/s103\n/s72/s61/s32/s53/s46/s53/s32/s100/s101/s103\n/s72/s61/s32/s53/s32/s100/s101/s103\n/s72/s40/s97/s41\n/s40/s98/s41/s87 /s91/s97/s46/s117/s46/s93 /s87 /s91/s97/s46/s117/s46/s93\n/s61/s32/s53/s46/s57/s32/s100/s101/s103\n/s72\n/s61/s32/s54/s32/s100/s101/s103\n/s72/s61/s32/s53/s46/s56/s32/s100/s101/s103\n/s72/s61/s32/s53/s46/s53/s32/s100/s101/s103\n/s72/s61/s32/s53/s32/s100/s101/s103\n/s72\n/s61/s32/s55/s32/s100/s101/s103\n/s72/s61/s32/s55/s32/s100/s101/s103\n/s72/s61/s32/s54/s46/s53/s32/s100/s101/s103\n/s72 /s72\nFIG. 7. FMR spectrum (absorbed power Was a function of the external field magnitude Hext)\nobtained numerically for NiFe/Co system. (a) FM interlayer interaction ˜J= 0.001 J/m2. (b) AFM\ninterlayer interaction ˜J=−0.001 J/m2. Differnet curves in the same plot correspond to different\ninclination angle of the external magnetic field θH. The curves for different θHare shifted with\nrespect to each other for better visibility.\ninteraction.\nIV. CONCLUSION\nWe considered the FMR resonance in two coupled magnetic layers. We showed that the\ninteraction between these layers leads to the occurrence ofthe s o-called Fano resonance. The\nFano resonance shows as a peculiarity in the absorption spectrum o f the coupled system. In\nparticular, the resonance peak becomes asymmetric. The asymme try type is defined by the\nsignoftheinteractionbetweenthelayers. Onecanusetheasymme trytodistinguishbetween\n11FM and AFM interlayer coupling. Using numerical simulations one can ev en estimate a\nmagnitude of the interaction fitting the asymmetric FMR peak.\nAs a final remark we would like to mention that in our work we considere d the isotropic\ninteraction Eq. (1). Such an equation describes the exchange cou pling. However, many\nexperiments evidence that in magnetic multilayer systems there is als o the magneto-dipole\ncoupling called the “orange-peel” effect. In contrast to the excha nge coupling, the “orange-\npeel” effect is anisotropic and described by a different equation [20]. T he anisotropy will\nlead to the angular dependence of the coupling constant J=J(θH). This, peculiarity can\nbe used for distinguishing between the exchange coupling and the “o range-peel” effect. This\nopportunity requires further investigation.\nV. ACKNOWLEDGMENTS\nThis research was supported was supported by the Russian Scienc e Foundation (Grant\n16-12-10340).\nREFERENCES\n1J. Lindner and K. Baberschke, J. Phys.: Condens. 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Baggio-Saitovitch,\nJournal of Applied Physics 109, 083917 (2011), https://doi.org/10.1063/1.3569690.\n15B. Khodadadi, J. B. Mohammadi, J. M. Jones, A. Srivastava, C. Mew es, T. Mewes, and\nC. Kaiser, Phys. Rev. Applied 8, 014024 (2017).\n16J. J. I. Wong, L. Ramirez, A. G. Swartz, A. Hoff, W. Han, Y. Li, and R . K. Kawakami,\nPhys. Rev. B 81, 094406 (2010).\n17J. Moritz, P. Bacher, S. Auffret, and B. Dieny,\nJournal of Magnetism and Magnetic Materials 323, 2391 (2011).\n18Y. S. Joe, A. M. Satanin, and C. S. Kim, Physica Scripta 74, 259 (2006).\n19R. Topkaya, M. Erkovan, A. Ozturk, O. Ozturk, B. Aktas, and M. Ozdemir, J. Appl.\nPhys.108, 023910 (2010).\n20M. F. Kuznetsov, O. G. Udalov, and A. A. Fraerman, arXiv:1807.055 90 (2018).\n13"    },    {        "title": "1607.03872v1.Surface_state_dominated_spin_charge_current_conversion_in_topological_insulator_ferromagnetic_insulator_heterostructures.pdf",        "content": "Surface state dominated spin-charge current conversion in\ntopological insulator/ferromagnetic insulator heterostructures\nHailong Wang1, James Kally,1, Joon Sue Lee,1Tao Liu,2\nHouchen Chang,2Danielle Reifsnyder Hickey,3Andre Mkhoyan,3\nMingzhong Wu,2Anthony Richardella,1Nitin Samarth1\u0003\n1Department of Physics, The Pennsylvania State University,\nUniversity Park, Pennsylvania 16802, USA\n2Department of Physics, Colorado State University, Fort Collins, CO 80523, USA and\n3Department of Chemical Engineering and Materials Science,\nUniversity of Minnesota, Minneapolis, Minnesota 55455, USA\n(Dated: July 14, 2016)\nAbstract\nWe report the observation of ferromagnetic resonance-driven spin pumping signals at room tem-\nperature in three-dimensional topological insulator thin \flms { Bi 2Se3and (Bi,Sb) 2Te3{ deposited\nby molecular beam epitaxy on Y 3Fe5O12thin \flms. By systematically varying the Bi 2Se3\flm\nthickness, we show that the spin-charge conversion e\u000eciency, characterized by the inverse Rashba-\nEdelstein e\u000bect length ( \u0015IREE), increases dramatically as the \flm thickness is increased from 2\nquintuple layers, saturating above 6 quintuple layers. This suggests a dominant role of surface\nstates in spin and charge interconversion in topological insulator/ferromagnet heterostructures.\nOur conclusion is further corroborated by studying a series of Y 3Fe5O12/(Bi,Sb) 2Te3heterostruc-\ntures. Finally, we use the ferromagnetic resonance linewidth broadening and the inverse Rashba-\nEdelstein signals to determine the e\u000bective interfacial spin mixing conductance and \u0015IREE.\n1arXiv:1607.03872v1  [cond-mat.mes-hall]  13 Jul 2016The development of next-generation spintronic devices has driven extensive studies of\nspin-to-charge conversion through measurements of the inverse spin Hall e\u000bect (ISHE)\nand/or the inverse Rashba-Edelstein e\u000bect (IREE) in both three-dimensional (3D) [1{6]\nand two-dimensional (2D) material systems [7{15]. Topological insulators (TIs) such as the\nBi-chalcogenides are naturally relevant in this context due to the large spin-orbit coupling\n(SOC) strength and the inherent spin-momentum \\locking\" in their surface states [9, 16, 17]\nwhich promise very e\u000ecient spin-charge conversion e\u000eciency. Previous studies of spin trans-\nfer in TI-based heterostructures have involved ferromagnetic metals that provide a shunting\ncurrent path, therefore introducing potential artifacts which complicate the picture and anal-\nysis [8, 9, 11, 12]. To circumvent these problems, we have grown and characterized bilayers\nof TIs on ferrimagnetic insulator Y 3Fe5O12(YIG) thin \flms with an exceptionally low damp-\ning constant [19]. Here, we report the ferromagnetic resonance (FMR)-driven spin pumping\nobserved in YIG/Bi 2Se3bilayers, showing robust spin pumping signals at room temperature.\nSystematic variation of the Bi 2Se3thickness allows us to unambiguously demonstrate that\nthe spin-charge conversion e\u000eciency, characterized by the inverse Rashba-Edelstein e\u000bect\n(IREE) length \u0015IREE in a 2D material system[7], dramatically increases from (1 :1\u00060:13)\npm to 35\u00064 pm as the Bi 2Se3thickness varies from 2 to 6 quintuple layers (QL). When the\ntop and bottom surface states with opposite spin polarizations decouple from each other,\n\u0015IREE saturates and is constant, providing clear evidence for the dominant role of surface\nstates in inducing spin-charge conversion in 3D TIs.\nWe \frst discuss the structural and interfacial characterization of the YIG/Bi 2Se3het-\nerostructure using high-resolution scanning transmission electron microscopy (HR-STEM).\nFigure 1(a) shows an atomically ordered 6 QL Bi 2Se3layer grown on an epitaxial 30-nm YIG\nthin \flm. We note that an amorphous layer of about 1 nm in thickness is observed at the\nYIG/Bi 2Se3interface, most likely due to the nucleation of the template layer in the two-step\ngrowth process (see Supplementary Material at [link to be added] for more details about the\ngrowth method). The atomic force microscopy (AFM) image in Fig. 1(b) shows a smooth\nsurface with a roughness of about 0.71 nm. A representative \u0012\u00002\u0012x-ray di\u000braction (XRD)\nscan of a 40 QL Bi 2Se3\flm shown in Fig. 1(c) indicates a phase-pure Bi 2Se3layer. Figure\n1(d) shows a representative FMR derivative absorption spectrum for a 30-nm YIG \flm used\nin this study taken at a radio-frequency (rf) f= 3 GHz with a magnetic \feld Happlied in\nthe \flm plane. The peak-to-peak line width (\u0001 HPP) obtained from the spectrum is 9.2 Oe,\n2and an e\u000bective saturation induction of 1.76 kOe is extracted from \ftting the frequency de-\npendence of the resonance \feld [19]. The spin pumping measurements are performed using\na microwave transmission line on the YIG/TI bilayers at room temperature (approximate\nsample dimensions of 1 mm \u00025 mm). During the measurements, a DC magnetic \feld His\napplied in the x-z-plane and the spin pumping voltage VSPis measured across the \u00185 mm\nlong TI layer along the y-axis, as illustrated in Fig. 2(a). At the resonance condition, the\nYIG magnetization Mprecesses around the equilibrium position and transfers angular mo-\nmentum to the conduction electrons in the TI \flms through interfacial exchange coupling\n[4]. The resulting pure spin current is injected along the z-axis with spin polarization \u001b\nparallel toM, and then converted to a charge current leading to the spin pumping signals.\nFigure 2(b) shows the temperature dependence of the resistivity of 6 QL Bi 2Se3and\n(Bi,Sb) 2Te3thin \flms grown on YIG. The metallic behavior (decrease in resistivity at low\ntemperature) is the typical behavior of Bi 2Se3due to Se vacancies [20]. For (Bi,Sb) 2Te3, the\nresistivity increases by \u001850% from room temperature to 2 K, consistent with surface state\ndominated transport in this thin \flm [21]. The carrier concentrations obtained from Hall\ne\u000bect measurements at room temperature are 4 \u00021013cm\u00002and 9:8\u00021012cm\u00002for 6 QL\nBi2Se3and (Bi,Sb) 2Te3, respectively.\nFigure 2(c) shows the observed VSPvs.Hspectra of the YIG/Bi 2Se3(6QL) bilayers at\nf= 2, 3, and 4 GHz using 100 mW microwave power. The observed spin pumping signals\nchange sign when the magnetic \feld His reversed from \u0012H= 90\u000eto 270\u000e, as expected\nfrom either IREE or ISHE. At 2 and 3 GHz, the observed signal is about 40 \u0016V, and for 4\nGHz, the signal decreases to about 20 \u0016V, which results from the variation of the microwave\ntransmission line performance at di\u000berent frequencies. Figure 2(d) shows the spin pumping\nspectra of a YIG/Bi 2Se3(6 QL) sample at microwave powers of 18, 32, 56, and 100 mW\nand an excitation frequency of 3 GHz. The upper inset shows the rf-power dependence of\nVSPat\u0012H= 90\u000e, indicating that the observed spin pumping signals are in the linear regime.\nTo probe the spin to charge conversion mechanism in TI layers, we systematically vary\nthe Bi 2Se3thickness from 2 to 60 QL. Figure 3(a) shows the spin pumping spectra when\n\u0012H= 90\u000efor 4, 6, 24, and 40 QL thicknesses of Bi 2Se3grown on YIG, respectively. The\nsigni\fcant enhancement of the spin pumping signal in the low Bi 2Se3thickness regime mainly\nresults from the increased resistivity. For a 2D material system, such as the TI surface states,\nthe spin to charge conversion is dominated by IREE [22, 23] and the injected spin current is\n3converted into a 2D charge current, Jc=\u0015IREEJs. The spin current density Jsis in units of A\nm\u00002, and the 2D charge current density Jcis in units of A m\u00001; the parameter \u0015IREE has the\ndimension of length and is introduced to characterize the spin to charge conversion e\u000eciency\nin 2D material systems [7, 23]. The observed spin pumping voltages VSPdominated by IREE\ndepend on several material parameters [7]:\nVSP=\u0000wR\u0015 IREEJs; (1)\nwherewandRare the sample width and resistance, respectively. Jsis the spin current\ndensity at the YIG/TI interface which can be expressed as [3, 5, 6]:\nJs=2e\n~g\"#hrf2~!2\u0014\n\r4\u0019Ms+q\n(\r4\u0019Ms)2+ 4!2\u0015\n2\u0019(\u0001Hpp)2\u0002\n(\r4\u0019Ms)2+ 4!2\u0003; (2)\nwhereg\"#is the e\u000bective interfacial spin mixing conductance [24], \u0001 Hppis the FMR peak-\nto-peak linewidth, hrfis the radio frequency \feld, !is the FMR angular frequency, and\nMsis the saturation induction of the YIG thin \flms. We can determine the e\u000bective spin\nmixing conductance g\"#from the FMR linewidth broadening of the YIG thin \flm [2, 3, 24]:\ng\"#=2\u0019p\n3Ms\rtYIG\ng\u0016B!\u0000\n\u0001HYIG=TI\u0000\u0001HYIG\u0001\n; (3)\nwhere\ris the absolute gyromagnetic ratio, tYIGdenotes the thickness of the YIG thin \flms,\ngis the Land\u0013 e factor, and \u0016Bis the Bohr magnetron.\nIf the spin pumping signal is dominated by the ISHE, spin di\u000busion should be taken into\naccount according to Jc=\u0012SH\u0015SDtanh\u0010\ntTI\n2\u0015SD\u0011\nJs, and the spin pumping signal will follow\n[2, 3, 6]:\nVSP=\u0000wR\u0012 SH\u0015SDtanh\u0012tTI\n2\u0015SD\u0013\nJs; (4)\nwhere\u0015SDis the spin di\u000busion length, tTIis the thickness of the TI thin \flm and \u0012SHis the\nspin Hall angle. The distinct di\u000berence between Eqs. (1) and (4) is whether the observed\nspin pumping signal is dominated by the spin momentum \\locking\" in the surface states\n[25{27] or by the SOC interaction.\nTo answer this question, Fig. 3(b) shows the Bi 2Se3thickness dependence of VSP(blue\npoints) and \u0015IREE (orJc=Js) (red points), where we de\fne Jc=VSP\nwR. Above 6 QL, Jc=Js\nalmost follows a constant value of about 35 pm. Below 6 QL, Jc=Jsdramatically decays by\na factor of 30 from 35 \u00064 pm to 1:1\u00060:13 pm when at 2 QL thickness. Earlier studies have\n4reported that the thickness of the Bi 2Se3surface states is approximately 2-3 nm [28, 29].\nAbove 6 QL, the top and bottom Bi 2Se3surface states decouple from each other; below 6\nQL, the interaction of the two surface states with opposite spin polarizations can decrease\nthe interfacial spin momentum \\locking\" e\u000eciency. This is consistent with angle-resolved\nphotoemission spectroscopy (ARPES) studies that show the opening of a gap in the Dirac\ncone when the Bi 2Se3thickness is below 6 QL, accompanied by a decrease in the spin\npolarization of the surface states [28, 29]. Qualitatively, our data shown in Fig. 3(b) follow\nthis trend and strongly indicate the key role played by the surfaces states in spin-charge\nconversion in Bi 2Se3. If we try to interpret the data in Fig. 3(b) with the spin di\u000busion\nmodel (Eq. 4), the \ft yields a value of \u0015SD\u00181:6 nm and also requires the presence of a\n\\dead\" layer at the interface (see Supplementary Material at [link to be added] for detailed\nanalysis using the spin di\u000busion model). This short vertical spin di\u000busion length suggests\nthat the spin polarized electron current is restricted to the bottom surface of the TI. Thus,\nwhile we cannot de\fnitively rule out the spin di\u000busion model, a more physically meaningful\npicture at this stage is that the surface states probably play a dominant role in the spin-\ncharge conversion. We note that the value we obtain for ( Jc=Js) (or\u0015IREE) is approximately\ntwo orders of magnitude smaller than the spin Hall angle reported using a spin torque FMR\nstudy at room temperature [9]. One possible reason for this discrepancy is the amorphous\nlayer at the interface shown in the HR-STEM \fgure, which potentially decreases the spin\ninjection e\u000eciency. Another reason may be the di\u000berence in the fundamental measurement\nmechanism between these two probing techniques. In a spin torque FMR experiment, as\nthe charge current \rows through the TI layers, the electrons can potentially have multiple\nscattering processes to transfer the spins to the ferromagnetic layers. However, in an FMR\nspin pumping measurement, this multiple scattering process may not be valid.\nTo further verify that the spin-charge conversion e\u000eciency is dominated by the sur-\nface states of TIs, we grew \fve di\u000berent TI heterostructures on YIG as control sam-\nples and measured their spin pumping signals. The \fve control samples are sample A:\nYIG/(Bi,Sb) 2Te3(6 QL); sample B: YIG/Bi 2Se3(1 QL)/(Bi,Sb) 2Te3(6 QL); sample C:\nYIG/Bi 2Se3(6 QL)/(Bi,Sb) 2Te3(6 QL); sample D: YIG/Bi 2Se3(1 QL)/Cr 0:2(Bi0:5Sb0:5)1:8Te3(6\nQL); and sample E: YIG/Bi 2Se3(6 QL)/Cr 0:2(Bi0:5Sb0:5)1:8Te3(6 QL). Figure 3(c) shows the\nspin pumping spectra of control samples B, C, D and E at 3 GHz radio-frequency and 100\nmW power. The enhancement of the spin pumping signal of samples D and E mainly results\n5from the larger resistivity of Cr 0:2(Bi0:5Sb0:5)1:8Te3compared to (Bi,Sb) 2Te3. Normalizing\nby the resistance and sample width, we obtained the spin charge conversion ratio of the \fve\ncontrol samples and compared them with the values for YIG/Bi 2Se3in Fig. 3(d). First, the\nvalues of\u0015IREEobtained for sample C and sample E are 37 \u00064 pm and 34\u00064 pm, respectively.\nBoth the values are quite close to 35 \u00064 pm measured for YIG/Bi 2Se3(6QL), indicating\nthat as long as the Bi 2Se3thickness is above 6 QL, the spin-charge conversion e\u000eciency is\nroughly constant and does not depend on the bulk properties: Cr doping and di\u000berent band\nstructures do not change the values. Second, for sample A, (Bi,Sb) 2Te3directly grown on\nYIG,\u0015IREE= 17\u00062 pm, about half of the value of Bi 2Se3. This is in sharp contrast with\nearlier results which reported a much larger spin Hall angle of the (Bi,Sb) 2Te3compared\nwith Bi 2Se3using a spin-polarized tunneling study [23]. This most likely results from the\ndi\u000berent interfacial quality and conditions that determine the spin momentum \\locking\"\ne\u000eciency. We expect that the bottom surface state condition at the YIG/(Bi,Sb) 2Te3in-\nterface [28] is not as good as the CoFeB/MgO/(Bi,Sb) 2Te3interface [23] for which TI was\ngrown on the commercial InP substrates with minimal lattice mismatch and the highest\nsample quality. In the end, we compare the values in samples B and D that both have\n1 QL Bi 2Se3seed layers. For sample D, we intentionally dope the (Bi,Sb) 2Te3with Cr,\nwhich can induce ferromagnetism at low temperature [30]. At room temperature, the Cr\ndoping mainly changes the transport properties and the SOC strength of the bulk states.\nThe values for samples B and D are \u0015IREE= 20\u00062 pm and 22\u00063 pm, respectively. Their\nsimilar spin-charge conversion e\u000eciencies demonstrate that the properties of the TI bulk\nstate do not play a signi\fcant role here, con\frming the interface-dominated spin pumping\nphenomena. It is also important to note that values of \u0015IREE for samples B and D are\nlower than the value for YIG/Bi 2Se3(6QL). As in other studies of spin pumping into TIs,\nthe interfacial condition presents a critical challenge for controlling the spin conversion\ne\u000eciency [8, 11, 12]; in sample B, both YIG/Bi 2Se3and Bi 2Se3/(Bi,Sb) 2Te3interfaces will\ncontribute to the formation of the surface states. Thus, structural defects and/or strain\ninduced dislocations in the trilayer heterostructures can potentially result in the observed\nlower values. A thorough understanding about the correlation of the interfacial conditions\nof TI surfaces states and the spin-charge conversion e\u000eciency requires further investigation.\nFinally, we compare the spin transfer e\u000eciency at YIG/Bi 2Se3to that at YIG/Pt. Note\nthat Pt is an ideal spin sink and a well-studied non-magnetic material with large SOC [3, 6].\n6Figure 4(a) shows the inverse spin Hall spectrum of a YIG(30nm)/Pt(5nm) bilayer sample\nunder 3 GHz and 100 mW microwave power when the H\feld is in plane. The observed\nsign change of the spin pumping signal with \feld reversal is expected for the ISHE in a 3D\nmaterial system [2, 3]. From the FMR linewidth broadening, the obtained YIG/Pt e\u000bective\nspin mixing conductance is (5 :19\u00060:6)\u00021018m\u00002, which lies in the range of the values\nreported by other groups using spin pumping [5, 6]. We compare this value with the obtained\nspin mixing conductance at various Bi 2Se3thicknesses in Fig. 4(b). When Bi 2Se3is 6 QL\nthick, the spin mixing conductance at the YIG/Bi 2Se3interface is (4 :13\u00060:5)\u00021018m\u00002.\nAlthough there are some variations, the reported values are in the range of 3 \u00007\u00021018m\u00002\nwhen the Bi 2Se3thickness varies from 2 to 60 QL, which is essentially comparable to the\ndetermined value at the YIG/Pt interface, demonstrating an e\u000ecient spin transfer in YIG/TI\nheterostructures. It is important to note that in the large Bi 2Se3thickness regime, we do\nnot observe an enhancement of g\"#, which is typically observed in the YIG/transition metal\nbilayers due to the decrease in back\row spin current caused by the spin di\u000busion in the\nbulk [24, 32]. This also con\frms the TI surface states dominated spin-charge conversion\nmechanism.\nIn conclusion, we report robust spin pumping at room temperature in YIG/Bi 2Se3bilay-\ners and other YIG/TI heterostructures. By measuring IREE voltages and interfacial spin\ncurrent density, we determine the value of \u0015IREE and reveal its systematic behavior with\nBi2Se3thickness, demonstrating the dominant role of surface states in spin-charge conver-\nsion. The inferred IREE length indicates the important role of interface conditions in spin\nHall physics in topological insulators. Further investigation is required for a thorough un-\nderstanding of the correlation between the formation of the surface states and the variation\nof spin-charge conversion e\u000eciency at the interfaces.\nThe work at Penn State, Colorado State and University of Minnesota is supported by\nthe Center for Spintronic Materials, Interfaces, and Novel Architectures (C-SPIN), a funded\ncenter of STARnet, a Semiconductor Research Corporation (SRC) program sponsored by\nMARCO and DARPA. NS and AR acknowledge additional support from ONR- N00014-15-\n1-2364. 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(b) Resistivity of 6 QL Bi 2Se3and (Bi,Sb) 2Te3thin \flms grown on YIG as a function of\ntemperature. (c) VSPvs.Hspectra of YIG/Bi 2Se3(6QL) atf= 2, 3, and 4 GHz using 100\nmW microwave power. (d) VSPvs.Hspectra of the YIG/Bi 2Se3(6QL) sample for the microwave\npower of 18, 32, 56, and 100 mW at f= 3GHz. Inset: rf-power dependence of the corresponding\nVSPat\u0012H= 90\u000e.\n11(a)(b)(c)(d)FIG. 3. (Color online) (a) VSPvs. Hspectra of YIG/Bi 2Se3(4QL), YIG/Bi 2Se3(6QL),\nYIG/Bi 2Se3(24QL), and YIG/Bi 2Se3(40QL) at f= 3 GHz using 100 mW microwave power.\nThex-axis is shifted by the resonance \feld ( HR) for clarity. (b) Dependence of VSP(blue\npoints) and the spin-to-charge conversion e\u000eciency Jc=Jsdetermined by \u0015IREE (red points) on the\nBi2Se3thickness. (c) VSPvs.Hspectra of control sample B: YIG/Bi 2Se3(1QL)/(Bi,Sb) 2Te3(6QL)\n(blue curve); sample C: YIG/Bi 2Se3(6QL)/(Bi,Sb) 2Te3(6QL) (green curve); sample D:\nYIG/Bi 2Se3(1QL)/Cr 0:2(Bi0:5Sb0:5)1:8Te3(6QL) (red curve), and sample E: YIG/Bi 2Se3(6\nQL)/Cr 0:2(Bi0:5Sb0:5)1:8Te3(6QL) (black curve) at 3 GHz and 100 mW. (d) Comparison of Jc=Js\nfor the control samples with the corresponding values for YIG/Bi 2Se3.\n12(a)(b)FIG. 4. (Color online) (a) V ISHE vs.Hspectra of YIG/Pt (5nm) bilayer at radio-frequency of 3\nGHz and 100 mW microwave power. (b) Dependence of the YIG/Bi 2Se3interfacial spin mixing\nconductance g\"#on Bi 2Se3thickness. The blue dashed line indicates the value of g\"#at the YIG/Pt\ninterface.\n13"    },    {        "title": "2307.12390v1.Unconventional_spin_polarization_at_Argon_ion_milled_SrTiO3_Interfaces.pdf",        "content": "1 \n Unconventional spin polarization at Argon ion milled SrTiO 3 Interfaces. \n \nAmrendra Kumar1, Utkarsh Shashank2, Suman Kumar Maharana1, John Rex Mohan2, Surbhi \nGupta3, Hironori Asada4, Yasuhiro Fukuma2,5* and Rohit Medwal1* \n1Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India \n2Department of Physics and Information Technology, Faculty of Computer Science and Systems \nEngineering, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan  \n3Department of Physics, Motilal Nehru National Institute of Technology, Barrister Mullah Colony, \nTeliarganj, Prayagraj, Uttar Pradesh, 211004, India \n4Department of Electronic Devices and Engineering, Graduate School of Science and Engineering, \nYamaguchi University \n5Research Center for Neuromorphic AI hardware, Kyushu Institute of Technology, Kitakyushu \n808-0196, Japan \n*Correspondence should be addressed to R.M.: rmedwal@iitk.ac.in  \nCo-correspondence should be addressed to Y.F..: fukuma@phys.kyutech.ac.jp   \n \n \nKeywords:  Spin torque ferromagnetic resonance, ion milling, two-dimensional electron gas, \nRashba field  \n \n \n \n 2 \n ABSTRACT  \nInterfacial two-dimensional electron gas (2DEG) formed at the perovskite-type oxide, such as \nSrTiO3, has attracted significant attention due to its properties of ferromagnetism, \nsuperconductivity, and its potential application in oxide-based low-power consumption electronics. \nRecent studies have investigated spin-to-charge conversion at the STO interface with different \nmaterials, which could affect the efficiency of this 2DEG interface.  \nIn this report, we presented a 𝐴𝑟ା ion milling method to create a 2DEG at STO directly by inducing \noxygen vacancies. To quantify the spin-to-charge conversion of this interface, we measured the \nangular-dependent spin torque ferromagnetic resonance (ST-FMR) spectra, revealing an \nunconventional spin polarization at the interface of of Argon ion milled STO  and NiFe . \nFurthermore, a micromagnetic simulation for angular dependent spin torque ferromagnetic \nresonance (ST-FMR) has been performed, confirming the large unconventional spin type spin \npolarization at the interface.   3 \n I. INTRODUCTION \nEfficient charge to spin conversion at interface has emerged as a promising research direction in the \nfield of spintronics for the development of next-generation spin based electronic devices1,2,3. Unlike \nconventional electronics, which rely solely on the charge of electrons, spintronics exploit both the \ncharge and the intrinsic spin of electrons to manipulate and store information at high speed and low \npower consumption.  In recent years it has been found out that there are two major mechanisms for \nthe conversion of charge to spin in nonmagnetic materials using spin hall effect and  Rsahba-\nEdelstein4,5,6,7 effect which are called Bulk effect and Interface effect respectively. A crucial element \nin spintronics research is the two-dimensional electron gas (2DEG)8,9,10,11, a unique physical system \nthat plays a pivotal role in enabling numerous spintronic applications. Therefore, 2DEG is a \nfoundational component in spintronics research, offering a versatile platform for investigating and \nharnessing the spin properties of electrons. Its unique properties, such as spin-polarized transport, \nefficient spin injection and detection, and strong spin Hall effect, make it a crucial building block \nfor various spintronic applications, including quantum computing, spin-FETs, and spin filters. The \nongoing research and development in 2DEG12,13,14 based spintronics holds great promise for the \nadvancement of future electronic and computing technologies.  \nIt has been reported that perovskite oxide-based interfaces have shown formation of 2D electron gas \nas STO/ALO39. Mainly STO has its 2D electron gas at its surface due to the vacant oxygen sites. In \nthis work, we present the formation of 2D electron gas using  𝐴𝑟ା ion milling by removal of oxygen \nfrom STO surfaces. This formed 2D electron gas shows inversion symmetry at the interface, creating \nRashba effect at the surface of STO resulting in out-of-plane electric field. So, when the charge \ncurrent 𝐽௫ flows through the surface, it generates a transverse spin current through Edelstein effect.  \nwhich upon diffusing to adjacent ferromagnetic materials exerts a torque on its magnetization.  4 \n Experimental:  \nThe 𝑆𝑟𝑇𝑖𝑂ଷ single crystal possesses a unique conducting surface state, which make it ideal for \napplications involving spin-dependent transport15,16,17. Additionally, 𝑆𝑟𝑇𝑖𝑂ଷ shows the large spin \npolarization due to strong spin-orbit coupling, unique electronic band structure, and quantum \nconfinement of the 2DEG. The 𝑆𝑟𝑇𝑖𝑂ଷ15,16,17 single crystal can host an unusual conducting surface \nstate which can be used for spin dependent charge transport and large spin polarization. To develop \nthe conducting surface of 𝑆𝑟𝑇𝑖𝑂ଷ, an ion milling process was used, where an ion gun produced a \nbeam of high energetic ions (typically noble gases like Ar+ ions) and bombarded its towards the \nsurface of crystal. To create conducting surface of 𝑆𝑟𝑇𝑖𝑂ଷ, accelerated ions, also called ion milling \nprocesses, were used. This bombardment of Argon ions on the surface of 𝑆𝑟𝑇𝑖𝑂3 with orientation \n(100) builds a conducting surface state by removing oxygen ions from 𝑆𝑟𝑇𝑖𝑂ଷ. The choice of inert \ngas helps us to prevent the 𝑆𝑟𝑇𝑖𝑂ଷ,  from additional impurities. It is expected that the thickness of \nthe surface conducting layer changes with increasing the milling time. This can be simply co-\nrelated with the conductance of the surface as a function of milling time. After confirming the \nformation of conducting surface states, ferromagnetic layer 𝑁𝑖଼𝐹𝑒ଶ  (hereafter refer as 𝑁𝑖𝐹𝑒 ) \nwas deposited on ion milled 𝑆𝑟𝑇𝑖𝑂ଷ. The designed 𝑆𝑟𝑇𝑖𝑂ଷ/2𝐷𝐸𝐺/𝑁𝑖𝐹𝑒 stack were subjected to \nmicro device fabrication using photolithography technique for spin transport. \nTo perform the spin dependent transport measurements, spin torque ferromagnetic resonance \n(ST-FMR)18,19 devices were prepared in the in-plane excitation geometry as shown in Fig. 1(b).   \nIn the ST-FMR measurement, a microwave current 𝐼  is applied in the longitudinal direction \nalong with an in-plane external magnetic field 𝐻௫௧ at an angle ( 𝜙) of 0° to the sample. The applied \n𝐼 current generates Oersted field (Ampere’s law) which thereby exerts an Oersted field torque 5 \n (𝜏ை) on magnetization vector of 𝑁𝑖𝐹𝑒 layer while injected 𝐼 into 2DEG layer will get convert \ninto oscillating transverse spin current. This results in a spin orbit torque ( 𝜏ை ) acting on the \nmagnetization vector of 𝑁𝑖𝐹𝑒. Thus, both Oersted field torque and spin orbit torque collectively \ndrives the magnetization precession in 𝑁𝑖𝐹𝑒 along the effective magnetic field ( 𝐻) as shown in \nFig. 1(b), when applied microwave frequency and external magnetic field satisfy resonance \ncondition. This precession leads to oscillation of resistance due to anisotropic magnetoresistance \n(AMR) of 𝑁𝑖𝐹𝑒.  \n     \nFigure 1. (a) Schematic showing (𝐴𝑟ା) ion milling in SrTiO 3 substrate (b) Schematic of ST-FMR \nmeasurement with lock-in setup to measure the  \n𝑉\n௫\n voltage across the device.  \nA rectified DC voltage across the SrTiO 3/2DEG/NiFe device from the mixing of  𝐼 and oscillating \nresistance is detected using lock-in and high frequency bias tee, while applying simultaneously the \nmicrowave. The rectifying voltage 𝑉௫  was measured using phase sensitive lock in technique \nwhile sweeping an external in plane magnetic field was swept at an angle of 𝜙. The rectifying \n6 \n voltage 𝑉௫ is fitted using the following equation20,21, \n𝑉௫=𝑉ௌ𝐹ௌ௬(𝐻௫௧)+𝑉𝐹௦௬(𝐻௫௧), (1) \nwhere,  𝑉ௌ௬(𝐻௫௧)=(௱ு)మ\n(ுೣିுೞ)మା(௱ு)మ , is the symmetric part of the 𝑉௫  spectrum, \n𝐹௦௬(𝐻௫௧) =∆ு(ுೣିுೞ)\n(ுೣିுೞ)మା(௱ு)మ , is the antisymmetric part, 𝛥𝐻  and 𝐻௦  are the half-width-at-\nhalf-maximum (linewidth) and the resonance field, and amplitude 𝑉ௌ  which is associated with the \nsymmetric part of the spectra and related to in plane torques (𝜏∥) , and 𝑉 related to antisymmetric \npart of the spectra which is due to out-of-plane torques (𝜏ୄ). Throughout the frequency range, we \nobserved a similar behavior of the symmetric and antisymmetric components of the ST-FMR \nspectra as shown in figure 2(a). The amplitude of the spectra decreases with increasing higher \nfrequency due to decrease in the precession cone angle of magnetization precession. A typical 𝑉௫ \nof STO/2DEG/NiFe sample measured at 5 GHz at 𝜙= 0°, as shown figure 2(b) clearly showing \nthe difference in the amplitude of the symmetric and antisymmetric spectra at opposite magnetic \nfield. The effective magnetization, 4𝜋𝑀, of STO/2DEG/NiFe  sample   is estimated from fitting \n𝑓 vs  𝐻௦ plot with in-plane Kittel equation22.  \n𝑓=𝛾\n2𝜋ට(𝐻௦+𝐻)൫𝐻௦+𝐻+4𝜋𝑀൯        (3) \nWhere, 𝐻 and γ  are the effective in plane magnetic anisotropic field and gyromagnetic ratio, \nrespectively. The estimated value of 4𝜋𝑀  and ఊ\nଶగ   are 951 𝑚𝑇  and 0.0288 𝐺𝐻𝑧/𝑚𝑇 , \nrespectively.  The broadening of ST-FMR has linear relation with frequency as shown figure 2 (d) . \nThe Gilbert damping parameter 𝛼 which depends on linewidth 𝛥𝐻, is estimated using, \n𝛥𝐻= ∆𝐻୭+2π𝑓\nγ𝛼, (2) \n Also ∆𝐻 and 𝛼 is estimated using linear fitting as shown in equation (2), is the inhomogeneous 7 \n linewidth broadening and Gilbert damping factor which is independent of 𝑓, and depends on the \nsample quality. The value of Gilbert damping factor obtained is 0.007. \n \nFigure 2. Frequency-dependent ST-FMR measurements: (a) Lorentzian fitting of 𝑉௫ voltage for \n5 to 11 GHz frequency (b) Lorentzian fitting of 𝑉௫  voltage for both negative and positive applied \nmagnetic field at 5 GHz where 𝑉ௌ௬ and 𝑉௦௬ represents the symmetric and antisymmetric part \nof the spectra (c) Linear fitting of fullwidth half maxima with frequency (d) Kittel fit of frequencies \nranging from 5 to 11 GHz at 𝜙 = 0°and the corresponding 𝑀. \n \nIn a two-dimensional electron gas (2DEG) interface exhibiting substantial spin-orbit coupling \n(SOC), the flow of a charge current (𝐽) gives rise to a transverse spin current 𝐽ௌ. This 𝐽௦ exerts a \ntorque on the magnetization vector of the adjacent layer NiFe. The exerted torque can perform \n8 \n ferromagnetic resonance in the adjacent layer while out balancing damping like torque (𝜏) in \nthe external applied magnetic field 𝐻௫௧ . The magnetic material NiFe exhibits a change in \nresistance when  𝐽 flows through it, known as anisotropic magnetoresistance (AMR). This effect \narises from spin-orbit coupling and depends on the angle between the 𝐻௫௧ and the applied 𝐽.  In-\nplane angular-dependent ST-FMR measurements were performed to fully comprehend the \nsymmetry torques concerning the applied external magnetic field (𝐻௫௧). It has been shown that \nwhen both the polarization  𝑝̂ of spin current and the rf field 𝐻  are parallel to y axis, and are \ntransverse to nonmetal/ferromagnetic material, the amplitudes of both the symmetric and \nantisymmetric components are known to be proportional to sin(2ϕ)cos(ϕ)23. This spin polarization \nconfiguration is called conventional spin polarization spin current and here sin (2𝜙) contribution \nis due to AMR of the magnetic layers, while the 𝑐𝑜𝑠(𝜙)  contribution describe the angular \ndependence of the torque amplitude exerted on magnetization by 𝐼(∝𝐽) . So, the overall angular \ndependence of symmetric and antisymmetric are proportional to sin(2𝜙)cos (𝜙) . For Pt/Py \nsample both symmetric and antisymmetric component of 𝑉௫  are proportional to \nsin(2𝜙)cos (𝜙)  as shown in figure S5, which can be attributed to very low magnetization of \nplatinum. If the symmetric and antisymmetric component of 𝑉௫  has additional dependence \nincluding the sin(2𝜙)cos(𝜙). This shows there are additional torque components working on the \nFM magnetization vector. These additional torques arises due when 𝑝̂  and 𝐻  contain 𝑥  and 𝑧 \ncomponents and this spin polarization configuration is called unconventional spin polarization. \nThe angular dependence of symmetric and antisymmetric components of 𝑉௫  is obtained at \n5 𝐺𝐻𝑧  for 25  minutes argon milled STO/2DEG/NiFe   sample as shown in figure 2 (a) and (b) \nrespectively. The symmetric component of is fitted using the following equation24,25  \n𝑉ௌ௬=𝑆௫𝑠𝑖𝑛(2𝜙)𝑆𝑖𝑛(𝜙)+𝑆௬𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙)+𝑆௭𝑠𝑖𝑛(2𝜙)      (4) 9 \n Where the 𝑆௫,  𝑆௬  and 𝑆௭  are the weightage of 𝑠𝑖𝑛(2𝜙)𝑠𝑖𝑛(𝜙) , 𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙)  and 𝑠𝑖𝑛(2𝜙) . \nSimilarly, the antisymmetric component is fitted using the following equation.  \n𝑉௦௬=𝐴௫𝑠𝑖𝑛(2𝜙)𝑠𝑖𝑛(𝜙)+𝐴௬𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙)+𝐴௭𝑠𝑖𝑛(2𝜙)    (5) \nWhere the 𝐴௫, 𝐴௬ and 𝐴௭ are the weightage of 𝑠𝑖𝑛(2𝜙)𝑠𝑖𝑛(𝜙), 𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙) and 𝑠𝑖𝑛(2𝜙). \nThe angular dependence has additional components in addition to 𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙) unlike in case \nof Pt/NiFe26. This indicates the breaking of twofold (180°+𝜙) and mirror symmetry (180°−𝜙).  \nThese additional contributions may arise due to non-uniform microwave current flow in the \ndevices, since NiFe has much lower resistivity compared to the 2DEG formed at STO interface, \nthat may give rise to the additional angular dependent components in the ST-FMR spectra. \nWe note that, in our devices, the angular dependence of symmetric and antisymmetric components \nis not purely 𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙) . Table in figure 3 (c) shows the weightage of the different symmetric \nand antisymmetric parts of the spectra at 5 GHz in the case of the STO/2DEG/NiFe sample. Here \nthe symmetric component has 16.79% sin (2𝜙)cos (𝜙)  dependence, 80.71% arising from \nsin (2𝜙)sin (𝜙)  dependence with the rest 2.50% arising from sin (2𝜙) . Meanwhile, its \nantisymmetric component has 2.93%, 60.58%, and 36.49% contributions from sin (2𝜙)cos (𝜙), \n𝑠𝑖𝑛(2𝜙)𝑠𝑖𝑛(𝜙), and 𝑠𝑖𝑛(2𝜙) dependence, respectively.  The large weightage of sin (2𝜙)sin (𝜙) \npresent in both symmetric and antisymmetric spectra in our sample, unlike Pt/NiFe where this \ncomponent has nearly zero percentage weightage due to only, SHE presents in Pt, here is shows \nthe presence of strong spin orbit coupling present at interface of STO and NiFe.    10 \n  \nFigure 3. In plane angle-dependent ST-FMR measurements: (a, b) Extracted component from \nLorentzian fitting of  𝑉௫ data for in plane angle  (ϕ) sweep (a) Symmetric contribution 𝑉ௌ௬   and \nits fitted components as function of in plane angle between applied 𝐼 and the external magnetic \nfield 𝐻௫௧  (b) Antisymmetric contribution  𝑉௦௬ and its component with the in plane angle (ϕ) (c) \nTable showing the percentage contribution of different type of torques in symmetric and in \nantisymmetric part of  𝑉௫ voltage. \n \nTo better understand the spin polarization dependence of symmetric and antisymmetric component \nof the ST-FMR signal, we have performed a simulation using MuMax3 27. The simulation uses the \nLandau-Lifshitz-Gilbert-Slonczewski equation 28,18 \n𝑑𝑚ሬሬ⃗\n𝑑𝑡= −𝛾𝑚ሬሬ⃗×𝐵ሬ⃗+𝛼𝑚ሬሬ⃗×𝑑𝑚ሬሬ⃗\n𝑑𝑡+𝛾|𝐽|ℏ𝜃ௌு\n2𝑒𝑡ிெ 𝜇 𝑀ௌ[𝑚ሬሬ⃗×(𝜎 ⃗ ×𝑚ሬሬ⃗)+𝜉ி 𝑚ሬሬ⃗×𝜎 ⃗]    (6) \nwhere  𝑚ሬሬ⃗  is the normalized magnetization, 𝐵ሬ⃗  is the effective magnetic field and 𝛼  is Gilbert \n11 \n damping factor, 𝑡ிெ  is the thickness of FM layer, 𝜇𝑀ௌ  is the saturation magnetization. Here \n𝜏ௌ்[𝑚ሬሬ⃗×(𝜎⃗×𝑚ሬሬ⃗)]  and (𝜏ௌ்×𝜉)𝜎⃗×𝑚ሬሬ⃗  are damping-like and field-like torques respectively. We \nsimulate NiFe/Pt and NiFe/STO  ferromagnet (FM) / nonmagnet (NM) bilayers.  \nWe use 𝑁𝑖𝐹𝑒 with a width of 160 nm and thickness of 5 nm. Simulation parameters are shown in \nthe table in figure 4 (b). The applied current density 𝐽   is in x direction exerting a magnetic \nfield 𝐵 on the NiFe and due to spin Hall effect in NM, generated spin current will exert SOT in \nNiFe. The charge to spin conversion efficiency (SHA) is taken as 100% for NiFe/STO  and and 5% \nin case of NiFe/Pt in the simulation. We have performed ST-FMR simulation for NiFe/Pt  sample \nusing conventional case of spin polarization, with 𝑝̂ of spin current and 𝐻, arising from applying \nthe 𝐼  in 𝑥 direction, both are in  𝑦 direction, transverse to the nonmetal/ferromagnetic material. \nThe simulation results are shown in (Supplementary figure S5) the angular dependence of ST-FMR \nspectra for both symmetric and antisymmetric spectra as 𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙). \nAs for STO(Arା)/NiFe  sample the in-plane angle dependence of symmetric and antisymmetric \ncomponents was not proportional to the 𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙) suggesting presence of additional torque. \nWe have simulated this using unconventional spin polarization case by keeping the  𝐻 field in 𝑦 \ndirection and taking polarization as (1,−0.3,−0.05) . The presence of x and z components in spin \npolarization invites additional torques due to which the symmetric and antisymmetric component \nwas no longer showing the 𝑠𝑖𝑛(2𝜙)𝑐𝑜𝑠(𝜙)  dependence. The studies have shown that angular \ndependence other than 𝑠𝑖𝑛(2𝜙)𝑠𝑖𝑛(𝜙)  appear in presence of anisotropic spin relaxation due to \nRashba-type spin orbit field or due to non-uniform distribution of 𝐼 . The simulation shows \nsymmetric part has additional terms 𝑠𝑖𝑛(2𝜙)𝑠𝑖𝑛(𝜙)  and 𝑠𝑖𝑛(2𝜙) , arising due to damping like \ntorque in x direction and field like torque in z direction due to Oersted field. Similarly, there are \nadditional terms 𝑠𝑖𝑛(2𝜙)𝑠𝑖𝑛(𝜙) and 𝑠𝑖𝑛(2𝜙), which arises from field like torque in x direction and 12 \n damping like torque in z direction.  \nThe simulation shows large dependence of symmetric and antisymmetric on 𝑠𝑖𝑛(2𝜙)𝑠𝑖𝑛(𝜙)  in \nunconventional spin polarization case, unlike conventional spin polarization case of Pt/Py where \nthis is completely zero. Since in simulation 𝐼 distribution is same for both, we can attribute this \neffect to anisotropic spin diffusion in ferromagnet due to strong Rashba like SOC present at the \nsubstrate and NiFe material interface.  \n   \nFigure 4. In plane angle-dependent ST-FMR data obtained from MuMax3: In plane angle \ndependent 𝑉௫  from ST-FMR simulation using MuMax3 at 5 GHz (a) schematic showing the \ngeometry of the sample, direction of charge current, spin current and applied external magnetic \nfield (b) table showing the parameters used in ST-FMR simulation (c) symmetric component within \nplane angle (d) antisymmetric component within plane angle . \nIn summary, we have demonstrated unconventional spin polarization at the argon milled STO \n13 \n interface interfacing with NiFe. The symmetric and antisymmetric components of rectification \nvoltage revealing the breaking of spin orbit torque symmetries. The micromagnetic simulation \nstudy under the similar experimental conditions further confirms the presence of the \nunconventional spin polarization at the interface. Our results emphasize the potential of modifying \nmaterial surface properties through ion milling to achieve efficient charge to spin conversion, with \npromising of development of a new kind of oxide-based spintronics. \nData availability statement  \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request. \n \n \n \n \n \n \n \n \n  14 \n References:  \n(1) Rojas-Sánchez, J.-C.; Oyarzún, S.; Fu, Y.; Marty, A.; Vergnaud, C.; Gambarelli, S.; Vila, \nL.; Jamet, M.; Ohtsubo, Y.; Taleb-Ibrahimi, A. Spin to Charge Conversion at Room \nTemperature by Spin Pumping into a New Type of Topological Insulator: α-Sn Films. Phys. \nRev. Lett.  2016, 116 (9), 96602. \n(2) Everhardt, A. S.; Mahendra, D. C.; Huang, X.; Sayed, S.; Gosavi, T. 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The Influence of Atomic Structure on the Formation of 16 \n Electrical Barriers at Grain Boundaries in SrTiO 3. Appl. Phys. Lett.  1999, 74 (18), 2638–\n2640. \n(16) Iglesias, L.; Sarantopoulos, A.; Magén, C.; Rivadulla, F. Oxygen Vacancies in Strained \nSrTiO 3 Thin Films: Formation Enthalpy and Manipulation. Phys. Rev. B  2017, 95 (16), \n165138. \n(17) Li, Y.; Phattalung, S. N.; Limpijumnong, S.; Kim, J.; Yu, J. Formation of Oxygen Vacancies \nand Charge Carriers Induced in the N-Type Interface of a LaAlO 3 Overlayer on SrTiO 3 \n(001). Phys. Rev. B  2011, 84 (24), 245307. \n(18) Liu, L.; Moriyama, T.; Ralph, D. C.; Buhrman, R. A. Spin-Torque Ferromagnetic \nResonance Induced by the Spin Hall Effect. Phys. Rev. Lett.  2011, 106 (3), 36601. \n(19) Liu, L.; Pai, C.-F.; Li, Y.; Tseng, H. W.; Ralph, D. C.; Buhrman, R. A. Spin-Torque \nSwitching with the Giant Spin Hall Effect of Tantalum. Science (80-. ).  2012, 336 (6081), \n555–558. \n(20) MacNeill, D.; Stiehl, G. M.; Guimaraes, M. H. D.; Buhrman, R. A.; Park, J.; Ralph, D. C. \nControl of Spin–Orbit Torques through Crystal Symmetry in WTe2/Ferromagnet Bilayers. \nNat. Phys.  2017, 13 (3), 300–305. \n(21) Stiehl, G. M.; MacNeill, D.; Sivadas, N.; El Baggari, I.; Guimarães, M. H. D.; Reynolds, N. \nD.; Kourkoutis, L. F.; Fennie, C. J.; Buhrman, R. A.; Ralph, D. C. Current-Induced Torques \nwith Dresselhaus Symmetry Due to Resistance Anisotropy in 2D Materials. ACS Nano  2019, \n13 (2), 2599–2605. https://doi.org/10.1021/acsnano.8b09663. \n(22) Kittel, C.; McEuen, P. Introduction to Solid State Physics, Vol 8 Wiley New York. 1976. 17 \n  "    },    {        "title": "0809.3859v1.Electrical_detection_of_spin_pumping__dc_voltage_generated_by_ferromagnetic_resonance_at_ferromagnet_nonmagnet_contact.pdf",        "content": "Electrical detection of spin pumping: dc voltage generated by ferromagnetic\nresonance at ferromagnet/nonmagnet contact\nM. V. Costache1;2, S. M. Watts1, C. H. van der Wal1, and B. J. van Wees1\n1Physics of Nanodevices Group, University of Groningen,\nNijenborgh 4, 9747 AG Groningen, The Netherlands.\n2Massachusetts Institute of Technology, Cambridge, MA 02139,USA.\n(Dated: November 16, 2021)\nWe describe electrical detection of spin pumping in metallic nanostructures. In the spin pumping\ne\u000bect, a precessing ferromagnet attached to a normal-metal acts as a pump of spin-polarized current,\ngiving rise to a spin accumulation. The resulting spin accumulation induces a back\row of spin\ncurrent into the ferromagnet and generates a dc voltage due to the spin dependent conductivities of\nthe ferromagnet. The magnitude of such voltage is proportional to the spin-relaxation properties of\nthe normal-metal. By using platinum as a contact material we observe, in agreement with theory,\nthat the voltage is signi\fcantly reduced as compared to the case when aluminum was used.\nFurtheremore, the e\u000bects of recti\fcation between the circulating rf currents and the magnetization\nprecession of the ferromagnet are examined. Most signi\fcantly, we show that using an improved\nlayout device geometry these e\u000bects can be minimized.\nPACS numbers: 72.25.Ba, 72.25.Hg, 73.23.-b, 85.75.-d\nI. INTRODUCTION\nDuring the last several years there has been a contin-\nuing interest in high frequency phenomena in spintronic\ndevices, as they are expected both to provide applications\nfor microwave signal-processing, and to become a power-\nful new tool for fundamental studies of spin dynamics in\nmagnetic nanostructures1,2,3,4,5.\nIt was predicted by Slonczewski1and Berger2that an-\ngular momentum is transferred from spin polarized cur-\nrents to the magnetization of the ferromagnets when\ncharge currents are sent trough spin valves with non-\ncollinear magnetizations (i.e. spin torque e\u000bect). This\ncan excite and even switch the magnetization direction\nof the softer ferromagnet. Experiments with pillar-type\nstructures6,7,8con\frmed these predictions.\nIt is natural to expect that if a spin current can in-\nduce magnetization motion the reciprocal process may\nalso be possible: a moving magnetization in a ferromag-\nnet can emit a spin current into an adjacent conductor.\nThis e\u000bect is the so-called spin pumping, proposed by\nTserkovnyak et al.9and Brataas et al.10. Spin pumping\nis a mechanism where a pure spin current, which does not\ninvolve net charge currents, is emitted at the interface\nbetween a ferromagnet with a precessing magnetization\nand a normal-metal region. It is an important mech-\nanism to generate spin currents, since other electronic\nmethods based on driving an electrical current through\na ferromagnet/semiconductor interface are strongly lim-\nited by the so-called conductance mismatch11. Berger12\nproposed a similar mechanism to generate a dc voltage\nby ferromagnetic resonance (FMR), based on spin-\rip\nscattering in the ferromagnet as induced by spin waves.\nRecently, spin pumping has been demonstrated in fer-\nromagnetic resonance experiments with thin multilayers,\nwhere it appears as an enhancement of the Gilbert damp-\ning constant of magnetization dynamics13,14,15,16,17, andusing time resolved magneto-optic Kerr e\u000bect18. Al-\nthough these experiments are very important in provid-\ning evidence for the spin pumping mechanism, the de-\ntection technique can be viewed as an indirect method\nto measure the spin pumping e\u000bect. Several experimen-\ntal methods have been proposed to electrically detect\nthe spin pumping mechanism19,20. The general prob-\nlem of these methods is the recti\fcation e\u000bects at the\nferromagnet/normal-metal contact, which can suppress\nor mimic the spin pumping signal21,22,23. Thus, the iden-\nti\fcation of these spurious e\u000bects is crucial and represents\none of the main themes of this paper.\nIn a recent paper,24we have demonstrated spin pump-\ning with a single permalloy strip in an electronic device,\nin which it is directly detected as a dc voltage signal. In\nthis paper, we describe additional experiments on spin\npumping e\u000bect, designed explicitly to eliminate the rec-\nti\fcation e\u000bects. We explain in more detail the theo-\nretical prediction for the voltage, and we identify and\nquantify di\u000berent contributions of the recti\fcation e\u000bect.\nImportantly, we show that by using an appropriate de-\nvice geometry, these e\u000bects can be minimized.\nII. SPIN PUMPING EFFECT\nAs discussed above, the emission of a spin current into\na conductor by a moving magnetization of an adjacent\nferromagnet is essentially the reciprocal of the spin torque\nmechanism in spin valves, where the magnetization is\nexcited by a spin current.\nA simpli\fed picture of the process is schematically\nshown in Fig. 1. We consider a F/N junction at equilib-\nrium, where in F exist a larger population of spins in the\ndirection of magnetization, than antiparallel. When the\nmagnetization direction is suddenly switched, the bands\ninstantaneously shift in energy. However, in order to goarXiv:0809.3859v1  [cond-mat.mes-hall]  23 Sep 20082\nFIG. 1: Simpli\fed picture of the spin pumping process. (a)\nPopulation of spin-up and spin-down bands in equilibrium.\n(b) Situation after sudden reversal of the magnetization di-\nrection. The arrows denote spin \row from one spin popula-\ntion to another one. (c) Equilibrium situation again but with\nmagnetization in opposite direction. [Adapted from ref.25]\nback to the equilibrium situation there has to be spin\ntransfer from one spin population to another (i.e. spin\nrelaxation). If F is in contact with N, this transfer of\nspins can go via N. Thus the spin relaxation process for\nF is modi\fed when it is in contact with an adjacent N,\nand depends on the spin relaxation properties on N. As\na result, an ac spin current is emitted into N when the\nmagnetization is switched back and forth under an oscil-\nlating magnetic \feld. Tserkovnyak et al.9analyzed the\ncase of circular precession of the magnetization and found\nthat in addition to the ac current, a dc spin current is\nalso emitted. A way to periodically change the magne-\ntization direction is to put F into FMR, where circular\nprecession of the magnetization can be resonantly excited\nby a small applied rf magnetic \feld21,26.\nThe transfer of spin angular momentum by the precess-\ning magnetization of F in contact with N (spin pumping)\nwas \frst described9using the formalism of parametric\ncharge pumping27developed in the context of mesoscopic\nscattering problems. The main points of this description\nare discussed below.\nSpin currents at the interface. As illustrated in\nFig. 2, a spin current Ipump\ns is pumped by the (resonant)\nprecession of a ferromagnet magnetization into an adja-\ncent normal-metal region. Assuming the F at FMR state,\nTserkovnyak et al.9have calculated the spin pumped cur-\nrent using a scattering matrix approach based on the mi-\ncroscopic details of the interface,\nIpump\ns =~\n4\u0019g\"#m\u0002dm\ndt: (1)\nwhere mrepresents the magnetization direction. g\"#\nis the real part of the mixing conductance28,29, a ma-\nterial parameter which describes the transport of spins\nthat are noncollinear to the magnetization direction at\nthe interface and is proportional to the torque acting on\nthe ferromagnet in the presence of a noncollinear spin\naccumulation in the normal metal30,31. This equation\nshows that the spin current, which goes into N, is per-\npendicular both to the magnetization direction mand\nto the change of min time. This current has ac and\ndc components, but in the limit !\u001cN\u001d1 (see laterdiscussion), the time-averaged pumping current reads10\njhIpump\nsitj=Idc=~!g\"#sin2\u0012=4\u0019.\nFIG. 2: The F/N structure in which the resonant precession\nof the magnetization direction mpumps a spin current Ipump\ns\ninto N. The spin pumping builds up a spin accumulation \u0016N\ns\nin N that drives a spin current Iback\nsback into the F. The com-\nponent of the Iback\nsparallel to mcan enter into F. Since the\ninterface and the bulk conductances of F are spin dependent,\nthis can result in a dc voltage across the interface.\nDepending on the spin related properties of the N, the\nspin current emission has two limiting regimes. When\nthe N is a good \\spin sink\" (in which spins relax fast),\nthe injected spin current is quickly dissipated and this\ncorresponds to a loss of angular momentum and an in-\ncrease in the e\u000bective Gilbert damping of the magnetiza-\ntion precession. This has been observed experimentally\nin nano-pillar structures13,14,15,16,17. The total spin cur-\nrent is given by Ipump\ns .\nThe opposite regime is when the spin-\rip relaxation\nrate is smaller than the spin injection rate. In this case,\na spin accumulation \u0016sbuilds up in the normal metal\n(Fig. 2). The spin accumulation can di\u000buse away from\nthe interface, but can also di\u000buse back into the F. This\nback \row current is given by\nIback\ns=g\"#\n2\u0019N[\u0016s\u0000m(m\u0001\u0016s)]; (2)\nwhereNis the one-spin density of states. The total spin\ncurrent in this case is IF\ns=Ipump\ns +Iback\ns.\nSpin battery. A spin battery operated by FMR has\nbeen proposed by the Brataas et al.10in the limit of weak\nspin-\rip scattering in the F. The spin accumulation in N\ncan be calculated by solving the spin di\u000busion equation\n@\u0016s\n@t=DN@2\u0016s\n@x2\u0000\u0016s\n\u001cN(3)\nwhere\u001cNis the spin-\rip time and DNis the di\u000bu-\nsion coe\u000ecient in N. We assume that the spin di\u000bu-\nsion length in N is much larger than the spin precession\nlengthl!\u0011p\nDN=!(!is precessional frequency), i.e.\n\u0015N=pDN\u001cN\u001dl!, or equivalent by !\u001cN\u001d1. This\nmeans that if the length of N is larger than l!, the x, y\ncomponents of spin accumulation are fully averaged (due\nto dephasing) and the remaining z component is constant3\nand along the static magnetic \feld direction10. The time-\naveraged spin accumulation h\u0016sit=\u0016zzin the N close\nto the interface reads10\n\u00160;z=~!sin2\u0012\nsin2\u0012+\u0011; (4)\nwhere\u0012is the precession cone angle and \u0011is a reduction\nfactor determined by the ratio between injection time and\nspin-\rip relaxation time.\nIntuitively, the spin accumulation can be measured\nelectrically using a second ferromagnet as a spin depen-\ndent contact, placed at a shorter distance compared to\nthe spin-\rip length10,32,33,34.\nVoltage at F/N interface. Importantly, Wang\net al.35have predicted a more direct way to detect the\nspin pump e\u000bect in which the precessing ferromagnet acts\nalso as the detector. We have to take into account that\nthe spin accumulation \u0016sin a di\u000busive metal drives the\nspin currentIback\nsback into the F. The component par-\nallel to mcan enter F. Moreover, since the interface and\nthe bulk conductances of F are spin dependent, this can\nresult in charge accumulation, close to the interface, and\nthereby a dc voltage across the interface. The chemical\npotential di\u000berence across the interface has been calcu-\nlated by Wang et al.35following the lines of the Brataas\net al.10model, but including the spin di\u000busion back into\nF and spin-relaxation in F. As mentioned above, the rel-\nevant length-scale for the averaging of the transverse (x,\ny) components of the spin current is l!. Therefore for a\ndevice with dimensions larger than l!the spin-up (down)\ne\u000bective conductances g\"(#)\n!of the interface are composed\nof the interface conductances g\"(#)in series with a con-\nductance of the bulk N over a length scale of l!. These\nrelations are given by g\"(#)\n!=g\"(#)=(1 +g\"(#)=g!) and\nthe mixing conductance g\"#\n!=g\"#=(1 +g\"#=g!), where\ng!= (\u001bNA)=l!(A is the area of the interface). Polariza-\ntionp!= (g\"\n!\u0000g#\n!)=(g\"\n!+g#\n!) is also introduced.\nIn the limit of large spin-\rip in F and the size of N \u001d\n\u0015Nand for small angle precession ( \u0012!0), the chemical\npotential di\u000berence is given by35\n\u0001\u00160=p!g\"#\n!\n2(1 +gN\ngF)(1\u0000p2!)(g\"\n!+g#\n!) + 2gN\u00122~!; (5)\nwheregN(gF) is the conductance of the bulk N (F) over\na length scale of \u0015N(\u0015F). For a thorough review of the\nabove discussion see ref.35.\nInterface currents matching. In this section, we\ndescribe a simple way to \fnd the voltage (similar to Eq.\n5) using spin-current matching at the interface. By writ-\ning all the currents involved in the process and matching\nthem at the interface, all components of the spin accumu-\nlation at the interface can be determined. It is convenient\nto transform the equations into a rotating frame of refer-\nence in which the uniform magnetization motion can be\nformally eliminated, and the unit magnetization vector is\n^m= (sin\u0012; 0;cos\u0012 ). Basically, for this problem we haveto consider three currents with their components. First,\nthe spin pumping current (Eq. 1) is given by\nIpump\ns;?=g\"#sin\u0012~!: (6)\nSecond, the back \row current consists of components par-\nallel and perpendicular to ^m, and can be written in terms\nof spin accumulation \u00160at the interface,\nIback\ns;k=gF\u00160;k;\nIback\ns;?=g\"#\u00160;?: (7)\nThe sum of Eqs. 6 and 7, gives the total spin current\non the F side of the interface. Third, the spin current on\nthe N side of the interface is found by solving the Bloch\nequations for the spin accumulation in N, from this the\ncurrent at the interface is given by36\nIN\ns=g!0\n@\u00160;x\u0000\u00160;y\n\u00160;x+\u00160;ygN\ng!\u00160;z1\nA; (8)\nin terms of\u00160at the interface. This current has three\ncomponents. The z component is determined only by the\nusual spin relaxation process. For the x and y compo-\nnents, two e\u000bects are important: precession, which re-\nsults in mixing of the two components, depending on the\ntime spent in N; and averaging, which reduces the total\namplitude of the components. The spin accumulation \u00160\nis determined by matching the currents at the interface\nIN\ns=IF\ns=Iback\ns+Ipump\ns . The dc voltage at the inter-\nface is proportional to the projection of \u00160onto ^m, and\nfor the limit g\"#\u0015g!is given by\nV=\u0000p\u00160\u0001^m'\u0000pg!\ngF\u0012\n1\u0000gN\ng!\u0013\ncos\u0012sin2\u0012~!: (9)\nThe simple form of Eq. 9 results from the relative in-\ndependence of the dc voltage on g\"#(physical argument\nof this result remains to be clari\fed). For our devices\n(N = Al and F = Py). Using \u001bF= 6:6\u0001106\n\u00001m\u00001,\n\u001bN= 3:1\u0001107\n\u00001m\u00001,\u0015F= 5 nm,\u0015N= 500 nm and\nl!= 300 nm we estimate the conductances at room tem-\nperature:\ngF=A=\u001bF=\u0015F'1\u00011015\n\u00001m\u00002\ng!=A=\u001bN=l!'1\u00011014\n\u00001m\u00002\ngN=A=\u001bN=\u0015N'8\u00011013\n\u00001m\u00002: (10)\nAnd according to Xia et al.37,g\"#=A'5\u00011014\n\u00001m\u00002.\nFor a quantitative assessment of the relations 5 and 9 we\nassume\u0012\u00195\u000e(sin2\u0012= 0:01) and!= 1011s\u00001(~!=\n65\u0016eV). First Eq. 9, by using p= 0:4 we \fnd dc voltage\n\u001920 nV. Second Eq. 5, by using p!= 0:06,g\"=A=\n0:31\u00021015\n\u00001m\u00002andg#=A= 0:19\u00021015\n\u00001m\u00002(from\nref.38), the dc voltage is of the same order of magnitude\n\u001920 nV.4\nFIG. 3: (a) Schematic diagram of the device. On the lower\nside, through the shorted-end of a coplanar strip a current Irf\ngenerates an rf magnetic \feld, denote by the arrows. The Py\nstrip in the center produces a dc voltage \u0001 V=V+\u0000V\u0000. H\ndenotes the static magnetic \feld applied along the strip. (b)\nScanning electron microscope pictures of the central part of\nthe devices.\nIII. EXPERIMENTAL PROCEDURES\nOur detection technique is based on the asymmetry\nin the spin pumping e\u000bect between two contacts in a de-\nvice geometry where a ferromagnet is contacted with two\nnormal-metal electrodes. The largest such asymmetry is\nobtained when one of the metal electrodes is a spin sink\nsuch as Pt, for which we expect a negligible contribution,\nwhile the other has a small spin \rip relaxation rate, such\nas Al. Therefore, we anticipate a net dc voltage across\na Py strip contacted by Pt and Al electrodes when the\nferromagnet is in resonance.\nAdditional, we studied control devices where the Py\nstrip is contacted by the same material Pt and Al. For\nthese devices we expected no signal because: (i) The volt-\nages for identical interfaces are the same and their con-\ntribution to \u0001 Vcancels. (ii) Pt has a very short spin\ndi\u000busion length, resulting in a small spin accumulation,\na small back\row and thus a lower signal.\nFigure 3(a) shows a schematic illustration of the lateral\ndevices used in the present study. The central part of the\ndevice is a ferromagnetic strip of permalloy (Ni 80Fe20, or\nPy) connected at both ends to normal metals, Al and/or\nPt (V\u0000andV+contacts). The devices are fabricated\non a Si/SiO 2substrate using e-beam lithography, mate-\nrial deposition and lift-o\u000b. A 25 nm thick Py strip with\n0.3\u00023\u0016m2lateral size was e-beam deposited in a base\npressure of 1x10\u00007mBar. Prior to deposition of the 30\nnm thick Al or/and Pt contact layers, the Py surface was\ncleaned by Ar ion milling, using an acceleration voltage\nof 500 V with a current of 10 mA for 30 sec, removing the\noxide and few nm of Py material to ensure transparent\ncontacts. We measured in total 17 devices (this includes\n4 devices with a modi\fed contact geometry, described\nlater in the paper). The di\u000berent contact material con-\n\fgurations are shown in Fig. 3(b).\nFigure 4(a) illustrates the experimental setup for the\nFIG. 4: Schematic diagrams of the experimental setup and\nof the microwave frequency modulation method. (a) A TTL\nsignal at a reference frequency fref(17 Hz) generated by a\nLock-in Ampli\fer (master device) is \frst fed into a frequency\ndoubler. Then, the TTL at 2x frefis fed into a CW Microwave\nGenerator. At each TTL input, the CW Generator provides\nfrequency hopping of the rf current switching between fhigh\nandflowatfref. The dc voltages produced by the device\nare ampli\fed and detected by the Lock-in Ampli\fer as a dif-\nference \u0001 V=V(fhigh)\u0000V(flow). (b) At the bottom, the\nresonant frequency dependence on the static magnetic \feld\nis shown. Next, the diagrams of the dc voltage vs. static\nmagnetic \feld corresponding to the resonance at high and\nlow frequencies. On top, the measured di\u000berence in dc volt-\nage between the two frequencies, \u0001 V=V(fhigh)\u0000V(flow) is\nplotted.\nmeasurements. We measured the dc voltage generated\nbetween the V+,V\u0000electrodes as a function of a slowly\nsweeping magnetic \feld ( H) applied along the Py strip,\nwhile applying an rf magnetic \feld ( hrf) perpendicular\nto the strip.\nWe have recently shown that a submicron Py strip\ncan be driven into the uniform precession ferromagnetic\nresonance mode21,26by using a small perpendicular rf\nmagnetic \feld created with an on-chip coplanar strip\nwaveguide39(CSW) positioned close to Py strip (similar\ngeometry as shown in Fig. 3). For the applied rf power 9\ndBm, an rf current of \u001912mA rms passes through the\nshorted-end of the coplanar strip waveguide and creates\nan rf magnetic \feld with an amplitude of hrf\u00191:6 mT\nat the location of the Py strip40. We con\frmed with\nanisotropic magnetoresistance (AMR) measurements21\nthat on-resonance the precession cone angle is \u00195\u000e.\nIn order to reduce the background (ampli\fer) dc o\u000b-5\nset and noise we adopted a lock-in microwave frequency\nmodulation technique. During a measurement where the\nstatic magnetic \feld is swept from -400 mT to +400 mT,\nthe rf \feld is periodically switched between two di\u000berent\nfrequencies and we measured the di\u000berence in dc voltage\nbetween the two frequencies \u0001 V=V(fhigh)\u0000V(flow)\nusing a lock-in ampli\fer. For all the measurements the\nlock-in frequency is 17 Hz and the di\u000berence between the\ntwo microwave frequencies is 5 GHz. A diagram of the\nmeasurement method is shown in Fig. 4(b).\nIV. RESULTS AND DISCUSSION\nA. Detection of Spin pumping\nHere, we describe precise, room-temperature measure-\nments of the dc voltage across a Py strip contacted by Pt\nand Al electrodes when the ferromagnet is in resonance.\nFigure 5 shows the electric potential di\u000berence \u0001 Vfrom\na Pt/Py/Al device. Sweeping the static magnetic \feld in\na range -400 mT to +400 mT, a peak and a dip like signal\nare observed at both positive and negative values of the\nstatic \feld. Since we measured the di\u000berence between\ntwo frequencies, the peak corresponds to the high reso-\nnant frequency ( fhigh) and the dip to the low resonant\nfrequency ( flow), see Fig. 4(b). For the opposite sweep\ndirection, the traces are mirror image. We measured 8\ndevices with contact material Pt/Py/Al. The measured\nresonances are all in the range +100 nV to +250 nV. No-\ntably, the dc voltages are all of the same sign (always a\npeak forfhigh), meaning that for Pt/Py/Al devices, the\nAl contact at resonance is always more negative than the\nPt contact.\nFirst, we look at the peak/dip position dependence of\nthe rf frequency. In Figure 6(a), the dc voltage in gray\nscale is plotted versus static \feld for di\u000berent high (low)\nfrequencies of the rf \feld. Figure 6(b) shows the \ftting of\nthe peak/dip position dependence of the rf \feld frequency\n(dotted curve) using Kittel's equation for a small angle\nprecession of a thin-strip ferromagnet42:\nf=\r\u00160\n2\u0019q\n(H+NkMS)(H+N?MS) (11)\nwhere\ris the gyromagnetic ratio, Nk,N?are in-plane\n(along the width of the strip) and out-of-plane demagne-\ntization factors and MSis the saturation magnetization.\nThe \ft to this equation (see Fig. 6(b)) gives \r= 176\nGHz/T, and Nk\u00160MS= 60 mT, N?\u00160MS= 930 mT,\nconsistent with earlier reports26,43. The \ft con\frms that\nthe dc voltage appears at the uniform ferromagnetic res-\nonance mode of the Py strip. The measured amplitude of\nthe dc voltage as a function of the square of the applied\nrf current, at 13 GHz and 139 mT is shown in Fig. 6(c).\nHere, we observe a linear dependence on the square of\nthe rf current, consistent with the prediction of the spin\npumping theory, see Eqs. 5 and 9.\nFIG. 5: The dc voltage \u0001 Vgenerated by a Pt/Py/Al de-\nvice in response to the rf magnetic \feld plotted as a function\nof the static magnetic \feld. The frequencies of the rf \feld\nare as shown. The peaks (dips) correspond to resonance at\nfhigh(flow). The data are o\u000bset vertically, for clarity.24\nFurther, we studied several control devices where both\nelectrodes are of the same non-magnetic material, Al (5\ndevices) or Pt (4 devices). Here we expected no signal be-\ncause of the reasons mentioned above. The results from\nAl/Py/Al devices show smaller signals than Pt/Py/Al\ndevices, with a large scatter in amplitude and both with\npositive and negative sign for the resonance at fhigh. Val-\nues for the 5 devices are -100 nV (shown in Fig. 7(a)),\n+25 nV, +30 nV, +75 nV and +110 nV. In contrast, all\n4 Pt/Py/Pt devices exhibit only weak signals less than\n20 nV, with resonance signals barely visible, as in Fig.\n7(b).\nThe overall values of the dc voltages as a function of\ndi\u000berent contact materials are shown in Fig. 8. We sum-\nmarize the results as follow:\nFirst, the Pt/Py/Al devices have signals that are always\npositive, on average 150 nV, and with a scatter compa-\nrable in amplitude to that of Al/Py/Al devices around\nzero. This scatter in the signal amplitude can be due to:\n(i) samples variation, due to di\u000berent interface quality,\nnot identical contacts (i.e. di\u000berent overlap between the\nN electrodes and the Py strip, see Fig. 3(b)) and a small\nvariation in distance between the Py strip and the CSW;\n(ii) di\u000berent rf power at the end of CSW, due to di\u000berent\npositions and contact resistance of the microwave probe\non the CSW. These characteristics are di\u000ecult to esti-\nmate for each device.\nSecond, we attribute the signals from Al/Py/Al devices\nto the asymmetry of the two contacts, possibly caused by\nsmall variation of the interfaces and in the contact geom-\netry. Depending on the asymmetry, the signals therefore6\nFIG. 6: (a) Gray scale plot of the dc voltage \u0001 V, measured\nfunction of static \feld for di\u000berent high (low) frequencies of\nthe rf \feld from the Pt/Py/Al device41. The dark (light)\ncurves denote resonance at flow(fhigh). (b) The static mag-\nnetic \feld dependence of the resonance frequency of the Py\nstrip (dots). The curve is a \ft to Eq. 11. (c) The amplitude\nof the dc voltage from a Al/Py/Al device as a function of the\nsquare of the rf current, at 13 GHz and 139 mT (dots). The\nline shows a linear \ft.24\nhave a scatter around zero.\nThird, in the Pt/Py/Pt devices, independent of possible\nasymmetry, we expected and found very small signals.\nTherefore, we conclude that the signals measured with\nthe Pt/Py/Al devices arise mainly from the Al/Py inter-\nface.\nB. Spin pumping vs. Recti\fcation e\u000bects\nWe now discuss the recti\fcation e\u000bects. As we have\nshown recently21, due to capacitive and inductive cou-\npling between the CSW and the Py strip, rf currents\n(I(t) =I0cos!t ) can be induced in the detection circuit.\nThe rf currents in combination with a time-dependent\nAMR (R(t)'\u0001R cos (!t+')) can give a dc e\u000bect\ndue to recti\fcation e\u000bect ( Vdc'hI\u0001Rit). However, for\nrecti\fcation to occur, the resistance R(t) must have \frst\nharmonic components, which is not true assuming circu-\nlar or even elliptical precession of the magnetization.\nThere are two ways to have \frst harmonic components:\n(i) an o\u000bset angle between the applied \feld and the long\naxis of the Py strip, namely bulk recti\fcation e\u000bect; (ii)\nan o\u000bset angle between the circulating rf currents and the\nmagnetization. When the rf circulating currents enter\nand leave the strip, they can pass through a large angle\nrelative to the magnetization. Asymmetry in the entry\nand exit paths, due to di\u000berent conductivities of the two\ncontacts, in combination with the time-dependent AMR,\ncan lead to a recti\fcation e\u000bect at the contacts, which\nFIG. 7: The dc voltage \u0001 Vgenerated across the Al/Py/Al (a)\nand Pt/Py/Pt (b) devices as a function of the static magnetic\n\feld. The frequencies of the rf \feld are as shown.24\nFIG. 8: Overall distribution of the amplitude of the dc volt-\nages as a function of di\u000berent contact materials. Di\u000berent\nsymbol represents di\u000berent batch of samples. This includes 4\ndevices with longitudinal electrode device geometry, indicated\nby symbol ( \u000e) and discussed in section IV B.\nwe call the contact recti\fcation e\u000bect44.\nEven if, we can accurately control the o\u000bset angle be-\ntween the applied \feld and the Py strip, we cannot rule\nout the contacts e\u000bect that may also contribute to the\ndata presented in the previous section, see Fig. 5. A\nsmall contribution from recti\fcation e\u000bects on top of spin\npumping signal can also explain the asymmetric peak/dip\nshape which does not have a Lorentzian shape as ex-\npected from Eq. 5. In order to study these e\u000bects we pre-\npared a new set of 4 devices very similar to the one shown\nin Fig. 3(b), but now with contacts at the ends of the Py\nstrip, extending along the long axis of the strip, see Fig.7\nFIG. 9: The dc voltage generated by Al/Py/Pt (a) and\nPt/Py/Pt (b) devices for a device geometry shown in Fig.\n(d). (c) The comparison of the signals for the two Pt/Py/Al\ndevices, one with longitudinal electrode device geometry (bold\nline) and other with transverse electrode device geometry. (d)\nSEM picture of an longitudinal electrode geometry, Pt/Py/Al\ndevice.\n9(d) for a SEM image. In this geometry, the induced rf\ncurrent \rows through the contacts predominantly paral-\nlel to the magnetization direction. This suppresses the\npossible contribution to the measured dc voltages from a\nrecti\fcation e\u000bect at the contacts45.\nWe \frst align the devices with Py strip parallel to the\napplied \feld and measure the dc voltage function of the\n\feld, as explained above. The measurements are shown\nin Fig. 9(a),(b) for Pt/Py/Al and a Pt/Py/Pt con\fgura-\ntion. These results are consistent with the above discus-\nsion, as the Pt/Py/Al devices show signals equal to the\naverage value measured in the previous device geometry,\nwhile the Pt/Py/Pt devices show no signal as expected.\nFigure 9(c) shows a comparison between the voltages of\ntwo Al/Py/Pt devices with longitudinal (bold line) and\ntransverse (normal line) contacts geometry, at fhigh =\n18.5 GHz,flow= 13.5 GHz. Particularly, devices with\nthe longitudinal contacts exhibit, in addition to the main\npeak, a series of peaks at higher \felds. An exact expla-\nnation of these observations is not yet clear. We assume\nthese are related to end-mode resonances, since in this\ncontacts geometry we are sensitive also to the magnetic\nstructure of the ends of the Py strip. Moreover, we found\nno signi\fcant di\u000berence in the measured dc voltages be-\ntween these two contacts geometries, Fig. 3(b) and Fig.\n9(d).\nIn order to con\frm the above assumptions and to quan-\nFIG. 10: The dc voltage at fhigh= 18 GHz, flow= 13 GHz for\n(a) Pt/Py/Al and (b) Pt/Py/Pt devices for di\u000berent angles\nbetween the static \feld and the long axis of the strip.\ntify the bulk recti\fcation e\u000bect, we misaligned the direc-\ntion of the static \feld with respect to the Py strip long\naxis by 5\u000e(and 10\u000e) and measured the voltage at fhigh\n= 18 GHz, flow= 13 GHz. The results for Pt/Py/Al\nand Pt/Py/Pt devices are shown in Fig. 10. Note that\nwe see signi\fcant contributions from the bulk recti\fca-\ntion e\u000bect only at o\u000bset angles larger than 5\u000e. This rules\nout that small o\u000bset angles which may be present in the\nother geometry caused signi\fcant e\u000bects in the results,\nat most 10-20 nV.\nIn the following, the above results are analyzed tak-\ning into account that the voltages measured in Pt/Py/Al\ndevices are due to two e\u000bects: (i) spin pumping, and\n(ii) bulk recti\fcation e\u000bect for a non-zero o\u000bset angle.\nOf these two e\u000bects only the bulk recti\fcation depends\non the sign of the o\u000bset angle. This means that if we\ntake the sum of the voltages measured at + =\u000010\u000e,\nV(10\u000e) +V(\u000010\u000e), we obtain two times the contribu-\ntion from the spin pumping e\u000bect with a Lorentzian peak\nshape. And in contrast we expect no signal if we do the\nsame operation for Pt/Py/Pt devices. These results are\nshown in Fig. 11(a).\nOn the other hand, if we subtract, V(10\u000e)\u0000V(\u000010\u000e),\nwe obtain two times the contribution from the bulk rec-\nti\fcation e\u000bect. Figure 11(b) shows the resulting data,\nwhich is practically the same for both devices, Pt/Py/Al\nand Pt/Py/Pt. Such a result is expected because the\nbulk recti\fcation e\u000bect does not depend on the contact8\nFIG. 11: To isolate the contribution from di\u000berent e\u000bects we\nperformed: (a) The sum between the voltage at o\u000bset angle of\n10\u000eand\u000010\u000e,V(10\u000e)+V(\u000010\u000e), vs. static \feld for Pt/Py/Al\nand Pt/Py/Pt devices. The result represents 2 times contri-\nbution from spin pumping e\u000bect. (b) The di\u000berence between\nthe voltages, V(10\u000e)\u0000V(\u000010\u000e), which represents 2 times\ncontribution from bulk recti\fcation e\u000bect, as explained in the\ntext.\nmaterial.\nWe now consider a quantitative assessment of possible\ncontribution to the measured signal from recti\fcation ef-\nfects, namely bulk and contact recti\fcation e\u000bect. Note,\nthe contact recti\fcation e\u000bect in principle should cancel\nfor equivalent contacts. Both recti\fcation e\u000bects depend\nprimarily on the rf circulating current, which varies from\ndevice to device, depending on position of the pico-probe\nand rf current frequency. In a similar device geometry21\nwe have estimated the rf currents, to be up to 30 \u0016A.\nWith this value we obtain:\n(i) Bulk: A rough estimate of an upper bound contribu-\ntion, assuming an o\u000bset angle of 2\u000e, gives 15 nV. The\ndata shown in Fig. 10(b) (for zero degree) is less than\nthis value.\n(ii) Contact: This contribution, which is present only in\ndevices with the transverse electrode geometry, is esti-\nmated at 30 nV44.\nThe sum of these contributions can have any value be-\ntween -45 and 45 nV, and thus can add or subtract to\nthe average spin pumping signal (150 nV), given the rise\nto extra scatter in the data, see Fig. 8.In addition to signal magnitude analysis, it is also im-\nportant to discuss the di\u000berence in signal shape due to\nthese e\u000bects. It should be noted that each of the recti-\n\fcation e\u000bects discussed above can have a signal shape\nwhich can be any combination between absorptive and\ndispersive peak shape. In contrast, the spin pumping\nsignal is only absorptive with a Lorentzian shape.\nV. CONCLUSION\nWe have presented dc voltage due to the spin pumping\ne\u000bect, across the interface between Al and Py at ferro-\nmagnetic resonance. We found that the devices where the\nAl contact has been replaced by Pt show a voltage close\nto zero, in good agreement with theory. We observed a\nquadratic dependence of dc voltage function of precession\ncone angle, in agreement with the discussed theory. The-\noretical predicted spin pumping voltage (20 nV) is less\nthan the values observed experimentally (in average 150\nnV). This underestimation might arise from the fact that\nthe model does not consider device geometry, disorder at\nthe interface and assumes an homogeneous magnetiza-\ntion in the ferromagnet.\nFurthermore, to rule out a possible contribution from\nrecti\fcation e\u000bects to the measured signal, we have stud-\nied devices with di\u000berent electrode geometries. We ob-\nserved that for a non-zero o\u000bset angle, between the static\n\feld and the Py strip, the measured voltages are due to\ntwo di\u000berent e\u000bects, namely spin pumping and recti\fca-\ntion e\u000bects. By using an appropriate device geometry,\nthese e\u000bects can be quanti\fed and for the zero o\u000bset an-\ngle the recti\fcation e\u000bects are minimized.\nThis work demonstrates a means of directly converting\nmagnetization dynamics of a single nanomagnet into an\nelectrical signal which can open new opportunities for\ntechnological applications.\nACKNOWLEDGMENTS\nWe thank M. Sladkov, J. Grollier, A. Slachter, G.\nVisanescu, J. Jungmann for discussion and assistance in\nthis project and B. Wolfs and S. Bakker for technical sup-\nport. This work was supported by the Dutch Foundation\nfor Fundamental Research on Matter (FOM), the Nether-\nlands Organization for Scienti\fc Research (NWO), the\nNanoNed and the DynaMax.\n1J. C. Slonczewski, J. Mag. Magn. Mater. 159, L1 (1996).\n2L. Berger, Phys. Rev. B 54, 9353 (1996).\n3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n425, 380 (2003).\n4A. A. Tulapurkar, Y. 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Kittel, Introduction to Solid State Physics, Ch. 16\n(John Wiley & Sons, New-York, 7th ed., 1996).\n43F. Giesen, J. Podbielski, T. Korn, M. Steiner, A. van Staa,\nand D. Grundler, Appl. Phys. Lett. 86, 112510 (2005).\n44M. V. Costache, Ph.D. thesis, University of Groningen\n(2007).\n45Giving the arguments discussed, it is also posible a contri-\nbution from the rf currents \row in/out of plane. We have\ncalculated this contribution and \fnd that is negligible."    },    {        "title": "1811.09348v3.Broadening_Frequency_Range_of_a_Ferromagnetic_Axion_Haloscope_with_Strongly_Coupled_Cavity_Magnon_Polaritons.pdf",        "content": "Broadening Frequency Range of a Ferromagnetic Axion Haloscope with\nStrongly Coupled Cavity-Magnon Polaritons\nGraeme Flower,1,a)Jeremy Bourhill,1Maxim Goryachev,1and Michael E. Tobar1,b)\nARC Centre of Excellence for Engineered Quantum Systems, Department of Physics, University of Western Australia,\n35 Stirling Highway, Crawley WA 6009, Australia\n(Dated: 1 April 2019)\nWith the axion being a prime candidate for dark matter, there has been some recent interest in direct detection\nthrough a so called `Ferromagnetic haloscope.' Such devices exploit the coupling between axions and electrons\nin the form of collective spin excitations of magnetic materials with the readout through a microwave cavity.\nHere, we present a new, general, theoretical treatment of such experiments in a Hamiltonian formulation for\nstrongly coupled magnons and photons, which hybridise as cavity-magnon polaritons. Such strongly coupled\nsystems have an extended measurable dispersive regime. Thus, we extend the analysis and operation of\nsuch experiments into the dispersive regime, which allows any ferromagnetic haloscope to achieve improved\nbandwidth with respect to the axion mass parameter space. This experiment was implemented in a cryogenic\nsetup, and initial search results are presented setting laboratory limits on the axion-electron coupling strength\nofgaee>3:7\u000210\u00009in the range 33 :79\u0016eV<ma<33:94\u0016eV with 95% con\fdence. The potential bandwidth\nof the Ferromagnetic haloscope was calculated to be in two bands, the \frst of about 1 GHz around 8 :24 GHz\n(or 4:1\u0016eV mass range around 34 :1\u0016eV) and the second of about 1 :6 GHz around 10 GHz (6 :6\u0016eV mass range\naround 41:4\u0016eV). Frequency tuning may also be easily achieved via an external magnetic \feld which changes\nthe ferromagnetic resonant frequency with respect to the cavity frequency. The requirements necessary for\nfuture improvements to reach the DFSZ axion model band are discussed in the paper.\nINTRODUCTION\nDark matter continues to be one of the major un-\nsolved problems in physics. Evidence of its existence\nis abundant but its identity remains a mystery. Many\nphysicists believe it consists of particles that come\nfrom extensions to the current Standard Model (SM).\nThe axion is one such particle that comes from an SM\nextension originally proposed to solve a problem in\nQuantum Chromodynamics known as the strong charge\nparity problem1. It has since been shown to make an\nexcellent candidate for dark matter, potentially able to\naccount for all of dark matter2,3. A key parameter in\naxion theory is the Pecci-Quinn symmetry breaking scale\n(fa) which determines the mass and coupling strengths\nto ordinary matter. The most known model parameter\nbands are KSVZ (Kim-Shifman-Vainshtein-Zakharov)4,5\nand DFSZ (Dine-Fischler-Srednicki-Zhitnitsky)6,7. De-\nspite existence of several theoretical proposals8,9, this\nparameter is unknown with limited bounds on it. As\nsuch there is a vast portion of the parameter space\nof axion mass and couplings to search making the\n\frst observation of axion or axion like particles very\nchallenging10.\nThe most successful implementation of axion direct\ndetection experiments11are, so called, axion halo-\nscopes, such as the Axion Dark Matter eXperiment\n(ADMX)12,13, HAYSTAC14, CULTASK15, ORGAN16,\na)Electronic mail: 21302579@student.uwa.edu.au\nb)Electronic mail: michael.tobar@uwa.edu.auACTION17and multi-cell cavity approaches18,19. These\nexperiments focus on photon coupling to galactic axion\nhalo probing the hypothesis that axions constitute the\nbulk of dark matter in the Milky Way as opposed to\nhelioscopes20,21focusing on axion and Axion Like Parti-\ncles sourced from the Sun. Mentioned above haloscopes\nexploit the inverse Primako\u000b e\u000bect22and strong DC\nmagnetic \felds as virtual photons to detect axions at\nRF and microwave frequencies putting limits on axion-\nphoton coupling parameter ga\r\r. The same coupling is\ninvestigated using a slightly modi\fed approach based on\ndielectric materials in experiments such as MADMAX23.\nOn the other hand, ferromagnetic haloscopes, considered\nin this work, probe axion coupling to electron spin,\ncoupling parameter gaee. Such experiments probe the\ninteraction between axions in the dark matter halo and\ncollective spin excitations in ferromagnetic materials,\nquantized version of which is widely known as a magnon.\nThis resultant signal is then readout through coupling\nthe magnetic mode to a photon mode by placing the\nmagnetic sample inside a microwave cavity. Similar in\nspirit to ferromagnetic haloscope, searches with nuclear\nspins, such as CASPER24, constitute another class of\ngalactic haloscope targeting another coupling parameter\ngaNN.\nThere has been some recent interest in this scheme with\nthe QUaerere AXion (QUAX) experiment25{27. Cur-\nrently, though, the QUAX experiment is largely static\nin frequency due to it operating in the, so called, fully\nhybridised regime, and so has limited capability in scan-\nning axion mass, a critical feature for an axion halo-\nscope. In the recent development of this setup25, two\nmethods to make the experiment frequency tunable arearXiv:1811.09348v3  [physics.ins-det]  29 Mar 20192\nproposed. The \frst approach is to tune both the cav-\nity and magnon mode equally to move the hybrid fre-\nquencies. This approach, while achievable in principle,\nadds signi\fcant complexity to the experiment. The sec-\nond tuning method for a ferromagnetic haloscope is to\ndetune the magnetic resonance from the cavity reso-\nnance. Though, this is practical and could be easily im-\nplemented, no method to determine how the experiment\nsensitivity changes in such a regime is provided. In the\npresent work, we cover this lack of knowledge about this\nmethod by considering a new approach to determining\nthe dynamics of such the cavity-magnon system.\nI. DETECTION PRINCIPLES\nA. Axion Electron Interaction\nTo understand working principles of magnetic axion\nhaloscopes, one needs to consider interaction between ax-\nions and an individual electron. The starting point here\nis the following Lagrangian for the DSVZ axion coupled\nto fermionic \feld:\nL=\u0016 (x)(i~\r\u0016@\u0016\u0000mc) (x)\u0000igaeea(x)\u0016 (x)\r5 (x);(1)\nwhere (x) is the spinor \feld of the fermion, mis its\nmass,a(x) is the axion \feld, \r\u0016are the Dirac matrices\nandgaeeis the axion to electron coupling strength. Based\non this Lagrangian and ignoring the, so called, parity\nviolating term one can derive a Hamiltonian operator for\nan electron in the non-relativistic limit:\n^H=\u0000~2\n2mer2\u0000gaee~2\n2me^\u001b\u0001ra; (2)\nwhereeis the electron charge, ^ \u001bis the Pauli matrices\nvector. This Hamiltonian can be compared to the one\nfor an electron in a magnetic \feld B:\n^H=\u0000~2\n2mer2\u0000e~2\nme^\u001b\u0001B: (3)\nThe similarity between Hamiltonians (2) and (3) allows\nto introduce the pseudo-magnetic \feld1,25,28{30:\nBaee=gaee\n2era: (4)\nThus, the axion e\u000bects on electrons could be regarded\nas an e\u000bect of an anomalous oscillating magnetic \feld.\nParameters of these oscillations as well as its direction\nare considered later in Section II. Thus far only the DFSZ\naxion has been considered. This is due to the suppression\nof the KSVZ axion to ordinary quark and lepton coupling\nat the tree level31,32. The coupling constant, gaee, for the\nDFSZ axion, however, is given by:\ngaee=me\n3facos2(\f); (5)\nwhere cot(\f) is the ratio of two Higgs vacuum expecta-\ntion values of this model6,33. For the following work, it\nis assumed cos( \f) = 1.B. Ferromagnetic Haloscopes\nIn order to maximise the axion induced signal in a\nrealistic detector, it is important to shift from a single\nspin system to a system with a very high spin number.\nThis number of spins is maximised in ferromagnetic\ncrystals, particularly Yttrium Iron Garnet (YIG), that\nhave been proposed as the core sensing element in pre-\nvious proposals25,26,28. In addition to this, high quality\nYIG is characterised by relatively narrow linewidths\nthat makes it a popular choice for research in the\n\feld of hybrid quantum system for future quantum\ninformation processing34{39. Also, it has been proposed\nto use a microwave cavity to read out axion induced\nmagnetisation. An excited ferromagnetic sample in free\nspace will dissipate energy by dipole radiation. Placing\nit in a microwave cavity, therefore, has the additional\nbene\ft of mitigating this dissipation by collecting the\nradiation in the cavity. Thus, the haloscope based on\naxion-electron coupling appears as the hybrid mode\ncavity-ferromagnetic quasi particle (or cavity-magnon\npolariton) widely considered for applications in quantum\ninformation processing40,41.\nTo model the dynamics of a ferromagnet, it is common\nto use the Bloch equations to determine the induced mag-\nnetization from the pseudo-magnetic \feld. In the con-\ntext of the QUAX proposal25,26, the solutions to these\nequations were modi\fed in conjunction with the cavity\nequations of motion in the fully hybridised regime as-\nsuming the magneto-static and cavity mode form coupled\nharmonic oscillators. Analysis of this system, however,\nneeds to be extended to the dispersive regime. To model\nthe system in a single framework, this work considers a\nHamiltonian approach. The Hamiltonian of the total de-\ntector system consists of its cavity and magnon parts, Hc\nandHmrespectively, as well magnon-cavity and axion-\nmagnon interaction parts, HintandHaeerespectively (for\nunits where ~= 1):\nH=Hc+Hm+Hint+Haee\nH=!ccyc+!mbyb+gcm(cy+c)(by+b) +Haee;(6)\nwherecy(c) is a creation (annihilation) operator for\nphoton,by(b) is a magnon creation (annihilation) opera-\ntor,!cis the cavity frequency, !mis magnon frequency,\nandgcmis the cavity-magnon coupling rate. These\nexpressions can been found from \frst principles42,43. It\ncan also be noted that the magnon Hamiltonian assumes\nthe demagnetizing and magneto-crystalline anisotropic\n\feld do not introduce non-linearities into the dynamics.\nThese assumptions are satis\fed using a spherical sample\nand small input \felds respectively43. The magnon reso-\nnance frequency is set by the external magnetic \feld B0\nvia the Zeeman e\u000bect !m=\r(B0+Bani), where\ris the\ngyromagnetic ratio and Baniis the linear contribution of\nthe magneto-crystalline anisotropy, typically determined\nexperimentally.3\nTo determine the magnon-axion interaction term in the\nHamiltonian, it is straightforward to write the interaction\nin terms of its Zeeman energy:\nHaee=\u0000Z\nVmM\u0001BaeedV; (7)\nwhere Mis the sample magnetization vector, Vmis its\nvolume. Given the uniform precession magneto-static\n(Kittel) mode, which has constant magnetisation over the\nsample volume, the expression can be easily integrated:\nHaee=\u0000\rS\u0001Baee; (8)\nwhere S=MVm\n\ris a macrospin operator. For this oper-\nator, raising and lowing spin operators are introduced as\nS\u0006=Sx\u0006iSy, where for the following, the zdirection is\nchosen to align with the external DC magnetic \feld. Us-\ning this set of spin operators, the spin-axion interaction\nHamiltonian is expanded as:\nHaee=\u0000\rBaeehsin(\u001e)\n2(S+e\u0000i\u0012+S\u0000ei\u0012) +Szcos(\u001e)i\n;\n(9)\nwhere\u0012and\u001eare the spherical coordinate azimuthal and\npolar angles of the pseudo-magnetic \feld vector respec-\ntively. Finally, the Primako\u000b-Holstein transformations\nare used to write the interaction it in terms of magnon\noperators introduced above44:\nS+= (p\n2S\u0000byb)b;\nS\u0000=by(p\n2S\u0000byb);\nSz=S\u0000byb;(10)\nwhereSis the total spin number of the macrospin\noperator. This number is determined by S=\u0016\ng\u0016BNs,\nwhere\u0016is the magnetic moment of the magnetic sample,\n\u0016Bis the Bohr magneton, gis the g-factor ( g= 2)\nandNsis the number of spins in the sample (given by\nNs=nsVmwithnsas spin density and Vmas volume).\nFor YIG, used in previous proposals as well as this work,\nit is estimated that\u0016\n\u0016B= 5:045and the spin density\nns= 2:2\u00021028m\u00003. From these values the total spin\nnumber for a 2mm diameter YIG sphere used in this\nwork may be estimated as S= 2:3\u00021020.\nFor low excitation numbers ( hbybi \u001c 2S), the\nmacrospin operators may be approximated by S+\u0019p\n2SbandS\u0000\u0019p\n2Sbygiving the \fnal expression for\nthe axion-magnon coupling term:\nHaee=\u0000\rBaeehr\nS\n2sin(\u001e)(be\u0000i\u0012+byei\u0012)+\nbybcos(\u001e)i\n;(11)where constant terms are ignored. It can be noted\nthe axion-magnon interaction Hamiltonian (11) includes\ntwo sets of terms: components relating to the pseudo-\nmagnetic \feld perpendicular to the external magnetic\n\feld and producing magnon excitations, and terms cou-\npled to the parallel component producing a modulation of\nthe magnon frequency. For an axion producing magnons\nin the GHz frequency range, it is shown in the follow-\ning discussion that the latter frequency modulation term\ncan be ignored under a Rotating Wave Approximation\n(RWA), but could be used to search for low mass axions\nusing frequency metrology techniques46,47. Thus, only\nthe former term producing excess of power in the magnon\nspectrum can be used for the current experiment.\nC. Detector Dynamics\nIn order to capture the dynamics of the total exper-\niment, one needs to include dissipation. This is done\nby deriving the Heisenberg-Langevin equations of mo-\ntion for open systems. In this case internal dissipation in\nthe cavity and magnon modes are modelled as coupling\nto internal thermal baths as well as the cavity coupling\nto an external bath through a probe/port. The general\nform of these equations is given by48:\ndf\ndt=\u0000i[f;H] +X\nn\u0010\u0010\u0014n\n2gy+ip\u0014nainy\nn\u0011\n[f;g]\u0000\n[f;gy]\u0010\u0014n\n2g+ip\u0014nain\nn\u0011\u0011\n;(12)\nwherefandgare arbitrary system operators, ain\nnand\n\u0014nis the input \feld and dissipation rate due to coupling\nof somea\feld to the nth thermal bath respectively. For\nderivation of axion induced output power, all noise terms\nare ignored, and it is assumed that the magnon mode\ndoes not couple to the external excitation port.\nThe resulting equations of motion for annihilation op-\nerators are then converted to the rotating frame ( c!\nce\u0000i!at;b!be\u0000i!at, where!ais the axion signal fre-\nquency) and a RWA is made. This procedure gives the\nequations of motion as follows in the lab frame:\ndc\ndt=\u0000i\u0010\n!c\u0000i\u0014c\n2\u0011\nc\u0000igcmb; (13)\ndb\ndt=\u0000i\u0010\n!m\u0000i\u0014m\n2\u0011\nb\u0000igcmc+ir\nS\n2\rBaeesin(\u001e)ei\u0012;\n(14)\nwhere\u0014c=\u0014int\nc+\u0014ext\ncand\u0014m=\u0014int\nmare the dissipa-\ntion rates of the cavity and magnon respectively. These\nquantities include internal and external (from the cavity\nprobe) dissipation where applicable.\nThe derived equations of motion (13) can be easily\nsolved for the cavity \feld in terms of the driving pseudo-4\nmagnetic \feld giving:\nc(!a) =igcmq\nS\n8\rBaeesin(\u001e)ei\u0012\n\u0014c\u0014m\u0001c\u0001m\u0000g2cm; (15)\nwhere \u0001n=!a\u0000!n\n\u0014n+i\n2. This result can be converted\nto the output \felds using the boundary conditions from\ninput-output theory cout+cin=p\n\u0014extcc, as well as the\noutput power from Pout=~!ahc\u0003\nout(t)cout(t)i:\nPout(!a) =~!a\u0014ext\ncg2\ncmS\n8\r2B2\naeesin2(\u001e)\nj\u0014c\u0014m\u0001c\u0001m\u0000g2cmj2: (16)\nNote should be taken that the only geometrical de-\npendence left in the output power is sin2(\u001e), which is\nmaximum when the pseudo-magnetic \feld from axion\ninteraction is perpendicular to the DC magnetic \feld.\nThis dependence is discussed in Section II.\nTo determine the axion induced power on resonance,\nthe resonant frequency of the coupled normal modes can\nbe used, where a simpli\fcation has been made under the\nassumption of a strongly coupled system ( gcm\u001d\u0014c;\u0014m):\n!\u0006=!c+!m\n2\u0006r\u0010\ncm\n2\u00112\n+g2cm; (17)\nwhere \ncm=!m\u0000!c. These expression give the on\nresonance ( !a=!\u0006) power expressions:\nP\u0006=~!a\u0014ext\ncS\n2\r2B2\naeesin2(\u001e)\n\u00142c\u00142m\n4g2cm+\u0010\n\ncm\n2gcm(\u0014m\u0000\u0014c)\u0006r\u0010\n\ncm\n2gcm\u00112\n+ 1(\u0014c+\u0014m)\u00112:\n(18)\nIt can be noted that this form of the power on reso-\nnance is a function of the detuning over coupling rate.\nThus, the range of frequencies where this experiment\nis sensitive is proportional to the coupling rate gcm.\nImportantly, this formula allows the determination of\nthe sensitivity of this experiment in the dispersive regime\nwhen!c6=!m. To demonstrate the response of the sys-\ntem in the dispersive regime a set of output powers are\ncalculated for a 2mm diameter YIG sphere for various\nrelationships between linewidths of subsystems. Given\nlinewidths\u00141= (2\u0019)10MHz and \u00142= (2\u0019)50MHz, three\ncases are considered: (1) for \u0014c> \u0014m,\u0014c=\u00141and\n\u0014m=\u00142, (2) for\u0014c=\u0014m=\u00141, (3) for\u0014c<\u0014m,\u0014c=\u00142\nand\u0014m=\u00141. In all cases \u0014ext\nc=\u0014c=2. The result for\ngcm= (2\u0019)745MHz,!a= (2\u0019)8:2GHz (corresponding\ntojBaeej= 3:4\u000210\u000023T) and sin( \u001e) = 1 is depicted\nin Fig. 1. As the magnon excitation is readout through\nthe cavity mode, this might imply that the output\nsignal will be strongest in the full hybridisation regime.\nHowever, Fig. 1 shows that this is not the case in\ngeneral. Particularly, when \u0014c6=\u0014m, the maximum\nsignal strength is obtained in the dispersive regime. This\nopens up new ways to optimise such an experiment,\noften by operating in the dispersive regime.\n(A)\n(B)FIG. 1: Detector output power as a function of scaled\ndetuning\ncm\ngcmfor an YIG sphere for various linewidths:\n(A) from the higher frequency normal mode ( P+), (B)\nfrom the lower frequency normal mode ( P\u0000).\nII. AXION WIND\nWith the detector dynamics determined in terms of\nthe driving pseudo-magnetic \feld, it is now necessary to\ndetermine its components in the laboratory frame. As\nsuch a classical axion \feld can be substituted (a step\nwhich will be justi\fed later). This is given by:\na(r;t) =a0sin\u0010\n!at\u0000mava\u0001r\n~\u0011\n; (19)\nwherea0=q\n2\u001aa\nc~\nmais the magnitude of the axion \feld,\n\u001aa\u00190:45GeV/cm3is the local dark matter density49,\nandvais the velocity of the axion wind.\nBaee=gaee\n2emajvaj\n~a0sin\u0010\n!at\u0000majvajx\n~\u0011\n^ x;(20)\nwhere ^ xis the direction of the axion wind. In the present\nwork, the velocity in the galactic frame of the axion wind\nis treated as a Maxwell distribution with mean velocity5\nof zero and velocity dispersion of \u001bv= 270km/s. Thus,\nin the laboratory frame it has the velocity of the galactic\nframe of approximately 220km/s in the direction of the\nconstellation Cygnus50. It should be noted that this ex-\nperiment being a directional detector of the axion dark\nmatter \feld makes it also sensitive to the so called `dark\nmatter hurricane' or S1 stream which provides an addi-\ntional source of axions as well as di\u000berent velocity distri-\nbution compared with those in the dark matter halo51.\nIn this work, however, we only consider the axion wind\nfrom the earth's movement through the dark matter halo\nwhich gives the following parameters25:\njBaeej= 3:4\u000210\u000023\u0010ma\n34\u0016eV\u0011\nT; (21)\n!a\n2\u0019= 8:2\u0010ma\n34\u0016eV\u0011\nGHz; (22)\ngaee= 1:0\u000210\u000015\u0010ma\n34\u0016eV\u0011\n; (23)\n\u0015ra= 0:74\u0015a= 0:75h\nmava= 8:8\u001034\u0016eV\nma\u0011\nm:(24)\nNote the de-Broglie wavelength of galactic axions is\nmuch larger than the scale of a haloscope experiment for\nthe frequencies considered. This justi\fes the treatment\nof the axion \feld a(r;t) as a classical \feld.\nThe next step is to determine the direction of the ax-\nion wind with the laboratory frame in Perth, Western\nAustralia. For this, the velocity due to the Sun's orbit\naround the galactic centre as well as the velocity due\nto the Earth's orbit around the Sun can be taken into\naccount. Appropriate coordinate transforms52give the\nvelocity of the axion wind in the laboratory frame. In\nthese calculations, only the components which are per-\npendicular to the external DC magnetic \feld oriented\nlocally upwards (from the centre of the Earth), are im-\nportant to the operation of the haloscope. The mag-\nnitude of the perpendicular components of the velocity\nBaee;?=q\nB2aee;x +B2aee;y is shown in Fig. 2 for the 27th\nof August 2018 (local time) in Perth where it is scaled to\nits absolute value. It can be seen that there is an 8 hour\nperiod per day which a ferromagnetic haloscope would\nbe sensitive and a 2 hour period where our signal should\ndisappear.\nIII. EXPERIMENTAL DESIGN\nFrom the expression of output power, Eq. (18), it\nfollows that in order to maximise the detector sensitivity\nit is required to maximise the number of spins (i.e. spin\ndensity and volume) as well as cavity and magnon mode\nQuality factors. For a scanning tunable experiment,\nit is also important to widen the range of sensitive\nfrequencies that can be done by improving the coupling\nrate between cavity and magnon modes. This has been\nachieved using two post re-entrant cavities allowing to\n0 5 10 15 20\nTime of Day (Hours)0.30.40.50.60.70.80.91.0Baee,/|Baee|\nFIG. 2: The daily modulation of the pseudo-magnetic\n\feld perpendicular to the direction of an external \feld\nat the University of Western Australia on the 27th of\nAugust 2018.\nreach the strong coupling regime34. As a system of\nessentially two coupled quasi-lumped resonators, such\ncavities exhibit two modes: the bright and the dark\nmodes. In the former mode, the magnetic \feld around\neach post as well as the electric \felds in the gaps are in\nphase. This phase relation concentrates the magnetic\nenergy in between the posts. So, by placing the magnetic\nsample in that space, one achieves large magnetic \flling\nfactors and thus a large coupling between the photon\nand magnon mode.\nTo achieve large spin density and magnon quality fac-\ntors, a 2mm diameter single domain crystal YIG sphere\nis used. The sphere is placed in a Te\ron holder at the\ncentre of the copper two post re-entrant cavity. This cav-\nity is a 30\u00028mm (diameter\u0002height) cylinder with two\nidentical posts of 2 \u00027mm at a separation of 2.5mm from\nthe cavity centre. A photo of this assembly is shown in\nFig. 3 (A). Finite Element Modelling (FEM) was used to\ndetermine the cavity resonance frequencies and the mode\nshapes which for the bright mode is shown in Fig. 3 (B).\nTo characterise the detector in terms of losses and\ncouplings, the system response in terms of scattering\nparameters is measured as a function of external mag-\nnetic \feld. A simpli\fed schematic of the corresponding\nexperimental setup is sketched in Fig. 4 (A). The cavity\nand sphere are placed inside the bore of a superconduct-\ning magnet held at T\u00195K. A source oscillator (SO)\nis attenuated and fed to a weakly coupled probe in the\ncavity for spectroscopic measurements. The readout\nchain consists of a cryogenic High Electron Mobility\nTransistor (HEMT) ampli\fer based low noise ampli\fer\nbolted directly to the cavity (AMP1), an isolator to\nprevent re\rected noise, a room temperature low noise\nampli\fer (AMP2) fed into the RF port of a microwave6\n(A)\n(B)\nFIG. 3: (A) A picture of the detector cavity without its\nlid. Visible is the loop probes, the re-entrant cavity\nposts, the Te\ron holder and the YIG sphere. (B)\nColour density plot of magnetic \feld strength jHjfor\nthe bright mode of the two post re-entrant cavity.\nmixer. The signal is then mixed with a Local Oscillator\n(LO) downconverting the signal to a few MHz and\n\fnally measured by a vector signal analyser (VSA).\nAMP1 has a gain of 41 :5 dB and a noise temperature\nof 4K at 8 :2 GHz quoted by the manufacturer, and\nAMP2 had a measured gain of 21dB. For the initial\nspectroscopic measurements, the SO, mixer, the LO and\nthe VSA are substituted by a Vector Network Analyser\nthat measures the corresponding S-parameters. The\nresulting spectroscopic data alongside with two mode\nsystem \ft are shown in Fig. 4 (B).\nFrom Fig. 4 (B), the cavity frequency is estimated as\nfc= 8:937 GHz, the magnon frequency as a function\nof external magnetic \feld could be approximated by\nfm(B0) = 27:99B0(GHz/T)+0 :660(GHz), and the\nmagnon-photon coupling rate is gcm= 751 MHz.The uncoupled linewidths can also be inferred from\nspectroscopic data giving \u0014m= (2\u0019)11:1 MHz and\n\u0014c= (2\u0019)7:53 MHz (from fully hybridised values\n\u0014\u0006= (2\u0019)9:32 MHz where in general \u0014c+\u0014m=\u0014++\u0014\u0000).\nThe input probe of the cavity is only weakly coupled\nfor transmission measurements which can be considered\na negligible source of loss. Prior to running this ex-\nperiment re\rection measurements of the output cavity\nprobe were performed to infer the level of coupling to\nthe photon mode as \u0014ext\nc= (2\u0019)5:48MHz. The output\nprobe needs to be near critically coupled to maximise\npower output. The measured parameter \u0014ext\nc, implies\ncoupling to the hybrid mode when fully hybridised as\n\f=\u0014ext\nc\n\u0014\u0006\u0000\u0014extc\u00191:4, where\f= 1 is the condition for\ncritical coupling.\n(A)\n(B)\n-20dB\nFIG. 4: (A) Experimental setup used to characterise\nthe detector (involving SO source) and \fnal data\nacquisition. (B) System response in terms of\nS-parameters as a function of external magnetic \feld.7\nIV. RESULTS\nThe axion signal is expected to appear as excess of\npower over background noise in a power spectra. It\nis therefore necessary to discriminate against arti\fcial\npeaks that could appear due to various spurious signals.\nFor this reason, two sets of data for each frequency\nrange are taken each day. The \frst is over an 8hr\nperiod where the velocity is \u001890% perpendicular to the\nexternal magnetic \feld (corresponding to \u001881% of the\nmaximum power) and then a second set of data is taken\nover the 2 hours period when the velocity is only \u001835%\nperpendicular to the external \feld (corresponding to\n\u001812% of the maximum power). The second dataset\nallows any persistent spurious signals to be identi\fed\nand eliminated.\nThe exclusion data was taken over a range of frequen-\ncies for a period of 6 days in August 2018. Each day of\nmeasurements consists of 320,000 individual power spec-\ntra measured over a sensitive period of time ( \u00188hrs in-\ntegration time) and 80,000 power spectra collected over\nthe minimally sensitive period ( \u00182hrs integration time).\nThe frequency of the hybrid resonance was tuned using\nthe external magnetic \feld allowing to scan for \u001836MHz\naround 8:2GHz. In this case only the lower hybrid mode\nwas used to attempt to measure an axion signal and the\na limited 36MHz tuning range used considered due to\navailability of equipment. The full potential range of\nthis haloscope can be calculated from equations 17 and 18\ngiving a full width at half maximum (FWHM) of 1 :6GHz\naround 10GHz for the upper hybrid mode and a FWHM\nof 1:0GHz around 8 :2GHz for the lower hybrid mode.\nThroughout the experiment the cavity and \frst ampli-\n\fer were kept at a temperature of \u00185K. The sum of the\ncavity physical temperature and the noise temperature\nof the \frst stage ampli\fer can be used to estimate the\nexpected noise temperature of the system as \u00189K at\n8:2GHz. Analysis of the spectra was done using a sim-\nilar method to E. Daw53. For each set of data, de\fned\nby the DC magnetic \feld B0used to set a central fre-\nquency of the lower hybrid mode, f\u0000, a span of\u00186MHz\n(reduced from the \u00189MHz hybrid linewidth due to spu-\nrious system noise) of data around the central frequency\nwas analysed. In each case, the power spectral density,\nwith bin width \u0001 f= 3:125kHz (chosen to be similar to\nthe axion signal width 4 :1kHz), was scaled by the ampli-\n\fer gains and a polynomial \ft was made. The residuals\nwere then analysed to con\frm the validity of the \ft. Next\nthe lowest possible cut is made in the sensitive data such\nthat any bins above this chosen power value also exist\nabove the same cut in the insensitive data. This allows\nthe identi\fcation of bad bins, as any bins above the cut in\nboth the sensitive and insensitive data are irrelevant spu-\nrious signals and thus can be removed from the analysis.\nThe remaining Gaussian noise can be analysed to deter-\nmine standard deviation ( \u001b) and e\u000bective noise temper-\nature of the system, making use of the Dicke radiometerB0(mT)f\u0000(GHz)\u001b(10\u000023W)Te\u000b(K)Cut (\u001b)Pexl(\u001b)\n298 8.2086 4.89 11.0 6.2 15.8\n297.5 8.2018 4.73 10.4 4.2 10.3\n297 8.1950 5.81 12.8 5.0 13.2\n296.5 8.1879 5.50 12.1 5.7 14.9\n296.1 8.1820 5.74 12.6 5.0 12.9\n295.6 8.1754 5.78 12.7 5.2 13.6\nTABLE I: Measured standard deviations of residuals\n(scaled by ampli\fer gains), cuts made and excluded\npower for each set of data.\nequation54:\n\u001b=kBTe\u000br\n\u0001f\nt; (25)\nwherekBis the Boltzmann constant, Te\u000bis the e\u000bective\nsystem noise temperature, and tis the integration time.\nA potential axion signal would appear as a residual bin\nwith an excess of power over the mean, with the cut\nmade earlier de\fning the largest distinguishable signal\namongst the noise. Gaussian statistics are then used\nto determine the excluded signal power ( Pexl) to a 95%\ncon\fdence based on the cut made53. These results are\nshown in Table I. From Table I, it can be seen that the\ne\u000bective temperature determined from the measured\nnoise is comparable to the estimate based on the physical\ntemperature and \frst stage ampli\fer, thus the measured\nresults are consistent with expectations. The larger\nmeasured e\u000bective noise temperature is due to the small\ncontribution of noise due to cables and second stage\nampli\fer.\nTo determine excluded couplings and magnetic \felds\nthe following relations were used:\nPexl\nP\u0000=g2\naee;exl\ng2aee; (26)\nwheregaee;exlis the excluded axion to electron coupling\nstrength. Here the excluded power is also scaled by a\nLorentzian line-shape and multiplied by 0 :81 to account\nfor the reduction in sensitivity due to the proportion of\naxion wind perpendicular to the external magnetic \feld\nat that time of day. This relation allows to put limits\non the axion-electron coupling strength gaeeas shown\nin Fig. 5 where also several predictions are made in the\nform of the dashed lines (see Section V for detailed dis-\ncussions).\nV. DISCUSSION AND PERSPECTIVES\nIt can be seen in Fig. 5 that the results of this analysis\nare still orders of magnitude from current astronomical\nlimits on axion electron coupling and expected DFSZ\nmodel predictions. They do, however, demonstrate8\nFIG. 5: The DFZS axion model band and exclusion plot\nfor axion-electron coupling strength gaeeas a function\nof axion mass: limits due to white dwarf cooling55,56are\nin light blue, this work limits are in dark blue, dashed\nlines show several predictions for the future work.\nhow such ferromagnetic haloscopes, in a single cavity\ncon\fguration, can search over a larger range than their\nlinewidth by tuning their hybrid frequencies. This\nlarger range isn't strictly experimental improvements\nbut rather from improved capabilities of the experiment\ndue to more general theoretical analysis. Additionally,\nwhilst this detector isn't sensitive enough to detect\nDFSZ axions, it does have the capability to detect\naxion-like particles (ALPs) which don't have a \fxed\nrelation between axion mass and axion-normal matter\ncoupling strengths.\nWhile this initial result demonstrates the experiments\nusability, further improvements, particularly in sample\nvolume and line-widths, are needed to improve the\noverall sensitivity. Single domain ferrimagnetic spheres\nwith a diameter of 5mm are easily available and in-\nferred magnon linewidths of around 2 :4MHz at central\nfrequency 14GHz can be seen for a Ga:YIG sample in\nthe QUAX experiment25. Additionally, their approach\nto increasing the e\u000bective volume of the magnetic ma-\nterial in the cavity by including multiple spheres could\nalso be utilised. Such improvements would boost the\noutput power from an axion signal. The corresponding\nimprovement is shown in Fig. 5 by the dark blue dashed\nline assuming a larger 5mm diameter sphere, improved\nlinewidths of \u0014m= (2\u0019)2:4MHz and a measurement\ntime of one day with existing ampli\fers. Alternatively,\none may consider materials with higher spin density such\nas Lithium Ferrite where the same order of magnetic\nlosses together with absence of spurious modes have\nbeen observed57.\nThe sensitivity would also be improved by reducing\nthe detector background noise, for example, by imple-\nmentation of quantum limited parametric ampli\fers\nbased on Josephson junctions. Successful operation ofsuch ampli\fers, viable in the considered frequency range,\nhas been demonstrated for axion-photon haloscopes14.\nThe result of these improvements changing sensitivity of\nthe detector is shown in Fig. 5 with the purple dashed\nline.\nIn order to improve the limits even further, one needs\nto build a viable detection scheme based on a single\nphoton or magnon counter, ideally using a Quantum\nNon-Demolition (QND) measurement to surpass the\nstandard quantum limit. Such devices would then only\nbe limited by shot noise due to a non-zero temperature\nof the cavity. Single photon counters in the context on\naxion haloscopes and are discussed by Lamoreaux, et\nal.58, showing superconducting qubits in cavities to be\na promising avenue for a QND single photon counter59.\nA single magnon counter could similarly be constructed\nby coupling a qubit to a magnon mode which was\nrecently achieved to resolve numbers state60,61. This is\none experimental advantage of ferromagnetic haloscopes\nover traditional photon haloscopes as the DC magnetic\n\feld need not extend over the entire cavity, thus a\nfocused magnetic \feld on the ferrimagnetic sample\nwould allow the presence of superconducting devices in\nthe cavity to aid measurement. Non-QND, single photon\ncounters have also been shown to be another promising\navenue for axion haloscopes62,63. The result of a QND\nmeasurement based experiment is predicted in Fig. 5\nwith both reasonable and optimistic improvements in\nthe signal-to-noise ratio. The red dashed line is the\nresult of the above assuming a perfect e\u000eciency QND\nmeasurement is achieved requiring a maximum physical\ntemperature of 12 :5mK to ensure a 95% con\fdence of\nno dark counts of the detector over the measurement\ntime. The \fnal black dashed line in Fig. 5 is a prediction\nof an extremely optimistic QND measurement scheme\nassuming the signal power can be further boosted with\na volume of the magnetic sample of Vm= 0:13 L and an\nimproved linewidth of \u0014m= (2\u0019)200 kHz. These esti-\nmations are done following the procedure by Lamoreaux\net al.58, where it is noted that in this case the limiting\nfactor is the minimum detectable power required to\ncount at least three photons over the measurement time.\nCONCLUSIONS\nA new theoretical perspective on ferromagnetic halo-\nscopes was presented which demonstrated methods of\nimproving the operation of a ferromagnetic haloscope.\nParticularly easy frequency tunability to search the ax-\nion mass parameter space is achievable by operating in\nthe dispersive regime. It also highlighted the importance\nof a large cavity-magnon coupling strength to produce\na large bandwidth and provided new methods to opti-\nmise experimental design and operation. The device was\nimplemented and set limits on axion-electron coupling of9\ngaee>3:7\u000210\u00009in the range 33 :79\u0016eV<ma<33:94\u0016eV\nwith 95% con\fdence. Limitations of this experiment, in\nthe form of electronic noise and small sample volume are\ndiscussed, and improvements are suggested in the form\nof quantum or shot noise limited measurement. These\npredictions highlight the need for shot noise limited mea-\nsurement in the form of single photon or magnon coun-\nters, as well as the requirement to dramatically improve\nsignal power.\nACKNOWLEDGEMENTS\nThis work was supported by the Australian Research\nCouncil grant number DP190100071 and CE170100009\nas well as the Australian Government's Research Train-\ning Program.\nREFERENCES\n1R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).\n2J. Ipser and P. Sikivie, Phys. Rev. Lett. 50, 925 (1983).\n3J. Jaeckel and A. 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Martin6\n1Institute for Solid State Physics, The University of Tokyo, Kashiwa, 277-8581, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing, 100190, China\n4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan\n6Aix Marseille Univ, Universit´ e de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France\n(Dated: October 31, 2023)\nWe theoretically study spin pumping from a ferromagnetic insulator (FI) into a carbon nanotube\n(CNT) . By employing the bosonization method, we formulate the Gilbert damping induced by the\nFI/CNT junction, which can be measured by ferromagnetic resonance. We show that the increase\nin the Gilbert damping has a temperature dependence characteristic of a Luttinger liquid and is\nhighly sensitive to the Luttinger parameter of the spin sector for a clean interface. We also discuss\nthe experimental relevance of our findings based on numerical estimates, using realistic parameters.\nI. INTRODUCTION\nSpin pumping induced by ferromagnetic resonance\n(FMR) [1, 2] is a fundamental technique in spintronics\nfor generating spin current from a ferromagnet to an ad-\njacent material [3, 4]. While spin pumping has been used\nfor injecting spin into various materials, it can also be uti-\nlized for detecting spin excitations in various systems [5–\n17]. Compared with bulk measurement techniques, such\nas nuclear magnetic resonance (NMR) and neutron scat-\ntering experiments, spin pumping has an advantage in\nsensitivity for nanostructured systems such as surfaces,\nthin films and atomic-layer compounds [5].\nThe study of exotic spin excitations which emerge in\nspecific materials is one of the forefront topics of con-\ndensed matter physics. A typical example is spin exci-\ntation in quasi-one-dimensional interacting electron sys-\ntems, whose low-energy excitation can be described by\nthe Tomonaga-Luttinger liquid [18–20]. Spin excitations\ninherent to the Tomonaga-Luttinger liquid have been\nstudied in carbon nanotubes (CNTs) by using NMR [21–\n23]. While NMR can detect the local spin susceptibility\nin CNTs, the use of spin pumping to detect spin excita-\ntions is expected to provide useful information reflecting\nthe exotic character of the Luttinger liquid, which can-\nnot be captured by NMR. It is thus important to clarify\nwhat kind of information about the Luttinger liquid can\nbe obtained from a spin pumping experiment.\nIn this work, we theoretically formulate the increase\nin the Gilbert damping due to spin pumping in a setup\nin which spin is injected into CNTs. We consider a\nmagnetic junction composed of a ferromagnetic insula-\ntor (FI) and a single-wall CNT (see Fig. 1) and take in-\nterfacial randomness into account with a simple model.\nWe derive an analytic expression for the increase in the\nGilbert damping by utilizing the bosonization method\nand second-order perturbation with respect to the inter-\nfacial exchange coupling.\nWe will focus on the two limiting cases, i.e., a clean in-\nterface and a dirty interface. We show that for both cases\nFIG. 1. Magnetic junction composed of a ferromagnetic in-\nsulator (FI) and a single-wall carbon nanotube (CNT). The\ndimension of the FI is W×W′×d′.\nthe temperature dependence of the increase of the Gilbert\ndamping shows a power-law behavior, with an exponent\nreflecting the Luttinger parameters. For a clean inter-\nface, the exponent includes information on the Luttinger\nparameters in the spin sector and is shown to be sensitive\nto small deviations from unity (which is the value of the\nSU(2) symmetric model in the spin sector). For a dirty\ninterface, the exponent depends on the Luttinger param-\neters of both the spin and charge sectors as in an NMR\nmeasurement. We estimate the increase of the Gilbert\ndamping using realistic parameters and discuss the ex-\nperimental feasibility.\nOur paper is organized as follows. We introduce the\nmicroscopic model of the FI/CNT magnetic junction in\nSec. II. We analytically calculate the increase in the\nGilbert damping in Sec. III and subsequently estimate\nit with realistic parameters in Sec. IV. Finally, we briefly\ndiscuss the experimental relevance of our findings in\nSec. V and summarize our results in Sec. VI. A detailedarXiv:2303.01343v2  [cond-mat.mes-hall]  28 Oct 20232\nderivation of the analytic expressions is given in the two\nAppendices.\nII. MODEL\nLet us consider a junction composed of a CNT and\nFI, whose Hamiltonian is given by H=HCNT+HFI+\nHint. Here, HCNTandHFIdescribe electrons in the CNT\nand FI, respectively, and Hintrepresents the interfacial\nexchange interaction between the CNT and FI. We will\ngive their explicit forms in the subsections that follow.\nA. Carbon nanotube\nThe low-energy Hamiltonian of electrons in CNTs is\ngiven by\nHCNT=HK+HC, (1)\nwhere HKandHCrepresent the kinetic energy and\nthe forward scattering potential due to the screened\nCoulomb interaction, respectively. Using standard con-\nventions [24], the Hamiltonians describing these energies\nof electrons in CNTs are given by\nHK=−ivFZ\ndxX\nrασrψ+\nrασ(x)∂xψrασ(x), (2)\nHC=1\n2Z\ndx dy ρ (x)V(x−y)ρ(y), (3)\nwhere ψrασ(x) is the slowly varying part of the\nfield operator of electrons, vFis the Fermi velocity,\nV(x) is the screened Coulomb potential, and ρ(x) =P\nrασψ†\nrασ(x)ψrασ(x) is the electron density operator.\nThe subscripts, r(=±),α(=±), and σ(=±), repre-\nsent the direction of propagation, the nanotube branch\n(the valley), and the spin orientation, respectively. Using\nthe bosonization method [19, 24], the annihilation oper-\nator describing fermions in the CNT can be expressed in\nterms of bosonic fields, θασ(x) and ϕασ(x), as\nψrασ(x) =ηrασ√\n2πaei(−rθασ(x)+ϕασ(x)), (4)\nwhere ηrασis the Klein factor, and ais a short-length\ncutoff which can be identified with the lattice constant of\nthe CNT. To diagonalize the Hamiltonian, we introduce\nnew bosonic fields for the charge and spin sectors, θjδ(x)\nandϕjδ(x) as\nθασ(x) =1\n2X\njδhjδ(α, σ)θjδ(x), (5)\nϕασ(x) =1\n2X\njδhjδ(α, σ)ϕjδ(x), (6)\nwhere δ(=±) represents symmetric/antisymmetric\nmodes, j(=c, s) indicates the charge/spin mode, hc+=1,hc−=α,hs+=σ, and hs−=ασ. The Hamiltonian\nof the CNTs can be written as\nHCNT=X\nj,δvjδ\n2πZ\ndx[K−1\njδ(∂xθjδ)2+Kjδ(∂xϕjδ)2],(7)\nwhere Kjδis the Luttinger parameter and vjδ=vF/Kjδ.\nB. Ferromagnetic insulator\nWe consider a bulk FI described by the quantum\nHeisenberg model and employ the spin-wave approxi-\nmation assuming that the temperature is much lower\nthan the magnetic transition temperature and the mag-\nnitude of the localized spin, S0, is much larger than\none [8, 9, 11, 15–17, 25]. In this situation, the Hamil-\ntonian for the FI is approximately written as a superpo-\nsition of magnon modes:\nHFI=X\nkℏωkb†\nkbk, (8)\nwhere bkis the annihilation operator of magnons, ℏωk=\nDk2+ℏγghdcis the magnon dispersion, Dis spin stiffness,\nγgis the gyromagnetic ratio, and hdcis the static mag-\nnetic field. We will only focus on uniform spin precession\ninduced by external microwaves. For this purpose, it is\nsufficient to consider the magnon mode of k=0with the\nsimplified Hamiltonian\nHFI=ℏω0b†\n0b0. (9)\nMicrowave absorption in FMR can be related to the\nimaginary part of the retarded spin correlation function,\nwhich is defined as\nGR(ω) =−i\nℏZ∞\n0dt ei(ω+iδ)t⟨[S+\n0(t), S−\n0]⟩, (10)\nwhere S+\n0=√2S0b0and S−\n0=√2S0b†\n0are spin\nladder operators of the FI for k=0andS+\n0(t) =\neiHt/ℏS+\n0e−iHt/ℏ. For an isolated bulk FI, the spin sus-\nceptibility is calculated as:\nGR\n0(ω) =2S0/ℏ\nω−ω0+iδ. (11)\nIn real experiments, the FMR linewidth is finite due to\nthe the Gilbert damping. To represent this finite spin\nrelaxation in the bulk FI, we introduce a phenomeno-\nlogical dimensionless parameter αGand express the spin\ncorrelation function as\nGR\n0(ω) =2S0/ℏ\nω−ω0+iαGω. (12)3\nC. Interfacial exchange interaction\nNow let us consider the interfacial exchange interaction\nbetween the FI and the CNT with the Hamiltonian,\nHint=S+\n0s−+S−\n0s+, (13)\nwhere s±is the spin ladder operator of the CNT, defined\nas\ns−=r\n1\nNFIX\nr,r′X\nα,α′ZW\n0dx J(x)\n×e−i(α−α′)kFx−i(r−r′)qFxψ†\nrα−(x)ψr′α′+(x),(14)\nands+= (s−)†. Here, Wis the length of the interface,\nJ(x) is the interfacial exchange coupling, NFIis the num-\nber of unit cells in the FI, kFis the Fermi wavenumber,\nandqF(≪kF) is the momentum mismatch associated\nwith the two modes. Because the interfacial exchange\ncoupling J(x), which is induced by quantum mechanical\nmixing between CNT and FI, is sensitive to distances of\natoms across the junction, we assumed that it depends\non the position xdue to random atomic configuration\nnear the interface. A simplified model for randomness in\nJ(x) will be accounted for in the next section.\nIII. FORMULATION\nA. Gilbert damping\nUsing second-order perturbation with respect to the\ninterfacial exchange coupling, the spin susceptibility is\ncalculated as\nG(iωn) =1\nG0(iωn)−1−Σ(iωn)(15)\nΣ(iωn) =−1\nℏZℏβ\n0dτeiωnτ⟨Tτs+(τ)s−(0)⟩ (16)\nwhere s±(τ) = eHCNTτ/ℏs±e−HCNTτ/ℏ. The retarded\nspin correlation function is obtained by analytic contin-\nuation iωn→ω+iδas\nGR(ω) =2S0/ℏ\nω−(ω0+δω0) +i(αG+δαG)ω0,(17)\nδω0\nω0≃2S0\nℏω0Re ΣR(ω0), (18)\nδαG≃ −2S0\nℏω0Im ΣR(ω0), (19)\nwhere ΣR(ω) is the retarded self-energy defined by\nΣR(ω) =Z\ndt eiωtΣR(t), (20)\nΣR(t) =−iθ(t)\nℏ⟨[s+(t), s−(0)]⟩, (21)θ(t) is the step function, and αG+δαG≪1 has been\nassumed. In our work, we focus on the increase in the\nGilbert damping due to the junction, δαG, which is writ-\nten in terms of the dynamic spin susceptibility of CNTs.\nB. Self-energy of electrons in CNTs\nBy substituting Eq. (14) into Eq. (21), we obtain\nΣR(t) =−i\nℏθ(t)2S0\nNFIX\nr,r′X\nα,α′ZW\n0dxZW\n0dy⟨J(x)J(y)⟩imp\n×e−i(kF(α−α′)+qF(r−r′))(x−y)Crαr′α′(x, y, t ),(22)\nCrαr′α′(x, y, t ) =⟨[ψ†\nrα,+(x, t)ψr′α′,−(x, t),\nψ†\nr′α′,−(y,0)ψrα,+(y,0)]⟩0. (23)\nHere, ⟨···⟩ impindicates a random average for the in-\nterfacial exchange coupling. For simplicity, we assume\nthat the exchange coupling follows a Gaussian distribu-\ntion whose average and variance are given by\n⟨J(x)⟩imp=J1, (24)\n⟨δJ(x)δJ(y)⟩imp=J2\n2aδ(x−y), (25)\nwhere δJ(x) =J(x)− ⟨J(x)⟩imp. Here, J1andJ2repre-\nsent respectively the average and the standard deviation\nof the distribution. The ratio J2/J1reflects the random-\nness of the interfacial exchange coupling. In particular,\nthe case of J2/J1= 0 corresponds to a clean interface\nwithout randomness.\nAccordingly, the self-energy is calculated as\nΣR(t) = ΣR\n1(t) + ΣR\n2(t), (26)\nΣR\n1(t) =−iθ(t)2S0J2\n1\nℏNFIX\nr,r′,α,α′ZW\n0dxZW\n0dy\n×e−i(kF(α−α′)+qF(r−r′))(x−y)Crαr′α′(x, y, t ),(27)\nΣR\n2(t) =−iθ(t)2S0J2\n2a\nℏNFIX\nr,r′,α,α′ZW\n0dx C rαr′α′(x, x, t ).\n(28)\nSince the integrand of ΣR\n1(t) includes a rapidly oscillating\npart as a function of ( x−y), the integral is negligibly\nsmall except for the case of α=α′andr=r′. There,\nwe obtain\nΣR\n1(t) =−iθ(t)2S0J2\n1\nℏNFIX\nr,αZW\n0dxZW\n0dy C rαrα(x, y, t ).\n(29)\nWe should note that ΣR\n1(t) corresponds to the process\nof electron creation and annihilation in the same branch\nand represents momentum-conserving spin relaxation for\na clean junction. In contrast, ΣR\n2(t) represents spin re-\nlaxation for a “dirty” junction that is independent of the4\nelectron momentum. Here, the word “dirty” means that\nduring spin exchange process the momentum of electrons\nin the CNT is not conserved and transitions between dif-\nferent branches of valleys and propagation directions are\nallowed. The following discussion will consider two lim-\niting cases for the interface. For the clean interface limit\n(J1≫J2), the magnon self-energy is represented with\nΣR\n1(t), while in the dirty interface limit ( J1≪J2), it is\nrepresented with ΣR\n2(t).\nC. Clean interface\nSince the correlation function Crαr′α′(x, y, t ) can be\ncalculated using the bosonization method (see Ap-\npendix A), the self-energy ΣR\n1(t) can be obtained ana-\nlytically. Therefore, the corresponding increase in the\nGilbert damping is obtained as\nδαG,1=−2S0\nℏω0Im ΣR\n1(ω0)\n=−4S0J2\n1\nℏ2ω0(2πa)2NFIZW\n0dxZW\n0dyZ∞\n0dtsinω0t\n×Im\"\u0012sinh(iπa/β ℏvF)\nsinh(π(ia−(x−y)−vFt)/βℏvF)\u0013γ−1\n×\u0012sinh(iπa/β ℏvF)\nsinh(π(ia+ (x−y)−vFt)/βℏvF)\u0013γ+1#\n,(30)\nγ≡Ks+\n4+Ks−\n4+1\n4Ks++1\n4Ks−. (31)\nAfter analytic integration with respect to t(see Ap-\npendix B for details), we obtain\nδαG,1=2\nπΓ(γ)2\nΓ(2γ)S0J2\n1Wa\nℏ2v2\nFNFI\u00122πa\nβℏvF\u00132γ−3\n×I(πW/β ℏvF, γ), (32)\nI(w, γ) =1\nwZw\n0dz′Zz′\n0dz e−2(γ−1)z\n×F(γ−1, γ,2γ; 1−e−4z), (33)\nwhere F(a, b, c ;x) is the hypergeometric function.\nD. Dirty interface\nThe self-energy ΣR\n2(t) can be obtained in a similar\nway as above. The corresponding increase in the Gilbertdamping is given by\nδαG,2=−2S0\nℏω0Im ΣR\n2(ω0)\n=−S0J2\n2aW\nℏ2ω0(πa)2NFIX\nr,r′,α,α′Z∞\n0dtsinω0t\n×Im\"\u0012sinh(iπa/β ℏvF)\nsinh(π(ia−vFt)/βℏvF)\u00132γrαr ′α′#\n,(34)\nγrαr′α′=γ1δr,r′δα,α′+γ2δr,−r′δα,α′\n+γ3δr,r′δα,−α′+γ4δr,−r′δα,−α′,(35)\nγ1= (Ks++Ks−+ 1/Ks++ 1/Ks−)/4, (36)\nγ2= (Kc++Kc−+ 1/Ks++ 1/Ks−)/4, (37)\nγ3= (Ks++Kc−+ 1/Kc++ 1/Ks−)/4, (38)\nγ4= (Kc++Ks−+ 1/Ks++ 1/Kc−)/4. (39)\nWe should note that δαG,2is proportional to W, since the\nspin relaxation rate is determined through spatially-local\nspin exchange in the dirty interface and is proportional\nto the number of spin-exchange channels. After analytic\nintegration with respect to t(see Appendix B for details),\nwe obtain\nδαG,2=1\n2πS0J2\n2aW\nℏ2v2\nFNFI\n×X\nr,r′,α,α′Γ(γrαr′α′)2\nΓ(2γrαr′α′)\u00122πa\nβℏvF\u00132γrαr ′α′−2\n.(40)\nIV. NUMERICAL ESTIMATE\nNext, we evaluate numerically the increase in the\nGilbert damping by using realistic experimental parame-\nters. While the increase was formulated for a single CNT\nin the previous section, to increase the signal, it would be\nmore useful if we considered a junction with a bundle of\nCNTs. Thus, in the following, we will consider a junction\ncomposed of a FI and a bundle of CNTs with an area of\nW×W′(see Fig. 1) and multiply δαG,1andδαG,2by\nthe number of CNTs in the junction, NCNT=W′/d(d:\nthe diameter of CNTs).\nThe parameters are given in Table I. The Fermi ve-\nlocity vF, lattice constant a, diameter d, Luttinger pa-\nrameters of CNTs, Kc+,Kc−, and Ks−, are taken from\nRefs. [20, 24, 26]. The value of Ks+is an experiment\nresult [22] under a magnetic field of 3 .6 T[27]. The spin\namplitude S0and the lattice constant a′are determined\nby assuming that the FI is made from yttrium iron gar-\nnet (YIG). The interfacial exchange coupling ( J1orJ2)\nis roughly estimated to be 2 K [28]. The number of unit\ncells is estimated as NFI=WW′d′/a′3, where d′is the\nthickness of the FI.5\nTABLE I. Parameters used for the numerical estimate.\nMicrowave frequency ω0 1 GHz\nFermi velocity of CNT vF 106m/s\nLattice constant of CNT a 2.46˚A\nDiameter of CNT d 1.5 nm\nAmplitude of spins of FI S0 10\nLattice constant of FI a′12.376˚A\nThickness of FI d′10 nm\nInterfacial exchange couplings J1,J2\nclean interface J1= 2 K, J2= 0\ndirty interface J1= 0 K, J2= 1,2,3 K\nLuttinger parameters Kc+ 0.20\nKs+ 1.07\nKc−,Ks−1\n100 1010-2\n10-3\n10-4\n3 300\nFIG. 2. Temperature dependence of increase in the Gilbert\ndamping, δαG,1, for a clean interface ( J1≫J2).\nA. Clean interface\nThe estimated increase in the Gilbert damping for a\nclean interface ( J1= 2 K ≫J2) is shown in Fig. 2 as\na function of temperature. While δαG,1is proportional\nto 1/Tat high temperatures, it is almost constant at\nlow temperatures. The crossover temperature for a fixed\nlength Wis given by T∗=g(γ)ℏvF/(kBW) (kB: Boltz-\nmann constant), which is proportional to 1 /W. The fac-\ntorg(γ), which depends only on γ, is explicitly shown\nlater. The increase in the Gilbert damping is shown as a\nfunction of the junction length Win Fig. 3. While δαG,1\nis proportional to Wfor a short junction, it is almost\nconstant for a long junction. The crossover length for a\nfixed temperature Tis given by W∗=g(γ)ℏvF/(kBT).\nIn the present estimate, the condition Lth≪vF/ω0\nalways holds, where Lth=ℏvF/kBTis a thermal length.\nUnder this condition, the increase in the Gilbert damping\nbecomes independent of ω0and is approximately given\n3K\n10K\n30K\n100K\n300K\n10-310-410-510-610-2\n10-510-3\n10-4\nFIG. 3. Junction-length dependence of increase in the Gilbert\ndamping, δαG,1, for a clean interface ( J1≫J2).\nby\nδαG,1=Γ(γ)2\nΓ(2γ)S0J2\n1a′3a\n(ℏvF)2dd′\u00122πa\nLth\u00132γ−3\nf(γ, πW/L th),\n(41)\nf(γ, w) =(\nw/π, (w/π≪g(γ)),\ng(γ),(w/π≫g(γ)),(42)\ng(γ) =2\nπZ∞\n0dz e−2(γ−1)zF(γ−1, γ,2γ; 1−e−4z).\n(43)\nFrom this analytic expression, we obtain\nδαG,1∝(\nT2γ−2W, (W≪g(γ)Lth),\nT2γ−3g(γ),(W≫g(γ)Lth).(44)\nThe exponent γ= (Ks++Ks−+K−1\ns++K−1\ns−)/4 cor-\nresponds to unity when Ks+=Ks−= 1. Even in the\npresent estimate employing Ks+= 1.07, the exponent is\nalmost unity ( γ= 1.00114). By setting γ= 1, we can\nreproduce the power in the temperature and junction-\nlength dependence of δαG,1shown in Figs. 2 and 3.\nFinally, let us discuss the factor g(γ). If γis slightly\nlarger than 1 as in the present estimate, the geometric\nfunction is approximated as F(γ−1, γ,2γ;x)≃1. Then,\nthe factor g(γ) is approximately given as\ng(γ) =1\nπ(γ−1). (45)\nThis expression indicates that the increase in the Gilbert\ndamping in the high-temperature limit ( T≫T∗) or the\nlong-junction limit ( W≫W∗) is highly sensitive to the\ndeviation of γfrom unity. The crossover temperature\nT∗and the crossover length W∗also include the factor6\n100 1010-5\n10-810-6\n10-7\n3 300\nFIG. 4. Temperature dependence of increase in the Gilbert\ndamping, δαG,2, for a dirty interface ( J2≫J1). The three\nlines correspond to J2= 1, 2, and 3 K, respectively.\ng(γ)∝(γ−1)−1. Thus, the increase in the Gilbert damp-\ning can be used to investigate small deviations of γfrom\nunity. Then, the Luttinger parameter Ks,+in the spin\nsector can also be determined from Eq. (31) if we know\nwhether it is greater or less than unity. We note that in\nthe NMR measurement [22] Ks,+decreases as the mag-\nnetic field increases. Using this experimental tendency,\nwe expect that Ks,+can be determined uniquely.\nB. Dirty interface\nNext, we consider a dirty interface ( J2≫J1). Figure 4\nshows the increase of the Gilbert damping, δαG,2, as a\nfunction of the temperature for J2= 1, 2, and 3 K. In\nthis case, δαG,2is proportional to T−0.43in the whole\ntemperature range and shows a nontrivial exponent in-\nherent to the Tomonaga-Luttinger liquid.\nThe condition Lth≪vF/ω0also holds for a dirty in-\nterface. Therefore, δαG,2can be approximated as\nδαG,2=1\n2πS0J2\n2aa′3\n(ℏvF)2dd′\n×X\nr,r′,α,α′Γ(γrαr′α′)2\nΓ(2γrαr′α′)\u00122πa\nLth\u00132γrαr ′α′−2\n.(46)\nNoting that a≪Lth, the factor (2 πa/L th)2γrαr ′α′in\nEq. (46) is largely reduced as γrαr′α′increases. There-\nfore, in the sum of Eq. (46), it is sufficient to keep the\nterms in which γrαr′α′takes a minimum value. In the\npresent estimate, γrαr′α′is given by Eq. (35) with\n(γ1, γ2, γ3, γ4) = (1 .001,0.784,1.001.0.784). (47)\nUpon setting the minimum exponent to be γmin= 0.784,\nwe obtain δαG,2∝T2γmin−2=T−0.432, which is consis-tent with the numerical results shown in Fig. 4. There-\nfore, the nontrivial exponent inherent to the Tomonaga-\nLuttinger liquid appears in spin pumping through a dirty\njunction. Note that the approximate expression is inde-\npendent of the junction length Wfor a fixed thickness,\nsince the W-linear factor in NFI=WW′d/a′3cancels\nout the factor of Win Eq. (34).\nThe equation for the increase in the Gilbert damping\nfor the dirty interface has almost the same form as that\nfor 1/T1Tin NMR experiments where T1is the longitu-\ndinal relaxation time of nuclear spins[21–23]. Therefore,\nthe power law of the temperature dependence for the\ndirty interface is the same as in NMR experiments. This\nis because the spin transfer occurs at a spatially localized\npoint due to the impurity average at the dirty interface,\nleading to the same situation as the NMR experiment in\nwhich 1 /T1Tis related to the local dynamic spin suscep-\ntibility.\nV. EXPERIMENTAL RELEVANCE\nWe estimated the increase in the Gilbert damping δαG\nin two limiting situations, i.e., clean and dirty interfaces.\nIf we choose YIG as the ferromagnet, δαGshould be\nroughly in the range 10−5–10−2, because it should be\ncomparable to the Gilbert damping of bulk YIG, αG,\nwhich is of order of 10−5–10−3. For a clean interface,\nδαGis large enough to be measured in FMR experiments\n(see Figs. 2 and 3). Note that δαGcan be reduced by in-\ncreasing the thickness of YIG (denoted by d′). On the\nother hand, for a dirty interface, δαGis too small for it to\nbe observable by spin pumping (see Fig. 4). However, we\nwill moderate judgement on the possibility of observing\nδαGfor a dirty interface, because detailed information\non the interfacial exchange coupling is still lacking. We\nshould note that in the present modeling of randomness,\nthe increase in the Gilbert damping is given by a sum of\nthese two contributions, i.e., δαG=δαG,1+δαG,2, for an\narbitrary strength of interfacial randomness.\nOur calculation can be applied straightforwardly to\nother one-dimensional electron systems such as quasi-\none-dimensional magnets, whose low-energy states are\nalso described by the Tomonaga-Luttinger liquid model.\nIn particular, the low-energy states of spin systems with\nin-plane anisotropy are characterized by a Luttinger pa-\nrameter Kssmaller than 1. If Ksis sufficiently smaller\nthan 1, δαGshould show nontrivial power-law behavior\nwith respect to the temperature even for a clean inter-\nface.\nVI. SUMMARY\nWe theoretically studied spin pumping from a ferro-\nmagnetic insulator into carbon nanotubes. First, we for-\nmulated the increase in the Gilbert damping in terms\nof the spin susceptibility and described the interfacial7\nexchange coupling with a simple model, in which two\ntypes of spin-flip process, i.e., momentum-conserving and\nmomentum-nonconserving processes, coexist. Then, we\nanalytically calculated the increase in the Gilbert damp-\ning by treating electrons in carbon nanotubes in the\nframework of the Luttinger liquid. For a clean interface,\nthe increase in damping is proportional to the inverse of\nthe temperature at high temperatures while it is almost\nconstant at low temperatures. The crossover tempera-\nture includes information on the Fermi velocity in carbon\nnanotubes. We also found that the increase in damping is\nhighly sensitive to the deviation of the Luttinger param-\neter in the spin sector from unity. For a dirty interface,\nthe increase in damping shows a power-law dependence\non the temperature with a nontrivial exponent reflecting\nthe nature of the Tomonaga-Luttinger liquid. We also\nestimated the increase of the Gilbert damping using re-\nalistic parameters. Our results indicate a possible appli-\ncation of spin pumping for detecting power-law behavior\nof spin excitation in low-dimensional systems. Detection\nof other types of spin excitation in exotic many-body\nstates will be left as a future study.\nACKNOWLEDGMENTS\nThis French-Japanese collaboration is supported by\nthe CNRS International Research Project “Excitations\nin Correlated Electron Systems driven in the Giga-\nHertz range” (ESEC). This work received support from\nthe French government under the France 2030 invest-\nment plan, as part of the Initiative d’Excellence d’Aix-\nMarseille Universit´ e - A*MIDEX. We acknowledge sup-\nport from the institutes IPhU (AMX-19-IET-008) and\nAMUtech (AMX-19-IET-01X). T. K. acknowledges sup-\nport from the Japan Society for the Promotion of Sci-\nence (JSPS KAKENHI Grants No. 20K03831). M.\nM. acknowledges support by a Grant-in-Aid for Sci-\nentific Research B (23H01839 and 21H01800) and A\n(21H04565) from MEXT, Japan, and by the Priority Pro-\ngram of Chinese Academy of Sciences, under Grant No.\nXDB28000000.\nAppendix A: Correlation functions\nHere, we briefly summarize the calculation of the corre-\nlation function Crαr′α′(x, y, t ) defined in Eq. (23). Using\nthe bosonic fields, the correlation function is written as\nCrαr′α′(x, y, t )\n=1\n(2πa)2\u0002\n⟨eAeBeCeD⟩0− ⟨eCeDeAeB⟩0\u0003\n, (A1)\nA=−i(−rθα+(x, t) +ϕα+(x, t)), (A2)\nB=i(−r′θα′−(x, t) +ϕα′−(x, t)), (A3)\nC=−i(−r′θα′−(y,0) +ϕα′−(y,0)), (A4)\nD=i(−rθα+(y,0) +ϕα+(y,0)), (A5)where we set r= +1 ( r=−1) for the left-going (right-\ngoing) branch. Using the formula,\n⟨eA1eA2···eAN⟩= exp\n1\n2X\ni⟨A2\ni⟩+X\ni<j⟨AiAj⟩\n,\n(A6)\nwhich holds when [ Ai, Aj] is a c-number, we obtain\n⟨eAeBeCeD⟩0≡eFrαr ′α′(x−y,t)\n= exph1\n2⟨(A2+B2+C2+D2)⟩+⟨AB⟩+⟨CD⟩\n+⟨AC⟩+⟨AD⟩+⟨BC⟩+⟨BD⟩i\n, (A7)\n⟨eCeDeAeB⟩0=eFr′α′rα(y−x,−t). (A8)\nThe correlation functions of the bosonic fields, which are\ndefined as GXY\njδ=⟨X(x, t)Y(y,0)⟩, are calculated as [19]\nGθθ\njδ(x, t) =Kjδ\n4(I(x, t) +I(−x, t)), (A9)\nGϕϕ\njδ(x, t) =1\n4Kjδ(I(x, t) +I(−x, t)), (A10)\nGθϕ\njδ(x, t) =Gϕθ\njδ(x, t) =1\n4(I(x, t)−I(−x, t)),(A11)\nI(x, t) =−log\u00142iβℏvF\nLsinh\u0012π(ia−x−vFt)\nβℏvF\u0013\u0015\n.\n(A12)\nUsing these correlation functions, we obtain\nFrαr′α′(x, t) =F1δr,r′δα,α′+F2δr,−r′δα,α′\n+F3δr,r′δα,−α′+F4δr,−r′δα,−α′,(A13)\nF1(x, t) =˜Gθθ\ns++˜Gθθ\ns−+˜Gϕϕ\ns++˜Gϕϕ\ns−\n−r(˜Gθϕ\ns++˜Gθϕ\ns−+˜Gϕθ\ns++˜Gϕθ\ns−),(A14)\nF2(x, t) =˜Gθθ\nc++˜Gθθ\nc−+˜Gϕϕ\ns++˜Gϕϕ\ns−, (A15)\nF3(x, t) =˜Gθθ\ns++˜Gθθ\nc−+˜Gϕϕ\ns++˜Gϕϕ\nc−\n−r(˜Gθϕ\ns++˜Gθϕ\nc−+˜Gϕθ\ns++˜Gϕθ\nc−),(A16)\nF4(x, t) =˜Gθθ\nc++˜Gθθ\ns−+˜Gϕϕ\ns++˜Gϕϕ\nc−, (A17)\nwhere ˜GXY\njδ(x, t)≡GXY\njδ(x, t)−GXY\njδ(0,0). Com-\nbining these results enables the correlation function\nCrαr′α′(x, y, t ) to be obtained analytically.8\nAppendix B: Analytic expressions of integrals\nFor a clean interface, the increase in damping is given\nas\nδαG,1=−4S0J2\n1\nℏ2ω0(2πa)2NFIIγ, (B1)\nIγ=v2\nF\u0012βℏ\nπ\u00133Zw\n0dx′Zw\n0dy′Z∞\n0dusin(˜ω0u)\n×Im(\u0014sinh(iα)\nsinh(iα+x−y−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−x+y−u)\u0015γ−1)\n, (B2)\nwhere ˜ ω0=βℏω0/π,u=πt/β ℏ,w=πW/β ℏvF,x′=\nπx/β ℏvF,y′=πy/β ℏvF, and α=πa/β ℏvF. Changing\nvariables from x′andy′with Z= (x+y)/2 and z=x−y,\nthe integral is modified as\nIγ=v2\nF\u0012βℏ\nπ\u00133Z∞\n0dusin(˜ω0u)\n×\"Zw/2\n0dZZ2Z\n−2Zdz+Zw\nw/2dZZ2(w−Z)\n−2(w−Z)dz#\n×Im(\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1)\n=−v2\nF\n4\u0012βℏ\nπ\u00133Zw/2\n0dZZ−2z\n2zdzZ∞\n−∞du\n×(ei˜ω0u−e−i˜ω0u)\n×\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1\n.\n(B3)\nIn the last equation, we have used the relation\n\u0012sinh(iα)\nsinh(iα±z−u)\u0013∗\n=sinh(iα)\nsinh(iα∓z+u)(B4)\nand the symmetry of the integrand with respect to Z↔\nw−Zandz↔ −z. At this stage, it is useful to introduce\nBγ(ζ, z) =Z∞\n−∞du eiζu\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1\n, (B5)\nso that\nIγ=−v2\nF\n2\u0012βℏ\nπ\u00133Zw/2\n0dZZ2Z\n−2Zdz\n×[Bγ(˜ω0, z)− Bγ(−˜ω0, z)]. (B6)Setting v= 2uand rearranging the hyperbolic sine func-\ntion, we obtain\nBγ(ζ, z) =1\n2(1−e−2iα)2γe−2zZ∞\n−∞dv\n×e−v(γ−iζ/2)\n[e−v+e−2z+i(π−2α)]γ+1[e−v+e−2z+i(π−2α)]γ−1,\n(B7)\nwhich can be computed analytically, invoking the formula\n3.315.1 in Ref. [29] as\nBγ(ζ, z)\n=1\n2(1−e−2iα)2γe−i(π−2α)(γ+iζ/2)e−2z(γ−1−iζ/2)\n×|Γ(γ+iζ/2)|2\nΓ(2γ)F(γ−1, γ,2γ; 1−e−4z), (B8)\nwhere F(a, b, c ;x) is the Gauss hypergeometric function.\nTo leading order in the small parameter α(≪1), this\nreduces to\nBγ(ζ, z) =1\n2(2α)2γeπζ/2+iζze−2z(γ−1)\n×|Γ(γ+iζ/2)|2\nΓ(2γ)F(γ−1, γ,2γ; 1−e−4z).(B9)\nIn practice, we are mostly interested in the regime where\n˜ω0=βℏω0≪1 so we can focus on values of ζsuch\nthat|ζ| ≪1. This allows us to expand Bγ(ζ, z) for small\nvalues of ζ, which yields, after substituting back into the\nexpression for Iγ,\nIγ=−π\n2v2\nFω0\u0012βℏ\nπ\u00134\n(2α)2γΓ(γ)2\nΓ(2γ)Zw/2\n0dZ\n×Z2Z\n0dze−2z(γ−1)F(γ−1, γ,2γ; 1−e−4z),(B10)\nwhere we have used the symmetry of the integrand with\nrespect to z↔ −z. Combining this result with Eq. (B1),\nEqs. (32) and (33) can be derived, after rewriting the\nintegral variable as z′= 2Z.\nIn the limiting case, the integral I(w, γ) given in\nEq. (33) can be approximated into a simple form. For the\nshort-junction limit ( w/π=W/β ℏvF=W/L th≪g(γ)),\nwe obtain\nI(w, γ)≃1\nwZw\n0dz′Zz′\n0dz,e−2z(γ−1)=w\n2.(B11)\nFor the long-junction limit ( w/π≫g(γ)),\nI(w, γ)\n≃1\nwZw\n0dz′Z∞\n0dz,e−2z(γ−1)F(γ−1, γ,2γ; 1−e−4z)\n=π\n2g(γ), (B12)9\nwhere g(γ) is defined by Eq. (43). These analytical ex-\npressions lead to Eqs. (41) and (42) in the main text.\nFor a dirty interface, the increase in damping is ex-\npressed as\nδαG,2=−S0J2\n2aW\nℏ2ω0(πa)2NFIX\nr,r′,α,α′I′\nγrαr ′α′, (B13)\nI′\nγ=βℏ\nπZ∞\n0dusin(˜ω0u)Im(\u0014sinh(iα)\nsinh(iα−u)\u00152γ)\n,\n(B14)\nwith the same dimensionless variables as for a clean in-\nterface. By a similar way as the clean case, the integral\nI′\nγis modified as\nI′\nγ=−βℏ\n4π[Aγ(−˜ω0)−Aγ(−˜ω0)], (B15)\nAγ(ζ) =Z∞\n−∞du e−iζu\u0014sinh(iα)\nsinh(iα−u)\u00152γ\n. (B16)\nSetting v= 2uand rearranging the hyperbolic sine func-tion, we obtain\nAγ(ζ) =1\n2(1−e−2iα)2γZ∞\n−∞dve−(γ+iζ/2)v\n(e−v+ei(π−2α))2γ.\n(B17)\nInvoking the formula 3.314 in Ref. [29], this can be com-\nputed as\nAγ(ζ) =1\n2(2 sin α)2γeαζe−πζ/2|Γ(γ+iζ/2)|2\nΓ(2γ),(B18)\nwhich then yields, to leading order in α(≪1),\nI′\nγ=−βℏ\n4π(2α)2γ|Γ(γ+iζ/2)|2\nΓ(2γ)sinh(π˜ω0/2).(B19)\nAssuming ˜ ω0=βℏω0/π≪1, this is further simplified as\nI′\nγ=−β2ℏ2\n8πω0\u00122πa\nβℏvF\u00132γΓ(γ)2\nΓ(2γ), (B20)\nwhich finally leads to Eq. 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Ryzhik, Table of Integrals, Series,\nand Products (Elsevier Academic Press, 2007)."    },    {        "title": "2204.03534v2.Voltage_driven_exchange_resonance_achieving_100___mechanical_efficiency.pdf",        "content": "Voltage-driven exchange resonance achieving 100% mechanical efficiency\nJun-Yu Tang1and Ran Cheng2, 1\n1Department of Physics and Astronomy, University of California, Riverside, California 92521, USA\n2Department of Electrical and Computer Engineering,\nUniversity of California, Riverside, California 92521, USA\nMagnetic resonances driven by current-induced torques are crucial tools to study magnetic materials but are\nvery limited in frequency and mechanical efficiency. We propose an alternative mechanism, voltage-induced\ntorque, to realize high efficiency in generating high-frequency magnetization dynamics. When a ferromagnet-\ntopological insulator-ferromagnet trilayer heterostructure is operated as an adiabatic quantum motor, voltage-\ninduced torque arises from the adiabatic motion of gapped topological electrons on the two interfaces and act\noppositely on the two ferromagnetic layers, which can excite the exchange mode where the two ferromagnetic\nlayers precess with a p-phase difference. The exchange mode resonance, bearing a much higher frequency\nthan the ferromagnetic resonance, is accompanied by topological charge pumping, leading to a sharp peak in\nelectrical admittance at the resonance point. Because the output current is purely adiabatic while dissipative cur-\nrent vanishes identically, the proposed voltage-driven exchange resonance entails a remarkably high mechanical\nefficiency close to unity, which is impossible in any current-driven systems.\nI. INTRODUCTION\nEfficient manipulation of magnetization dynamics with\nlow-energy dissipation has been a lasting inquiry in spintron-\nics for decades [1–3]. Recently, there are growing interests in\nexploiting topological insulators (TIs) to achieve this goal [4–\n6]. For example, in a ferromagnet-topological (FM-TI) het-\nerostructure, surface electrons are subject to spin-momentum\nlocking, which can efficiently convert an applied charge cur-\nrent to a transverse spin polarization that exerts spin-orbit\ntorques (SOTs) on adjacent magnetic moments [7–16].\nWhile there are plenty of studies on current-induced mag-\nnetization switching [11–21] and spin-torque ferromagnetic\nresonance [8, 13, 18, 22] in FM-TI heterostructures, the oper-\nating speed is limited by the ferromagnetic spin dynamics—\ntypically a few to a few tens of gigahertz. Recently, it has been\nproposed that in trilayer FM-TI-FM heterostructures, the in-\nterlayer exchange coupling mediated by topological electrons\ncan be as strong as its counterpart in conventional spin valves\nwith normal metal spacer [23, 24]. Such a trilayer system has\ntwo resonance modes: a ferromagnetic mode in which the two\nFM are in phase, and an exchange mode in which the two FM\nprecess with a p-phase difference as illustrated by Fig. 1(a).\nThe eigenfrequency of the latter, amplified by the interlayer\nexchange interaction, can be much higher than that of the for-\nmer, opening a new pathway towards ultrafast spin dynamics\ndriven by topological electrons.\nHowever, leveraging the high-frequency exchange mode\nresonance (EMR) calls for driving forces of extremely high\nmechanical efficiency such that the concomitant waste heat\nwill not damage the device. In traditional current-driven sys-\ntems, the transport current inevitably incurs substantial Joule\nheating while only a small fraction of the input power eventu-\nally goes into magnetization dynamics. In TIs with surface\nstates gapped by the proximity of magnetization, one usu-\nally needs to tune the Fermi level into the conduction band\nto retrieve a metallic interface so that an applied charge cur-\nrent can flow and generate non-equilibrium spin accumula-\ntion [3, 9, 10, 16, 21]. Consequently, the conducting surfacebecomes dissipative, which significantly jeopardizes the me-\nchanical efficiency. On the other hand, when the Fermi level\nlies in the gap ( i.e., insulating surface), an applied voltage can\ngenerate the adiabatic motion of valence electrons devoid of\ndissipative currents [7, 15, 17, 25, 26], which can then drive\nthe magnetization as an adiabatic quantum motor (or Thou-\nless motor) without Joule heating [27, 28]. When a FM-TI\nheterostructure is operated this way, the output current will\nbe solely attributed to the magnetization precession (namely,\ntopological charge pumping) [29]. As a result, the system\nbarely consumes any energy when it remains stationary. In the\nvicinity of magnetic resonance, all input power goes into mag-\nnetization dynamics to overcome Gilbert damping, whereas\nOhmic loss is eliminated. In such circumstances, the magneti-\nzation dynamics is actually driven by voltage-induced torques\nrather than current-induced torques. However, it remains elu-\nsive whether voltage-induced torques are able to drive the de-\nsired high-frequency EMR in FM-TI-FM heterostructures.\nIn this paper, we first develop a semiclassical theory to\nquantify the voltage-induced SOT arising from the topolog-\nical surface states of a TI exchange coupled to multiple in-\nsulating FMs, which, for Fermi level lying in the gap, is de-\ntermined by the Berry curvature crossing the momentum and\ntime spaces. Different from that in FM-TI bilayer systems, the\neffective magnetic field responsible for the SOT in a FM-TI-\nFM trilayer system has not only a parallel but also transverse\ncomponents with respect to the direction of voltage drop. The\nparallel component, being the dominant contribution, is of\nopposite sign on opposite interfaces, which can directly ex-\ncite the EMR when driving frequency matches the eigenfre-\nquency. With the onset of EMR, the reciprocal effect of SOT,\ntopological charge pumping, will generate a secondary adi-\nabatic current on top of the driving force, forming a sharp\npeak of electrical admittance. Because the output current is\npurely adiabatic, which does not lead to Joule heating, all in-\nput power is cost by sustaining the magnetization dynamics,\nyielding a theoretical mechanical efficiency of 100% unless\nthe system is not perfectly insulating ( i.e., in the presence of\nleakage current). The considered setting can be represented by\nan effective circuit with identical electrical responses. With allarXiv:2204.03534v2  [cond-mat.mes-hall]  13 Aug 20222\nthese unique properties combined, the voltage-driven EMR in\ntrilayer FM-TI-FM heterostructures opens an exciting possi-\nbility to achieve high-frequency magnetization dynamics with\nalmost zero dissipation.\nThe paper is structured as the following. In Sec. II, we\nformulate a Berry phase theory of the coupled dynamics of\ntopological electrons and magnetization. In Sec. III, we ap-\nply our formalism to an FM-TI-FM trilayer and numerically\ncalculate the voltage-induced SOT. In Sec. IV, we study the\nvoltage-driven EMR by calculating the dynamical susceptibil-\nity. In Sec. V, we investigate the topological charge pumping\neffect associated with the EMR to quantify the overall electri-\ncal response of our system, and then discuss an experimental\nscheme to detect the EMR. In Sec. VI, we evaluate the me-\nchanical efficiency of the proposed setup and discuss the in-\nfluence of imperfections (such as leakage current) our theory\nhas ignored. The whole paper is summarized with conclud-\ning remarks in Sec. VII. Mathematical details about several\nderivations are clarified in the Appendices.\nII. FORMALISM\nLet us consider free electrons moving in a multi-layer sys-\ntem where each magnetic layer is described by a uniform\nmagnetization (macrospin model). In the presence of exter-\nnal electromagnetic field represented by a scalar potential j\nand a vector potential A, the electronic Hamiltonian can be\nwritten as\nH=H0(¯hk+eA;m1;m2;\u0001\u0001\u0001)\u0000ej; (1)\nwhere H0(A!0)is the unperturbed Hamiltonian of topologi-\ncal electrons on the surfaces of a TI, e>0 is the absolute value\nof the electron charge, ¯hkis the crystal momentum with ¯hthe\nreduced Planck constant, and mjis the (dimensionless and\nunit) magnetization vector of the j-th magnetic layer. Here\nwe allow H0to depend on all mjbecause topological elec-\ntrons on different interfaces are coupled, which will become\nclear in Sec. III [see Eq. (9)] while here we keep the formal-\nism general without specifying the form of H0. Under the\nadiabatic regime where inter-band transition is negligible, an\nindividual electron can be described by a semiclassical wave\npacketjW(rc;kc)iwithrcandkcthe center-of-mass posi-\ntion and momentum, respectively [30]. We then construct\nthe Lagrangian for the wave packet as Le=hWj(i¯hd=dt\u0000\nH)jWi[31]. By adding the effective Lagrangian for each\nmagnetic layerLj=\u0000Nj(¯hSfjcosqj+ej)[32] with Sthe\nspin magnitude, Njthe total number of magnetic atoms in\nlayer j(for cubic lattice, Nj=Nj\nxNj\nyNj\nz), and ejthe magnetic\nfree energy as a functional of mj(rc;t)[and othermidirectly\ncoupled tomj], we obtain the effective Lagrangian for an in-\ndividual electron wave packet interacting with all magnetic\nlayers as (see details in Appendix A)\nL=Le+å\njLj=\u0000hWjHjWi\u0000å\njNjej\n\u0000¯h\u0000\nSNj˙fjcosqj\u0000˙rc\u0001kc\u0000˙mj\u0001Amj\u0000˙kc\u0001Ak\u0001\n;(2)\nΔ!Δ\"1234(b)\nFMFMTI!\"#\n(a)FIG. 1. (a) Illustration of the exchange mode in a FM-TI-FM trilayer\nand the coordinate system. (b) Band structure of the FM-TI-FM tri-\nlayer around the Gpoint, where D1\u0018Jsd\u0000b0andD2\u0018b0.\nwhere the layer index jis summed wherever repeated. All\nterms are evaluated at location rc. In Eq. (2), the ˙fjcosqj\nterm accounts for the magnetization dynamics of layer j\nwhere fjandqjare the two spherical angles specifying\nthe orientation of mj;Amj=ihuj(¶=¶mj)juiandAk=\nihuj(¶=¶k)juiare the Berry connections in the magnetiza-\ntion and momentum spaces, respectively, where juiis the peri-\nodic part of the Bloch wave function. By taking the functional\nderivatives ofLwith respect to rc,kc, andmj, we obtain a\nset of coupled equations of motion:\n¯h_kc=\u0000eE; (3a)\n_rc=1\n¯h¶E\n¶kc+˙kc\u0001 !Wkk+˙mi\u0001 !Wmik; (3b)\nNjS˙mj\u0002mj=¶E\n¯h¶mj+˙mi\u0001 !Wmimj+˙kc\u0001 !Wkmj;(3c)\nwhere iis summed while jis a dangling variable specify-\ning the magnetic layer under consideration, E=hWjHjWi+\nåjNjej,E=\u0000rcj\u0000¶A=¶tis the applied electric field at\ncenter of the wave packet rc, and !Wmn=¶mAn\u0000¶nAmis the\nBerry curvature tensor with m,nrunning through each com-\nponent ofkcandmj. The inner product of the tensor with a\nvector contracts the first sub-index while the second sub-index\nis dangling. For example, the n-component of ˙kc\u0001 !Wkkrefers\nto˙kc;mWkmknwhere mis summed while nspecifies the spatial\ncomponent of the resulting vector.\nEquation (3) is purely general as multiple physical effects\nare captured by different components of the Berry curvature\ntensor. The !Wkkterm is the anomalous velocity responsible\nfor the quantum anomalous Hall conductivity. The real-space\ncomponent !Wrr=0 because we do not consider inhomoge-\nneous magnetization texture. The cross term !Wmikin Eq. (3b)\nis the Berry curvature connecting momentum kto the mag-\nnetizationmi, hence, time t[asmi=mi(t)]. It character-\nizes how magnetization dynamics affects the electron motion,\nnamely, topological charge pumping. Reciprocally, the !Wkmj3\nterm in Eq. (3c) accounts for the back-action of the electron\nmotion on the magnetization dynamics, which refers to the\nvoltage-induced SOT. Finally, the !Wmimjterm couples the\nmagnetization of layer iwith that of layer jthrough topolog-\nical surface electrons; the i=jcomponent renormalizes the\ngyromagnetic ratio in a specific layer which will become clear\nin the following.\nTo derive the voltage-driven magnetization dynamics, we\ninsert Eq. (3a) into Eq. (3c) to eliminate ˙kc. To account for all\nBloch electrons, we also integrate the electron-related terms\noverkcwithin the first Brillouin zone (BZ) and rcover the\narea of the interface:RdrcR\nBZLef(kc)dkc, where f(k)is the\nFermi–Dirac distribution. For simplicity, we restrict to the\nlow-temperature regime such that f(k) =1 fore(k)<eFand\nf(k) =0 otherwise. By taking the cross product mj\u0002on both\nsides of the equation, we obtain the effective magnetization\ndynamics as (see details in Appendix A)\n˙mj=\u0000gmj\u0002(Hm\nj+HSOT\nj\u00001\nNj\nzSå\ni6=j˙mi\u0001Wmimj);(4)\nwhere\nHm\nj=\u0000a2\n0\n(2p)2¯hSZ\nBZ¶ej\n¶mjdkc (5)\nis the effective magnetic field acting on mjin the absence of\nelectric field with Nj\nzdenoting the number of atomic layers in\nlayer j(note that Nj=Nj\nxNj\nyNj\nz), and\nHSOT\nj\u0011eE\nNj\nz¯hS\u0001Wkmj (6)\nis the effective field for the voltage-induced SOT exerting\nonmjwith the overline representing an integration over the\nfirst BZ: Wkmj\u0011a2\n0=(2p)2R\nBZWkmjdkc. In deriving Eqs. (5)\nand (6), we assumed that the whole system (TI and all mag-\nnetic layers) has a simple cubic lattice with lattice constant a0,\nwhich does not sacrifice any essential physics. Since both mj\nandgare dimensionless in our scaling convention, HSOTis\nexpressed in the frequency dimension.\nThe last term of Eq. (4), as mentioned above, represents the\ninterlayer coupling mediated by topological electrons, which\nturns out to be a dissipative coupling [33]. Compared with\nthe interlayer Heisenberg exchange interaction [23, 24], this\ndissipative coupling is negligible as estimated in Appendix B.\nIn our convention, the bare gyromagnetic ratio is scaled to\nunity, while in Eq. (4),\ng\u0011\u0014\n1+1\nNj\nzSmj\u0001\n\u0015\u00001\n(7)\nis the effective gyromagnetic ratio [26] renormalized by the\nintra-layer component of the Berry curvature\nWl\u00111\n2el mna2\n0\n(2p)2Z\nBZdkcWmj\nmmj\nn; (8)where el mn is the Levi-Civita symbol and Wmj\nmmj\nnrefers to\nthemnelement of the Berry curvature tensor !Wmjmjlying in\nthe configurational phase space ( i.e., magnetization direction).\nAn intuitive understanding of the renormalization of gis that\nthe topological electrons are polarized by the local magneti-\nzation, thus contributing partially to the total magnetic mo-\nment. By numerically evaluating the intra-layer components\nof !Wmjmj, we estimate that the renormalization is only 10\u00005,\nso in the following we will use g=1 for simplicity.\nIII. VOLTAGE-INDUCED SOT IN FM-TI-FM TRILAYER\nWe now apply the formalism above to a FM-TI-FM trilayer\nsystem as schematically illustrated in Fig. 1(a), assuming that\nboth FM layers are insulating and having the same thickness\n(soNt\nz=Nb\nz=Nz). On the interfaces, the topological electrons\ncouple to the adjacent magnetic moments via the exchange in-\nteraction Jsd. Under the basis (ct\nk\";ct\nk#;cb\nk\";cb\nk#)Twhere ct(b)\nks\nannihilates an electron with wave vector k=fkx;kygand spin\nsat the top (bottom) interface, the Hamiltonian H0in Eq. (1)\ncan be written as [34, 35]:\nH0=¯hvf(kxtz\nsy\u0000kytz\nsx)+A(k)tx\nI\n+Jsd(mt\u0001\u001b)\b(mb\u0001\u001b); (9)\nwhere\n(\b) denotes direct product (direct sum), vfis the\nFermi velocity, \u001band\u001care the vectors of Pauli matrices rep-\nresenting the spin and layer ( i.e., top and bottom) degrees of\nfreedom, respectively [36]. A(k) =b0+b1(k2\nx+k2\ny)describes\nthe overlap between the top and bottom surface states in the\nthin TI film, where b0andb1depend on the TI thickness.\nWe first look into a simple case where the two FM layers\nare collinear (but not necessarily along the ˆ zdirection) and\nare described by a single magnetization vector m. The energy\ndispersion can then be solved from Eq. (9) as\ne=\u0006q\nA(k)2+(¯hvfk)2+J2\nsd\u0006x; (10)\nwhere\nx=2Jsdq\n(¯hvf)2[^z\u0001(k\u0002m)]2+A(k)2: (11)\nFigure 1(b) shows the band structure around the Gpoint,\nwhere aside from the exchange gap determined by Jsd, the\noverlap between the top and bottom interfaces b0opens a hy-\nbridization gap D2µb0. There exists a topological phase tran-\nsition at b0=Jsd[34, 37], where the band gap D1µ(Jsd\u0000b0)\ncloses. The topologically nontrivial (trivial) regime corre-\nsponds to b0<Jsd(b0>Jsd).\nTo obtain the effective SOT field HSOTdefined in Eq. (6)\nas a function of the mdirection, we now perform numerical\ncalculation using typical material parameters: Jsd=100 meV,\nb0=50 meV, b1=1 eVnm2and¯hvf=1 eVnm [38–40]. The\nFermi level lies in the surface gap in the absence of doping\nand gating. Figures 2(a)-(c) plot the angular dependence of4\n020/2\n-0.200.2\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.5\n020/2\n-2-101210-3\n020/2\n-6-42\n020/2\n-6-4-20246\n020/2\n-0.200.2\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.5\n020/2\n-2-101210-3\n020/2\n-6-42\n020/2\n-6-4-20246\n020/2\n-0.200.2\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.5\n020/2\n-2-101210-3\n020/2\n-6-42\n020/2\n-6-4-20246\n020/2\n-0.200.2\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.5\n020/2\n-2-101210-3\n020/2\n-6-42\n020/2\n-6-4-20246\n020/2\n-0.200.2\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.5\n020/2\n-2-101210-3\n020/2\n-6-42\n020/2\n-6-4-20246\n020/2\n-0.200.2\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.5\n020/2\n-2-101210-3\n020/2\n-6-42\n020/2\n-6-4-20246\nɅ\n\u0006ൌͳ\u0006ൌǦͳ\n\u0006ൌǦͳ\u0006ൌͳ\u0006ൌͲ\n([+(+\nɅ\nɅɅɅɅ)0)07,\n)07,)0\u000bD\f\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u000bE\f\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u000bF\f\n\u000bG\f\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u000bH\f\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u000bI\f\nWP\nWP\nEPEP+(\nɅ\n\u0006ൌͳ\u0006ൌǦͳ\n\u0006ൌǦͳ\u0006ൌͳ\u0006ൌͲ\n([+(+\nɅ\nɅɅɅɅ)0)07,\n)07,)0\u000bD\f\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u000bE\f\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u000bF\f\n\u000bG\f\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u000bH\f\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u000bI\f\nWP\nWP\nEPEP+(\nθθ\nϕϕϕHSOTxHSOTyHSOTz\n×10−2×10−3\nC = 1C = 1C = -1\nC = -1C = 0(a)(d)(b)(c)(e)(f)\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.50.00.51.01.5\n020/2\n-2-101210-3\n020/2\n-6-420246\n020/2\n-6-4-20246×10−2\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.50.00.51.01.5\n020/2\n-2-101210-3\n020/2\n-6-420246\n020/2\n-6-4-20246×10−2\n020/2\n-0.200.2\n020/2\n-0.3-0.2-0.100.10.20.3\n020/2\n-1.5-1.0-0.5\n020/2\n-2-101210-3\n020/2\n-6-42\n020/2\n-6-4-20246mtmb\nmbmt\nFIG. 2. The SOT field acting on the top FM (in x,yandzcomponents) as a function of the polar angle qand the azimuth angle fwhenmtand\nmbare (a)-(c) in phase, and (d)-(f) in the exchange mode configuration. The coordinate systems are illustrated in the left panels. The blank\nstrips mark the boundary dividing distinct topological phases, on which the band gaps close. HSOTtis plotted in the unit (ea0Ex)=(4p2Nz¯hS)\n(frequency dimension) where Exis the applied electric field. Parameters: Jsd=100meV, b0=50meV, b1=1eV\u0001nm2and¯hvf=1eV\u0001nm.\nHSOTexerting onmt, whileHSOTacting onmbis exactly\nthe opposite (thus not plotted). The corresponding SOT is\nt=gHSOT\u0002m. We observe that HSOTis largely pointing\nin the ˆ xdirection, i.e., collinear with the applied electric field\nE. However, different from the case of FM-TI bilayer where\nHSOTis exactly along the direction of E, here the SOT field\nalso has small but finite projections on the yandzaxes, the\nexistence of which can be attributed to the surface mixing term\nA(k)in the Hamiltonian.\nBecauseHSOTis opposite for mtandmb, the two FM\ninitially polarized along zwill evolve towards opposite di-\nrections, which subsequently brings about noncollinearity and\ntriggers the interlayer exchange interaction, leading to a pre-\ncessional motion of the two FM with a p-phase difference.\nThis means that the high-frequency exchange mode can be\nexcited by an AC electric field through the voltage-induced\nSOT. However, in the exchange mode, mtandmbare non-\ncollinear and have opposite in-plane components. Therefore,\nthe band structure, the Berry curvature, hence HSOT, all de-\npending on the spin configuration, must be re-calculated based\non the new configuration: mt\nx=\u0000mb\nx,mt\ny=\u0000mb\nyandmt\nz=mb\nz.\nFigures 2(d)-(f) plot HSOTfor the exchange mode, where we\nfind that HSOT\nx is still the dominant component. Just as in\nthe collinear configuration, mtandmbare subject to oppo-\nsite SOT fields. So, for small-angle precessions, the voltage-\ninduced SOT is able to sustain the exchange mode, which will\nbe further discussed in Sec. IV.\nComparing to Fig. 2(a) in which two topological phases\nwith Chern number C=\u00061 are adjacent (given that b0<Jsd),\na topological trivial phase ( C=0) is identified in Fig. 2(d)\nbetween q=55\u000eandq=125\u000e(these values depend on b0).In the C=0 phase, HSOT\nx is small but not zero, suggesting\nthat the voltage-induced SOT involves the bulk (surface) ef-\nfect and is not a pure edge effect. By contrast, the HSOT\nyand\nHSOT\nzcomponents are larger in the C=0 phase than in the\nC=\u00061 phases. While both HSOT\ny andHSOT\nz exhibit sinu-\nsoidal dependence on f, the former (latter) has a period of p\n(2p). Moreover, HSOT\ny(HSOT\nz) is an odd (even) function of\n(q\u0000p=2).\nTo conclude this section, we finally evaluate the ”torkance”\ndefined as the SOT field per applied electric field: gHSOT=E.\nWith the parameters used in plotting Fig. 2, the torkance is\nabout 1 GHz =(V=mm)forNzS=10 (see Appendix B). As\na comparison, in a typical current-driven ferromagnetic reso-\nnance [19, 20], the torkance is on the same order of magnitude\nwhen platinum (Pt) is used as the spin generator.\nIV . SOT-DRIVEN EXCHANGE MODE RESONANCE\nTo study the excitation of EMR by the voltage-induced\nSOT derived above, we resort to the coupled Lan-\ndau–Lifshitz–Gilbert (LLG) equations:\n˙mt= (wEmb+wAmt\nzˆz+HSOT)\u0002mt+amt\u0002˙mt;\n(12a)\n˙mb= (wEmt+wAmb\nzˆz\u0000HSOT)\u0002mb+amb\u0002˙mb;\n(12b)\nwhere wAandwEare the angular frequencies of the easy-axis\nanisotropy and interlayer exchange interaction, respectively,\nandais the dimensionless Gilbert damping constant. Here5\n0204060801000.5151050100\nPrinted by Wolfram Mathematica Student Edition/uni007Cmtx/uni007C/uni007Cmty/uni007C/uni03C9/2/uni03C0(GHz)\n/uni03C9/2/uni03C0(GHz)/uni0302x/uni0302y/uni0302x/uni0302y/uni0302x/uni0302y/uni0302x/uni0302y(c)4045505560-0.15-0.10-0.050.000.05\nPrinted by Wolfram Mathematica Student Edition/uni03C7(GHz−1)Re[˜/uni03C7/uni2225]Im[˜/uni03C7/uni2225](a)\n4045505560-0.15-0.10-0.050.000.05\nPrinted by Wolfram Mathematica Student Edition/uni03C9/2/uni03C0(GHz)Re[˜/uni03C7/uni22A5]Im[˜/uni03C7/uni22A5](b)\nFIG. 3. Real and imaginary parts of the dynamical susceptibility\nfor: (a) parallel component ˜ckand (b) perpendicular component\n˜c?as functions of wwith a resonance frequency wr=2p=50 GHz\nand a driving SOT field gHSOTx=0:628 GHz. (c) Logarithmic plot\nofjmtx=mtyj, where at the EMR point (dashed line) the ratio is 1.\nSchematics of the polarization is illustrated at different frequencies.\nwe assume the two FM layers are identical and share the same\nanisotropy, but our theory can easily be generalized to account\nfor non-equivalent FM layers. While wEmay be either fer-\nromagnetic or antiferromagnetic depending on the TI thick-\nness [23, 24], here we assume that wE>0 (i.e., ferromagnetic\ncoupling). As we are considering insulating FM and gapped\nsurface states, there is no transport current and the Oersted\nfield in ordinary spin-torque resonance experiments does not\nexist. Nonetheless, the applied AC electric field amounts to a\ndisplacement current proportional to ¶E=¶t, which, accord-\ning to Maxwell’s equation, generates a stray field similar to\nthe Oersted field. We estimate that the stray field stemming\nfrom the displacement current is negligible for thin films (see\nAppendix B), thus to a good approximation, only the SOT\nfieldHSOTneeds to be taken into consideration. Linearizing\nthe LLG equations around the ground state mt=mb=ˆzwith\nrespect to the phase vectors ˜ mt(b)\n?= (˜mt(b)\nx;˜mt(b)\ny)eiwt, we ob-\ntain ˜mt=b\nx=\u0006˜ck˜HSOT\nxand ˜mt=b\ny=\u0006˜c?˜HSOT\nx. Here, the driv-\ning field ˜HSOT\nx=HSOT\nxeiwtwhile the small HSOT\nyandHSOT\nz\ncomponents are ignored, and\n˜ck(w) =\u0000(iaw+wr)\nw2\u0000(iaw+wr)2; (13a)\n˜c?(w) =iw\nw2\u0000(iaw+wr)2; (13b)\nwhere wis the driving frequency of the applied electric field,\nwr=2wE+wAis the bare EMR frequency for negligible\na[41]. Depending on wE, the EMR frequency could be much\nhigher than that of ordinary ferromagnetic resonances which\nis on the order of wAin the absence of external magnetic fields.\nFigures 3(a) and (b) plot ˜ckand ˜c?for an EMR frequency ofwr=2p=50 GHz, where we see that the real (imaginary) part\nof˜ckis antisymmetric (symmetric) around wrand the sym-\nmetry pattern of ˜c?is just the opposite. For very small a,\nthe phase difference between ˜ckand ˜c?is about p=2 and al-\nmost independent of frequency (unless w=wris as large as\n1=a), which indicates a persistent counterclockwise preces-\nsion ofmt(b)regardless of the driving frequency, as illustrated\nin Fig. 3(c). Nonetheless, when going off-resonance, the mag-\nnitudej˜ckjdecays faster (slower) than j˜c?jforw>wr(for\nw<wr). Therefore, the actual polarization becomes elliptical\nand evolves towards opposite linear oscillations when moving\naway from the EMR point on which a circular polarization is\nachieved.\nV . TOPOLOGICAL CHARGE PUMPING\nWith the onset of EMR, the precessing magnetization gen-\nerates an adiabatic current through topological charge pump-\ning, which is the reciprocal (reverse) effect of voltage-induced\nSOT [29]. This picture remains valid so long as the adiabatic\ncondition ¯hw\u001cD1is satisfied ( i.e., the motion of mt(t)and\nmb(t)does not agitate inter-band transitions of electrons). In\ncontrast to transport currents accompanied by Joule heating,\nan adiabatic current is non-dissipative and incurs no Ohmic\nloss [27–29], which is a salient advantage of voltage-driven\nsystems. At the same time, topological charge pumping pro-\nduces an output AC current that directly signals the magneti-\nzation dynamics, through which the EMR can be experimen-\ntally detected. Inserting ˙ rcexpressed in Eq. (3b) into the sur-\nface current density I=\u0000e=(2p)2R˙rcf(kc)dkcgives [42]\nI=\u0000e\n(2p)2Z\nBZ\u0010\n˙mt\u0001 !Wmtk+˙mb\u0001 !Wmbk\u0011\ndkc: (14)\nWhen being applied to a simple FM-TI bilayer, Eq. (14)\nhas only one term, yields exactly the same result derived in\nRef. [43]. In the ferromagnetic mode of our FM-TI-FM tri-\nlayer, however, the in-phase precession of mtandmbcancel\neach other so that Ivanishes. By contrast, in the exchange\nmode they have a pphase difference, thus constructively con-\ntributing to the output current. At the EMR point, the sys-\ntem becomes circularly polarized, which reflects also in the\npumped current as plotted in Fig. 4(a), where the xandycom-\nponents are both sinusoidal and differ by a phase of p=2.\nAs the reverse effect of SOT, topological charge pumping\nshould also depend on the amplitude of EMR characterized\nbyqand the band topology of electrons characterized by the\nChern number. In this regard, we numerically compute Ias\na function of qandb0=Jsdand plot the result in Fig. 4(b).\nWe see that the pumped current vanishes in the topological\ntrivial phase ( C=0); the maximum ratio of b0=Jsdto keep\nthe system topologically non-trivial decreases with increasing\nq(i.e., growingjm?j) until it vanishes at a critical value qc.\nFrom the shape of the phase boundary in Fig. 4(b), we read\noff a good tolerance on the resonance amplitude: the system\nremains in the C=1 phase even when qis no longer small at\nEMR unless b0=Jsd!1. For example, if b0=0:8Jsd,qcan6\n0234-505\n0/6/3/20.51.01.5\n05010000.511.522.510-7\n--005010000.511.522.510-7\n--0/uni007C˜/uni03C3xx/uni007CArg[˜/uni03C3xx]0234-505\n0/6/3/20.51.01.5\n05010000.511.522.510-7\n--005010000.511.522.510-7\n--0b0Jsd\nArg[˜/uni03C3yx]/uni007C˜/uni03C3yx/uni007C/uni03B8/uni03D5\nAmplitude of  ×10−2C=1C=0(a)(b)\n(c)(d)\n/uni03C9/2/uni03C0(GHz)/uni03C9/2/uni03C0(GHz)20406080102030405060\n00.20.40.60.8\nyx\n/uni03C02/uni03C02/uni03C02/uni03C02/uni03C3(A/V)246810246810×10−8×10−8\n(A/m)\n(A/m)\nFIG. 4. (a) Surface current density in the xandydirections as a\nfunction of the precessional angle fat the EMR for wr=2p=50\nGHz. For visual clarity, the amplitude of oscillation is exaggerated\ntojmt\nx(y)j=0:1. (b) The amplitude of surface current density Ias a\nfunction of the polar angle qand the ratio b0=Jsd(Jsd=100 meV).\nThe phase boundary is marked by a white dashed line. (c) and (d):\nAmplitude and phase of electrical admittance ˜sxxand ˜syxas a func-\ntion of the driving frequency w. Parameters: a=0:01 and NzS=10.\ngrow up to p=6 (which is certainly not a small-angle) without\nlosing the band topology.\nBecause the SOT and the topological charge pumping take\nplace simultaneously, their combined effect manifests as a lin-\near relation between the output current density and the driving\nelectric field\n˜Im(w) =˜smn(w)˜En(w); (15)\nwhere ˜smnis the electrical admittance (or AC conductivity).\nSubstituting the LLG equations (12) into Eq. (14) to eliminate\n˙mband ˙mt, and considering Eq. (6), we obtain:\n˜sxx(w) =izw˜ck(w)W2\nkxmtx; (16)\n˜syx(w) =izw˜c?(w)W2\nkymty; (17)\nwhere z=2e2\nNz¯hSa2\n0(see details in Appendix C). The amplitude\nand phase of ˜sxx(w)and ˜syx(w)are plotted in Figs. 4(c) and\n(d), where the amplitude of both components exhibits a sharp\npeak around the EMR point wr=50 GHz. Since zscales\ninversely with NzS, the thicker the FM films, the smaller the\npeak value of electrical admittance. As a comparison, we note\nthat the quantized Hall conductivity sH=e2=his on the or-\nder of 10\u00005A=V, so the peak value of ˜syx(w)forNzS=10 is\nabout two orders of magnitude smaller. Therefore, the trans-\nverse component of the pumped current is overwhelmed by\nthe anomalous Hall effect, and is thus very difficult to detect.\nBy contrast, the longitudinal output current determined by\n˜sxx(w)is a conspicuous effect, because the whole system isinsulating and the output current solely originates from the\nmagnetization precessions. Basing on the form of ˜sxx(w), we\nconceive an experimental scheme as illustrated in Fig. 5 to\nmeasure topological charge pumping, and hence the EMR,\nof the FM-TI-FM trilayer heterostructure. We connect the\ntrilayer device in series with a reference impedance Z0and\npower the whole circuit by an AC voltage source. The volt-\nage drop across Z0measured by a voltmeter vanishes (reaches\nmaximum) for off-resonance (on-resonance) conditions as the\nimpedance of our device, 1 =˜sxx(w), goes to infinity (reaches\nminimum). To better characterize the electrical response as-\nsociated with the voltage-driven EMR, we further map our\ndevice to an effective circuit whose longitudinal admittance\nis identical to Zeff(w) =1=˜sxx(w). The effective circuit, as\nillustrated in the top panel of Fig. 5, consists of a capacitor,\nan inductor, and two resistors. The physical parameters in the\neffective circuit are solved as (see Appendix D)\nR=Zm\na;r=Zma;L=Zm\nwr;C=1\nZmwr; (18)\nwhere\nZ\u00001\nm=W2\nkxmtxzw\nl(19)\nwith landwthe length and width of the FM-TI-FM trilayer.\nFor a square sample with w=l=1,Zmis about 5\u0002108W. For\nGilbert damping a=0:01 and resonance frequency wr=2p=\n50 GHz, the two resistances, the inductance, and the capac-\nitance appearing in the effective circuit are evaluated to be\nR=5\u00021010W,r=5\u0002106W,L=1:59\u000210\u00003H and\nC=0:64\u000210\u00008pF, respectively. It is easy to see from the\neffective circuit that our trilayer device behaves as a capacitor\n(an inductor) for w\u001cwr(w\u001dwr); right at the EMR point,\nthe device behaves as a resistor because the impedance from\nCandLcancel. We emphasize that Randrin Eq. (18) do not\nproduce real Joule heating; they simply represent magnetic\ndissipation (Gilbert damping) in the form of effective resis-\ntors.\nVI. MECHANICAL EFFICIENCY\nAn adiabatic quantum motor has a theoretical mechanical\nefficiency of 100% because its output current is purely adi-\nabatic, incurring zero Ohmic loss (Joule heating) [27, 28].\nTo confirm that our considered FM-TI-FM trilayer system is\nindeed operated as an adiabatic quantum motor, we need to\nquantify how much input power is eventually converted to the\nmagnetization dynamics, which is reflected by the mechan-\nical efficiency h\u0011PM=PJ, where PM=2aj˙m2jMsVm=gis\nthe magnetic dissipation rate with Vmthe volume of each FM\nlayer ( Vm=a3\n0NxNyNz) and PJ=jIxExjwlis the total elec-\ntric power consumed by the trilayer device. The overline here\nrepresents a time average over one period of oscillation ( e.g.,\nIxEx\u0011RT\n0IxExdt=Twith T=2p=w). In the absence of leak-\nage currents, we obtain after a lengthy derivation (see details7\n!!V\n\"#$%!!\"\"\"\nFIG. 5. Detection scheme of the EMR and effective circuit for the\nFM-TI-FM trilayer. Z0is a fixed impedance, the voltage drop on\nwhich is monitored by a voltmeter.\nin Appendix C)\nPM=aE2\nxwlz\n2W2\nkxmtxw2\u0002\nj˜ck(w)j2+j˜c?(w)j2\u0003\n; (20)\nPJ=E2\nxwlz\n2W2\nkxmtxw\f\fIm\u0002˜ck(w)\u0003\f\f; (21)\nwhich, when Eqs. (13a) and (13b) are inserted, leads to\nh=PM\nPJ=1; (22)\nconfirming that the considered setup indeed functions as an\nadiabatic quantum motor. Here, the mechanical efficiency\nshould not be confused with the “spin-torque efficiency” de-\nfined exclusively for current-induced torques[44].\nWhile the ideal 100% mechanical efficiency is an intrin-\nsic characteristic of adiabatic quantum motor, real systems\nalways have imperfections that inevitably jeopardize the me-\nchanical efficiency. In our case, the TI may not be a perfect\ninsulator even when the surface states are gapped by the ad-\njacent magnetization [45], resulting in leakage currents that\ndo incur Ohmic loss. The leakage effect can be represented\nby a very large resistance Rleakin parallel with a very small\ncapacitance Cleak, which should be added in parallel to the ef-\nfective circuit drawn in Fig. 5. Under an AC voltage drive,\nhowever, Cleakdoes not contribute to any time-averaged dis-\nsipation. Therefore, the leakage-induced Ohmic loss makes\nup a dissipation power Pleak=E2\nxl2=2Rleak, hence the actual\nmechanical efficiency becomes h=PM=(PJ+Pleak). Using\ntypical material parameters ( e.g., Bi 2Te2Se [46] for the TI and\nCrI3for the FM [47]), we estimate Pleakto be somewhere be-\ntween 10% to 50% of PJdepending on several geometrical\nfactors of the device (such as the TI thickness), so the actual\nmechanical efficiency is still very high. With improved mate-\nrial qualities ( e.g., more insulating TI) and device engineering,\nwe are optimistic that hcan be pushed towards its theoretical\nlimit ( i.e.100%) in the future.\nTo benchmark the radical boost of mechanical efficiency byusing voltage-induced SOT, we now estimate hfor the well-\nestablished current-driven resonances. Consider a YIG/Pt bi-\nlayer [48, 49] where the SOT is generated by the spin Hall\neffect in the Pt. As calculated in Appendix C, his below 1%\nat the ferromagnetic resonance point [50]. In current-driven\nsystems, the mechanical efficiency is limited by three major\nfactors: (1) only a small fraction of the driving current is con-\nverted into spin current by the spin Hall effect; (2) the back-\nflow effect [51, 52] suppresses the effective spin-mixing con-\nductance governing the spin transmission across interfaces;\n(3) the Ohmic loss in a metallic spin generator such as Pt is\nenormous. If the FM layer is changed into a permalloy, the\nshunting current effect will further reduce hdown to the order\nof 0:1%. By sharp contrast, none of these mechanisms exist in\nour system. First, the voltage-induced SOT does not involve\nbulk spin diffusion as in the spin Hall effect. Second, spins do\nnot transmit across the interface but drive the magnetization\ndirectly as an adiabatic quantum motor. Third, the output cur-\nrent stems from the precessing motion of magnetization ( i.e.,\ntopological charge pumping) which does not generate Joule\nheating as that for a dissipative current.\nVII. CONCLUDING REMARKS\nIn conclusion, an insulating FM-TI-FM trilayer het-\nerostructure can be operated as an adiabatic quantum motor\nvia the combined effect of voltage-induced spin-orbit torque\nand its reverse effect, topological charge pumping, which of-\nfers a theoretical mechanical efficiency of 100% as Joule heat-\ning is obviated. This mechanism is particularly suitable for\nachieving the high-frequency exchange resonance in which\nthe two FM layers are out of phase by p. Even in the pres-\nence of leakage currents, our proposed setup can still function\nwith a mechanical efficiency two orders of magnitude higher\nthan that in current-driven resonances. We anticipate that our\nfindings will facilitate the development of ultrafast spintronic\ndevices with extremely low-energy dissipation.\nIf the two FM layers are antiferromagnetically aligned, the\ninstantaneous SOT acting on mtis opposite to that acting on\nmb, whereas the associated SOT field HSOTis of the same\nsign. Consequently, an AC voltage drive can excite the an-\ntiferromagnetic resonance in such a system. Since the anti-\nferromagnetic resonance is composed of two degenerate chi-\nral modes in the absence of magnetic fields, the topological\ncharge pumping, hence the overall electrical response embed-\nded in ˜smn, becomes more complicated than that in the EMR.\nThis will be left for future studies.\nWhile we have demonstrated our findings in an FM-TI-FM\ntrilayer, the essential mechanism is equally applicable to an\ninsulating FM-TI bilayer, where the resonance frequency is\nlower than the EMR studied above.\nWe also mention that our calculations can be generalized to\nantiferromagnet-TI interfaces so long as the interface is un-\ncompensated ( e.g., using the layered antiferromagnet CrI 3).\nHowever, our model is invalid for compensated interfaces\n(e.g., using MnPS 3) where the compensating magnetic order\ncannot open gaps of the Dirac electrons.8\nThe authors acknowledge fruitful discussions with H.\nZhang, Y . Li and E. der Barco. This work is supported by\nthe Air Force Office of Scientific Research under Grant No.\nFA9550-19-1-0307.\nAppendix A: Lagrangian and equation of motion\nWe first verify that Ljgives rise to the correct LLG\nformj. The magnetization vector can be parametrized\nby the polar angle qand azimuthal angle fasmj=\n(sinqjcosfj;sinqjsinfj;cosqj). In spherical coordinates,\ndL\ndq=cosq\u0012\ncosfdL\ndmx+sinfdL\ndmy\u0013\n\u0000sinqdL\ndmz=ˆeq\u0001dL\ndm;\n(A1)\ndL\ndf=sinq\u0012\n\u0000sinfdL\ndmx+cosfdL\ndmy\u0013\n=sinqˆef\u0001dL\ndm;\n(A2)\nwhere the unit vectors ˆ eq= (cosqcosf;cosqsinf;\u0000sinq)\nand ˆef= (\u0000sinf;cosf;0). Consequently,\n0=dLj\ndmj=dLj\ndqjˆeq+1\nsinqjdLj\ndfjˆef\n=\u0012¶Lj\n¶qj\u0000d\ndt¶Lj\n¶˙qj\u0013\nˆeq+\u0012¶Lj\n¶fj\u0000d\ndt¶Lj\n¶˙fj\u0013ˆef\nsinqj:(A3)\nApplying Eq. (A3) to Lj=Nj(\u0000¯hS˙fjcosqj\u0000ej)gives\n\u0000¯hSmj\u0002˙mj\u0000¶ej\n¶mj=0; (A4)\nwhere ˙mj=˙qjˆeq+˙fjsinqjˆef. By multiplying mj\u0002on both\nsides, Eq. (A4) reproduces the LLG equation\n˙mj=g¶ej\n¯hS¶mj\u0002mj: (A5)\nNote that in our convention, gis positive so the magnetic mo-\nment is\u0000gSjwithSj=\u0000¯hSmj(Sis the quantum number\nof total angular momentum). Then, by applying the Euler-\nLagrangian equations to the total Lagrangian L=Le+åiLi\nwith theLipart taken care of by Eq. (A3) and following the\nstandard procedure in Ref. [26, 30], we obtain Eq.(3).\nFor uniform motion of a three-dimensional FM driven by\ntwo-dimensional electrons, a factor Nj\nzappears in the SOT\nterm in Eq. (4) because the transferred spin angular momenta\nis evenly distributed to all magnetic moments. The renor-\nmalized gyromagnetic ratio Eq. (7) originates from the i=j\ncomponent from the second term on the right-hand side of\nEq. (3c).\nAppendix B: Dissipative coupling and displacement current\nWe calculate the Berry curvature in the configurational\nphase space, \nmtmb, with respect to angle ffor the exchange\n0/23/24-1-0.8-0.6-0.4-0.200.20.40.60.8110-6\n/uni03A9mtxmbx,/uni03A9mtymby/uni03D51.63.2\n−1.6−3.20/23/24-1-0.8-0.6-0.4-0.200.20.40.60.8110-6\n0/23/24-1-0.8-0.6-0.4-0.200.20.40.60.8110-6/uni03A9mtxmby/uni03A9mtxmbz/uni03A9mtymbx/uni03A9mtymbz\n0/uni03C0/2/uni03C03/uni03C0/22/uni03C00.0×10−7FIG. B1. Relevant components of \nmtmbas functions of fformtx=\n\u0000mbx=sinqcosf,mty=\u0000mby=sinqsinfandmtz=mbz=cosq.\nParameters: q=6\u000eandNj\nzS=10.\nmode. The results are shown in Fig. B1, where only the six\nrelevant components are shown because of ˙ mz=0. Then the\neffective field stemming from mtcan be directly obtained by\n˙mt\u0001\nmtmb. The effective field from mbcan be obtained in a\nsimilar way. While the Gilbert aranges from 10\u00002to 10\u00004,\nthe dissipative coupling due to Berry curvature Wmtmbis on the\norder of 10\u00007as shown in Fig. B1. Therefore, we can ignore\nthis dissipative coupling term.\nFor precessions in the linear response regime, we require\njm?j=p\n(j˜mxj2+j˜myj2)=2\u00140:1. At the EMR point, we\nobtain from the LLG equations\njm?(wr)j=gHSOT\nxq\n(2+a2)=[2w2r(4a2+a4)]\n\u0019gHSOT\nx\n2awr\u00140:1; (B1)\nwhich imposes an upper limit of the applied electric field\nEx. For wr=2p=50GHz and a=0:01, the maximum value\nofgHSOT\nx is 0:628 GHz. The torkance, i.e., SOT field (in\nfrequency unit) per applied electric field, is evaluated to be\nejWkxmtxj=Nj\nzS¯h\u00191 GHz =(V=mm). Without breaking linear\nresponses, the maximum electric field at gHSOT\nx=0:628GHz\n(jm?j=0:1) is estimated to be Ex=6:28\u0002105V=m=\n0:628 V =mm.\nWhen an AC electric field Eis applied to a uniform dielec-\ntric thin film (thickness dand width w), a displacement current\nis generated, which generates a magnetic field according to\nthe Maxwell equation Ñ\u0002H=¶tD=e0er¶tE, where e0is\nthe vacuum permittivity and er\u00191 is the relative permittivity.\nWhenE(t) =Ex(t)ˆxandw\u001dd(very thin films), we have\nH(t)\u0019\u0000 ˆye0erwEx(t)d=2. Even for d=100nm (thick film)\nand high frequency w=2p=wr=2p=50GHz, the magnitude\nof the stray field is only about jHj\u00191:39\u000210\u00002A=m when\nEx=0:628 V =mm (maximum value allowed, see the preced-9\ning paragraph), which is converted to a negligible magnetic\nflux densityjBj=m0jHj=1:74\u000210\u00008T. As a comparison,\nthe SOT field gHSOT\nxgenerated by the same electric field is\n0:628 GHz, which corresponds to an effective flux density of\n3:569\u000210\u00003T that is 5 orders of magnitude larger than the\nstray field. By contrast, in current-driven resonances [20], the\nstray field is comparable to the SOT field.\nAppendix C: Energy dissipation and mechanical efficiency\nAccording to the LLG equation ˙M=gHeff\u0002M+\na\nMsM\u0002˙MwithM=Msm, the magnetic dissipation power\nper volume is\npm=\u0000dM\ndt\u0001Heff=\u0000a\nMs(M\u0002˙M)\u0001Heff\n=\u0000a\nMs(Heff\u0002M)\u0001˙M=\u0000a\ngMs˙M2: (C1)\nThe time-averaged total magnetic dissipation power is then\nPM=2jpmVmj\u00192aVmj˙M2j=gMs=2aMsVmj˙m2j=gwhere\nthe factor 2 accounts for both the top and bottom FMs.\nSince E(t) =Re[Exeiwt]andI(t) =Re[Ixeiwt+if(w)], the\ninstantaneous electrical power is\nPJ(t) =I(t)E(t)wl=IxExwlcos(wt)cos(wt+f)\n=IxExwl\n2[cos(2wt+f)+cos(f)]: (C2)\nAfter taking the time average PJ=RT\n0PJ(t)dt=Twith T=\n2p=w, the cos (2wt+f)term drops out, so\nPJ=IxExwl\n2cos(f) =1\n2Re[˜Ix˜E\u0003\nx]wl; (C3)\nwhere ˜Ix=Ixeiwtand ˜Ex=Exeiwt. Because ˜Ix=˜sxx˜Exand\n˜sxx=j˜sxxjeif(w), we have\nPJ=E2\nxwl\n2j˜sxxjcos(f) =E2\nxwl\n2Re[˜sxx]: (C4)\nNote that, at resonance ˜sxxis real and ˜sxx=10\u00007A=V accord-\ning to Fig. 4(c). With typical device geometry w=l=10mm\nandEx=6:28\u0002105V=m,PJis estimated to be at the order\nof 10\u00006W which is three order smaller than that of current-\ninduced ST-FMR system, indicating a relatively low heating\neffect.\nNext we relate PMandPJto the Berry curvature \nkm.\nConsider ˜ mt\n?= (˜mt\nx;˜mt\ny)eiwtwhere ˜ mt\nx=˜ck˜HSOT\nxand ˜mt\ny=˜c?˜HSOT\nxwith ˜HSOT\nx=e˜Ex\nNz¯hSWkxmtx, we know\nmt\nx(t) =Re[˜mt\nx] =Re[˜ck˜HSOT\nx] =eWkxmtx\nNz¯hSRe[˜ck˜Ex]\n=eWkxmtx\nNz¯hSj˜ckjExcos(wt+fm\nk); (C5)\nmt\ny(t) =Re[˜mt\ny] =Re[˜c?˜HSOT\nx] =eWkxmtx\n¯hNzSRe[˜c?˜Ex]\n=eWkxmtx\nNz¯hSj˜c?jExcos(wt+fm\n?); (C6)\nwhere fm\nk(?)=fm\nk(?)(w)is the phase of ˜ck(?)(w). Considering\nEqs. (13a) and Eq. (13b), we know the phase difference\nfm\nk\u0000fm\n?=Arg\u0002˜ck=˜c?\u0003\n=Arg[\u0000a+iwr=w]; (C7)\nwhich is about p=2 and almost independent of wunless w=wr\nis comparable to 1 =a. Using Eqs. (C5) and (C6), we know:\nj˙mtj2=˙mt2\nx+˙mt2\ny\n= \neExwWkxmtx\nNz¯hS!2\nå\nm=k;?j˜cmj2sin2(wt+fm\nm);(C8)\nwhich, after averaging over time, gives\nj˙mtj2= \neExwWkxmtx\nNz¯hS!2j˜ckj2+j˜c?j2\n2: (C9)\nThe same expression applies to mbas well. Therefore, the\ntotal magnetic dissipation power is\nPM= \neExwWkxmtx\nNz¯hS!2aMsVm(j˜ckj2+j˜c?j2)\ng; (C10)\nwhich proves Eq. (20) when we substitute Ms=gS¯h=a3\n0and\nVm=a3\n0NxNyNz, and wl=NxNya2\n0.\nIntegrating out kcin Eq. (14) gives\nIx(t) =\u00002e=a2\n0(˙mt\nx\u0001Wmtxkx+˙mt\ny\u0001Wmtykx); (C11)\nwhere the factor 2 ascribes to the constructive relation be-\ntween the top and bottom FM, as ˙ mt=\u0000˙mbandWmtk=\n\u0000Wmbk. Numerically, we find jWkxmtyj\u001cjWkxmtxjfor small\nq(see Fig. 2), so the second term on the right-hand side of\nEq. (C11) can be neglected [as we neglect HSOT\nyin Eq. (12)\nand only focus on dominant HSOT\nx], and then\nIx(t) =\u00002e2Ex(Wkxmtx)2\nNz¯hSa2\n0wj˜ckjsin(wt+fm\nk); (C12)10\nfrom which we can read off the conductivity\n˜sxx=2ie2(Wkxmtx)2\nNz¯hSa2\n0w˜ck: (C13)\nLikewise, by finding Iy(t)from Eq. (C11), we obtain\n˜syx=2ie2(Wkymty)2\nNz¯hSa2\n0w˜c?; (C14)\nwhich proves Eqs. (16) and (17). Inserting Eq. (C13) into\nEq. (C4) gives\nPJ=\u0000e2(Wkxmtx)2wl\nNz¯hSa2\n0wj˜ckjE2\nxsin(fm\nk)\n=\u0000E2\nxwlz\n2W2\nkxmtxwIm(˜ck); (C15)\nwhich proves Eq. (21) as Im (˜ck)<0. Note that we have con-\nsidered an open circuit condition in the ydirection, so PJdoes\nnot have any contribution from Iy.\nWith the expressions for PMandPJ, we obtain the mechan-\nical efficiency as\nh=PM\nPJ=aw(j˜ckj2+j˜c?j2)\njIm(˜ck)j: (C16)\nFrom Eqs. (13a) and (13b), we know\nj˜ckj2=a2w2+w2\nr\n(1+a2)2w4+2(a2\u00001)w2w2r+w4r;(C17)\nj˜c?j2=w2\n(1+a2)2w4+2(a2\u00001)w2w2r+w4r;(C18)\nIm[˜ck] =\u0000aw[(1+a2)w2+w2\nr]\n(1+a2)2w4+2(a2\u00001)w2w2r+w4r:(C19)\ninserting which into Eq. (C16) yields h=1.\nAs a comparison, in current-driven magnetization dynam-\nics,PMandPJare both proportional to the driving current den-\nsityjcwhich can be fixed by external circuits. For example,\nin a YIG/Pt bilayer, the spin Hall effect converts jcinto a pure\nspin current density jsflowing in the perpendicular direction.\nRegardless of the magnetization dynamics, the (fixed) current\ndensity generates an Ohmic dissipation power\nPJ= (jcwdn)l\nswdn=j2\ncVn\ns; (C20)\nwhere dnandsare the thickness and conductivity and Vn=\nwld nis the volume of the Pt layer. The spin current density js\ndelivers spin angular momenta to the YIG by a damping-like\ntorque, which is characterized by the real part of the spin-\nmixing conductance Gr. Similar to what we have done above\nfor the voltage-induced torque, we assume a simple cubic lat-\ntice for the whole system. Let jcbe applied along x, then thespin polarization is along y, and hence the effective SOT field\nis along xagain, which makes it possible to use the same sus-\nceptibility ˜ckand ˜c?derived in Eqs. (13a) and (13b). Using\nthe convention in Ref. [53] to account for spin diffusion and\nspin backflow, Eq. (C1) gives rise to\nPM=aq2\nsj2\ncx2¯hwla3\n0\ne2Sdmw2\u0000\nj˜c?j2+j˜ckj2\u0001\n; (C21)\nwhere qs=js=jcis the spin Hall angle, dmis the thickness of\nYIG, and\nx=lGrtanhdn\n2l\ns+2lGrcothdn\n2l(C22)\nis a renormalization factor due to spin diffusion and spin back-\nflow with lthe spin diffusion length of Pt. The mechanical\nefficiency is then\nh=PM\nPJ=as¯hq2\nsx2\nSe2\u0012a3\n0\ndndm\u0013\nw2\u0000\nj˜c?j2+j˜ckj2\u0001\n:(C23)\nAs a typical estimate, Gr\u00181018m\u00001for YIG/Pt. For dn\u0018l,\nxis on the order of 1%. At the resonance point, regardless of\nwr(whose value is much lower in ferromagnetic resonances),\nw2\u0000\nj˜c?j2+j˜ckj2\u0001\nis roughly 1 =a. Therefore, hturns out to\nbe at most 1%. If a metallic FM is used, the shunting effect\nwill further reduce the mechanical efficiency.\nAppendix D: Effective circuit\nWe seek for an effective circuit to better reflect the electrical\nresponse of the considered FM-TI-FM trilayer. To this end,\nwe notice that the electrical impedance of our system is\nZsys=l\n˜sxxw=Zm\u0014\u0000(1+a2)w2+w2\nr+2iaww r\niww r\u0000aw2\u0015\n;(D1)\nwhere Z\u00001\nm=W2\nkxmtxzw=l. To find hints for the desired effec-\ntive circuit, we further decompose Zsysas\nZsys=Zm(aw\u0000iwr)2+w2\nw(aw\u0000iwr)\n=Zm\u0010\na+wr\niw\u0011\n+Zm\u0000\na+wr\niw\u0001; (D2)\nwhere the first term can be represented by a resistor connect-\ning in series with a capacitor and the second term can be rep-\nresented by a resistor connecting in parallel with an induc-\ntor. 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Comparatively, h=1 is\nindependent of frequency (if leakage currents are ignored) in\nour system.\n[51] Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Spin pumping\nand magnetization dynamics in metallic multilayers, Phys. Rev.\nB66, 224403 (2002).\n[52] H. Jiao and G. E. W. Bauer, Spin Backflow and AC V oltage\nGeneration by Spin Pumping and the Inverse Spin Hall Effect,\nPhys. Rev. Lett. 110, 217602 (2013).\n[53] R. Cheng, J.-G. Zhu, and D. Xiao, Dynamic Feedback in\nFerromagnet–Spin Hall Metal Heterostructures, Phys. Rev.\nLett. 117, 097202 (2016)."    },    {        "title": "1407.1174v1.Temperature_Dependent_Ferromagnetic_Resonance_via_the_Landau_Lifshitz_Bloch_Equation__Application_to_FePt.pdf",        "content": "Temperature Dependent Ferromagnetic Resonance via the Landau-Lifshitz-Bloch\nEquation: Application to FePt\nT. A. Ostler and M. O. A. Ellis\n1Department of Physics, University of York, York YO105DD, U. K.\nD. Hinzke and U. Nowak\n2Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany\nUsing the Landau-Lifshitz-Bloch (LLB) equation for ferromagnetic materials, we derive analytic\nexpressions for temperature dependent absorption spectra as probed by ferromagnetic resonance\n(FMR). By analysing the resulting expressions, we can predict the variation of the resonance fre-\nquency and damping with temperature and coupling to the thermal bath. We base our calculations\non the technologically relevant L10FePt, parameterised from atomistic spin dynamics simulations,\nwith the Hamiltonian mapped from ab-initio parameters. By constructing a multi-macrospin model\nbased on the LLB equation and exploiting GPU acceleration we extend the study to investigate the\neffects on the damping and resonance frequency in \u0016m sized structures.\nI. INTRODUCTION\nThe magnetic properties of ferromagnetic structures such\nas thin films, nanowires and nanoparticles have been\nstudied extensively both experimentally1,2and theoret-\nically3,4. The interest in these particles is driven by\nfundamental features on the one hand and technological\nperspectives on the other5–7. Ferromagnetic resonance\n(FMR),whichhasbeenappliedwithgreatsuccesstothin\nferromagnetic films in the past8, can be used to measure\nimportant material properties, such as the damping, gy-\nromagnetic ratio and anisotropy constant. The tempera-\nture dependence of these properties for large or complex\nstructures are often difficult to predict using analytical\ntreatments, especially when temperature effects are in-\ncluded4,9. As well as being difficult to calculate analyt-\nically, temperature dependent calculations of (for exam-\nple) FMR can be slow to converge. The convergence can\nbecome particularly troublesome if thermal fluctuations\nare accounted for. A specific motivation for this work is\nthe interest in L10FePt materials, which is a promising\ncandidate for ultrahigh density magnetic recording10,11.\nThe ability to tune magnetic properties, such as the\ndamping is important for example, in devices based on\nspin transfer torque where a low damping of a free layer\nis essential for reducing the power consumption and can\naffect the signal to noise ratio12. In some cases such as in\nGMR read sensors, high damping is preferred to improve\nthermal stability13.\nFor technologies based on Heat Assisted Magnetic\nRecording (HAMR), understanding temperature effects\nand fluctuations in strongly anisotropic materials, will\nbe crucially important. In this article we present ana-\nlytical and numerical calculations of the material prop-\nerties of strongly anisotropic materials at elevated tem-\nperatures. We do so by utilizing the formalism of the\nLandau-Lifshitz-Bloch (LLB) equation of motion for fer-\nromagnetic particles, which has an intrinsic temperature\ndependence via various input functions. In the first partof the paper we present the derivation of the temperature\ndependent analytic expression for the power absorbed\nby the particle. This analytic expression allows us to\nlook at the effect of temperature on FMR curves for sin-\ngle domain particles. The temperature dependent input\nfunctions that enter into the LLB formalism have been\nparameterised from atomistic spin dynamics with the ex-\nchange parameters calculated directly from ab-initio cal-\nculations14. We have tested the expressions with a sin-\ngle spin and multispin (with exchange) LLB numerical\nmodel, byshowinganumberofresonancecurvesatdiffer-\nent temperatures against the derived expressions (with-\nout demagnetizing fields). The analytic expressions for\nthe damping and resonance frequency show the overall\ntrend of the temperature dependent behaviour.\nIn the second half of the paper we extend the scope of\nour analysis using a multi-macrospin model based on the\nLLB formalism with large number of exchange coupled\nmacrospins. We present numerical calculations of FMR\nin2Dand3Dstructureswiththeinclusionofdemagnetiz-\ning effects and (stochastic) thermal fluctuations. Specif-\nically, we have investigated the effects of the anisotropy\nconstant and film thickness and anisotropy on the mea-\nsured damping in out-of-plane films at high tempera-\ntures. Our findings show that, depending on thickness or\nanisotropy, there is a competition between the demagne-\ntizing and anisotropy energy that can modify the damp-\ning significantly. We have implemented this large scale\nmodel on the CUDA GPU platform so that even with the\ninclusion of the stochastic thermal terms, it is possible to\nobtain good averaging of the FMR power spectra.\nThere are limited experimental ferromagnetic studies of\nchemically ordered FePt due to it’s large magnetocrys-\ntalline anisotropy15. However, it is possible to perform\nso-called optical FMR with the use of laser pulses16. In a\ntheoretical work by Butera17the resonance spectra were\ncalculated using a computational model for disordered\nnanoparticles of FePt. This study showed that the mea-\nsured damping depended strongly on the amount of dis-arXiv:1407.1174v1  [cond-mat.mtrl-sci]  4 Jul 20142\norder. To our knowledge there are no systematic studies\non the temperature dependence of the properties such as\ndamping due to the limited fields in typical FMR setups,\nour results provide insight into this complex issue.\nII. LANDAU LIFSHITZ BLOCH EQUATION\nThe LLB equation for magnetic macrospins describes the\ntime evolution of an ensemble of atomic spins and al-\nlows for relaxation of the magnitude of the magnetiza-\ntion. The equation was originally derived by Garanin18\nwithin a Mean-Field approximation from the classical\nFokker-Planck equation for atomistic spins interacting\nwith a heat bath. The resulting LLB equation has been\nshown to be able to describe linear domain walls, a do-\nmain wall type with non-constant magnetisation length.\nThese results are consistent with measurements of the\ndomain wall mobility in YIG crystals close to the Curie\npoint (Tc)19and with atomistic simulations20. Further-\nmore, the predictions for the longitudinal and transverse\nrelaxation times have been successfully compared with\natomistic simulations21. Consequently, we use this equa-\ntion in the following for the thermodynamic simulations\nof macro-spins. The use of the LLB formalism has the\nadvantage over traditional micromagnetics that it auto-\nmatically allows for changes in the modulus of the mag-\nnetisation. In theory it is indeed possible to calculate\ntemperaturedependentFMRusingtheatomisticspindy-\nnamics (ASD) model however, such an approach would\nbe extremely computationally expensive. This compu-\ntational expense in the ASD model arises because, for\nFMR calculations, large system sizes are required to re-\nduce the effects of thermal noise. Whilst large systems\nare possible to calculate, the FMR calculations also re-\nquire averaging over many cycles of the driving field, up\nto 100’s of nanoseconds. These two restrictions combined\nmeans that this method is not suitable, even with GPU\nacceleration or a (for example MPI) distributed memory\nsolution22.\nA further challenge for accurate calculation of magnetic\nproperties is the accounting of the long-ranged exchange\nin materials such as FePt. Through proper parameter-\nisation of the LLB equation14one can account for such\nlong ranged interactions in the so-called multiscale ap-\nproach14. Via this multiscale approach we can then\nbridge the gap between electronic structure calculations\nto large scale (of the order of micrometres) calculations\nof material properties. With this in mind the LLB model\nis then ideally placed to describe temperature dependent\nferromagnetic resonance.\nThe LLB equation, without the stochastic term, can be\nwritten in the form:\n_ m=\u0000\r[m\u0002He\u000b] +\r\u000bjj\nm2(m\u0001He\u000b)m\n\u0000\r\u000b?\nm2[m\u0002[m\u0002He\u000b]]: (1)Besides the usual precession and relaxation terms, the\nLLB equation contains another term which controls lon-\ngitudinal relaxation (second term in equation 1). Hence\nmis a spin polarisation which is not assumed to be of\nconstant length and even its equilibrium value, me(T), is\ntemperature dependent. The value of mis equal to the\nratio of the magnetization of the macrospin normalised\nby the magnetization at saturation ( M=Ms).\u000bkand\u000b?\nare dimensionless longitudinal and transverse damping\nparameters (defined below) and \ris the gyromagnetic\nratio taken to be the free electron value. The transverse\ndamping parameter in this equation is related to what is\nusually measured in experiments (the Gilbert damping,\n\u000bg) by the expression:\n\u000bg=\u000b?\nm(2)\nThe LLB equation is valid for finite temperatures and\neven above Tc, though the damping parameters and ef-\nfective fields are different in the two regions. Throughout\nthis paper, we are only interested in the case T\u0014Tcwith\nthe damping parameters \u000bk=2\u0015T\n3Tcand\u000b?=\u0015\u0010\n1\u0000T\n3Tc\u0011\n.\nThe single particle free energy (without demagnetizing\nfields) is given by:\nf=\u0000BM s0mz+Ms0\n2~\u001f?(m2\nx+m2\ny)\n+Ms0\n8~\u001fkm2e(m2\u0000m2\ne)2; (3)\nand the effective fields, He\u000b=\u00001\nM0s\u000ef\n\u000emgiven by18:\nHe\u000b=B+HA+1\n2~\u001fk\u0012\n1\u0000m2\nm2e\u0013\nm;(4)\nwhere Brepresents an external magnetic field and HA=\n\u0000(mxex+myey)=~\u001f?an anisotropy field. Here, the sus-\nceptibilities ~\u001flare defined by ~\u001fl=@ml=@Hl, whereHl\nis thel=k;?. In these equations, \u0015is a microscopic\nparameter which characterizes the coupling of the indi-\nvidual, atomistic spins with the heat bath. For the pur-\npose of testing the model we use a thermal bath coupling\nconstant of \u0015=0.05, consistent with Ref. 15. There are\ndiffering values of the damping constant in the literature,\nfor example, for granular FePt Becker et almeasured a\ndamping constant of 0.1 using an optical FMR technique,\nwhereas Alvarez et alfound a value of 0.055 using stan-\ndard FMR in a broad frequency range15. It should be\npointedoutherethatwhilst \u0015isacouplingtothethermal\nbath equivalent to that used in atomistic spin dynamics\nand is assumed to be temperature independent.\nAt this point we should take some time to define the\ndifferent constants related to the damping and their dif-\nferences. The parameters, \u0015,\u000b?,\u000bkand\u000bgcorrespond\ntothethermalbathcoupling, thetemperaturedependent\ntransverse and longitudinal damping parameters and the3\ndamping parameter that one would measure experimen-\ntally, respectively. The thermal bath coupling is temper-\nature independent and is a phenomenological parameter\nthat is the same as that used in atomistic spin dynam-\nics. The transverse and longitudinal damping parame-\nters that enter into the LLB equation define the relax-\nation rates of the transverse and longitudinal magnetiza-\ntion components. Finally, the parameters, \u000bg, is equal\nto the perpendicular damping ( \u000b?) that enters into the\nequation of motion, divided by the magnetization and is\nwhat one would find in an FMR measurement from the\nlinewidth.\nFor the application of this equation one has to know a-\npriorithespontaneousequilibriummagnetisation me(T),\nthe perpendicular ( ~\u001f?(T)) and parallel ( ~\u001fk(T)) suscep-\ntibilities. In this work these are calculated separately\nfromaLangevindynamicssimulationofanatomisticspin\nmodel, however it is possible to calculate these proper-\nties from mean field calculations23. We use a model for\nFePt which was introduced earlier and which is mean-\nwhile well-established in the literature24–27. Since this\nmodel was derived from first principles, a direct link is\nmade from spin dependent density functional theory cal-\nculations, viaaspinmodel, toourmacrospinsimulations.\nThe calculation of these parameters is discussed in more\ndetail in Ref.14.\nIII. ANALYTIC SOLUTION FOR THE FMR\nABSORBED POWER SPECTRUM P( !)\nThe focus of this section is on the derivation of an an-\nalytical solution for the power spectrum P(!)using the\nLLB equation for a single macro spin. The power, P(!),\nabsorbed in an FMR experiment is given by4:\nP(!) =hM\u0001@B\n@ti=\u0000!\n2\u0019Z2\u0019=!\n0MSVmx_Bxdt;(5)\nwhereVis the volume of the macrospin and !is the\nfrequency of the driving field. The right hand side of\nequation 5 assumes that the time-varying field is applied\nin thexdirection with the static applied field in z. The\ntime-dependence of the x-component of the magnetiza-\ntion can be derived from Eq. 1. Using the assumptions\nthatm2is constant and mxas well as myare small,\nleading leads to the approximation mz\u0019m. Under this\nassumption Eq. 1 can be written in linearised form. To-\ngether with the linearised form of the effective field the\nsolution of the resonance frequency, !0, and transverse\nrelaxation time, \u001c, can be obtained (for full details see\nappendix A):\n!0(T) =\r\u0010\nBz+m(T)\n~\u001f?(T)\u0011\n; (6)\u001c(T) =m(T)\n\u0015\u0000\n(1\u0000T\n3Tc)!0(T)\u00002\n3\rT\nTcHe\u000b(T)z\u0001:(7)\nHerem=me+ ~\u001fkBzis an approximation written to\nfirst order of the susceptibility for the purposes of the\nanalytic calculation. In the zero temperature case under\nthe conditions that \u000b=\u000b?,\u000bk= 0andm=me= 1,\n!0and\u001care the same as for the Landau Lifshitz Gilbert\n(LLG) equation, !0=\r(Bz+1\n~\u001f?)and\u001c=1\n\u0015!0.\nThe analysis of equations 6 and 7 show that there is little\nvariation of the measured damping, \u000bg, with the applied\nfield as one would expect28. Also, at low temperature,\nas expected, the measured damping equal to the input\ncoupling to the thermal bath, \u0015. The temperature de-\npendence of \u000bgshows that (for a chosen value of \u0015) there\nis an increase with temperature, diverging at the Curie\npoint. In a recent article28the measured damping as a\nfunction of applied field (up to 7 T) was shown to be\nalmost independent of temperature. In the same study,\nthe damping was measured at two temperatures; 170 K\nand 290 K. Between these two temperatures the damping\nwas shown to be around 0.1 with a slight increase as one\nwould expect. Figure 1 shows the analysis of equation 6\nand 7 for the physical input parameters for FePt. In the\nfigure the measured damping is calculated as \u000bg= 1=!0\u001c\nand is shown to increase with temperature, diverging at\nthe Curie point. The contours show lines of constant\ndamping explicitly. This demonstrates that if one as-\nsumes no temperature dependence of the thermal bath\ncoupling,\u0015, the measured damping will not be constant.\n0.50.40.30.2αg=0.1\nαg0.750.500.250.00λ0.200.150.100.050.00T [K]6005004003002001000\u0000T[K]\n↵g\nFigure 1: (Color online) Analytically derived Gilbert damp-\ning as a function of temperature and the intrinsic coupling to\nthe thermal bath, \u0015, valid for a single macrospin without de-\nmagnetizing effects. For each value of \u0015the damping is shown\nto increase with temperature consistent with other works21,28.\nThe lines are contours of constant measured damping.\nThe solution of the resulting inhomogeneous differential\nequation A4, combined with equation 5 leads us to the\nanalytic solution for the power absorbed during ferro-\nmagnetic resonance as a function of the frequency of the4\ndriving field:\nP(!;T) =\u0016s!2\n4\r\u000b?B2\n0\n1\n\u001c2+ (!\u0000!0)2; (8)\nwhere the temperature dependence comes from !0and\u001c\n(see equations 6 and 7) and B0is the amplitude of the\ndriving field. In the zero temperature case this solution\nreduces to that from the LLG equation.\nThe analytic solution given by equation 8 can be com-\npared to the numerical results, by integration of the LLB\nequation and using equation 5. By applying an alternat-\ning driving field in the xdirection and averaging equa-\ntion 5 until convergence we can compare the results of a\nsingle spin to the analytic expression. For FePt there is\na strong uniaxial exchange anisotropy, therefore in the\nabsence of any static applied field we still see a very\nstrong FMR line for single domain particles. Throughout\nthe calculations we use a driving field amplitude ( B0) of\n0.005T and a static applied field ( Bz) of 1 T. We inte-\ngratetheLLBequationusingtheHeunnumericalscheme\nwith a 5 fs timestep.\n650K500K300K100K0K\nω/γ[T]P/MsV [T2s]\n25201510501.2e-048.0e-054.0e-050.0e+00650K500K300K100K0K\nω/γ[T]P/MsV [T2s]\n25201510501.2e-048.0e-054.0e-050.0e+00\nFigure 2: (Color online) Power spectrum vs frequency in a\n1 T applied field. The data points are from LLB simulations\nfor a single macrospin and the solid lines are given by Eq. 8.\nFigure 2 shows the calculated absorbed power as a func-\ntion of frequency for a range of temperatures using the\nsinglespinLLBmodel. AswecanseefromFigure2there\nis a large decrease in the resonance frequency, given by\nequation 6, which we would expect to occur because of\nthe decrease in the anisotropy field. The analytic solu-\ntion agrees perfectly with the numeric model, except as\nwe approach the Curie temperature. This is because in\nthe analytical treatment we approximate the magnetisa-\ntion in a field, Bz, to depend on the parallel susceptibil-\nity (m=me+ ~\u001fkBz) which diverges as we approach the\nCurie temperature. This point has been discussed in ap-\npendix A and is an error in the analytic treatment only,not in the form of the LLB equation.\nP(!) =P0!2\n(!~\u000bg)2+ (!\u0000~!0)2(9)\nwhere ~\u000bg,P0and~!0are free fitting parameters and we\nuse the tilde to distinguish the resonance frequency and\ndamping from the analytically derived values. This use\nof this fitting procedure is allows us to compare with\nexperimental observations as this would be the kind of\nanalysis required to extract the damping parameter ( \u000bg).\nIV. MULTI-MACROSPIN NUMERICAL\nRESULTS\nIn the following section we introduce the stochastic LLB\nequation that introduces thermal fluctuations. As well as\nthe normal terms in the LLB described by equation 4, we\nalso include exchange coupling between the macrospins\nand the magnetostatic fields. The LLB equation with\nstochasticthermaltermsincludediswrittenforeachspin,\ni, in the form:\n_ mi=\u0000\r[mi\u0002Hi\ne\u000b] +\u0010i;k\n\u0000\r\u000b?\nm2\ni\u0002\nmi\u0002\u0002\nmi\u0002(Hi\ne\u000b+\u0010i;?)\u0003\u0003\n+\r\u000bjj\nm2\ni\u0000\nmi\u0001Hi\ne\u000b\u0001\nmi: (10)\nThe stochastic fields, \u0010i;?and \u0010i;khave zero mean and\nthe variance29:\nh\u0010\u0011\ni;?(0)\u0010\u0012\nj;?(t)i=2j\rjkBT(\u000b?\u0000\u000bk)\nMsV\u000b2\n?\u000eij\u000e\u0011\u0012\u000e(t)\nh\u0010\u0011\ni;k(0)\u0010\u0012\nj;k(t)i=2j\rjkBT\u000bk\nMsV\u000eij\u000e\u0011\u0012\u000e(t) (11)\nwherekis the additive noise, \u0011and\u0012represents the\nCartesian components. As well as the stochastic field,\nthe exchange is also included in the form:\nHi\nex=A(T)\nm2e2\nM0s\u00012X\nj2neigh (i)(mj\u0000mi)(12)\nwhereA(T)is the exchange stiffness, \u0001is the cell length\nandM0\nsis the saturation magnetization.\nIt should be pointed out that the inclusion of the stochas-\ntic term into the LLB equation leads to a slightly re-\nducedTCas compared to the LLB without the stochastic\nterm29.\nFigure 3 shows the power spectrum as a function of fre-\nquency for multi-macrospin calculations (coupled by ex-\nchange) for a system size of (100 nm)3with a unit cell\ndiscretizationof(6.25nm)3, thoughwehavecheckedunit5\n650K600K500K400K300K200K100K50K\nω/γ[T]P/MsV [T2s]\n201510501.4e-041.2e-041.0e-048.0e-056.0e-054.0e-052.0e-050.0e+00650K600K500K400K300K200K100K50K\nω/γ[T]P/MsV [T2s]\n201510501.4e-041.2e-041.0e-048.0e-056.0e-054.0e-052.0e-050.0e+00\nFigure 3: (Color online) Power spectrum vs frequency in a 1T\napplied field. The data points are from LLB simulations for\nmany exchange coupled macrospins including the stochastic\nfields and exchange and the solid lines are given by Eq. 8 (no\nmagnetostatic fields are included here).\ncell sizes down to (3.125 nm)3, i.e. below the typical do-\nmainwallsizeof4-6nm. Thesolidlinesaretheanalytical\nsolution, equation 8.\nAs discussed in the introduction, we have also introduced\ninto our model demagnetizing effects to extend the an-\nalytic study to more realistic materials. We have taken\nthe approach of that of Lopez-Diaz et al.used in the\nGPMagnet software30. In this approach, we write the\nmagnetostatic field in a (cubic) cell, i(Hd;i), as:\nHd;i=\u0000MsX\njN(ri\u0000rj)\u0001mj (13)\nwhere Nis the 3\u00023 symmetric demagnetizing tensor.\nThe sum runs over all cells at positions ri;j. The de-\nmagnetizing tensor is given by:\nN(ri\u0000rj) =1\n4\u0019I\nSiI\nSjdSi\u0001dS0\nj\njr\u0000r0j;(14)\nSi(Sj) are the surface of cell i(j) respectively, randr0\nare the points on the surface iandj. This sum requires\na summation from all cells and requires integration over\neach of the surfaces iandj, making it extremely compu-\ntationally expensive. If one were to perform the integra-\ntion 14 numerically for each surface of each cell the cal-\nculation is extremely time-consuming and converges very\nslowly with the number of mesh points on each surface.\nTo that end we have employed the method of Newell31,\nwhereby the surface integration of the cubes is calculated\nanalytically as in the OOMMF code32. Some further de-\ntails can be found in appendix B.V. FERROMAGNETIC RESONANCE IN THIN\nFILMS OF FEPT\nIn this section we present calculations of thin films of\nFePt. We begin by looking at the effect of temperature\non the damping of 2 nm thin films using the stochas-\ntic form of the LLB equation with demagnetizing fields\n(equation 13). We compare this to the results for the\nsingle spin analytic results. The thin films show a large\nincrease in the predicted damping over the analytic re-\nsultsduetotheinclusionofthedemagnetizingtermaswe\napproach the Curie temperature. The thickness depen-\ndence of the films is also calculated using the multi-spin\nmodel, showing that at low temperatures there is little\nvariation in damping with film thickness though, at tem-\nperatures approaching the Curie point there is a large\nreduction with increasing thickness.\nαFilm\nαBulkT [K]60030002.001.501.00T=600KT=450KT=300KT=150K\nω/γ[T]P/MsV [T2s]\n3025201510501.2e-048.0e-054.0e-050.0e+00αFilm\nαBulkT [K]60030002.001.501.00T=600KT=450KT=300KT=150K\nω/γ[T]P/MsV [T2s]\n3025201510501.2e-048.0e-054.0e-050.0e+00!/\u0000[T]P/MsV[ T2s]˜↵Film↵g\nFigure 4: (Color online) Ferromagnetic resonance curves in\nthin films of FePt for a range of temperatures below the Curie\ntemperature. Thepointsherearesimulateddataandthelines\nare the fits to equation 9. The inset shows the ratio of the\ndamping as measured in our 2D film to the damping calcu-\nlated analytically for a single macrospin. For low tempera-\ntures the two are equivalent, however, at higher temperatures\nthere is an enhanced damping in the thin films due to the\neffect of the magnetostatic field.\nBy systematically varying the anisotropy we have shown\nthat the this increase in damping occurs when the de-\nmagnetizing field dominates over the anisotropy term.\nFinally, this modification in the damping is shown to af-\nfect the switching times as we transition from one regime\nto another.\nThexandydimensions of the thin films in this sec-\ntion are 0.4 \u0016m\u00020.4\u0016m. Thez-dimension is initially one\ncell (2 nm) thick, i.e. a 2D film. Our cell discretiza-\ntion is 2 nm\u00022 nm\u00022 nm, below the domain wall width.\nWe apply the fields in the same orientation as discussed\nabove. The resonance curves are shown on figure 4 for a\nrange of temperatures for the 2D (2 nm thick) film. From6\neach FMR curve we have used a fitting procedure, as in\nfigure 2, to calculate the damping in the 2D structures\n(solid lines). The inset of figure 4 shows then the ratio of\nthe damping that we calculate for the 2D structures to\nthe analytically derived damping for single domain par-\nticles in section III. In the low temperature limit this\nratio is consistent with the analytic solution for a single\nmacrospin (i.e. it is 1), for high temperature however the\ndamping is increased as the demagnetizing effects start\nto dominate over the anisotropy.\n300 K500 K600 K\nThickness [nm]αg\n181410620.130.110.090.070.051814106214.012.010.08.06.04.0Thickness[nm]˜!0/\u0000˜↵gThickness[nm]\nFigure 5: (Color online) Damping as a function of film thick-\nness for a range of temperature. In the low temperature\nregime there is a slight increasing damping as a function of\nthickness. As the Curie temperature is approached there is a\nlarge decrease in the damping with film thickness. The inset\nshows the variation of the resonance frequency with thick-\nness. The resonance frequency shows an overall increase over\nall temperatures due to the decrease in the effective magne-\ntostatic field.\nNext, we consider the effects of film thickness on the\ndampingandresonancefrequency. Weincreasethethick-\nness of the film from 2 nm to 20 nm (1 cell to 10 cells)\nand calculate the ferromagnetic resonance curve for each\nthickness(amaximumof400,000cellsforaround100ns).\nThe resulting FMR curves were again analysed to extract\nthe damping and resonance frequencies.\nFigure 5 shows the variation of the damping and res-\nonance frequency as a function of the thickness of the\nthin film. The largest variation in the damping is shown\nclose to the Curie temperature (blue square, dotted line).\nForT=500 K, there is a small increase in the damping\nwith film thickness when going from 2 nm to 4 nm. Af-\nter 4 nm the curve shows little variation, consistent with\nthe T=300 K (red circles, dot-dash line) line. The vari-\nation in the damping, with film thickness, close to the\nCurie point will have a large effect on the magnetiza-\ntion dynamics in heat assisted magnetic recording (for\nwhich FePt is a promising candidate), that operates at\nelevated temperatures to allow for the reduction in the\nanisotropy at write head field of around 1-2 T. This re-duction in damping for thick layers of FePt would lead to\nlonger switching times (as we show below), limiting the\nwrite times.\nIn Ref. 33, Liu et al.showed that the damping in a mag-\nnetic tunnel junction consisting of a FeCoB free layer in-\ncreased with decreasing thickness. The mechanism was\nsaid to be caused by spin pumping and nonlocal back-\nground effects. Our results, whilst are not calculated for\nFeCoB, show that it is not required to invoke a mecha-\nnism via spin pumping but can arise due to an interplay\nbetween the anisotropy and the demagnetizing fields.\nAs well as looking at the effect of the film thickness on\nthe damping parameter we have also performed a sys-\ntematic variation of the anisotropy constant. In FePt,\nthe anisotropy can be modified, for example, by inducing\nlattice distortion or chemical disorder34. For the 2 nm\nthick films we have calculated the FMR spectra at three\ndifferent temperatures (300 K, 400 K and 500 K) for a\nrange of anisotropy values below the bulk value (vertical\ndashed grey line in fig. 5). From these calculations we\nhave measured the effective damping parameters using\nthe method described above. The overall trend shows a\ndecrease in the measured damping, the result of which is\nshown in figure 6.\nT=500KT=400KT=300K\nAnisotropy Energy [J/m3]˜αg\n8e+06 6e+06 4e+06 2e+06 0e+000.075 0.070 0.065 0.060 0.055 0.050\nFigure 6: (Color online) Dependence on the damping in FePt\nfor a range of anisotropy constants for three values of temper-\nature (300 K blue circle points, 400 K green triangle points\nand 500 K red square points). In the lower anisotropy range\nthe damping increases, consistent the results of figure 5. The\nlines are fits to exponential decays to give a guide to the eye.\nThe overall trend in figure 6 shows a decrease in the\ndamping when the anisotropy becomes dominant over\nthe demagnetizing field, consistent with the results of\nfigure 5.\nFigure 7 shows the calculated switching times for four\ntemperatures (610 K, 620 K, 630 K and 640 K) as a\nfunction of the thickness of the film. To calculate the\nswitching times we equilibrated the system at the tem-\nperature shown in the figure, we then applied a field with7\na step function to 2 T to reverse the magnetization in the\nzdirection. The switching times were then averaged over\n25 runs per point. The errors in the switching times are\nquite small so 25 runs seems to be a sufficient number to\ntake a good average.\nT=640 KT=630 KT=620 KT=610 K\nThickness [nm]Switching Time [ps]2018161412108642140120100806040200Switching Times [ps]Thickness[nm]Thickness[nm]\nFigure 7: (Color online) Switching times for thin films of FePt\nof differing thicknesses for a range of temperatures. Consis-\ntent with the result of figure 5 the reduction in the damp-\ning with increasing film thickness leads to an increase in the\nswitching time. A Heaviside step function of 2 T was applied\nto the field to reverse the magnetization after equilibration\nand the runs were averaged over 25 realizations of the ran-\ndom number seed. With the inclusion of the stochastic term\nthere is a reduced TCso the T=640 K line is already above\nthe transition temperature.\nThe thickness dependence of the switching times shown\nin figure 7 are consistent with the calculations of the\ndamping presented in figure 5. As the thickness is in-\ncreased there is an observed decrease in the damping\nwhich leads to the reduced switching times seen in fig-\nure 7. It should be pointed out that the field that we\napply is not sufficient to switch the magnetization below\naround 610 K, consistent with Ref. 35. The large reduc-\ntion in the switching time seen for the T=640 K case is\ndue to the fact that with the inclusion of the stochastic\nterm there is a slight reduction in the Curie temperature\nas shown in Ref. 29.\nVI. CONCLUSION\nWe have derived, using the LLB formalism an analytic\nsolutiontothepowerfrequencyspectrumfornanometre-\nsized, single domain ferromagnets during ferromagnetic\nresonance. Using the technologically relevant FePt, this\nanalytic solution agrees well with numerical simulations\nof both single spin and exchange-coupled multi-spin cal-\nculations including the stochastic thermal term.Analysis of the resulting FMR expressions for a single\nmacrospin show that the analytically derived damping is\nconsistent with those of extended thin films up to quite\nhigh temperatures. At temperatures close to TC, the\nanisotropy decreases quickly and the demagnetizing ef-\nfects play a more dominant role for the tin films. This\nmeans that our analytic expressions for thin films of\nmagnetically soft materials would not hold, however, the\nanalysisisstillvalidforsinglemacrospins(orsmallstruc-\ntures) of soft materials.\nWe have extended the calculations to include the thermal\nstochastic term and demagnetizing effects to explore the\neffect this plays on the temperature dependent ferromag-\nnetic resonance curves. By calculating FMR spectra as a\nfunction of film thickness, we have shown that there is an\nincreased damping for thinner films due to the interplay\nbetween the demagnetizing fields and the anisotropy. For\nthe thinner films there is more of a tendency for the films\nto want to lie in-plane due to the demagnetizing field.\nFor highly anisotropic materials (shown here for FePt)\nthis effect is more dominant at elevated temperatures.\nWe have verified that this increase in damping can be\nexplained by change in the dominance of the demagne-\ntizing energy by varying the anisotropy constant for the\nthin films. As the anisotropy constant is decreased the\ndamping increases, consistent with the results of varying\nthe film thickness.\nFinally, we have shown that this reduction in the damp-\ning has an effect on the switching times. This conclusion\ncouldhaveimportantconsequencesforheatassistedmag-\nnetic recording, which operates at elevated temperatures,\nand require sufficiently thick grains to have sufficient ma-\nterial for good read back of the magnetic signal.\nAcknowledgements\nThis work was supported by the European Commission\nunder contract No. 281043, FemtoSpin . The financial\nsupportoftheAdvancedStorageTechnologyConsortium\nis gratefully acknowledged.\nAppendix A: Details of Analytic Derivation of P( !)\nThis section gives some more detail regarding the deriva-\ntion of the key equations discussed in section III. The\nlinearised equations of motion for the LLB equation 1\nare written:8\n_mx\u0019 \u0000\r(myHz\ne\u000b\u0000mHy\ne\u000b)\n+\r(\u000bjj\u0000\u000b?)\nm(mxHz\ne\u000b)\n+\r\u000b?\nm(mHx\ne\u000b)\n_my\u0019 \u0000\r(mHx\ne\u000b\u0000mxHz\ne\u000b)\n+\r(\u000bjj\u0000\u000b?)\nm(myHz\ne\u000b)\n+\r\u000b?\nm(mHy\ne\u000b)\n_mz= 0; (A1)\nwith the linearised effective fields then written as:\nHx;y\ne\u000b=Bx;y\u0000mx;y\n~\u001f?+1\n2~\u001fk\u0012\n1\u0000m2\nm2e\u0013\nmx;y\nHz\ne\u000b=Bz+1\n2~\u001fk\u0012\n1\u0000m2\nm2e\u0013\nm: (A2)\nIn equilibrium the z-component of the effective field\nvanishes, Hz\ne\u000b= 0. Using the linearised form of m,\nm=me(1 + \u0001m)as well asm2=m2\ne(1 + 2\u0001m), we\narrive at an expression for the z-component of the ap-\nplied magnetic field\nBz\u0000me\u0001m+me\u0001m2\n~\u001fk= 0:\nUsingthelinearisedformofthisequation, Bz\u0000me\u0001m\n~\u001fk= 0\naswellastheapproximation \u0001m= (m\u0000me)=mewehave\nan approximation for mduring FMR that is both field\nand temperature dependent:\nm(T;Bz) = ~\u001fk(T)Bz+me(T): (A3)\nAs is discussed in the main text the approximation A3\nleads to errors in the analytic treatment if the resonance\ncurve is calculated in an applied field. This is due to\nthe fact that the susceptibility diverges as we approach\nthe Curie temperature. This does not occur in the nu-\nmerical simulations and is only a problem in the analytic\ncalculations due to the above approximation A3.\nIn order to calculate the resonance frequency ( !0) as well\nas the transverse relaxation time ( \u001c) for the power spec-\ntrumP(!)one has to solve the linearised LLB equation\n(see Eq. A1). Using the notation ~m=mx+ imyand\n~He\u000b=Hx\ne\u000b+ iHy\ne\u000bleads to the differential equation,\n_~m\n\r= ~m\u0012\ni +\u000bjj\u0000\u000b?\nm\u0013\nHz\ne\u000b+m\u0010\u000b?\nm\u0000i\u0011\n~He\u000b:\nAs can be easily seen from Eq. A2, ~He\u000bis also ~m-\ndependent. Writing the effective field as, ~He\u000b=~B+A~m,withA=\u00001\n~\u001f?+1\n2~\u001fk(1\u0000m2\nm2e)and ~B=Bx+iBywe arrive\nat an inhomogeneous differential equation:\n_~m\n\r= ~m\u0012\n(i +\u000bjj\u0000\u000b?\nm)Hz\ne\u000b\u0013\n+ ~m\u0010\nm(\u000b?\nm\u0000i)A\u0011\n+m(\u000b?\nm\u0000i)~B: (A4)\nIn the first step, we solve the homogeneous part of the\ndifferential equation A4, using the Ansatz ~mhom(t) =\nexp (!t)whose solution leads to the expressions for !0\nand\u001c:\n!0=\r\u0010\nBz+m\n~\u001f?\u0011\n(A5)\n\u001c=m\n\u0015\u0000\n(1\u0000T\n3Tc)!0\u00002\n3\rT\nTcHz\ne\u000b\u0001:(A6)\nIn the next step, we solve the inhomogeneous differen-\ntial equation A4 under the assumption that the applied\nmagnetic field Bhas the form B= (B0exp(i!t);0;Bz),\nwhereB0\u001cBz. These lead to the following simplifica-\ntion of the right hand side of Eq. A4,\nm(\u000b?\nm\u0000i)~B=m(\u000b?\nm\u0000i)B0exp(i!t):(A7)\nUsing the Ansatz ~m(t) =u(t) ~mhom(t)whereu(t)is given\nby\nu(t) =Zt\nt0m(\u000b?\nm\u0000i)B0exp(i!t)\nexp(\u0000t\n\u001c) exp(i!0t)dt;\nand assuming t0= 0andt!1Eq. A4 has the solution\n~m(t) =(\u0000i +\u000b?\nm)\rmB 0(1\n\u001c\u0000i(!\u0000!0))\n1\n\u001c2+ (!\u0000!0)2exp(i!t):\nFrom this general solution, mxcan easily be derived\nmx=\rmB 0\n1\n\u001c2+ (!\u0000!0)2 \u0010\u000b?\n\u001cm\u0000(!\u0000!0)\u0011\ncos(!t)\n+\u00121\n\u001c+\u000b?\nm(!\u0000!0)\u0013\nsin(!t)!\n;\n(A8)\nand substituted into the definition for the power spec-\ntrumP(!)(seeEq.5). Thisleadstotheanalyticsolution\nfor the power spectrum P(!):\nP(!) =\u0016s!2\n4\r\u000b?B2\n0\n1\n\u001c2+ (!\u0000!0)2: (A9)\nAs we can see from equation A9, the analytic solution\nfor the absorbed power depends on the magnetization,9\nwhich in-turn depends on the longitudinal susceptibility.\nAs mentioned above, we approximate the magnetization\nin the presence of an applied field (equation A3) in terms\nof the zero field susceptibility. Therefore, equation A3 is\nonly strictly correct in the zero field limit. Away from\nthe critical temperature the zero field susceptibility is\nsmall, therefore in this limit the approximation holds. As\nwe approach the critical temperature the susceptibility\ndivergesasweapproachthephasetransition. Thismeans\nthat our analytic expression shows a deviation from the\nnumerically calculated result.\nA plot of the magnetization as a function of temperature\nusing equation A3 and data from numerical simulations\ncan be seen in Figure 8. For small values of the applied\nfield this error reduces as the susceptibility is defined for\nsmall changes in the applied field.\nBz= 10TBz=b5TBz=b0TLines: me+ ˜χ/bardblBzPoints: Numerical Data\nT[K]m[Red.]\n660 620 580 540 5001.0 0.8 0.6 0.4 0.2 0.0\nFigure 8: (Color online) Equilibrium magnetization vs tem-\nperature in different applied fields. The (red) solid line rep-\nresents the zero-field equilibrium magnetization, me, gained\nfrom atomistic FePt simulations, a fit to which defines the in-\nput function, me(T)14. The dashed (blue) and dotted (black)\nlines represent expression A3 for different applied fields. The\nsymbols represents the equilibrium magnetization in the pres-\nence of an applied field, Bz= 0;5;10T, from the numeri-\ncal simulations of a single macrospin without demagnetizing,\nstochastic or exchange fields.\nFigure 8 shows the equilibrium magnetization (red solid\ncurve), initially calculated from atomistic spin dynam-\nics simulations14, which is used as an input to the nu-\nmeric simulation. As well as the equilibrium magnetiza-\ntion, the figure also shows the magnetization as a func-\ntion of temperature in 5 and 10T applied fields, which is\nof course not zero at the (zero field) Curie temperature.\nThe dashed and dotted line is the analytic solution to the\nmagnetization (also in 5 and 10T fields), diverging across\nthe Curie temperature. As we can see, the magnetizationin an applied field from the analytic expression shows a\ndiverging behaviour as we approach the Curie tempera-\nture because of the diverging susceptibility, whereas the\nnumerical simulations (points) show no such divergence.\nAppendix B: Magnetostatic Fields\nFor efficient calculation of the magnetostatic fields we\nwrite the convolution 13 as:\nH\u0011\nd;i=X\n\u0012;jW\u0011\u0012\nijm\u0012\nj (B1)\nwhere the Greek symbols \u0011,\u0012again denote Cartesian\ncomponents x,y,zand Latin symbols i,jdenote the lat-\ntice sites.W\u0011\u0012\nijare interaction matrices which only de-\npendonthestructureofthematerial(cubicinthiswork).\nSince we are considering a translationally invariant lat-\ntice one can apply the discrete convolution theorem and\ncalculate the fields in Fourier space:\nH\u0011\nd;k=X\n\u0012W\u0011\u0012\nkm\u0012\nk: (B2)\nIt should be pointed out here that we have absorbed the\nprefactor,Msinto the interaction matrix, W\u0011\u0012\nij. Further-\nmore to write the fields in terms of units of Tesla to be\nconsistent with the form of the fields above, we have mul-\ntiplied equation 13 by \u00160. The Fourier transform of the\ninteraction matrix only has to be performed once and\nthus stored in memory.\nThere are a number of conditions that must be met in or-\nder to utilize the convolution theorem. In terms of signal\nprocessing theory the interaction matrix is seen as the re-\nsponse function and the magnetization data is the signal.\nWe should not that there are two conditions that must be\nsatisfied to utilize the convolution theorem. The first is\nthat the signal (spin system) must be periodic in space.\nThe second is that the range of the response function\nshould be the same as the signal36. The magnetic sys-\ntem is usually not periodic and the demagnetizing effects\nare long ranging and cannot be cut-off at a reasonable\ndistance due to the slow decay36. 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Gorn, Physical Review B 57, 14332\n(1998)."    },    {        "title": "1407.0737v1.Element_Specific_Depth_Profile_of_Magnetism_and_Stoichiometry_at_the_La0_67Sr0_33MnO3_BiFeO3_Interface.pdf",        "content": "arXiv:1407.0737v1  [cond-mat.str-el]  2 Jul 2014Element-Specific Depth Profile of Magnetism and Stoichiomet ry\nat the La 0.67Sr0.33MnO3/BiFeO 3Interface\nJ. Bertinshaw,1,2S. Br¨ uck,1,2D. Lott,3H. Fritzsche,4Y. Khaydukov,5O. Soltwedel,5\nT. Keller,5E. Goering,6P. Audehm,6D. L. Cortie,2W. D. Hutchison,7Q. M.\nRamasse,8M. Arredondo,9R. Maran,10V. Nagarajan,10F. Klose,2,11and C. Ulrich1,2\n1School of Physics, University of New South Wales, Sydney, NS W 2052, Australia\n2Australian Nuclear Science and Technology Organisation,\nLucas Heights, NSW 2234, Australia\n3Institute for Materials Research, Helmholtz\nZentrum Geesthacht, 21502 Geesthacht, Germany\n4Canadian Neutron Beam Centre, Chalk River Laboratories, On tario K0J 1J0, Canada\n5Max-Planck-Institute for Solid State Research,\noutstation FRM II, D-70569 Stuttgart, Germany\n6Max-Planck-Institute for Intelligent Systems, D-70569 St uttgart, Germany\n7School of Physical, Environmental and Mathematical Scienc es,\nUniversity of New South Wales, Canberra, ACT 2600, Australi a\n8SuperSTEM Laboratory, STFC Daresbury Campus,\nKeckwick Lane, Daresbury WA4 4AD, UK\n9School of Mathematics and Physics,\nQueen’s University Belfast, Belfast BT7 1NN, UK\n10School of Materials Science and Engineering,\nUniversity of New South Wales, Sydney, NSW 2052, Australia\n11Department of Physics and Materials Science,\nCity University of Hong Kong, Hong Kong, SAR China\n(Dated: April 5, 2022)\n1Abstract\nDepth-sensitive magnetic, structural and chemical charac terization is important in the under-\nstanding and optimization of novel physical phenomena emer ging at interfaces of transition metal\noxide heterostructures. In a simultaneous approach we have used polarized neutron and reso-\nnant X-ray reflectometry to determine the magnetic profile ac ross atomically sharp interfaces of\nferromagnetic La 0.67Sr0.33MnO3/multiferroic BiFeO 3bi-layers with sub-nanometer resolution. In\nparticular, the X-ray resonant magnetic reflectivity measu rements at the Fe and Mn resonance\nedges allowed us to determine the element specific depth profi le of the ferromagnetic moments in\nboth the La 0.67Sr0.33MnO3and BiFeO 3layers. Our measurements indicate a magnetically diluted\ninterface layer within the La 0.67Sr0.33MnO3layer, in contrast to previous observations on inversely\ndeposited layers [1]. Additional resonant X-ray reflection measurements indicate a region of an\naltered Mn- and O-content at the interface, with a thickness matching that of the magnetic diluted\nlayer, as origin of the reduction of the magnetic moment.\n2Novel electronic states in transition metal oxide (TMO) thin film multila yered structures\nhave created significant attention recently [1–6]. Unexpected pr operties, not present in the\nrespective bulk constituents, were demonstrated such as metallic conductivity at the inter-\nface between two insulators, or even superconducting behavior [4 ]. Fundamentally, these\nstates are a consequence of the symmetry breaking at the interf ace between dissimilar oxide\nmaterials [6]. The resulting interface-near electronic states are de fined by spin exchange\ncorrelations, orbital reconstructions, band bending, Coulombic, magnetic, or superconduct-\ning penetration into the adjacent layer, and epitaxial strain acros s the interface [5–8]. As\npointed outby H.Y. Hwang etal.[6], their detailed originis still under debate since standard\nexperimental techniques do not allow for the separation of intrinsic interface effects from\nmodifications in the chemical composition, in particular oxygen deficie ncies. This requires\nnew experimental techniques, which in particular can determine the chemical depth profile\nacross the interfaces with sub-nanometer resolution.\nTwo particularly interesting TMO materials are the multiferroic Bismut h Ferrite, BiFeO 3\n(BFO), and the ferromagnetic (FM) half-metal La 0.67Sr0.33MnO3(LSMO), which constitutes\nan almost completely spin polarized electron system. While for bulk LSM O the surface de-\npolarizes the spins [9] the problem can be largely overcome in TMO thin fi lm structures.\nBFO in itself is an exciting multiferroic compound, since it exhibits spont aneous magnetic\n(TN= 643 K) and electric (T C= 1143 K) polarization well above room temperature [10].\nBoth order parameters are indirectly coupled, i.e. the spin state ca n be controlled through\nan electric field [11]. LSMO/BFO/LSMO trilayers exhibit a large tunnel m agnetoresistance\n(TMR). Interestingly, voltage-induced changes in the TMR behavio r were recently demon-\nstrated in refs. [12, 13]. This opens possibilities for multi-level memor y state applications\nfor next-generation information storage devices [14]. It is unclear , however, if the observed\nTMR effect is at its optimum. Any magnetic interface modification leadin g to a pertur-\nbation of the spin-polarized flow at the LSMO interface would decrea se the efficiency of\nsuch a device [3]. Therefore it is imperative to precisely identify the ch emical and magnetic\nproperties of the interface states.\nIn this letter we demonstrate the influence of changes in the chemic al composition on the\ninterface magnetism of LSMO/BFO bi-layers. The combination of com plementary X-ray\nand neutron techniques provided a new and unique insight into the ma gnetic and chemical\ndepth profile of the bilayer interface, not accessible with standard techniques. The simulta-\n3neous analysis of polarized neutron reflectivity (PNR) and element s pecific X-ray resonant\nmagnetic reflectometry (XRMR) provided a means of accurately de termining the magnetic\nproperties with sub-nanometer resolution across the interface. Element specific resonant soft\nX-ray reflectivity (XRR) and XRMR measurements performed at th e Fe L 2,3and Mn L 2,3-\nedges allowed for the determination of the magnetic moments in the in dividual layers and\nindicated an altered stoichiometry at the interface as the reason f or an observed reduction\nof the magnetic moment.\nLSMO (300 ˚A)/BFO (200 ˚A and 300 ˚A) bi-layers were grown epitaxially on a SrTiO 3\n(STO) (001) substrate using a Neocera Pulsed laser Deposition (PL D) system with a 248nm\nwavelength KrF excimer Laser. The atomic precision of the layer-by -layer growth was cal-\nibrated by RHEED (reflection high electron energy diffraction). Oxy gen partial pressures\nof 100/10mTorr and substrate temperatures of 900 /850◦C were held for the deposition of\nthe LSMO/BFO layers, respectively. The deposited bi-layer was coo led in a partial oxygen\npressure of 200Torr at a rate of 20K /min (for further details see ref. 13). The bi-layers\nwere initially characterized by laboratory X-ray diffraction and X-ra y reflection (instrument\nX’Pert Pro), which reveal an epitaxial growth with a root-mean squ ared (RMS) interface\nroughness of 5(1) ˚A. This result is consistent with scanning transmission electron micro scopy\n(STEM) images on samples grown under identical conditions, which sh ow a sharp coherent\ninterface (see Supplemental Material). These results demonstra te the excellent quality of\nour films, andrules out interdiffusion larger than5 ˚Aacross the interface. To compensate for\nin-plane stress caused by the lattice mismatch, the c-axis lattice pa rameter was elongated\nin the 200 ˚A thick BFO layer from 3.965 ˚A to 4.081(5) ˚A, i.e. by 3.0 %, and compressed for\nLSMO from 3.871 ˚A to 3.850(5) ˚A, i.e. -0.55 %. This indicates a partial back-relaxation\nfrom the expected change in the c-axis lattice parameter, as obse rved in ref. [15].\nMagnetization measurements performed using Quantum Design PPM S and MPMS sys-\ntems revealed that the LSMO layer orders ferromagnetically below T C= 345(5)K (see Sup-\nplemental Material). This corresponds to a reduction of the expec ted bulk value of 370K\n[16]. In-plane hysteresis curves collected at T = 150K showed a low co ercivity of ∼1.7mT\nand a saturated magnetic moment of 3 .0µB/Mn-ion, which is also smaller than the bulk\nvalues of 3 .5µB/Mn-ion [16]. This is in accordance to previous experiments [15, 17, 18 ]\nand is primarily caused by the in-plane epitaxial strain. The change in t he c-axis lattice\nparameter, i.e. a modified c/a ratio, results in a biaxial tilt and deform ation of the MnO 6\n410 −610 −510 −410 −310 −210 −110 0R(a) \ni. \nii. \nR++ \nR−− \nModel I \nModel II \n0.02 0.04 0.06 0.08 0.10 \nQZ(˚A−1)−80 −40 040 80 Asymmetry (%) (b) /;#/;##/;###/;####/;#####/;######/;#######/;####### \n/;########/;#########/;#/;########## /;###########/;############/;#############/;####/;########## /;##############\n/;#########/;###############/;################/;############/;########## /;##############/;#######/;#################/;##################/;###################/;######/;#/;##/;### /;#/;##/;###/;####/;#####/;######/;####### \n/;########/;#########/;#/;########## /;###########/;############/;#############/;####/;########## /;##############\n/;#########/;###############/;################/;############/;########## /;##############\nFIG. 1. (color online) (a) Specular PNR reflectivity for the R++andR−−channels of a (i)\n300˚A LSMO/200 ˚A BFO sample taken at T = 150K on the instrument NREX and (ii) of a 300˚A\nLSMO/300 ˚A BFO sample taken at T = 300K on the instrument Platypus (the d ata are shifted\nin intensity). (b) Asymmetry between both channels of exper iment (i). The dashed lines depict\nthe less accurate two layer simulation without any interfac ial layers (Model I). The solid lines\ncorrespond to the fit of the final model (Model II), which is cha racterized by a 26 ˚A interface layer\nwith a 40% suppressed magnetic moment.\noctahedra, which affects the orbital arrangement. This weakens the in-plane hopping inte-\ngral and reduces the ferromagnetic double exchange interaction . No further effects, such as\nmodified Mn- or O-content in the bulk of the LSMO layer, are required to be accounted for,\ndemonstrating the excellent quality of the films.\nPolarized Neutron Reflectometry is a powerful technique for the in vestigation of magnetic\nthin film systems [19]. An accurate depth profile of the absolute ferr omagnetic moment can\nbe determined through changes in the specular reflectance betwe en spin-polarized neutron\nbeams [20]. Reflectivity curves were measured using the instrument s NREX, located at\nFRM-II, Munich, Germany, D3 at NRU, Chalk River, Canada, and on P latypus at ANSTO,\nAustralia. Figure 1 shows characteristic reflectivity data taken wit h a Qz-resolution of ∼4%.\nThespin-up( R++)andspin-down( R−−)neutronspinchannelswere measuredtoinvestigate\nthe magnetic moment of the bi-layer sample aligned parallel to an exte rnal applied field.\n5Additional scans of the spin-flip channels R+−andR−+, which probe the magnetic moment\nperpendicular to the neutron polarization, showed no measurable s ignal, indicating that\nthe LSMO magnetization was fully aligned within the plane of the film. Fig. 1a) shows\ntheR++andR−−spin-dependent specular reflectivities of two different bi-layer sam ples\nmeasured at 150K (i) and 300K (ii). The spin asymmetry ( A=R++−R−−\nR+++R−−) for the 150K\ndata is plotted in Fig. 1b). The temperature of 150K was chosen sinc e it is well below T C\nof the LSMO layer and above a structural phase transition of the S TO substrate at 110K\n[21]. To ensure full saturation of the Mn-ion moments, the sample wa s field-cooled under\nan applied magnetic field of 0 .5T parallel to the neutron polarization. Measurements were\nperformed with an external magnetic field of 0 .7mT, required to maintain the polarization\nof theneutron beam. Inthis way, the effect of anexternal magne tic field onthe interface was\nminimized while maximizing the contrast between spin channels. The Simu lReflec software\npackage was used to perform a simultaneous least squares fit [22]. S pecific parameters\nalready determined by XRR, i.e. layer thicknesses, density and non- magnetic interface\nroughness, were fixed during the fitting process. The dashed lines in Fig. 1 correspond to\nthe fitting of a two-layer system of LSMO and BFO without an interfa cial layer (Model I).\nA significant improvement to the fit, with a reduction of the residual error from 6.094 to\n3.147 (see Supplemental Material), was obtained when including an ad ditional magnetically\ndiluted interface layer of 26(5) ˚A with a 40% suppressed magnetic moment of 1.8(2) µB/Mn-\nion instead of the 2.6(3) µB/Mn-ion of the bulk of the LSMO layer (Model II). Additional\nmagnetic layers at the STO/LSMO interface or at the BFO/air interf ace did not yield any\nconvincing improvement of the fit.\nIn order to firmly determine whether the ferromagnetic diluted inte rface layer is located\nwithintheLSMO,theBFO,oracrosstheinterface, wehaveperfor medelement specificX-ray\nresonant magnetic reflection (XRMR) measurements at the Fe L 3and Mn L 3edges. XRMR\nis an extension of standard reflectometry with the use of X-ray ma gnetic circular dichroism\n(XMCD) [23]. Measurements were performed at the UE56/2-PGM1 b eamline at the BESSY\nII synchrotron of the Helmholtz Center Berlin, Germany at the ERN St endstation of the\nMax Planck Institute for Intelligent Systems, Stuttgart, German y [23]. A representative Fe\nL2,3X-ray absorption spectrum (XAS) of the sample in total electron y ield (TEY) is shown\nin Fig. 2 together with XRMR curves taken at the Fe L 3edge (E = 708 .4eV) for parallel\nand antiparallel alignment of the magnetization vector of the sample and the X-ray beam\n6700 710 720 730 740 \nEnergy (eV)0.00.51.0Absorption (a)\n708.4 eVL3\nL2\n0.0 0.1 0.2 0.3 0.4\nQZ(˚A−1)10 −610 −410 −210 0R(b)\n0.246 ˚A−1\nRnorth\nRsouth\n700 710 720 730 740 \nEnergy (eV)0.00.51.01.5R (10 −4)(c)\nRnorth\nRsouth/;#/;##/;###/;####/;#####/;####/;######/;#######/;########/;######### \n/;##########/;####/;#####/;####/;###########/;############/;#############/;##############/;###############/;######### /;#/;##\n/;#/;##\n/;#/;##/;###/;##/;####/;#####/;######/;#######/;########/;######### \n/;##########/;###########/;##/;###/;##/;############/;#############/;##############/;######### \n/;#/;##/;### /;##/;####/;#####/;######/;####### \n/;########/;#########/;##/;###/;##/;####/;#####/;######/;####### /;##########/;###########/;##/;###/;##/;############/;#############/;##############/;###############/;################/;####### \nFIG. 2. (color online) a) XAS inthevicinity of theFe L 2,3edges forunpolarized X-rays. b)XRMR\ncurves (parallel and antiparallel) measured at an energy of E = 708.4 eV using positive circular\npolarized X-rays and a flipped external magnetic field of 80mT . No visual difference between the\ntwo curves is apparent, implying the absence of a sizable mag netic signal. c) Reflected intensity\nenergy scan at Q z= 0.246˚A−1, a point of the reflectivity curve that would show a significan t\nsplitting in the presence of FM contribution.\npolarizationvector. The XRMR curves inFig. 2b) andc) show that th emeasurements taken\nfor both polarizations are quasi-identical. This already indicates the absence of any sizable\nferromagnetic moment in the BFO layer. Figure 3a) shows the magne tic asymmetry ratio\nof theQzmeasurement taken at the Fe L 3edge (from the data of Fig. 2b). Simulations\nof hypothetical FM signals in BFO are shown for comparison (progra m ReMagX [24]). A\nmagnetic polarization of 0 .2µB/Fe3+in the first 10 ˚A (grey area) and 20 ˚A (yellow area) of\nBFO at the LSMO interface were assumed. Using literature data for the magneto-optical\nconstants [25] and for the scaling [26], the simulations clearly show t hat even such a small\nmoment would result in a sizable magnetic asymmetry. Our measureme nts do not show any\nferromagnetic signal originating from the Fe-ions in the BFO layer do wn to a noise level\nof 0.3%, indicating that the Fe magnetic moments are not larger than 0.04µB/Fe3+in the\nBFO layer.\nAs the next step, measurements were performed at the Mn L 3edge at E = 643 .1eV (see\n7FIG. 3. (color online) a) XRMR magnetic asymmetry signal at t he Fe L 3edge at 708 .4eV\nconclusively indicates no Fe ferromagetic signal. The shad ed areas show simulated asymmetries for\na hypothetical magnetic moment of 0 .2µB/Fe in the first 10 ˚A (grey area) and 20 ˚A (yellow area).\nb) XRMR magnetic asymmetry measured at the Mn L 3edge at 643 .1eV. The best agreement is\nobtained for a model with a LSMO interfacial layer with reduc ed magnetization (black line). For\ncomparison, a homogenous magnetized LSMO layer does not rep roduce the periodicity for smaller\nQz(grey area). c) Resulting magnetic depth profile. The relati ve optical constant obtained from\nresonant XRR, which indicates the change in the chemical pro file, is shown as well.\nFig. 3b). Qualitative XAS was measured via fluorescence emission and used to calibrate\nliterature reference data for LSMO [27–29] to obtain the optical a nd magneto-optical index\nof refraction for LSMO. The degree of circular polarization of 90% f or the present mea-\nsurements was taken into account when calculating the magneto-o ptical part of the index\nof refraction. Figure 3b) shows a significant magnetic asymmetry f or the Mn L 3edge, in-\ndicating a spontaneous FM polarization. By simulating the Mn magnetic asymmetry we\nderived the Mn-specific magnetic depth profile. The magnetic reflec tivity calculations were\nbased on a magneto-optical approach to fit to the measured refle ctivity data [30], where\nthe polarization dependence of the incident X-ray beam upon the dir ection of the dielectric\nsusceptibility tensor is taken into account. We again applied Model I ( see Fig. 1) without an\ninterfacial layer between the LSMO and BFO layers (grey area). In this case the oscillations\nof the asymmetry signal cannot be reproduced. The best fit to th e data is obtained when an\n8interface layer with a gradually reduced magnetic moment in the top 1 8(4)˚A of the LSMO\nlayer was introduced (see solid line in Fig. 3b). The resulting magnetic p rofile is shown in\nFig. 3c). This is in good agreement to the result obtained by PNR. The combined PNR and\nXRMR analysis indicates a depleted FM region extending about 20 ˚A into the LSMO at the\nbi-layer interface.\nOur results are in contrast to previous XMCD measurements on an in verse BFO/LSMO\nbi-layer, whichyieldedaninducedFMinterfacelayerwithamagneticmo mentof0 .6µB/Fe3+-\nion within the BFO [1]. A similar result was obtained on a 180 ˚A BFO layer with a top layer\nof 75˚A of CoFeB [19]. PNR measurements on this bi-layer indicated a 20 ˚A interfacial layer\nwith a FM moment of 1 µB/Fe3+extending into the BFO layer. The induced FM magnetic\nmoment was explained by orbital or spin reconstructions at the inte rface.\nIn order to elucidate the origin of the reduction of the magnetic mom ent at the\nLSMO/BFO interface, we have performed intrinsic element specific r esonant X-ray re-\nflectivity measurements with linear polarization at the ERNSt endsta tion at BESSY-II.\nThis did yield valuable information on the local chemistry and valences a t the LSMO/BFO\ninterface. Using the results of the XAS study (see Fig. 2), XRR cur ves were collected at\nT=150K for the Mn L 3edge (maximum XAS signal at 644.1 eV) and off-resonance at\n900 eV. Similarly to the XRMR analysis, simulations of the XRR experimen tal data were\nperformed with an algorithm we developed. The XRR curves and the fi tted results are\nshown in Fig. 4. It is important to note that our experiments indicate that the bulk of the\nLSMO and BFO layers exhibit ideal stoichiometry, and that RMS interf acial roughnesses\nbetween the SrTiO 3substrate, LSMO, and BFO were found to be no larger than 5(1) ˚A.\nThe simulations did indicate that the XRR data collected in off-resonan ce condition are\nhighly sensitive to modifications within the BFO layer. A significant impro vement of the\ntheoretical fit to the data was achieved by including a 18(5) ˚A thick top layer with a 10%\nchanged optical constant (black line versus dotted line). This indica tes a modification of\nthe BFO stoichiometry towards impurity phases at the BFO interfac e to air.\nResonant XRR measurements performed at the Mn L 3edge offer the opportunity to\ndetect small alterations in the depth profile of the Mn stoichiometry . The analysis of the\nMn XRR data indicated a 17(5) ˚A layer inside the LSMO layer at the interface between\nLSMO and BFO with a variation of the optical constants (OC) of abou t 5% (see black\nsolid line in Fig. 4 in contrast to the purple dotted line for the simpler mod el assuming\n90.05 0.10 0.15 0.20 0.25 0.30 0.35 \nQZ(˚A−1)10 −710 −610 −510 −410 −310 −210 −110 0R\nT = 150 K(a)\nModel I\nModel II \nOff-resonant (900.0 eV)\nMn L 3-edge (644.1 eV)\n0 200 400 \nZ-pos. above substrate ( ˚A)0.00.51.0Rel. OC La 0.67 Sr0.33 MnO 3 BiFeO 3(b)Chemical profile\nMagnetic profile0.20 0.25 0.30 0.35 RQZ4E = 644.1 eV\n0.00.51.0Rel. XMCD \nFIG. 4. (color online) a) X-ray reflectivity data measured at 644.1eV and 900.0eV, i.e. resonant\nat the Mn L 3edge and off-resonant. The data was fitted using two different mod els as described\nin the text. The inset shows RQ4\nzin the most sensitive high Qzregion. b) Resulting chemical\n(through XRR) and magnetic (PNR and XRMR) depth profiles.\nhomogeneous layers). As the optical properties vary strongly clo se to the resonance peak\nof the Mn L 3edge, the absolute change in stoichiometry cannot be precisely qua ntified.\nHowever, the data points towards a modification of the interface s toichiometry with an\naltered oxygen and Mn-content in this region [27, 31, 32]. Even a sligh t modification in\nthe oxygen and Mn-concentration would result in a significant deviat ion of the Mn3+/Mn4+\nratio. This would have a direct impact on the concentration of the ch arge carriers which\nmediate the double exchange interaction between the ferromagne tically coupled ions. As\nconsequence, the magnetic phase transition temperature and th e ordered magnetic moment\ndecrease [16]. Indeed, an extrapolation of the temperature depe ndent PNR data indicates\nthat the interfacial layer has a reduced TCof 316(10)K (see Supplemental Materials). The\nthickness of this chemically altered interfacial layer matches exact ly with the region of the\nreduced magneticmoment asdetermined independently byPNR andX RMR. Thisreduction\nover 5-6 octahedral sites cannot be explained by epitaxial strain e ffects alone. Spin and\norbital reconstruction right at the interface will also not fully acco unt for this effect since\nboth would only take place in the topmost atomic layer. This indicates t hat the modified\n10magnetic properties in the interface region between the LSMO and B FO layers is mainly\ncaused by an altered oxygen and Mn concentration.\nIn conclusion, we have demonstrated that the combination of comp lementary reflectom-\netry techniques allowed for the precise determination of the magne tic and chemical depth\nprofile in LSMO/BFO heterostructures. Resonant element-specifi c X-ray reflection mea-\nsurements indicated an interface region of modified oxygen or Mn-c ontent with the same\nthickness as the magnetically diluted interfacial layer within the LSMO film as determined\nby the magnetic XRMR and PNR measurements. This magnetically dilute d layer hinders\nthespinpolarizedcurrent acrosstheinterfaceandhencedeterio ratesthefunctionalityoftun-\nneling junctions. Therefore, our result underlines the importance of the precise knowledge\nof the chemical composition at the interface. Modifications in the st oichiometry can occur\nduring the growth interruption, when the LSMO and the BFO layers w ere completed [19].\nThis is a common problem for all transition metal oxide thin film systems . Previous in-\nvestigations have shown that the atomic concentrations at the int erface can be controlled\nsystematically by the growth process [2, 15, 17]. Changing the dep osition conditions would\nallow for the engineering of the interface structure and thus enab le the growth of artificial\nheterostructures with specifically desired properties.\nThe authors would like to thank D. Manske at the MPI-FKF, Stuttga rt, Germany, for\nfruitful discussions and careful reading of the manuscript. This w ork was supported by the\nAustralianResearch Council throughthegrant DP110105346and byAINSE. V.N. andR.M.\nacknowledge the support of the ARC (grant LP0991794). Electro n microscopy was carried\noutattheNational Center forElectron Microscopy, Lawrence Be rkeley NationalLaboratory,\nsupported by the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.\n[1] P. Yu, et al.Phys. Rev. Lett. 105, 027201 (2010).\n[2] A. Ohtomo and H. Y. Hwang, Nature 427, 423 (2004).\n[3] H. Yamada, Y. Ogawa, Y. Ishii, H. Sato, M. Kawasaki, H. Ako h, and Y. Tokura, Science 305,\n646 (2004).\n[4] N. Reyren et al., Science 317, 1196 (2007).\n[5] J. Mannhart and D. G. Schlom, Science 327, 1607 (2010).\n11[6] H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Nagaosa, and Y. Tokura, Nature Mater.\n11, 103 (2012).\n[7] J. Chakhalian, J. W. Freeland, H.-U. Habermeier, G. Cris tiani, G. Khaliullin, M. van Veenen-\ndaal, and B. Keimer, Science 318, 1114 (2007).\n[8] N. Driza et al., Nature Mater. 11, 675 (2012).\n[9] M. Konoto, T. Kohashi, K. Koike, T. Arima, Y. Kaneko, Y. To mioka, and Y. Tokura, Appl.\nPhys. Lett. 84, 2361 (2004).\n[10] G. Catalan and J. F. Scott, Adv. Mater. 21, 2463 (2009).\n[11] T. Zhao et al., Nature Mater. 5, 823 (2006).\n[12] H. B´ ea, M. Gajek, M. Bibes, and A. Bart´ el´ emy, J. Phys. : Condens. Matter 20, 434221 (2008).\n[13] M. Hambe, A. Petraru, N. A. Pertsev, P. Munroe, V. Nagara jan, and H. Kohlstedt, Adv.\nFunct. Mater. 20, 2436 (2010).\n[14] W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 442, 759 (2006).\n[15] M. Angeloni, G. Balestrino, N. G. Boggio, P. G. Medaglia , P. Orgiani, and A. Tebano, J.\nAppl. Phys. 96, 6387 (2004).\n[16] A. Urushibara, Y. Moritomo, T. Arima, A. Asamitsu, G. Ki do, and Y. Tokura, Phys. Rev. B\n51, 14103 (1995).\n[17] P. Lecoeur, A. Gupta, P. R. Duncombe, G. Q. Gong, and G. Xi ao, J. Appl. Phys. 80,\n513(1996).\n[18] Y. Konishi, Z. Fang, M. Izumi, T. Manako, M. Kasai, H. Kuw ahara, M. Kawasaki, K. Ter-\nakura1, and Y. Tokura, J. Phys. Soc. Jpn. 683790 (1999).\n[19] H. B´ ea et al., Phys. Rev. Lett. 100, 017204 (2008).\n[20] M. R. Fitzsimmons et al., J. Magn. Magn. Mater. 271, 103 (2004).\n[21] G. Shirane and Y. Yamada, Phys. Rev. 177, 858 (1969).\n[22] “SimulReflec - Reflectivity Curves Simulations and Fitt ing,” Institute Laue Langevin, (2010).\n[23] S. Br¨ uck, S. Bauknecht, B. Ludescher, E. Goering, and G . Sch¨ utz, Rev. Sci. Instr. 79, 083109\n(2008).\n[24] S. Macke, S. Br¨ uck, and E. Goering, “ReMagX - x-ray magn etic reflectivity tool,” (2010).\n[25] H. B´ ea et al., Phys. Rev. B 74, 020101 (2006).\n[26] L. W. Martin et al., Nano Lett. 8, 2050 (2008).\n[27] E. Pellegrin, L. H. Tjeng, F. M. F. de Groot, R. Hesper, G. A. Sawatzky, Y. Moritomo, and\n12Y. Tokura, J. El. Spec. and Rel. Phenomena 86, 115 (1997).\n[28] J. W. Freeland et al., J. Phys.: Cond. Matter 19, 315210 (2007).\n[29] S. Br¨ uck et al., New J. Phys. 13, 033023 (2011).\n[30] J. Zak, E. R. Moog, C. Liu, and S. D. Bader, Phys. Rev. B 43, 6423 (1991).\n[31] K. Horiba, et al., Phys. Rev. B 71, 155420 (2005).\n[32] S. Picozzi et al., Phys. Rev. B 75, 094418 (2007).\n13"    },    {        "title": "2308.03701v2.Buzdin__Shapiro_and_Chimera_Steps_in___varphi_0__Josephson_Junctions.pdf",        "content": "Buzdin, Shapiro and Chimera Steps in φ0Josephson Junctions\nYu. M. Shukrinov1,2,3, E. Kovalenko4, J. Teki´ c5, K. Kulikov1,2, and M. Nashaat1,6\n1BLTP, JINR, Dubna, Moscow region, 141980, Russia\n2Dubna State University, Dubna, Russia\n3Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia\n4Center for the Development of Digital Technologies, Krasnogorsk, Russia\n5”Vinˇ ca” Institute of Nuclear Sciences, Laboratory for Theoretical\nand Condensed Matter Physics - 020, University of Belgrade, PO Box 522, 11001 Belgrade, Serbia\n6Department of Physics, Faculty of Science, Cairo University, 12613, Giza, Egypt\n(Dated: August 14, 2023)\nThe unique resonance and locking phenomena in the superconductor-ferromagnet-superconductor\nφ0Josephson junction under external electromagnetic radiation are demonstrated when not just\nthe electric but also the magnetic component of external radiation is taken into account. Due to\nthe coupling of superconductivity and magnetism in this system, the magnetic moment precession\nof the ferromagnetic layer caused by the magnetic component of external radiation can lock the\nJosephson oscillations, which results in the appearance of a particular type of steps in the current-\nvoltage characteristics, completely different from the well-known Shapiro steps. We call these steps\nthe Buzdin steps in the case when the system is driven only by the magnetic component and the\nChimera steps in the case when both magnetic and electric components are present. Unlike the\nShapiro steps where the magnetization remains constant along the step, here it changes though\nthe system is locked. The spin-orbit coupling substantially contributes to the amplitude, i.e., the\nsize of these steps. Dramatic changes in their amplitudes are also observed at frequencies near\nthe ferromagnetic resonance. Combinations of the Josephson and Kittel ferromagnetic resonances\ntogether with different types of locking pronounced in dynamics and current-voltage characteristics\nmake the physics of this system very interesting and open up a series of new applications.\nThe possibility to combine superconductivity and mag-\nnetism in hybrid Josephson structures holds promise to\nincrease the technological applications of superconduc-\ntors and superconducting nanostructures in the recent\nrapid development of spintronics and superconducting\nlogic devices. One particular structure that demonstrates\ntransport properties with disrupting scientific and tech-\nnological potential is the superconductor-ferromagnet-\nsuperconductor (SFS) φ0Josephson junction (JJ) [1, 2].\nIt belongs to a special class of anomalous JJs with a non-\ncentrosymmetric ferromagnetic layer and broken time-\nreversal symmetry, which results in a particular current-\nphase relation I=Icsin(φ−φ0) with the phase shift\nproportional to the magnetic moment [3]. Experimen-\ntal observations of this anomalous phase shift in differ-\nent systems [4–6] open up several new opportunities for\nsuperconducting spintronics [7]. The presence of bidirec-\ntional coupling between the magnetic moment of the bar-\nrier and the superconducting phase difference allows su-\nperconductivity to control magnetism and vice versa, to\ninfluence Josephson current via magnetic moment, which\ncould lead to a series of new applications [1–3, 7–9].\nIn the studies of ordinary superconductor-insulator-\nsuperconductor (SIS-type) Josephson junctions driven by\nexternal radiation, the influence of the magnetic compo-\nnent of radiation is usually neglected, and the description\nof the effect is reduced to adding the term Asin (ωt) to\nthe bias current. As was mentioned in Ref.[2], in the φ0\nJJ the microwave magnetic field generates an additional\nmagnetic precession with the microwave frequency which\nmight lead to a series of unusual effects. However, so\nfar these predictions have not been verified and detailedstudies of the interaction of electromagnetic radiation\nwith the φ0-junction taking into account the magnetic\ncomponent have not been carried out. Here we eliminate\nthis shortcoming and include the direct interaction of\nthe magnetic component of the microwave magnetic field\nwith the magnetic moment of the ferromagnetic layer in\nour study of radiation effects in the φ0junction. The\nconsidered geometry is demonstrated in Fig.1(a).\nIn this Letter, we demonstrate the effects of microwave\nradiation on the dynamics and the IV-characteristics\nof the SFS φ0Josephson junction taking into account\nboth the magnetic and electric components of radia-\ntion. This leads to two different mechanisms of locking\nthe Josephson oscillations and the ferromagnetic moment\nprecessions (see Fig.1(b)). These mechanisms of indirect\nlocking are investigated at different parameters of the\nφ0junction and microwave field. Additionally to the\n(a)\n   Electric \ncomponent   Magnetic \ncomponent\nJosephson \noscillations\nMagnetic \n moment(b)\nMagnetic \n moment\nJosephson \noscillations\nFIG. 1. (a) Geometry of the system; (b) Demonstration of\ntwo locking mechanisms in the φ0Josephson junction under\nan external electromagnetic field (see the text).\nShapiro steps caused by the electric component of radi-\nation, we observe a unique step created by the periodicarXiv:2308.03701v2  [cond-mat.supr-con]  11 Aug 20232\nfield of the magnetic component. By locking the mag-\nnetic moment precession, one can also lock the Josephson\noscillations due to their coupling with the ferromagnetic\nmoment. We call this step the Buzdin (BS) step since it\nwas first predicted in Ref.[2] to stress its different origin\nand properties from the Shapiro step (SS). When both\nradiation components are taken into account, we observe\nanother type of step named the composite or Chimera\nstep due to its creation by two different mechanisms. Be-\nlow we will call this step the C-step (CS). We show that\nCS is not a trivial sum of BS and SS; however, it has\nits own specific features reflecting its origin from two dif-\nferent locking mechanisms. While the electric component\ninteracts with the superconducting current, the magnetic\ncomponent, by driving the ferromagnetic moment, inter-\nacts also with the superconducting subsystem due to the\ncoupling of Josephson oscillations and ferromagnetic mo-\nment precession. We also show that when either Joseph-\nson or radiation frequencies approach the ferromagnetic\none the interplay of the Josephson and Kittel ferromag-\nnetic resonances appears.\nThe model is described by the system of equations ob-\ntained from the Landau-Lifshitz-Gilbert (LLG) equation,\nthe Josephson relation for the phase difference and volt-\nage, and the equation for the biased current of the resis-\ntively and capacitively shunted junction (RCSJ) model:\ndM\ndt=−γM×Heff+α\nM0\u0012\nM×dM\ndt\u0013\n, (1)\n˙φ=V(t),\n˙V= [I+Asin(ωRt)−V(t) +r˙my−sin(φ−rmy)]/βc,\nwith the effective magnetic field Heff:\nHeff=K\nM0Grsin\u0012\nφ−rMy\nM0\u0013\nˆy+ (2)\n+\u0012K\nM0Mz\nM0+HRsin(Ω Rt)\u0013\nˆz,\nwhere βcis McCumber parameter, G=EJ/(KV) repre-\nsents the ratio of the Josephson to magnetic anisotropy\nenergy, ris the Rashba spin-orbit coupling (SOC). The\nsecond term inside the sine function represents the phase\nshift φ0=rMy/M0. The gradient of the spin-orbit po-\ntential is along the easy axis, which is taken to be along\nz. In the LLG equation γis the gyromagnetic ratio, αis\nthe Gilbert damping, M0=|M|.\nWe use the dimensionless variables mi=Mi\nM0,(i≡\nx, y, z ), t→tωc, ω R=ΩR\nωc, where ωc= 2eIcR/ℏ\nis a characteristic frequency of the junction, the ferro-\nmagnetic resonance frequency Ω F=Kγ/M 0and the\namplitudes of the electric ER= (ERsin(Ω Rt),0,0), and\nmagnetic HR= (0,0, HRsin(Ω Rt) components are then\nnormalized to ωc, so that ωF=ΩF\nωc, hR=γ\nωcHR.\nFirst, we examine the effects of the magnetic compo-\nnent of external radiation (MCR) on the magnetization\ndynamics and the IV-characteristics in the ferromag-\nnetic resonance region when ferromagnetic resonance fre-\nquency is close to the Josephson one, i.e., when ωF≈ωJ.Unlike the electric component, the magnetic one can in-\nteract directly with the magnetic moment of the ferro-\nmagnetic barrier, which further leads to the appearance\nof the Buzdin step in the IV-characteristics. To get an\ninsight into the origin of BS, we switch off the term\nAsin (ωRt), describing the electric component of radia-\ntion, and investigate the effect of hRonly, i.e., concen-\ntrate on the features, produced by MCR.\nThe average voltage V, the maximum value of the mag-\nnetic moment, mmax\ny, and the superconducting current Is\nas functions of decreasing biased current Iare presented\nin Fig. 2(a). It shows the ferromagnetic resonance and\nits manifestation in these characteristics. The magnified\nFIG. 2. The average voltage V, the maximum value of the my\nmagnetic component mmax\nyand the superconducting current\nIsas a function of decreasing biased current IforA= 0, hR=\n1, r= 0.5, G= 0.01, α= 0.01, ωR= 0.485, ωF= 0.5; (b)\nMagnified view of (a) showing the Buzdin step. (c) mmax\nyat\ndifferent values of r.\nview of Fig. 2(a) in the FMR region with the Buzdin step\nis presented in Fig.2(b). Its appearance is a result of the\nlocking of Josephson oscillations by the magnetic compo-\nnent of external radiation. Namely, the MCR creates the\nperiodic precession of the magnetic moment, which then,\nthrough coupling with Josephson oscillations, also locks\nthe Josephson oscillations. So, due to their bidirectional\ncoupling, both the Josephson oscillations and the mag-\nnetic precession are locked with MCR. The specific man-\nifestation of this locking is also seen in the mmax\ny(I) and\nIs(I) in the current interval corresponding to the Buzidn\nstep, where the magnified view reveals a ”bubble-like”\nfeature due to changing of mmax\nyalong the step. So we\nobserve the locking of the magnetic moment precession\nwith a changed amplitude but fixed frequency, which is\ndemonstrated below.\nAn interesting question is related to the variation of\nthe parameters characterizing the interaction of Joseph-\nson oscillations with the magnetic moment, in particu-\nlar, the spin-orbit coupling r. This parameter plays a\nkey role in the appearance of the Buzdin steps since it is\nthe coupling between the magnetic moment and Joseph-\nson oscillation through which the locking of the magnetic3\nmoment transfers to the superconducting subsystem. Of\ncourse, if r= 0 and there is no coupling, and the Buzdin\nsteps do not exist. In Fig. 2(c), the variation mmax\ny(I) at\ndifferent parameters of spin-orbit coupling is presented,\nwhere the focus was on the changes of the bubble struc-\nture under r. We know from Fig.2(b) that the width\nof the bubble exactly corresponds to the width of the\nBuzdin step, and as the rdependence shows the bubble,\ni.e., the Buzdin step increases with the increase of the\nspin-orbit coupling.\nThe proof of the locking by MCR can be seen in Fig. 3\n(a), where the voltage dependence of mmax\nyis presented.\nAs we can see, there is a sharp minimum at the same\nFIG. 3. (a) mmax\nyas a function of V; (b) and (c) The time\ndependence of Vandmy, and the corresponding FFT analysis\nin the center of the bubble, respectively. Other parameters\nare as in Fig. 2.\nvalue of voltage (i.e., frequency) corresponding to the\nBuzdin step. The time dependence of Vandmyand\nthe corresponding Fast Fourier Transform (FFT) analysis\nin Fig. 3(b) and (c), respectively, further confirm the\nlocking by MCR.\nLet us now discuss the effects of both electric and mag-\nnetic components. In this case, the electromagnetic irra-\ndiation of φ0JJ leads to the realization of two different\nmechanisms of locking. The electric component of radi-\nation locks the Josephson oscillation, and it, due to the\ncoupling, locks the precession of the magnetic moment of\nthe ferromagnetic barrier [10]. In turn, the periodic field\nof the magnetic component through the interaction with\nthe magnetic moment locks the Josephson oscillations\nleading to the Buzdin steps in the IV-characteristics.\nThe combined effect of both components results in the\nappearance of a unique step different from BS and SS. As\nmentioned above, since it comes from both mechanisms\nof locking we call this step the composite or Chimera\nstep.\nThe effect of both the electric and magnetic compo-\nnents of radiation at r= 0.2,hR= 1, and A= 0.01\nis presented in Fig.4(a). It shows the IV-characteristics\nandmmax\ny(I) dependence for three cases: i) in the pres-\nence of two components; ii) the electric component only;\nand iii) the magnetic component only. We stress that\nthe C-step is not a trivial sum of BS and SS. In particu-\nlar, we see in its left part that it does not coincide with\nSS and BS under separate actions. The corresponding\nmmax\ny(I) dependence shows the bubble structure, which\nindicates that in this case, we observe also a locking with\nthe changed amplitude of the magnetization precession.\nFIG. 4. The effects of both radiation components. (a) Parts\nof IV-characteristics with the Buzdin, Shapiro, and composite\nsteps at r= 0.2; (b) The same at r= 0.4.\nIt is surprising that the bubble structure continues along\nthe whole CS because, in the second and third cases, we\nhave seen the bubble structure for the BS, and the locking\nstep for SS, where mmax\nystays constant along the step.\nIn this figure, the scale of the mmax\ny(I) dependence for the\nfirst, and third cases is the same, while in the second case,\nwe have made an arbitrary shift to demonstrate clearly\nthe manifestation of SS. An important fact here is that\nthe maximal amplitude of the magnetization precession\nis the same in BS and CS, i.e., CS conserves this charac-\nteristic of the magnetic component effect. We stress also\nthat the size of the Shapiro step does not change with\nthe spin-orbit parameter.\nAs we see, BS is smaller than SS at this model pa-\nrameters. We show in Fig. 2(c) that the value of BS is\ngrowing with r. So by variation of the φ0junction and\nexternal radiation parameters, we expect the case when\nBS is larger than SS. This case is shown in Fig.4(b) at\nr= 0.4. Additionally, SS calculated at hR= 0 is in\nthe current interval, which is out of the corresponding\ninterval for the C-step. Like in Fig.4(a), we have shifted\narbitrarily the mmax\ny(I) dependence for clarity.\nSo far we have only considered the region near FMR\nωF= 0.5 atωR= 0.485, which leads to the question of\nhow the behavior changes when the frequency of external\nradiation is equal to the ferromagnetic resonant one. In\nFig. 5(a) the IV-characteristics and mmax\nyare presented\nfor the magnetic component only ( A= 0 ) at ωR=ωF=\n0.5. The Buzdin step in the IV-characteristics and its\nlocking signature (the bubble structure) in the mmax\ny(I)\ndependence can be seen clearly.\nIn Fig. 5(b), the IV-characteristics and mmax\nyare plot-\nted when both radiation components are switched on for\nthree cases of different amplitudes of electric components.\nWe observe a fast increase in the C-step width with the\namplitude of the electric component.\nLocking is not the only effect that the microwave mag-\nnetic field may have on the system. Namely, its direct\ninfluence on the magnetic moment of the ferromagnetic\nlayer leads to the Kittel ferromagnetic resonance [11].\nDue to the coupling of the magnetic moment with the\nJosephson phase, it plays a specific role in the φ0junc-\ntion. The effective field in the case of the Kittel reso-\nnance in normalized units has the following components:4\nFIG. 5. (a) The IV-characteristics and mmax\nyas a function of\nIin the case of the magnetic components only for ωR=ωF=\n0.5. (b) The IV-characteristics and mmax\nyfor three cases of\ndifferent amplitudes of the electric components are indicated\nby numbers. Other parameters are G= 0.01,r= 0.2 and\nhR= 1.\nFIG. 6. Manifestation of the ferromagnetic Kittel resonance\ndue to the external electromagnetic wave with frequency 0 .5\natr= 0.2 and different values of A, hRandG.\nhx= 0,hy=Grsin(φ−rmy) +hRsin(ωRt),hz=mz.\nThe competition between the Kittel and Josephson\nFMR at different model parameters is presented in Fig.6.\nAt rather small values of the magnetic component am-\nplitude hRand Gilbert damping α, the Kittel resonance\n(KR) and Josephson resonance (JR) are centered at ωR=\nωFandωJ=ωF, respectively (see Fig.6 (a)). There is a\nmanifestation of a subharmonic peak at ωJ=ωF/2 pro-\nnounced as the horizontal line. As discussed before, close\nto the resonance condition ωJ=ωF, we observe a lock-\ning of the Josephson oscillation and magnetic moment\nprecession. With an increase in the coupling constant G,\nthe JR region dominates and the KR region becomes faint\n(see Fig.6(b)). The situation is reversed with increasing\nhRwhen the KR dominates.\nAlso, we observe a shift in the resonance frequency\nrelated to the effect of magnetic anisotropy. Moreover,\nthe KR region becomes more broadening and nonsym-\nmetric around the resonance frequency. The situation\nchanges dramatically when we take into account both\nthe electric and magnetic components of the microwave\nfield. In this case, a crossed resonance region appearswhich corresponds to subharmonics and resonances with\na combination of ωFandωRfrequencies. In addition\nto this, by comparing Fig.6(b) and Fig.6(d), the effect\nof A is manifested in the JR linewidth. So by chang-\ning the frequency and amplitude of the external elec-\ntromagnetic radiation, one can tune and manipulate the\nKittel and Josephson resonances in the hybrid Joseph-\nson junction systems. Experimentally, FMR in the SFS\nstructures has attracted much attention recently [12–15].\nThe Kittel regime [16] can be used to realize a magnetic\nlogic gate through the superconducting phase transition.\nA gate-controlled time-dependent spin-orbit coupling is\nproposed and demonstrated by the authors in Ref.[17]. A\ndramatic change in current-phase relations and Joseph-\nson energy can be seen in this case, even when there is\nno bias current. Using this and the experimental work\nRef.[6], one can tune the Chimera step by tuning SOC in\na hybrid SFS system.\nOne of the ways of testing our results experimentally\nwould be in a superconductor-ferromagnetic insulator-\nsuperconductor on a 3D topological insulator which has\na strong enough spin-orbit coupling needed for the φ0\nJJ [18]. The value of the Rashba-type parameter rcan\nvary in the range 0 .1−1 in such materials as permalloy\ndoped with Pt[19] or the ferromagnets MnSi andFeGe .\nTherefore, in the material with weak magnetic anisotropy\nK∼4×10−5KA−3[20], and a junction with a relatively\nhigh critical current density of (3 ×105−5×106)A/cm2\n[21] the value of the product Grcan easily be in the\nrange 0 .01−100. This makes it possible to reach the\nvalues used in our numerical calculations for the possible\nexperimental observation of the predicted effects.\nAs a summary, we have found that the magnetic com-\nponent of radiation brings a series of unique effects in the\ndynamics of the φ0Josephson junction. In particular,\nunlike the electric component which leads to the Shapiro\nsteps, the magnetic one can interact directly with the\nmagnetic moment of the ferromagnetic layer and due to\nits coupling with the superconducting phase can lock the\nJosephson oscillations. This leads to the appearance of\nthe Buzdin steps in the IV-characteristics. The lock-\ning of the magnetization precession is characterized by\na bubble structure along the step. The spin-orbit cou-\npling substantially contributes to the amplitude of the\nBuzdin steps, which exhibits a dramatic increase at fre-\nquencies near the ferromagnetic resonance. If both com-\nponents drive the system, the presence of two locking\nmechanisms leads to the Chimera steps. We stress that\nBS, SS, and CS have very different properties (ampli-\ntude dependence, etc.), which will be studied in detail as\na separate work. We consider that combinations of the\nJosephson and Kittel ferromagnetic resonances and the\ndifferent types of locking in the φ0Josephson junctions\ncould open up a series of new applications.\nWe thank A. Buzdin, I. Rahmonov, T. Belgibaev for\nfruitful discussion. Numerical simulations were funded\nby the Russian Science Foundation Project No. 22-71-\n10022. The work was partially funded by the Ministry5\nof Education, Science and Technological Development of\nthe Republic of Serbia, grant number 451-03-47/2023-01/200017 (”Vinˇ ca” Institute of Nuclear Sciences, University\nof Belgrade).\n[1] A. Buzdin, Direct Coupling Between Magnetism and Su-\nperconducting Current in the Josephson φ0 Junction,\nPhys. Rev. Lett. 101, 107005 (2008).\n[2] F. Konschelle and A. Buzdin, Magnetic Moment Manip-\nulation by a Josephson Current, Phys. Rev. Lett. 102,\n017001 (2009).\n[3] Yu. M. Shukrinov, Anomalous Josephson effect, Phys.\nUsp. 65, 317 (2022).\n[4] D. B. Szombati, S. Nadj-Perge, D. Car, S. R. Plissard,\nE. P. A. M. Bakkers and L. P. Kouwenhoven, Josephson\nφ0-junction in nanowire quantum dots, Nat. Phys., 12,\n568572 (2016).\n[5] A. Assouline, C. Feuillet-Palma, N. Bergeal, T. Zhang,\nA. Mottaghizadeh, A. Zimmers, E. Lhuillier, M. Eddrie,\nP. Atkinson, M. Aprili, H. Aubin, Spin-Orbit induced\nphase-shift in Bi2Se3Josephson junctions, Nat. Com-\nmun. 10, 126 (2019).\n[6] W. Mayer, M. C. Dartiailh, J. Yuan, K. S. Wickramas-\ninghe, E. Rossi, and J. Shabani, Gate controlled anoma-\nlous phase shift in Al/InAs Josephson junctions, Nat.\nCommun., 11, 212 (2020).\n[7] J. Linder and J. Robinson, Superconducting spintronics,\nNat. Phys. 11, 307 (2015).\n[8] A. S. Mel’nikov, S. V. Mironov, A. V. Samokhvalov,\nA. I. Buzdin, Superconducting spintronics: state of the\nart and prospect, Uspekhi Fizicheskikh Nauk, 192, 1339\n(2022); Phys. Usp., 65, 1248 (2022).\n[9] I. B. Bobkova, A. M. Bobkov, and M. A. Silaev, Magneto-\nelectric effects in Josephson junctions. J. Phys.: Condens.\nMatter 34, 353001 (2022).\n[10] S. A. Abdelmoneim, Yu. M. Shukrinov, K. V. Kulikov, H.\nElSamman, and M. Nashaat, Locking of magnetization\nand Josephson oscillations at ferromagnetic resonance in\naφ0junction under external radiation, Phys. Rev. B 106,\n014505 (2022).\n[11] C. Kittel, Introduction to solid state physics, (John Wi-\nley and Sons, 2005).\n[12] L. -L. Li, Y. -L. Zhao, X. -X. Zhang, and Y. Sun, Possible\nevidence for spin-transfer torque induced by spin-triplet\nsupercurrents, Chin. Phys. Lett. 35, 077401 (2018).\n[13] I.A. Golovchanskiy, N.N. Abramov, O.V. Emelyanova,\nI.V. Shchetinin, V.V. Ryazanov, A.A. Golubov, andV.S. Stolyarov, Magnetization Dynamics in Proximity-\nCoupled Superconductor-Ferromagnet-Superconductor\nMultilayers. II. Thickness Dependence of the Su-\nperconducting Torque, Phys. Rev. Appl. 19, 034025\n(2023).\n[14] I. A. Golovchanskiy, N. N. Abramov, V. S. Stol-\nyarov, V. I. Chichkov, M. Silaev, I. V. Shchetinin,\nA. A. Golubov, V. V. Ryazanov, A. V. Usti-\nnov, and M. Y. Kupriyanov, Magnetization Dynam-\nics in Proximity-Coupled Superconductor-Ferromagnet-\nSuperconductor Multilayers, Phys. Rev. Appl. 14,\n024086 (2020).\n[15] K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen,\nX. Montiel, M. Eschrig, T. Wagner, S. Komori, A. Sri-\nvastava, J. W. A. Robinson, and M. G. Blamire, Effect of\nMeissner Screening and Trapped Magnetic Flux on Mag-\nnetization Dynamics in Thick Nb/Ni 80Fe20/Nb Trilay-\ners, Phys. Rev. Appl. 11, 014061 (2019)\n[16] C. Kittel, On the Theory of Ferromagnetic Resonance\nAbsorption, Phys. Rev. 73, 155 (1948).\n[17] D. Monroe, M. Alidoust, and I. ˇZuti´ c, Tunable planar\nJosephson junctions driven by time-dependent spin-orbit\ncoupling, Phys. Rev. Applied 18, L031001 (2022).\n[18] I. V. Bobkova , A. M. Bobkov, I. R. Rahmonov, A. A.\nMazanik , K. Sengupta, and Yu. M. Shukrinov, Mag-\nnetization reversal in superconductor/insulating ferro-\nmagnet/superconductor Josephson junctions on a three-\ndimensional topological insulator, Phys. Rev. B 102,\n134505 (2020).\n[19] A. Hrabec, F. J. T. Goncalves, C. S. Spencer, E. Aren-\nholz, A. T. N’Diaye, R. L. Stamps, and C. H. Mar-\nrows, Spin-orbit interaction enhancement in permalloy\nthin films by Pt doping, Phys. Rev. B 93, 014432 (2016).\n[20] A. Yu. Rusanov, M. Hesselberth, J. Aarts, and A. I.\nBuzdin, Enhancement of the superconducting transition\ntemperature in Nb/permalloy bilayers by controlling the\nDomain State of the ferromagnet, Phys. Rev. Lett. 93,\n057002 (2004).\n[21] J. W. A. Robinson, F. Chiodi, M. Egilmez, G. B. Hal´ asz,\nand M. G. Blamire, Supercurrent enhancement in Bloch\ndomain walls, Scientific Report 2, 699 (2012)."    },    {        "title": "1211.2701v1.Spin_polarization_and_magnetoresistance_through_a_ferromagnetic_barrier_in_bilayer_graphene.pdf",        "content": "Spin polarization and magnetoresistance through a ferromagnetic barrier in bilayer\ngraphene\nHosein Cheraghchi\u0003and Fatemeh Adinehvand\nSchool of Physics, Damghan University, 6715- 364, Damghan, Iran\n(Dated: June 30, 2021)\nWe study spin dependent transport through a magnetic bilayer graphene nanojunction con\fgured\nas two dimensional normal/ferromagnetic/normal structure where the gate-voltage is applied on the\nlayers of ferromagnetic graphene. Based on the fourband Hamiltonian, conductance is calculated by\nusing Landauer Butikker formula at zero temperature. For parallel con\fguration of the ferromag-\nnetic layers of bilayer graphene, the energy band structure is metallic and spin polarization reaches\nto its maximum value close to the resonant states, while for antiparallel con\fguration, the nanojunc-\ntion behaves as a semiconductor and there is no spin \fltering. As a result, a huge magnetoresistance\nis achievable by altering the con\fgurations of ferromagnetic graphene especially around the band\ngap.\nPACS numbers: 73.23.-b,73.63.-b\nI. INTRODUCTION\nSince spin-orbit coupling in graphene1is very weak2\nand also there is no nuclear spin3, spin \rip length is so\nlong about 1 \u0016min dirty samples and room temperature4.\nClean samples are expected to have longer spin co-\nherency. This is good opportunity for spintronic ap-\nplications based on candidatory of graphene. On the\nother hand, graphene has not intrinsically ferromagnetic\n(FM) properties, however, it is possible to induce ferro-\nmagnetism externally by doping and defects5, Coulomb\ninteractions6or by applying an external electric \feld\nin the transverse direction in nanoribbons7. Recently,\nHaugen8proposed the FM correlations due to strong\nproximity of magnetic states close to graphene. The over-\nlap between the wave functions of the localized magnetic\nstates in the magnetic insulator and the itinerant elec-\ntrons in graphene induces an exchange \feld on itiner-\nant electrons in graphene giving rise spin splitting of the\ntransport. The exchange splitting which is induced by\nFM insulator Euo in graphene was estimated to be of\norder of 5meV . This splitting which is e\u000bectively simi-\nlar to a Zeeman interaction has so large magnitude that\ncan have important e\u000bects. Such spin splitting can be\ndirectly evaluated from the transmission resonances or\nmagnetoresistance of FM graphene junction. The ferro-\nmagnetism leads to a spin splitting e\u000bectively similar to\na Zeeman interaction but of much larger magnitude. The\ninduced exchange \feld is tunable by an in-plane external\nelectric \feld9. The possibility of controlling spin conduc-\ntance in FM monolayer graphene insulator has also been\nstudied by Yokoyama10. It was found that the spin con-\nductance has an oscillatory behavior in terms of chemical\npotential and the gate voltage.\nBilayer graphene on the other hand has shown to have\ninteresting properties for application in nanoelectronic\ndevices such as transistors based on graphene substrate.\nThe new type of integer quantum Hall e\u000bects11and also\nthe electronic band gap controllable by vertically applied\nelectric \feld are of its unusual properties in compared\nFIG. 1: Schematic view of normal/ferromagnetic/normal bi-\nlayer graphene. Two gate electrodes can be coated on top of\nthe magnetic insulator strips which are located on the upper\nand lower layers.\nto monolayer graphene12{15. Moreover, the parabolic\nband structure close to the Dirac points transforms to a\nMexican-hat like dispersion when an electric \feld is ap-\nplied on graphene. Optical measurements and theoretical\npredictions propose a 200 meV gap in bilayer graphene.\nThis controllable gap makes bilayer graphene as an ap-\npropriate candidate for spintronic devices. An e\u000bective\ntwo-band Hamiltonian can describe the low energy exci-\ntations of a graphene bilayer in the regime of low barrier\nheights16. However, the four-band Hamiltonian is known\nto give a better agreement with both experimental data\nand theoretical tight-binding calculations12,17. Very re-\ncently, spin splitting of conductance in bilayer graphene\nhas been investigated by using an e\u000bective two-band\nHamiltonian emerging from low energy approximation18.\nThis approximation is valid when energy of electrons hit-\nting on the potential barrier is about the barrier height.\nApplication of a potential di\u000berence between upper and\nlower layers intensi\fes the failure of this approxima-\ntion. On the other hand, magnetoresistance in bilayer\ngraphene has been studied in Ref.19by using 8\u00028 Hamil-arXiv:1211.2701v1  [cond-mat.mes-hall]  12 Nov 20122\ntonian when the induced exchange \felds are laid in plane\nof each layer with a rotation in their orientations aginst\neach other. By using Kobo-Greenwood formula, they\nhave investigated the dependence of conductivity and\nmagnetoresistance on temperature and induced exchange\n\feld.\nMotivated by these studies, based on the four-band\nHamiltonian and close to the Dirac points, we study\nspin current through a magnetic barrier creating by use\nof the proximity of a ferromagnetic insulator on bilayer\ngraphene. Conductance is calculated by use of Landauer-\nButtiker formula at zero temperature. The parameters\nof the barrier, energy and angle of incident electrons can\na\u000bect transport through a magnetic barrier classifying in\nthe propagating or evanescent modes. The dependnce\nof resonant peaks in transmission on system parameters\nis proposed to follow a resonance condtion. We have\nfound in some resonant energies and barrier parameters\nand also around the gapped region that a remarkable spin\npolarization and also magnetoresistance can be achieved.\nModel : We consider a normal/ferromagnetic/normal\nbilayer graphene nanojunction. The model we have used\nfor a bilayer graphene sandwiched between two ferromag-\nnetic insulator is shown schematically in Fig. 1. The\nexchange \felds induced by the ferromagnetic insulators\nis supposed to be perpendicular to the graphene plane.\nTherefore, Hamiltonian of the spin up detaches from the\nspin down. Two gate electrodes can be attached to the\nferromagnetic graphene from the upper and lower layers\nwhich control the barrier height in each of layers. This\nset-up is di\u000berent from the systems studied by Refs.19,20.\nThe exchange \feld splits this potential depending on the\nspin parallel (+) or antiparallel ( \u0000) to the exchange \feld.\nSo in the ferromagnetic part of bilayer graphene, we have\nV\u0006=V0\u0007\u0001 where \u0001 and V0are the exchange \feld and\nthe potential barrier made by the gate voltage, respec-\ntively. So the two spins are scattered from the barriers\nwith di\u000berent heights. It means that energy shift of the\ntop of the valance band in the barrier is di\u000berent for par-\nallel and antiparallel spins to the exchange \feld. This\nspin splitting causes to shift conductance as a function\nof energy for each spin resulting in magnetoresistance.\nTo investigate spin polarization and also magnetoresis-\ntance, we have considered two di\u000berent con\fgurations so\nthat the exchange \feld inducing by the magnetic insula-\ntors on each layer are parallel or antiparallel with respect\nto each other. The con\fguration prepared for observa-\ntion of magnetoresistance di\u000bers from the con\fguration\nconsidered by Ref.19,20. The parallel con\fguration has\na metallic behavior while the antiparallel con\fguration\ninduces a potential di\u000berence between upper and lower\nlayers concluding that the system has a semiconductor\nbehavior with the band gap of 2\u0001.\nThis paper is organized as follows: we brie\ry explain\nthe formalism which is used for calculating of transmis-\nsion based on the four-band Hamiltonian. Before present-\ning our results, it is so important to have a short review in\nsection III on transport through a barrier deposited on bi-\nFIG. 2: Schematic view of the potential barrier with height\nV0and width wand the wave number at normal incidence\ndirected in the x-axis for three regions.\nlayer graphene and its dependence on the system param-\neters such as energy of incident quasi-particles and their\nangle hitting into the barrier and also barrier parame-\nters. The method presented in section II is a detailed\nanalysis accompanied with some small corrections on the\nmethod used by Ref.17. We will present spin polarization\nin the parallel con\fguration in section IV. Magnetoresis-\ntance and its dependence on energy of incident particles\nand also induced magnetic \feld will be investigated in\nsection V. Finally, the last section concludes our results.\nII. FORMALISM\nIn the unit cell of bilayer graphene, we suppose that\ntwo independent sublattices A and B related to each\nmonolayer graphene are connected to each other in the\nBernal stacking. Close to the Dirac points and in the\nnearest neighbor tight binding approximation, the four\nband Hamiltonian and also its eigenfunction is written\nas the following:\nH=0\nBB@V1\u0019 t?0\n\u0019yV10 0\nt?0V2\u0019y\n0 0\u0019 V 21\nCCA;\t =0\nB@ A\n B\n B0\n A01\nCA (1)\nwhere\n\u0019= (px+ipy)vf=\u0000i~vf(@x\u0000ky)\nand in the above formula ky=ksin\u0012where\u0012andkare\nthe incident angle and wave number of quasi-particles\nhitting on a barrier which is created by applying a bias\nto a metallic strip deposited on bilayer graphene. More-\nover,V1andV2are the gate potentials applied on the\nupper and lower layers of bilayer graphene. Such a gate\npotential can be manipulated by applying a perpendic-\nular electric \feld on graphene sheet. Here, the barrier\nis approximated by a square potential of barrier with\nsharp variation. By solving the eigenvalue equation of3\nH\t =E\t, the four band spectrum can be concluded as\nthe following:\n(\"0)2=k2+\u000e2+(t0)2\n2\u0006t0r\n4k2\u000e2=(t0)2+ (t0\n2)2+k2(2)\nwhere the above parameters are de\fned as the follow-\ning:\n\"0= (E\u0000V0)=~vf=\"\u0000\u001d0; V 0= (V1+V2)=2\n\u000e= (V1\u0000V2)=2~vf; t0=t=~vf:(3)\nIn the case of \u000e= 0 andk << t0, in low energy limit,\nenergy spectrum behaves as E\u0000V0=\u0006~2k2=2mwhere\nm=t=(2v2\nf) is an e\u000bective mass. This approximation\nwhich results in a e\u000bective two-band Hamiltonian is valid\nwhen energy of incident electrons is close to the barrier\nheight. On the other word, in the case of zero poten-\ntial di\u000berence between two layers, the absolute value of\n(E\u0000V0) should be much smaller than the interlayer cou-\npling strength (0.4eV). For \u000e6= 0, this approximation\nmay fail for large potential di\u000berences. So one should\ncare to choose valid energy and potential ranges. How-\never, in spite of Ref.18, in this paper, we use four-band\nHamiltonian in which the only approximation is the Dirac\ncone.\nIf we assume plane wave solution for the Schroedinger\nequation, the wave function in each region with a con-\nstant potential is written as the following matrix prod-\nuct.\n\t =GM0\nB@a\nb\nc\nd1\nCA (4)\nwhere matrix elements of the matrices GandMare de-\n\fned as the following:\nG=0\nBB@1 1 1 1\nf+\n+f+\n\u0000f\u0000\n+f\u0000\n\u0000\nh+h+h\u0000h\u0000\ng+\n+h+g+\n\u0000h+g\u0000\n+h\u0000g\u0000\n\u0000h\u00001\nCCA\nM(x) =0\nBB@ei\u000b+x0 0 0\n0e\u0000i\u000b+x0 0\n0 0 ei\u000b\u0000x0\n0 0 0 e\u0000i\u000b\u0000x1\nCCA(5)\nf+\n\u0006=\u0006\u000b+\u0000iky\n\"0\u0000\u000e; f\u0000\n\u0006=\u0006\u000b\u0000\u0000iky\n\"0\u0000\u000e\ng+\n\u0006=\u0006\u000b++iky\n\"0+\u000e; g\u0000\n\u0006=\u0006\u000b\u0000+iky\n\"0+\u000e\nh\u0006=(\"0\u0000\u000e)2\u0000\u000b2\n\u0006\u0000k2\ny\nt0(\"0\u0000\u000e)(6)\nFIG. 3: Transmission at normal incidence \u0012= 0 as a function\nof energy di\u000berence between energy of incident particles and\nthe barrier height ( E\u0000V0)=~vFfor potential di\u000berence of\nupper layer against lower layer \u000e= 0 and 10meV .\nHere\u000bis the wave number in the x direction and is de-\n\fned as:\n\u000b2\n\u0006= [\u000e2+(\"0)2\u0000k2\ny\u0006p\n4(\"0)2\u000e2+ (t0)2((\"0)2\u0000\u000e2)] (7)\nIn the special case of normal incident angle and zero gate\npotential where ky=\u000e= 0, the wave number \u000b+is real\nin the energy range of 0 < \"0< t0and\u000b\u0000is real if\nthe energy of incident particles is in the range of \"0<\n0;\"0>\u0000t0. In this paper, studied system contains a\nmagnetic or electrostatic barrier of potential as shown in\nFig.2. The barrier width is w. The electrostatic potential\nwhich plays the role of the gate voltage is set to be V0\nin the barrier part and zero in the \frst and last regions.\nWe suppose that the energy range of incident particles\nis limited to the range of 0 < \"0< t0. Consequently\nin the barrier part we have \u0000V0< \"0\n2< t0\u0000V0. The\nwave numbers behind and in front of the barrier \u000b(1)\n+and\n\u000b(3)\n+are real while \u000b(1)\n\u0000and\u000b(3)\n\u0000are imaginary. In the\nbarrier part, for \"0\n2<0, the wave numbers \u000b(2)\n+and\u000b(2)\n\u0000\nare imaginary and real, respectively, while for \"0\n2>0\nthey behave vice versa. A schematic view of the barrier\nat normal incidence and wave numbers in each part are\nshown in Fig.2.\nBy applying continuity of the wave functions on the\nboundaries of the barrier, one can connect coe\u000ecient ma-\ntrix of the wave function for the last region A3to the\ncoe\u000ecient matrix for the \frst region A1.\nA1=NA 3\nN=M\u00001\n1(0)G\u00001\n1G2M2(0)M\u00001\n2(w)G\u00001\n2G3M3(w)(8)\nwhere N is called as transfer matrix. Since \u000b(1)\n\u0000and\u000b(3)\n\u0000\nare imaginary in the interested energy range, that part of4\nFIG. 4: Contour plot of transmission in plane of the incident\nangle and energy di\u000berence of \"0\n2= (E\u0000V0)=~vFaccompanied\nwith the band structure for potential di\u000berence of the upper\nlayer against the lower layer to be as a) \u000e= 0 and b) 10 meV .\nwave function which are associated by such wave numbers\nare exponentially a growing or decaying function. So we\nhave to set the coe\u000ecient of plane wave ei\u000b(1)\n\u0000x(cin Eq. 4)\nto be zero for the \frst region, because this part of wave\nfunction grows exponentially when x!\u00001 . Therefore,\ncoe\u000ecients matrix in the \frst region is supposed to be as\nA1= [1;r;0;eg]T, where the superscript Trefers to the\ntranspose of a matrix and egis the coe\u000ecient of growing\nevanescent state and ris the coe\u000ecient of the re\rected\npart of the wave function. In the last region, we have\nto set the coe\u000ecient ( din Eq. 4) of e\u0000i\u000b(3)\n\u0000xto be zero\nbecause this part of the wave function increases exponen-\ntially when x!1 . Therefore, the coe\u000ecients matrix in\nthe last regions is supposed to be as A3= [t;0;ed;0]T\nwheretis the coe\u000ecient of the transmitted part of wave\nfunction and edis the coe\u000ecient of decaying evanescent\nstate. In this region, there is no re\rected wave. How-\never, in equation 8 of Ref.17, matricesA1andA3have\nbeen considered to be completely displaced which leas to\ndi\u000berent results.\nBy rearrangement of the transfer matrix elements of\nEq. 8, the coe\u000ecient of transmitted part of wave function\nis derived in terms of transfer matrix elements as the\nfollowing;\nt= [N11\u0000N13N31=N33]\u00001: (9)\nSince the \frst and last regions possess similar wave num-\nbers, transmission probability is given as T=jtj2. Be-\nfore presenting our results, in the next section, we will\nshortly review transport properties through a potential\nbarrier by using the mentioned formalism.III. TRANSMISSION THROUGH A BARRIER\nON BILAYER GRAPHENE\nThe Klein tunneling in monolayer graphene results\nin a complete transmission through a barrier poten-\ntial in normal incident. However, in contrast to mono-\nlayer graphene, as a result of chiral symmetry in bilayer\ngraphene, transmission is zero for quasiparticles with en-\nergies lower than the barrier height. In the special case\nof\u000e=ky= 0, transmission through a potential barrier\ncan be analytically calculated in normal incident and for\ntwo ranges of energy \"0<0 and\"0>0.\nt=ei\u000b(1)w[cos(\u000b(2)w)\u0000iQsin(\u000b(2)w)]\u00001 (10)\nwhere\nQ=1\n2(\"0\n1\u000b(2)\n\"0\n2\u000b(1)+\"0\n2\u000b(1)\n\"0\n1\u000b(2))\nwhere in the above formula, parameters are de\fned as\n\"0\n1=\"=~vf;\"0\n2= (\"\u0000V0)=~vf. So the real part of the\nwave numbers inside and outside of the barrier part are\nde\fned as \u000b(1)= [(\"0\n1)2+\"0\n1t0]1=2and\u000b(2)= [(\"0\n2)2+\n\"0\n2t0]1=2, respectively. The energy of incident particle is\nsupposed to be always \"0\n1>0. So\u000b(1)is always real.\nFor the energy range of E <V 0, the wave number in-\nside the barrier part \u000b(2)and consequently Qare imag-\ninary so that \u000b(2)=i\u0014andQ=iqwhere\u0014andqare\nreal. As a result, transmission tends to zero as a function\nof the system parameters such as wand\"0\n2.\nT(\u0012= 0;\"0\n2<0) =tt\u0003= [cosh2(\u0014w) +q2sinh2(\u0014w)]\u00001\n(11)\nThis behavior is the trace of chiral symmetry in bilayer\ngraphene. However, if the incident angle is nonzero,\nsome resonant peaks appear in the transmission curve\n(see Fig. 4). For the energy range of E >V 0, all param-\neters such as \u000b(2)andQare real. Thus transmission has\nan oscillatory behavior as a function of \"0\n2as the following\nform,\nT(\u0012= 0;\"0\n2>0) = [cos2(\u000b(2)\n+w) +Q2sin2(\u000b(2)\n+w)]\u00001\n(12)\nIn the high energies limit E\u001dV0, we haveQ\u0000!1\nand so transmission is complete ( T\u0000!1). By apply-\ning a vertically electric \feld in the barrier part, a band\ngap is opened in the band structure of bilayer graphene\nwhich is proportional to the potential di\u000berence between\npotentials of each layers. In this case, chiral symme-\ntry is failed and therefore transmission in normal inci-\ndence is nonzero for energies lower than the barrier height\n(E < V 0). Transmission at normal incidence is repre-\nsented in Fig. 3 as a function of \"0\n2for\u000e= 0 and 10meV .\nApplication of a vertically electric \feld causes to emerge\nsome resonant tunneling states for energies of E <V 0. In5\nFIG. 5: Contour plot of transmission in plane of the incident\nangle and the barrier width for the barrier height 50 meV , the\nenergy of incident angle 17 meV and for the case of \u000e= 0.\nFIG. 6: Energy band structure for parallel and antiparallel\ncon\fgurations. In the parallel con\fguration, the direction of\nthe exchange \felds inducing in each layers are parallel with\nrespect to each other. a) for spin parallel and antiparallel to\nthe exchange \feld direction in the parallel con\fguration. b)\nfor both spins up and down in the antiparallel con\fguration.\nthis energy range, resonant states originates from inter-\nference of the incident and scattered waves. For all cases\nsuch as nonzero incident angles and \u000e6= 0, the resonant\npeaks are interpreted by the resonance condition relation\nproposed as\n\u000bbw=n\u0019 (13)\nwhere\u000bbis the x-component of the wave number inside\nthe barrier which can be calculated by Eq. 7.\nTo have more complete view, we prepare a contour\nplot of transmission in plane of the incident angle and \"0\n2\nwhich is shown in Fig. 4 for a \fxed width of the barrier.\nFor the normal incidence ( \u0012= 0), transmission behavior\nis compatible with the results shown in Fig. 3.\nFor energies higher than the barrier height \"0\n2>0,\ntransmitting channels are opened over all ranges of en-\nergies. However, transmitting window for the incident\nFIG. 7: a) Conductance and b) spin polarization as a func-\ntion of\"0\n2for di\u000berent induced exchange \felds \u0001 in the par-\nallel con\fguration. Here, the barrier height and width are\nconsidered to be as 50 meV and 40 nm, respectively.\nangles is limited with the condition that \u000b(2)\n+(in Eq. 7)\nis real. In the case of \u000e= 0, the range of incident an-\ngle in which transmission is high can be extracted as\n\u0000sin\u00001p\n(\"0\n2)2+\"0\n2t0\nk\u0014\u0012\u0014sin\u00001p\n(\"0\n2)2+\"0\n2t0\nk. Therefore,\nby increasing \"0\n2, the range of angles with high trans-\nmission becomes more extended. In the energy range of\n\"0\n2<0, independent of the value of \u000e, resonant peaks\nemerge for nonzero incident angles ( \u00126= 0) which obey\nthe resonance condition \u000bbw=n\u0019. So additional to some\nresonant energy states, we have some resonant widths in\nwhich transmission is high. Fig. 5 shows transmission\nin plane of the incident angle and the barrier width for\n\"0<0 and\u000e= 0. It is shown that based on the reso-\nnance condition (Eqs. 13,7), in large incident angles, \u000bb\nreduces and so in a \fxed resonance order ( n), the reso-\nnance condition is satis\fed for wide barriers. Therefore,\nthe resonance strips with complete transmission shown\nin Fig. 5, depend strongly on the incident angle in the\nrange of wide barriers.\nBy applying a vertically electric \feld in the barrier\npart, a band gap is opened around \"0\n2= 0. This band\ngap also has a trace in transmission as a transport gap\nshown in Fig. 4b.\nIV. RESULTS\nBy application of an averaged gate voltage V0, band\nstructure in the barrier part is shifted by V0value. Fig. 6\nshows band structure of parallel and antiparallel con\fgu-\nration magnetic insulators when a gate voltage is applied\non the barrier part. In case the exchange \felds inducing\nin each layers of bilayer graphene are parallel, particles\nwith spin parallel (spin up) and antiparallel (spin down)\nto the exchange \felds are scattered from barriers with dif-6\nFIG. 8: a) Conductance and b) spin polarization as a func-\ntion of barrier width for di\u000berent induced exchange \felds \u0001\nin the parallel con\fguration. Here, the barrier height and in-\ncident energy are considered to be as 50 meV and 17 meV,\nrespectively.\nferent heights. In the parallel con\fguration, spin splitting\nof the barrier potential in the ferromagnetic graphene is\nwritten asV\u0000\u0000V+= 2\u0001. Such spin splitting is also seen\nin the band structure that is shown in Fig. 6a. It is seen\nthat the top of valance band are shifted to lower/higher\nenergies for spins up/down. However, in the antiparallel\ncon\fguration, the band structure shown in Fig. 6b is the\nsame for both up and down spins. A band gap which is\nproportional to 2\u0001 appears in the band structure of the\nantiparallel con\fguration.\nA. Spin Polarization\nHere, there is a correspondence between the band\nstructure and transmission. According to the band struc-\nture, we expect to emerge spin polarization just for par-\nallel con\fguration because energy bands for up and down\nspins are shifted by 2\u0001 with respect to each other. How-\never, since the band structure for antiparallel con\fgu-\nration is the same for both spins, it is not expected to\nhave spin polarization for this con\fguration. The spin\npolarization is de\fned as:\nP=Gup\u0000Gdown\nGup+Gdown(14)\nwhereGupandGdown are conductance for up and down\nspins. The conductance is calculated by using Landauer\nformalism in the linear regime. Therefore, conductance\nis proportional to angularly averaged transmission pro-\njected along the current direction.\nG=Z\u0019=2\n\u0000\u0019=2T(E;cos (\u0012))cos\u0012d\u0012It is clear that additional to the transmission curves\n(Fig.4), resonance peaks also appears in conductance.\nSince up and down spins in the parallel con\fguration see\nbarriers with di\u000berent heights, resonance peaks in con-\nductance as a function of Fermi energy Eare shifted to\nhigher energies as \u0001 for spin down and to lower energies\nas\u0000\u0001 for spin up. This mismatching of conductance\npeaks for two spins causes to a large spin polarization at\nresonance states. Fig. 7 displays conductance and spin\npolarization as a function of \"0\n2for the parallel con\fgu-\nration. It is shown that conductance peaks and conse-\nquently spin polarization appears in the energy range of\n\"0\n2<0. It is seen that by inducing an exchange \feld, con-\nductance peaks in Fig. 7a split into two peaks which are\nrelated to each spin. This spin splitting is about 2\u0001. Spin\npolarization shown in Fig. 7b has an oscillatory behavior\nwith energy of incident particles for energies lower than\nthe barrier height \"0\n2<0. The amplitude of spin polar-\nization increases with the induced exchange \feld \u0001 and\nreaches to its maximum value. However, spin polariza-\ntion tends to zero for energies greater than the potential\nheight\"0\n2>0 except at E\u0018V0.\nIn the parallel con\fguration and for \"0\n2<0, Fig. 8a\nshows that conductance in the resonance widths has a\npeak. These peaks which are also seen in the transmission\ncurves of Fig. 5 are explained by the resonance condition\nof Eq. 13. It is shown that spin splitting of conductance\npeaks also appears in the resonance widths which is orig-\ninated from di\u000berent barrier heights for two spins up and\ndown. It should be noted that the conductance at res-\nonance widths decreases for wide barriers. In the wide\nrange of widths, the angularly window for transmitting\nchannels shown in Fig. 5 decreases with the widths.\nFig. 8b shows spin polarization as a function of the\nbarrier width. Again, spin polarization has an oscilla-\ntory behavior with the barrier width. The amplitude of\nspin polarization strongly increases by an increase of the\ninduced exchange \feld. Therefore, to manifest this spin\npolarization, we should manufacture the ferromagnetic\ngraphene part with the special widths in which spin po-\nlarization reaches to the value of unity.\nB. Magnetoresistance\nIn this section, we will show that by switching between\nparallel and antiparallel con\fgurations, one can obtain\nlarge magnetoresistance. Magnetoresistance is de\fned as\nthe following:\nMR =Gp\u0000Gap\nGp+Gap(15)\nwhereGp=Gp\nup+Gp\ndownandGap=Gap\nup+Gap\ndownare\nconductance for parallel and antiparallel con\fgurations.\nFig. 9 displays conductance in the parallel and antipar-\nallel con\fgurations and also magnetoresistance as a func-\ntion of\"0\n2and the barrier width for a \fxed exchange \feld7\nFIG. 9: Conductance in the parallel and antiparallel con\fg-\nurations as a function of a) \"0\n2= (E\u0000V0)=~vFfor a barrier\nwith the width of 40 nm and, c) the barrier width for a barrier\nwith the height of 50 meV. Magnetoresistance as a function\nof b)\"0\n2for a barrier with the width of 40 nm, d) the barrier\nwidth for a barrier with the height of 50 meV. The induced\nexchange \feld is considered to be as \u0001 = 5meV.\n\u0001 = 5meV . As we before expressed, spin splitting at\nthe resonance states (for \"0\n2<0) emerges in conductance\npeaks in the case of the parallel con\fguration. This be-\nhavior is clear in Fig. 9a and 9c. However, this splitting\nwill not occur for the case of antiparallel con\fguration.\nTherefore, large magnetoresistance appears around the\nconductance resonance peaks. In the parallel con\fgura-\ntion, a band gap appears around the barrier edge in the\nintervalV0\u0000\u0001<E <V 0+\u0001. This band gap has a trace\nin transmission and consequently conductance. Zero con-\nductance region around the barrier edge \"0\n2\u00180 which is\nseen in Fig.9a, is a result of the band gap. Since there is\nno such a band gap in the parallel con\fguration, magne-\ntoresistance as shown in Fig.9b reaches to its maximum\nvalue in the energy band gap. In the energy range greater\nthan the barrier height \"0\n2>0, there is no spin splitting\nand therefore, magnetoresistance tends to zero.\nAs we showed before, conductance has peak at reso-\nnant widths. Similar to the previous case, spin splitting\noccurs just for the parallel con\fguration. So magnetore-\nsistance increases around the resonance widths. The os-\nFIG. 10: a) Conductance in the parallel and antiparallel con-\n\fguration and b) magnetoresistance as a function of the in-\nduced exchange \feld \u0001 for a barrier with the height of 50\nmeV and energy of incident particles as 40 meV. Here the\nbarrier width is 20 nm.\ncillatory behavior of magnetoresistance as a function of\nthe barrier width is represented in Fig.9d.\nAs we showed, there is a large magnetoresistance\naround the barrier edge E\u0019V0. In this range of ener-\ngies, we investigate the dependence of magnetoresistance\nto the induced exchange \feld. This exchange \feld of\ngraphene can be controlled by an in-plane external elec-\ntric \feld9. Fig. 10b represents that magnetoresistance\nincreases monotonically by increasing the exchange \feld\n\u0001. It is interesting by increasing the exchange \feld up\nto 10meV , magnetoresistance reaches to its maximum\nvalue.\nTo explain this behavior, we investigate the depen-\ndence of conductance on the exchange \feld in the par-\nallel and antiparallel con\fgurations. In the antiparal-\nlel con\fguration, the band gap which is limited in the\ninterval of V0\u0000\u0001< E < V 0+ \u0001, enhances by in-\ncreasing the exchange \feld. Therefore, conductance in\nthe antiparallel con\fguration goes to zero when the ex-\nchange \feld is increased. Suppression of the conductance\nwith the exchange \feld in the antiparallel con\fguration\nis shown in Fig. 10a. However, in the parallel con\fg-\nuration, conductance increases by enhancement of the\nexchange \feld. The reason of this enhancement comes\nback to have larger angularly transmitting windows for\nlarger\"0\n2(see Fig. 4). In fact, e\u000bective potential for spins\nupV+=V0\u0000\u0001 is decreased by an increase in \u0001. So\n\"0\n2= (E\u0000V0)=~vFfor a \fxed energy is increased and\nconsequently Gupand soGpis increased by \u0001. As a\nconclusion, for the exchange \felds up to 10 meV, sup-\npression of Gapand an increase of Gpresults in a large\nmagnetoresistance which is so useful for designing spin\nmemory devices.8\nV. CONCLUSION\nWe have studied spin polarization and magnetoresis-\ntance of a normal/ferromagnetic/normal junction of bi-\nlayer graphene by using transfer matrix method and\nbased on the four-band Hamiltonian. Transport prop-\nerties simultaneously is controlled by two gate electrodes\n(V0), which are applied on the ferromagnetic graphene.\nTwo con\fgurations of the exchange \feld is considered\nperpendicular to the graphene sheet. This exchange \feld\nis induced by the proximity of a localized magnetic or-\nbital in a magnetic insulator coating on top of each layers\nof bilayer graphene in the barrier part. In the parallel\ncon\fguration which graphene has a metallic behavior, a\nspin splitting 2\u0001 occurs for the conductance at the reso-nant states just for energies lower than the barrier height\nE < V 0. However, there is no spin splitting in the an-\ntiparallel con\fguration. A band gap of 2\u0001 is opened\nin the antiparallel con\fguration which makes it a semi-\nconductor. As a result of spin splitting in the parallel\ncon\fguration, an oscillating spin polarization emerges for\nenergies lower than the barrier height. Furthermore, an\noscillatory of magnetoresistance with large amplitude is\nachievable for E <V 0when we are able to switch between\ntwo con\fgurations. There is also a large magnetoresis-\ntance in the energy range around the barrier edge origi-\nnating from the band gap which is openned by a vertically\nelectric \feld. In this range of energy, magnetoresistance\nreaches to its maximum value when the exchange \feld is\nincreased by an in-plane external electric \feld.\n\u0003Electronic address: cheraghchi@du.ac.ir\n1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.\nZhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov,\nScience 306, 666 (2004).\n2D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B. 74, 155426 (2006); Hongki Min, J. E. Hill, N. A.\nSinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald,\nPhys. Rev. B. 74, 165310 (2006); Y. Yao, F. Ye, X. L.\nQi, S. C. Zhang, and Z. Fang, Phys. Rev. B. 75, 041401,\n(2007).\n3B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Nat.\nPhys. 3, 192 (2007).\n4N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, B.\nJ. van. Wees, Nature, 448, 571 (2007).\n5O. V. Yazyev and L. Helm, Phys. Rev. B. 75, 125408\n(2007); T. O. Wehling, K. S. Novoselov, S. V. Morozov,\nE. E. Vdovin, M. I. Katsnelson, A. K. Geim and A. I.\nLichtenstein, Nano Lett. 8, 173, (2008); N. M. R. Peres, F.\nGuinea and A. H. Castro Neto, Phys. Rev. B. 72, 174406\n(2005).\n6N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys.\nRev. B. 72, 174406 (2005).\n7Y. W. Son, M. L. Cohen and S. G. Louie, Nature, 444,\n347 (2006).8H. Haugen, D. Huertas-Hernando, A. Brataas, Phys. Rev.\nB.77, 115406 (2008).\n9E.-J. Kan, Z. Li, J. Yang and J. G. Hou, Appl. Phys. Lett.\n91, 243116 (2007)); S. Dutta and S. K. Pati, J. Phys.\nChem. B. 112, 1333 (2008).\n10T. Yokoyama, Phys. Rev. B. 77, 073413 (2008).\n11K. S. Novoselov, et al. , Nature Physics 2, 177 (2006).\n12T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Roten-\nberg, Science 313, 951 (2006).\n13E. Castro, et. al. , Phys. Rev. Lett. 99, 216802 (2007).\n14J. Oostinga, H. Heersche, X. Liu, A. Morpurgo, L. Van-\ndersypen, Nat. Mater. 7, 151 (2008).\n15Y. Zhang, et. al , Nature, 459, 820 (2009).\n16E. McCann, V. I. Falko, Phys. Rev. Lett. 96, 086805\n(2006).\n17M. Barbier, P. Vasilopoulos, F. M. Peeters and J. Milton\nPereira, Phys. Rev. B. 79, 155402 (2009).\n18Y. Yu, Q. Liang, J. Dong, Phys. Lett. A. 375, 2858 (2011).\n19Y. G. Semenov, J. M. Zavada, K. W. Kim, Phys. Rev. B.\n77, 235415 (2008); Phys. Rev. Lett. 101, 147206 (2008).\n20V. Hung Nguyen, A. Bournel, P. Dollfus, J. Appl. Phys.\n109, 073717 (2011)."    },    {        "title": "0706.1702v1.Theory_of_the_Pseudospin_Resonance_in_Semiconductor_Bilayers.pdf",        "content": "arXiv:0706.1702v1  [cond-mat.mes-hall]  12 Jun 2007Theory of the Pseudospin Resonance in Semiconductor Bilaye rs\nSaeed H. Abedinpour,1Marco Polini,1,∗A.H. MacDonald,2B. Tanatar,3M.P. Tosi,1and G. Vignale4\n1NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy\n2Department of Physics, The University of Texas at Austin, Au stin, Texas 78712, USA\n3Department of Physics, Bilkent University, Bilkent, 06800 Ankara, Turkey\n4Department of Physics and Astronomy, University of Missour i-Columbia, Columbia, Missouri 65211, USA\n(Dated: October 29, 2018)\nThe pseudospin degree of freedom in a semiconductor bilayer gives rise to a collective mode\nanalogous to the ferromagnetic resonance mode of a ferromag net. We present a theory of the\ndependence of the energy and the damping of this mode on layer separation d. Based on these\nresults, we discuss the possibility of realizing transport -currentdriven pseudospin-transfer oscillators\nin semiconductors.\nPACS numbers: 73.21.-b, 71.10.Ca, 76.50.+g, 85.75.-d\nIntroduction —The layer degree of freedom in semicon-\nductor bilayers is often regarded [1] as an effective spin-\n1/2 pseudospin degree-of-freedom in which electrons in\nthe top layer are assigned one pseudospin state, and elec-\ntrons in the other layerthe opposite one. In the quantum\nHall regime, electron bilayers are sometimes [2] pseu-\ndospin ferromagnets. The appearance of these broken-\nsymmetry states motivates an interest in phenomena\nwhicharepseudospinanalogsoftheveryrobustmagneto-\nelectric effects which underpin spintronics in ferromag-\nnetic metals, and therefore might underpin a useful\npseudospintronic technology. Unfortunately it appears\nlikely [3] that pseudospin-ferromagnetismin semiconduc-\ntor bilayers is a phenomenon that is limited to strong\nmagnetic fields or, possibly, systems with extremely low\ndensities. This Letter is motivated by the observation\nthat one important spintronic device, the spin-transfer\noscillator [4], requires only collective spin-dynamics and\nnot spontaneous magnetic order. In a spin-transfer oscil-\nlator transport, currents drive ferromagnetic-resonance\ncollective spin-dynamics in the presence of applied fields\nstrong enough to oppose hysteretic switching. Since\npseudospin paramagnets do support a pseudospin res-\nonance collective mode, as we explain below, pseudospin\npolarized transport currents in a semiconductor bilayer\n(easily realizable using individual-layer contacting tech-\nniques [5]) could, provided that the resonance is suffi-\nciently sharp, drive collective pseudospin dynamics and\nyield a device with similar functionality. In this Letter\nwe report on a theory of the damping of the pseudospin\nresonance which suggests that it is possible to design bi-\nlayers with sharp pseudospin resonances.\nIn the absence of tunneling a semiconductor bilayer\nsupports two types of collective excitations [6, 7]: (i)\nan optical mode (the ordinary plasmon) with a long-\nwavelength dispersion relation ∝q1/2and weak damp-\ning (∝q2), in which electrons in the two layers oscillate\nin phase, and (ii) an acoustic plasmon with linear dis-\npersion and strong Landau damping ( ∝q) by particle-\nhole excitations, in which the electrons in the two layersoscillate out of phase. The pseudospin resonance is a\nq= 0 collective mode which develops from the out-of-\nphase plasmon when interwell tunneling is enabled. In\nthe pseudospin language interwell tunneling favors sym-\nmetric bilayerstates and thereforeacts like a pseudospin-\nmagnetic field, which we take to act in the ˆ zdirection.\nThepseudospinresonancethen involvescollectivepreces-\nsion around this pseudospin field, with ˆ y-direction pseu-\ndospins representing current flowing between the layers\nand ˆx-directionpseudospins representingchargeaccumu-\nlation in one of the layers. Theoretical treatments of the\npseudospin resonance have so far relied on the random\nphase approximation [8] (RPA), sometimes with Hub-\nbard or local-density-approximation corrections [8, 9].\nThese papers demonstrate that interactions shift the res-\nonance away from the position of the q= 0 particle-hole\nexcitations, eliminating the Landau damping process. In\nthis Letter we present a theory of the pseudospin trans-\nfer resonance that is based on a systematic expansion\nin powers of the difference V−between intra- and inter-\nlayer electron-electron interaction. We obtain an expres-\nsion for the interaction-induced resonance position shift\nwhich is exact to leading order in V−, and an expression\nfor the leading order damping contributions which ap-\npears at second order in V−. Damping of the pseudospin\nresonance is similar to damping of the ferromagnetic res-\nonance [10, 11] in a metal, except that it is intrinsic and\ndriven by electron-electron interactions rather than dis-\norder. The physical mechanism of damping is the pro-\nduction of two (or more) electron-hole pairs with zero\ntotal momentum. The phase space for these processes\nimplies that the damping rate is proportionalto the cube\nof the resonance frequency, implying that the resonance\nis sharp whenever its energy is small compared to the\nFermi energy of the bilayer system.\nThe model — In a semiconductor bilayer, electrons in the\nsame layer interact through the two-dimensional (2 D)\nCoulomb interaction Vs(q) = 2πe2/(ǫq) (ǫis the dielec-\ntric constant), while electrons in different layers are cou-\npled through the interlayer Coulomb interaction Vd(q) =2\nVs(q)e−qd. We assume a spatially constant inter-layer\ntunneling amplitude which we denote by ∆ SAS/2 and\npresent our theory using a pseudospin representation in\nwhich the tunneling term is diagonal, i.e.the represen-\ntation in which |↑∝angbracketrightrefers to the symmetric combination\nof single-layer states and |↓∝angbracketrightto the antisymmetric com-\nbination. The total Hamiltonian is then ( /planckover2pi1= 1)\nˆH=−∆SASˆSz\ntot+/summationdisplay\nk,α,σk2\n2mˆc†\nk,α,σˆck,α,σ\n+1\n2S/summationdisplay\nq/negationslash=0V+(q)ˆρqˆρ−q+2\nS/summationdisplay\nq/negationslash=0V−(q)ˆSx\nqˆSx\n−q.(1)\nHereσis the real-spin label, αis the pseudospin label, S\nis the sample area, ˆ ρq=/summationtext\nk,α,σˆc†\nk−q/2,α,σˆck+q/2,α,σand\nˆSa\nq=/summationtext\nk,α,β,σˆc†\nk−q/2,α,σ(τa\nαβ/2)ˆck+q/2,β,σare the total\ndensity and the pseudospin operators ( τabeing Pauli\nmatrices with a=x,y,z),ˆSa\ntot=ˆSa\nq=0, and, finally,\nV±(q) = [Vs(q)±Vd(q)]/2.\nTheory—The theory we develop in this Letter is based\non the observation that the difference between the intra-\nand inter-layer interaction V−(q) =πe2(1−e−qd)/(ǫq)\nis always smaller than πe2d/ǫ, which becomes a small\nperturbation when d≪max(rsaB,aB/r2\ns). Here rs=\n(πna2\nB)−1/2is the Wigner-Seitz density parameter and\naB=ǫ/(me2) is the Bohr radius. The above inequal-\nity guarantees that the third term in the Hamiltonian\n(1) is a small perturbation either compared to the ki-\nnetic energy [ ∼e2/(r2\nsǫaB)] which dominates in the high-\ndensity limit, or compared to the interaction energy\n[∼e2/(rsǫaB)] which dominates in the low-density limit.\nWe will therefore perform a systematic expansion forthe pseudospin resonance frequency and damping rate\nin powers of V−(q). Our approach will be asymptotically\nexact in the limit d≪aB, and is expected to be qualita-\ntively correct for d∼aB.\nWe determine the properties of the pseudospin res-\nonance by evaluating the transverse pseudospin re-\nsponse function χSxSx(q,ω) =∝angbracketleft∝angbracketleftˆSx\nq;ˆSx\n−q∝angbracketright∝angbracketrightω/S, where\nwe have introduced the Kubo product ∝angbracketleft∝angbracketleftˆA;ˆB∝angbracketright∝angbracketrightω=\n−ilimǫ→0+/integraltext+∞\n0dteiωte−ǫt∝angbracketleftΨGS|[ˆA(t),ˆB(0)]|ΨGS∝angbracketright[12].\nSince our aim is to calculate the transverse mode at\nq= 0 we will focus on the response function χSxSx(ω)≡\nχSxSx(q= 0,ω) [13]. The in-plane pseudospin operators\nsatisfy the Heisenberg equations of motion,\n\n\n∂tˆSx\ntot= ∆SASˆSy\ntot\n∂tˆSy\ntot=−∆SASˆSx\ntot−2\nS/summationdisplay\nkV−(k)(ˆSz\nkˆSx\n−k+ˆSx\nkˆSz\n−k);\n(2)\nˆSx\ntot, which measures the difference between charges in\nthe two layers, is a good quantum number when ∆ SAS→\n0 whereas ˆSy\ntotis not conserved even in this limit because\nof the pseudospin-dependent interactions. When d→0\nthese equations reduce to a pseudospin version of Lar-\nmor’s theorem, in which the precession is undamped and\nits frequency is given exactly by the non-interacting par-\nticle value ∆ SAS.\nOur theory starts by making repeated use of Eqs. (2)\nin the Kubo product identity [12, 14]: ∝angbracketleft∝angbracketleftˆA;ˆB∝angbracketright∝angbracketrightω=\n∝angbracketleftΨGS|[ˆA,ˆB]|ΨGS∝angbracketright/ω+i∝angbracketleft∝angbracketleft∂tˆA;ˆB∝angbracketright∝angbracketrightω/ω.After some alge-\nbraic manipulations we arrive at the following exactex-\npression for χSxSx(ω)\nχSxSx(ω) =Mz∆SAS\nΩ2+4∆2\nSAS\nΩ4S2/summationdisplay\nkV−(k)f(k)+2iω∆SAS\nΩ4S2/summationdisplay\nkV−(k)g(k)+4∆2\nSAS\nΩ4S3/summationdisplay\nk,k′V−(k)V−(k′)L(k,k′,ω).(3)\nHere Ω2=ω2−∆2\nSAS,Mz=∝angbracketleftΨGS|ˆSz\ntot|ΨGS∝angbracketright/S\nis the ground-state pseudospin magnetization per unit\narea,f(k) =∝angbracketleftΨGS|ˆSz\nkˆSz\n−k|ΨGS∝angbracketright − ∝angbracketleftΨGS|ˆSx\nkˆSx\n−k|ΨGS∝angbracketright,\ng(k) =∝angbracketleftΨGS|ˆSx\nkˆSy\n−k|ΨGS∝angbracketright+∝angbracketleftΨGS|ˆSy\nkˆSx\n−k|ΨGS∝angbracketright, and\nL(k,k′,ω) =∝angbracketleft∝angbracketleft[ˆSz\nkˆSx\n−k+ˆSx\nkˆSz\n−k];[ˆSz\nk′ˆSx\n−k′+ˆSx\nk′ˆSz\n−k′]∝angbracketright∝angbracketrightω.\nNotice that f(k) is purely real, g(k) is purely imaginary,\nandL(k,k′,ω) has both a real and an imaginary part.\nThe symmetric interaction V+does not appear explicitly\nin Eq. (3). When V−is set to zero ( d→0), the inter-\naction part of the Hamiltonian is pseudospin invariant.\nLarmor’s theorem then applies to the pseudospin degree-\nof-freedom and only the first term on the right hand-side\nof Eq. (3) survives. We refer to the Hamiltonian ˆHatV−= 0 as the reference system (RS), on which the per-\nturbative scheme outlined below is based.\nThe key idea now is to expand χSxSx(ω) in powers of\nV−. For example, the ground-state pseudospin magneti-\nzationMzis expanded as Mz=Mz\n0+Mz\n1+Mz\n2+...,\nwhere the n-th term Mz\nnisO(Vn\n−). The quantities f,g,\nandLare similarly expanded. Note that the zero-th or-\nder off(k), denoted by f0(k), is a non-zero difference\nbetween longitudinal and transverse pseudospin struc-\nture factors. On the other hand, the zero-th order of\ng(k), denoted by g0(k) vanishes because the RS Hamil-\ntonian is invariant under rotations by 90 degrees about\nthe ˆz-axis in pseudospin space which map ˆSx→ˆSyand\nˆSy→ −ˆSx.3\nThe pseudospin resonance frequency is the solution of\nthe equation ℜe[χ−1\nSxSx(ω⊥)] = 0. The inverse of χSxSx\ncanbeexpandedwiththehelpoftheformula χ−1\nSxSx(ω) =\nχ−1\n0(ω)−χ−2\n0(ω)χ1(ω)+χ−2\n0(ω)[χ−1\n0(ω)χ2\n1(ω)−χ2(ω)]+\n..., whereχn(ω) is then-th order term in the expansion\nforχSxSx. To appreciate the power of Eq. (3) we first\nuse it to find ω⊥to first order in ¯d=d/aB. Keeping only\nterms up to first order in Eq. (3) and making use of the\nformula above for the power series of χ−1\nSxSx(ω) we obtain\nχ−1\nSxSx(ω) =ω2−∆2\nSAS\nMz\n0∆SAS/parenleftbigg\n1−Mz\n1\nMz\n0/parenrightbigg\n−4V−(0)\n(Mz\n0)21\nS2/summationdisplay\nkf0(k)+O(¯d2),(4)\nwhich implies immediately that\nω2\n⊥= ∆2\nSAS+4∆SASV−(0)\nMz\n01\nS2/summationdisplay\nkf0(k)+O(¯d2).(5)\nThis equation is exact to all orders in the intralayer\nCoulomb interaction Vs. In the high-density (non-\ninteracting) limit one can find simple analytical expres-\nsions for Mz\n0andf0(k),Mz\n0= (nS−nAS)/2 and\nS−2/summationtext\nkf0(k) = (Mz\n0)2/2. Herenα=k2\nFα/(2π) are the\nband occupation factors, kFαbeing the Fermi wavenum-\nber for band α. In this limit Eq. (5) simplifies to\nω2\n⊥= ∆2\nSAS+2∆SASMz\n0V−(0)+O(¯d2). The secondterm,\nwhich supplies the interaction induced shift in the pseu-\ndospin resonance position, is a factor of two smaller thanin RPA theory [8]. The source of this difference is easy\nto understand: our calculation includes the first-order\nexchange corrections to the resonance frequency which\nare absent in the RPA. Since V−is independent of qat\nfirst order in d, corresponding to a δ-function interaction\nin real space, the like-real-spin contribution to the res-\nonance position shift present in the RPA is canceled by\nexchange interactions.\nThe main object of this work is to estimate the reso-\nnance decay rate, which appears first at second-order in\nV−andiszerointheRPA.(Additionalinteractioncorrec-\ntions to the resonance position ℜe(ω⊥), which we do not\ndiscuss at length, also appear at second order [15].) The\nlinewidth of the pseudospin resonance [= −2ℑm(ω⊥)] is\ngiven, up to second order in d, by\nΓ⊥=−4V2\n−(0)∆SAS\nMz\n0×lim\nω→∆SASℑm ℓ0(ω)\nω.(6)\nwhereℓ0(ω) is the wavevector sum of the four-spin cor-\nrelation function L(k,k′,ω). This quantity can be evalu-\natedanalyticallyin the high-densitykinetic-energydomi-\nnated limit in which it is dominatedby a decayprocessin\nwhich two particle-hole pairs are excited out of the Fermi\nsea, one involvinga pseudospin-flip. The second particle-\nhole excitation is diagonal in pseudospin and absorbs the\nmomentum emitted by the first. We find that\nℑm ℓ0(ω) =−π\n2(Mz\n0)3δ(ω−∆SAS)−π\n2S3/summationdisplay\nk,k′,k′′/summationdisplay\nα,βδ(ω−Ωα(k,k′))nk′′,αnk′′−k+k′,β(1−nk′′+k′,β)(1−nk′′−k,¯α)(7)\nwherenk,α= Θ(kFα− |k|) and Ω α(k,k′) =k·k′/m+\nα∆SAS. The first term on the right-hand-side of Eq. (7)\ndoes not contribute to the life-time of the shifted res-\nonance. In Fig. 1 we illustrate the dependence of\nℑm ℓ0(ω) onω. Theω3dependence at small ωis the\ndouble-particle-hole excitation manifestation of the fa-\nmiliar Pauli-blocking reduction in the excitation density-\nof-states in a Fermi sea which underlies Fermi liquid\ntheory; damping drops much more rapidly at low en-\nergies than for ferromagnetic resonance [11] dominated\nby single-particle decay processes. Eqs. (3), (5), (6), and\n(7)constitutethemostimportantresultsofthisworkand\nprovide, to best of our knowledge, the first microscopic\ntheory of the pseudospin resonance linewidth.\nNumerical results and discussion — Typicalnumerical re-\nsultsforΓ ⊥, calculatedfromEqs.(6)and(7)areshownin\nFigs. 2 and 3. In Fig. 2 we showΓ ⊥as a function of ∆ SASfor a bilayer with density n= 8.3×1010cm−2and inter-\nlayer distance d=L+w= 50˚A. Here L= 40˚A is the\nwidth of each quantum well and w= 10˚A is the barrier\nwidth (we have chosen material parameters correspond-\ningtoaGaAs/AlGaAsbilayer). Thenon-analyticbehav-\nior of Γ ⊥for ∆SAS∼3 meV is due to the transition from\nthesituationinwhichbothsymmetricandantisymmetric\nbands are occupied to that in which only the symmetric\nband is occupied. In Fig. 3 we illustrate the dependence\nof Γ⊥on density for a fixed value of ∆ SAS= 1.48 meV.\nSince the resonance frequency is close to ∆ SAS, these cal-\nculations predict that the pseudospin resonance can be\nvery sharp, especially when ∆ SASis small compared to\nthe Fermi energy of the bilayer. On physical grounds\nwe expect that the main effect of going to higher order\nindwill be to replace the bare interlayer interaction in\nEq.(6)byaweakerscreenedinteraction, furtherreducing4\n00.0050.010.0150.02\n0 5 10 15 20−ℑm ℓ0(ω)\nω(meV)012345\n0 0.1 0.2 0.3 0.4 0.5−ℑm ℓ0(ω)×105\nFIG. 1: (Color online) Imaginary part of the dynamical re-\nsponse function ℓ0(ω) (in units of eV−1nm−6) as a function\nofωfor a bilayer electron gas with n= 8.3×1010cm−2and\n∆SAS= 1.48 meV. The δ-function contribution at ω= ∆SAS\n[see first term in Eq. (7)] has been subtracted. The red solid\nline is the asymptotic result ℑm ℓ0(ω→ ∞) =−mn2/32.\nInset: a zoom of the low-energy region. The solid red curve is\nthe expression ℑm ℓ0(ω) =−γω3withγ≃3.41×10−4[16].\n00.020.040.060.080.1\n0 1 2 3 4 5Γ⊥(meV)\n∆SAS(meV)S2D\nQ2D\nFIG. 2: (Color online) Intrinsic linewidth Γ ⊥of the pseu-\ndospin resonance as a function of ∆ SASfor a bilayer with\ndensityn= 8.3×1010cm−2andd= 50˚A. TheS2 Dcurvewas\nevaluated using the bare 2 Dinteractions Vs(q) andVd(q) de-\nfined above whereas the Q2 Dresult was evaluated with more\nrealistic interactions weakened by form factors [17] which ac-\ncount for typical quantum well widths.\nthe damping.\nOur theory of the resonance amounts to the deriva-\ntion of an anisotropic , linearized pseudospin Landau-\nLiftshitz-Slonczewski equation:\n\n\n∂tMx= ∆SASMy−I\ne\n∂tMy=−ω2\n⊥+Γ2\n⊥\n∆SASMx−2Γ⊥\n∆SAS∂tMx; (8)\nwhereMais the average macroscopic pseudospin po-\nlarization, which becomes equal to ∝angbracketleftˆSa\ntot∝angbracketrightin the limit\nI→0. In the first line of Eq. (8) we have added a Slon-\nczewski [18] pseudospin transfer term proportionalto the\ntunnel current I, which is injected in one layer and ex-\ntracted from the other. As in the ferromagnetic case, it00.020.040.060.08\n0 5 10 15 20Γ⊥(meV)\nn(1010cm−2)S2D\nQ2D\nFIG. 3: (Color online) Intrinsic linewidth Γ ⊥of the pseu-\ndospin resonance as a function of nfor a bilayer with tunnel-\ning gap ∆ SAS= 1.48 meV and d= 50˚A. The labels S2 Dand\nQ2Dhave the same meaning as in Fig. 2.\nis the reaction counterpart of the torque which acts on\nthe transport quasiparticles to enable their transfer be-\ntween layers upon moving through the sample, and must\nbe present because of the nearly exact conservation of\npseudospin by interactions. In the second line of Eq. (8)\nwe haveadded a Gilbert-like damping term ∝∂tMx(the\nanisotropy of the Gilbert damping in the present prob-\nlem derives from the stronglyanisotropiccharacterof the\ninteraction part of the Hamiltonian). These equations\n[which describe a damped pseudospin precession of fre-\nquencyω⊥and damping rate Γ ⊥about the steady state\nvaluesMy(t→ ∞) =I/(e∆SAS),Mx(t→ ∞) = 0]\nare similar to those which describe spin-transfer torque\noscillators [4] in ferromagnets and suggest that simi-\nlar, and possibly more flexible, devices could be real-\nized in semiconductor bilayers. We anticipate that the\npseudospin resonance in ferromagnets will have negative\nrather than positive dispersion, because of the qdepen-\ndence of V−(q). The roles of this property, and the fact\nthat the single-particle and collective excitation frequen-\ncies are not widely separated, are difficult to fully antici-\npate. Nevertheless, this work suggests that experimental\nstudies of non-linear transport in bilayers have great po-\ntential.\nAcknowledgments — We thank Vittorio Pellegrini for\nhelpful discussions. M.P. acknowledges the hospitality\nof the Department of Physics and Astronomy of the Uni-\nversity of Missouri-Columbia. A.H.M. was supported by\nthe Welch Foundation, the ARO, and SWAN-NRI. G.V.\nwas supported by NSF Grant No. DMR-031368.\n∗Electronic address: m.polini@sns.it\n[1] A.H. MacDonald, P.M. Platzman, and G.S. Boebinger,\nPhys. Rev. Lett. 65, 775 (1990).\n[2] J.P. Eisenstein and A.H. MacDonald, Nature 432, 6915\n(2004).\n[3] S.ContiandG.Senatore, Europhys.Lett. 36, 695(1996);\nL. Zheng, M.W. Ortalano, and S. Das Sarma, Phys. Rev.\nB55, 4506 (1997).\n[4] S.I. Kiselev et al., Nature 425, 380 (2003); W.H. Rip-\npardet al., Phys. Rev. Lett. 92, 027201 (2004); A.A.\nTulapurkar et al., Nature 438, 339 (2005).\n[5] J.P. Eisenstein, L.N. Pfeiffer, and K.W. West, Appl.\nPhys. Lett. 57, 2324 (1990).\n[6] S. Das Sarma and A. Madhukar, Phys. Rev. B 23, 805\n(1981); G.E. Santoro and G.F. Giuliani, ibid.37, 937\n(1988).\n[7] R. Decca et al., Phys. Rev. Lett. 72, 1506 (1994); A.S.\nPlautet al., Phys.Rev. B 55, 9282 (1997); D.S.Kainth et\nal., J. Phys.: Condens. Matter 12, 439 (2000) and Phys.\nRev. B59, 2095 (1999); S. Holland et al., Phys. Rev. B\n66, 073305 (2002).\n[8] S. Das Sarma and E.H. Hwang, Phys. Rev. Lett. 81,\n4216 (1998). Notice that within this RPA calculation the\nfrequency of transverse plasmon at q= 0 is a strictly\nlinear function of the interlayer separation d.\n[9] P.G. Bolcatto and C.R. Proetto, Phys. Rev. Lett. 85,\n1734 (2000).\n[10] See for example V. Korenman and R.E. Prange, Phys.\nRev. B6, 2769 (1972); J. Sinova et al., Phys. Rev. B69, 085209 (2004); Y. Tserkovnyak, A. Brataas, G.E.W.\nBauer, and B.I. Halperin, Rev. Mod. Phys. 77, 1375\n(2005) and other related works cited in these papers.\n[11] E.M. Hankiewicz, G. Vignale, andY. Tserkovnyak, Phys.\nRev. B75, 174434 (2007).\n[12] G.F. Giuliani and G. Vignale, Quantum Theory of\nthe Electron Liquid (Cambridge University Press, Cam-\nbridge, 2005).\n[13] The other in-plane pseudospin susceptibilities are co m-\npletely determined by χSxSx(ω): we have χSySx(ω) =\n−iωχSxSx(ω)/∆SASandχSySy(ω) =−M/∆SAS+\nω2χSxSx(ω)/∆2\nSAS, whereMis defined after Eq. (3).\n[14] R. Nifos` ı, S. Conti, and M.P. Tosi, Phys. Rev. B 58,\n12758 (1998); I. D’Amico and G. Vignale, ibid.62, 4853\n(2000); Z. Qian and G. Vignale, Phys. Rev. Lett. 88,\n056404 (2002).\n[15] The expression for the frequency of the transverse pole\nup to order ¯d2is cumbersome and will be presented else-\nwhere.\n[16] The analytical expression for γis rather cumbersome and\nwill be presented elsewhere.\n[17] See e.g.Eqs. (15) and (16) in R. Cˆ ot´ e, L. Brey, and A.H.\nMacDonald, Phys. Rev. B 46, 10239 (1992).\n[18] J.A. Slonczewski, J. Magn. Magn. Mater. 159, L1(1996)."    },    {        "title": "0910.4678v1.Spin_torque_driven_ferromagnetic_resonance_in_a_nonlinear_regime.pdf",        "content": "arXiv:0910.4678v1  [cond-mat.mes-hall]  24 Oct 2009Spin-torque driven ferromagnetic resonance in a nonlinear regime\nW. Chen,1G. de Loubens,1J-M. L. Beaujour,1J. Z. Sun,2and A. D. Kent1\n1Department of Physics, New York University, New York, NY 100 03\n2IBM T. J. Watson Research Center, Yorktown Heights, NY 10598\n(Dated: September 14, 2009)\nSpin-valve based nanojunctions incorporating Co |Ni multilayers with perpendicular anisotropy\nwere used to study spin-torque driven ferromagnetic resona nce (ST-FMR) in a nonlinear regime.\nPerpendicular field swept resonance lines were measured und er a large amplitude microwave cur-\nrent excitation, which produces a large angle precession of the Co|Ni layer magnetization. With\nincreasing rf power the resonance lines broaden and become a symmetric, with their peak shifting\nto lower applied field. A nonhysteretic step jump in ST-FMR vo ltage signal was also observed at\nhigh powers. The results are analyzed in in terms of the foldo ver effect of a forced nonlinear oscil-\nlator and compared to macrospin simulations. The ST-FMR non hysteretic step response may have\napplications in frequency and amplitude tunable nanoscale field sensors.\nSpin-torque driven ferromagnetic resonance (ST-\nFMR) has proven to be a powerful technique to in-\nvestigate magnetic anisotropy, damping and spin-torque\ninteractions in nanometer scale metallic ferromagnets\n[1, 2, 3, 4]. The method involves injecting an rf cur-\nrent into a laterally confined giant magnetic resistance\nor magnetic tunnel junction device that drives one of the\nmagnetic layers into resonance. Homodyne mixing of the\nrf current and device resistance oscillations results in a\ndc voltage across the device.\nIn our earlier studies, ST-FMR was conducted on spin\nvalve junctions that incorporate Co |Ni multilayers that\nhave perpendicular magnetic anisotropy and a thick Co\nreference magnetic layer[5, 6]. The Co |Ni layer magnetic\nanisotropy was measured with rf currents in the sub-\nmA range, well below the devices’ spin-transferswitching\nthreshold of ≃8 mA. In these studies the rf current was\nmodulated on and off at a low frequency ( ∼kHz), and a\nlock-in amplifier was used to measure the mixing voltage,\nV. Measurements were conducted in a linear response\nregime in which the homodyne signal was proportional\nto the rf power, i.e. where the field swept resonance line,\nV(H)/I2\nrfwas independent of Irf. HereIrfis the ampli-\ntude of the rf current. The resonance lines were sym-\nmetric and magnetization precession angles were small\n(typically around 2◦and no more than 6◦).\nIn this Letter we present ST-FMR studies in a non-\nlinear regime, where the amplitude of the rf current is\nmuch larger. The resonance field and the linewidth of\nthe Co|Ni layer have been determined with increasing rf\ncurrent amplitude and the rf current is not modulated,\nenablinganinvestigationofthe effect offieldsweepdirec-\ntion onthe resonancelineshape. We analyzethese results\nwith a foldover theory of nonlinear oscillators that pre-\ndicts a shift in resonance field and a hysteretic response,\nwithalineshapethatdependsonthefieldsweepdirection\n(Fig. 1(b)). We also compare our results to macrospin\nsimulations.\nPillar junctions with a lateral dimension of 50 ×150\nnm2(Fig. 1(a)) were patterned on a silicon wafer us-\ning a nanostencil process [7]. Junctions were deposited\nusing metal evaporation with the layer structure /bardbl1.5(b) (a)\nI\nˆmpV\nM\n Mz\nxωMt\ny\nˆ m pM2\nt\n(H−Ho)/∆H\nFIG. 1: (a) Sample layer structure and ST-FMR circuit. (b)\nThe foldover effect: precession amplitude vs field at fixed\nrf frequency for three different normalized rf power levels,\nβ= 0.2βc,βcand 3βc(βc= 8/(3√\n3)). In the last case the\ncurve is multivalued, to the left of the resonance there is a\nlow and high amplitude branch, leading to field sweep direc-\ntion dependent hysteresis (red arrows). The dashed part of\nthe curve is an unstable oscillation. Inset shows the magne-\ntization configuration. The applied field is in the zdirection\nand the magnetization of the Ni |Co layer precesses about this\ndirection at an angular frequency ωand with a transverse\nmagnetization Mt.\nnm Cr|100 nm Cu |20 nm Pt |10 nm Cu |[0.4 nm Co |0.8\nnm Ni]×3|10 nm Cu |12 nm Co |200 nm Cu /bardbl, where\nthe thin (free) layer is composed of a Co |Ni multilayer\nand the thick (fixed) layer is pure Co.\nST-FMR measurements were conducted with a mag-\nnetic field applied nearly perpendicular (within ∼2◦)\nto the sample surface at room temperature. An rf cur-\nrent generated by a high frequency source is amplified by\n20 dB and is added to a dc current using a bias-T (Fig.\n1(a)). Positivedccurrentsaredefined suchthatelectrons\nflow from the free layer to the fixed layer. The magne-\ntoresistance is 2.1% and the pillar resistance is 1.53 Ω in\nthe parallel state (Fig. 1(b) of Ref. [5]). The inset of\nFig. 1(b) shows the magnetization configuration. With\nan applied field greater than 0.3 Tesla, the Co |Ni multi-\nlayeris magnetized along z, perpendicular to the plane of\nthe layers. However, since the Co layer has a large mag-\nnetization, µ0Ms≃1.5 T, it remains largely magnetized\nin plane, as represented by the vector ˆ mP. rf currents2\n0 2 4 6 8 10345670.550.600.65\n(c)\nVpeak/Irf [m:]\nIrf[mA]\nP0Hres [Tesla](b)\n0.5 0.6 0.7 0.8406080100120140-10 0 10\n4.54.6Current Bias [mA]\ndV/dI [Ohm]\nX 2V [PV]\nMagnetic Field [Tesla](a)\nFIG.2: (a)ST-FMRsignal as afunctionoftheappliedfieldat\ndifferentrfcurrentslevelsfor f= 16GHz. Irfwas 1.4, 2.1, 3.3,\n4.6, 6.2, 7.8, and 9.0 mA respectively. Each adjacent curve\nis offset by 10 µV. The resonance at Irf=1.4 mA is amplified\nby a factor of 2 above the background of 37 µV. Inset: dV/dI\nvsI of atµ0H=0.675 T, which is the resonance field at f=16\nGHz in the linear regime. (b) Magnetic field where the step\njump occurs vsIrf. (c)Vpeak/IrfvsIrf. The peak voltage is\nmeasured with respect to the baseline background voltage.\ncause the magnetization of the Co |Ni multilayer, M, to\nprecess with a transverse amplitude Mtat an angular\nfrequency ω.\nST-FMR was conducted with a constant rf power, and\na dc voltmeter was used to measure the mixing voltage.\nThe applied field was swept up and down at a rate 0.12\nT/min. Fig. 2(a) shows an example of such measure-\nments. In this case, the rf frequency fis 16 GHz and\nIrfranges between 1.4 and 9.0 mA. Each curve is offset\nby 10µV for clarity. At the lowest rf amplitude ( Irf=1.4\nmA), there is a 37 µV non-resonant background, which\nwe associate with nonlinearities in our measurement cir-\ncuit. This offset varies with the rf amplitude, which we\nattribute to other parasitic rectification processes in our\nmeasurement circuits, such as slightly non-ohmic con-\ntacts and junction heating effects. At Irf=1.4 mA we are\nin the linear response regime, the lineshape is symmetric,\nand the resonance field is 0.675 T. The critical current\nforspin-torque-induceddynamics atthis field is ∼8.5mA\n(inset of Fig. 2(a)). The lineshape begins to become\nasymmetric at Irf=2.1 mA. Starting from Irf=3.3 mA a\nstep jump of the mixing voltage occurs on the lower-field\nside of the resonance. The maximum ST-FMR voltage\nincreases as Irfincreases, and the step-jump field contin-\nues to decrease.\nThe shift of the resonance peak with increasing power\ncan be understood in terms of forced oscillator in which\nthe frequency of oscillator depends on the oscillation am-\nplitude, i.e. the system is a driven anharmonic oscillator.\nThe micromagnetic energy of the free layer can be writ-\nten as:\nE=−µ0M0mzH+1\n2µ0M0Meffm2\nz,(1)wheremz=Mz/M0andMeff=M0−2K1/(µ0M0) is\nthe effective easy-plane anisotropy. The first term is\nthe Zeeman energy and the second term is written in\nterms of Meff, which contains contributions due to shape\nanisotropy ( M0) and the layer perpendicular magnetic\nanisotropy ( K1). The resonance condition in this ge-\nometry (applied field along z) is given by [8], 2 πf=\nγµ0(Hres−Meffmz), where γis the gyromagnetic ratio.\nFor a fixed frequency the resonance field is given by:\nHres=2πf\nµ0γ+Meffmz (2)\nAs the precession amplitude mt=Mt/M0increases,\nthe longitudinal component of magnetization, mz≃\n1−m2\nt/2, decreases resulting in a shift of the resonance\nto lower fields ( µ0Meff≃0.2 T for our Co |Ni multilayer),\nas observed in our experiments.\nFor small rf current amplitudes the lineshape is\nLorentzian:\nm2\nt=aI2\nrf\n(H−H0)2+(∆H)2(3)\nwith a half width at half maximum of ∆ H=αω/(γµ0),\nwhereais aconstantthat depends onthe spin-torqueper\nunit current and the magnetization direction of the fixed\nmagnetic layer, ˆ mP.αis the Gilbert damping constant.\nWith increasingprecessionamplitude one finds [9, 10, 11,\n12] to first order in m2\nt:\n˜m2\nt=1\n(˜ǫ+β˜m2\nt)2+1. (4)\nHere ˜mt=mt/mtmaxis the transverse component of\nmagnetization normalized to the maximum precession\namplitude, m2\ntmax=aI2\nrf/(∆H)2; ˜ǫ= (H−H0)/∆H.\nThe resonance peak position is given by: ˜ ǫ=−β=\n−aMeffI2\nrf/[2(∆H)3]. So the resonance peak position is\npredicted to shift with the rf power, I2\nrf.\nFig. 1(b) shows a plot of ˜ mversus field for three\ndifferent rf amplitudes. There is a critical value of\nβc= 8/(3√\n3)≃1.54, which corresponds to the foldover\nthreshold for the excitation amplitude Irf, for which the\nleftsideoftheresonancecurvehasaverticaltangent. For\nsufficiently large current, β > βc, the resonance curve is\nmultivalued, with two stable/metastable branches differ-\ning in amplitude that overlap on the low field side of the\nresonance in Fig. 1(b). The dashed part of the curve in\nFig. 1(b) is an unstable oscillation. As a result hysteresis\nis predicted in field swept measurements, as indicated by\nthe red arrows in the figure. Hysteresis begins when the\nshift of the resonance peak is greater than approximately\nthe resonance linewidth ( Hres−H0=βc∆H).\nThe experimental data (Fig. 2(a)) have features qual-\nitatively consistent with the foldover model described\nabove: (1) The resonance lines become asymmetric with\nincreasing power; (2) There is a step jump in ST-FMR\nvoltage above a critical rf amplitude; and (3) The reso-\nnance peak position decreases with increasing rf power3\n0.5 0.6 0.7 0.8 020406080100\n4.5\n 5.6\n 6.3\n 7.1\n 7.9\n 8.9\n 10\n 11.2\n 12.6\n 14.1\n 15.8\n 17.8 mAV [PV]\nMagnetic Field [Tesla]Irf=(a)\n0.550.600.650.70\n0 5 10 15 0246\nP0Hres [Tesla](b)\n(c)\nVpeak/Irf [m:]\nIrf [mA]\nFIG. 3: Macrospin micromagnetic simulations of ST-FMR.\n(a)Voltageversusappliedfieldforvariousrfcurrents, sho wing\nthe high amplitude branch of the curve. (b) Resonance peak\nposition and (c) Vpeak/IrfvsIrf. The black squares refer to\nthe low amplitude branch and the red circles refer to the high\namplitude branch of the resonance curve.\n(Fig. 2(b)). However, there are discrepancies with the\nmodel, particularly the fact that little hysteresis is ob-\nserved at large rf amplitudes, when the resonance shift\nfar exceeds the resonance linewidth, µ0∆H= 0.02 T.\nAlso, the resonance field shifts nearly linearly with in-\ncreasing rf current.\nTo further understand the data we have conducted\nmacrospin simulations of the magnetization dynamics\nwith an applied rf current. We take the micromagnetic\nenergy given by Eq. 1 and an angular dependence of\nthe spin-torque and magnetoresistance determined using\ncontinuousrandommatrixtheory(CRMT)forourdevice\nlayer stack [13]. This model has been found to give spin-\ntorquesinagreementwiththosefoundinourexperiments\nconducted in the linear response regime [5]. The torque\nis parameterized by τST=Ik(θ)(/planckover2pi1/2e)ˆm×(ˆm׈mp)\nwithk(θ) = (a+bsinθ)/(c+dcosθ) (a= 1.98, b=\n7.11, c= 28.74, d= 25.95). The results of our simula-\ntions are shown in Fig. 3. The high amplitude branch,\nwith extends to lower field, is shown in Fig. 3(a). The\nresonance lines shapes are similar to those observed in\nexperiment. Fig. 3(b) shows the shift of the peak posi-\ntion for both of high and low amplitude branches as well\nas the peak voltage. For large Irfthe resonance peak po-sition shifts linearly with IrfandVpeak/Irfsaturates at\nlargeIrf(Fig. 3(c)), asseen in experiment. The main dis-\ncrepancy is that larger rf currents are needed to produce\nthe resonance shifts observed.\nIt also appears that the resonance at high rf currents\nfollows the high amplitude branch independently of the\nfield sweep direction. The evidence for this is the fol-\nlowing. First, little to no hysteresis is observed. Second,\nthe resonance shift far exceeds the linewidth as well as\nthe resonanceshift found for the low amplitude branch in\nmicromagnetic simulations (Fig. 3(b)). We suspect that\nthere are thermally driven transitions between these dy-\nnamical states that excite the high amplitude branchand\nreduce the hysteresis. The rf current may also heat the\ndevice reducing the magnetization of the free layer, in-\ncreasingthermal fluctuations and leading to alargershift\nof the resonance peak position with rf current than seen\nin the macrospin model. We also note that our analy-\nsis and macrospin simulations assume that the damping\ndoes not depend on precession amplitude. However, it\nhas been noted that the damping may be nonlinear and\nthis mayplayaroleinunderstandingthe largeamplitude\nST-FMR driven magnetization dynamics [14].\nIt is interesting to estimate the maximum precession\nangle,θmax. This can be done using the voltage peak\nas well as maximum resonance shift. The peak mixing\nvoltage is given by V=1\n4∆RIrfsinηsinθmax, where ∆ R\nis the junction magnetoresistance and ηis the angle be-\ntween the layers with no rf current applied ( η≃70◦, in\nthese experiments). The precession angle can also be es-\ntimated from the shift of the resonance field using Eq.\n2. Both approaches give θmax∼65◦atIrf=9.3 mA, the\nlargest rf current applied in the junction.\nIn sum, these results illustrate that large rf currents\ncan drive nonlinear magnetization dynamics, character-\nistic of any driven anharmonic oscillator. The observed\nnonhysteretic step response may prove useful for rf fre-\nquency and amplitude tunable nanometer scale field sen-\nsors. It will also be interesting to compare these re-\nsults to full micromagnetic simulations as well as to fur-\nther explore the role of thermal fluctuations on ST-FMR\nfoldover phenomena.\nWe thank A. N. Slavin for useful discussions and X.\nWaintal for the CRMT code used in our macrospin sim-\nulations. This research is supported by NSF-DMR-\n0706322 and the ARO-W911NF0710643.\n[1] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub-\nota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira,\nN. Watanabe, and S. Yuasa, Nature 438, 339 (2005).\n[2] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N.\nKrivorotov, R.A. Buhrman, andD. C. Ralph, Phys. Rev.\nLett.96, 227601 (2006).\n[3] J.N.Kupferschmidt, S.Adam, andP.W. Brouwer, Phys.\nRev. B74, 134416 (2006).[4] X. Wang, G. E. W. Bauer, B. J. van Wees, A. Brataas,\nand Y. Tserkovnyak, Phys. Rev. Lett. 97, 216602 (2006).\n[5] W. Chen, J.-M. L. Beaujour, G. de Loubens, A. D. Kent,\nand J. Z. Sun, Appl. Phys. Lett. 92, 012507 (2008).\n[6] W. Chen, G. de Loubens, J.-M. L. Beaujour, A. D. Kent,\nand J. Z. Sun, J. Appl. Phys. 103, 07A502 (2008).\n[7] J. Z. Sun, Appl. Phys. Lett. 81, 2202 (2002).\n[8] C. Kittel, Introduction to Solid State Physics (John Wi-4\nley & Sons, New York, 2005), 8th ed.\n[9] E. Schl¨ omann, Technical Report No. R-48 (unpublished)\n(1959).\n[10] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon,\nOxford, 1976).\n[11] Y. K. Fetisov, C. E. Patton, and V. T. Synogach, IEEE\nTrans. Magn. 35, 4511 (1999).[12] P. W. Anderson and H. Suhl, Phys. Rev. 100, 1788\n(1955).\n[13] V. S. Rychkov, S. Borlenghi, H. Jaffres, A. Fert, and\nX. Waintal, Physical Review Letters 103, 066602 (2009).\n[14] V. Tiberkevich and A. Slavin, Phys. Rev. B 75, 014440\n(2007)."    },    {        "title": "1007.3577v1.Exchange_anisotropy_pinning_of_a_standing_spin_wave_mode.pdf",        "content": "APS/unknown\nExchange anisotropy pinning of a standing spin wave mode\nR. Magaraggia, M. Kostylev, K. Kennewell, and R. L. Stamps\nSchool of Physics, University of Western Australia,\n35 Stirling Highway, Crawley, Western Australia 6009, Australia\nM. Ali, D. Greig, B. J. Hickey, and C. H. Marrows\nSchool of Physics and Astronomy, E. C. Stoner Laboratory,\nUniversity of Leeds, Leeds LS2 9JT, United Kingdom\n(Dated: November 6, 2018)\nStanding spin waves in a thin \flm are used as sensitive probes of interface pinning induced by an\nantiferromagnet through exchange anisotropy. Using coplanar waveguide ferromagnetic resonance,\npinning of the lowest energy spin wave thickness mode in Ni 80Fe20/Ir25Mn75exchange biased bilayers\nwas studied for a range of IrMn thicknesses. We show that pinning of the standing mode can be\nused to amplify, relative to the fundamental resonance, frequency shifts associated with exchange\nbias. The shifts provide a unique `\fngerprint' of the exchange bias and can be interpreted in terms\nof an e\u000bective ferromagnetic \flm thickness and ferromagnet/antiferromagnet interface anisotropy.\nThermal e\u000bects are studied for ultra-thin antiferromagnetic Ir 25Mn75thicknesses, and the onset\nof bias is correlated with changes in the pinning \felds. The pinning strength magnitude is found\nto grow with cooling of the sample, while the e\u000bective ferromagnetic \flm thickness simultaneously\ndecreases. These results suggest that exchange bias involves some deformation of magnetic order in\nthe interface region.\nPACS numbers: 75.30.Gw, 75.70.Cn, 76.50.+g\nI. INTRODUCTION\nExchange bias is an e\u000bect which has consequences\nfor the bulk of a ferromagnet as exhibited by hystere-\nsis loop o\u000bset. However its bulk e\u000bects arise from cou-\npling processes across a ferromagnetic/antiferromagnetic\ninterface[1, 2]. Directly probing these types of buried\ninterfaces to gain information on coupling is quite chal-\nlenging. Ferromagnetic resonance (FMR) is a powerful\ntool for studying magnetic parameters in ferromagnetic\nstructures through frequency shifts of the fundamental\nresonance mode. It is possible to also use FMR to detect\nstanding spin waves which provide, at least in principle,\ninformation about surfaces and buried interfaces[3{5]. In\nthis paper standing spin waves (also referred to as \\thick-\nness modes\") are used to probe interface properties due\nto exchange anisotropies in exchange biased bilayers. We\nshow that a useful measure for characterising exchange\nbias can be obtained from these modes, and this measure\ncan provide unique information about magnetic ordering\nin the interface region.\nNearly all studies of ferromagnetic resonance and spin-\nwaves in exchange biased structures have, to date, made\nuse exclusively of the fundamental resonance or zone cen-\nter spinwaves[4, 6, 7]. The frequencies of these excita-\ntions are governed primarily by local magnetocrystalline\nand shape anisotropies, magnetization, and applied \feld.\nThe resonance conditions for a ferromagnetic thin \flm\nwith no intrinsic anisotropies, and magnetised in plane,\nis given by[8]:(!\n\r)2= (Hf(\u0012) +Dk2\ny(\u0012))(Hf(\u0012) +\u00160Ms+Dk2\ny(\u0012))\n(1)\nThe spin wave frequency is !,\ris the gyromagnetic ra-\ntio,Msis the saturation magnetisation, Hfis the \feld\napplied to cause resonance, and \u0012is the direction of the\napplied \feld relative to the cooling \feld direction. A \fxed\nspin wave frequency is assumed and \u0012is varied, so that\nHfbecomes the experimentally meaured quantity. The\nwavevector component in the direction normal to the \flm\nplane isky. The\u00160Msterm originates from dynamic de-\nmagnetisation \felds in thin \flm geometry, and D=2A\nMsis the exchange coupling strength. In traditional treat-\nments of FMR as applied to exchange-bias the funda-\nmental FMR mode corresponds to k= 0. E\u000bective \felds\noriginating at the interface with the antiferromagnet are\nthen, as far as the FMR response is concerned, averaged\nover the ferromagnetic \flm thickness and are seen as an\ne\u000bective anisotropy \feld. In a resonance experiment us-\ning a \fxed frequency, these e\u000bective \felds appear in the\nmeasured value of Hf, the applied \feld for which res-\nonant absorption is observed. It is important to note\nthat the frequency shifts of the FMR associated with ex-\nchange bias do not contain direct information about the\ninterface region per se. Questions concerning the pene-\ntration depth of the interface \felds, or asymmetries asso-\nciated with di\u000berent boundaries, can only be addressed\nindirectly by varying \flm thicknesses within a series of\nsamples. A disadvantage of this approach is that samples\ncan vary substantially, even within the same series due\nto details of growth processes[2, 9].\nThe FMR mode averages local interface \felds laterallyarXiv:1007.3577v1  [cond-mat.mtrl-sci]  21 Jul 20102\nbecause it is a long wavelength excitation, though in re-\nality it does experience deformation due to the interfacial\npinning. In some cases, short wavelength spin waves can\nbe observed with conventional resonance techniques as\nstanding wave thickness modes con\fned by \flm geome-\ntry. It is access to these modes which allows a measure\nof interface pinning. Recently we have shown theoreti-\ncally and experimentally that broadband FMR driving\ntechniques that make use of stripline or coplanar waveg-\nuides can couple e\u000bectively to thickness modes in metal-\nlic multilayers[10, 11]. These thickness modes have some\ndiscrete wavevector ky(\u0012), and therefore involve contribu-\ntions from exchange. Hereafter these modes are referred\nto as \\FEX modes\". These will each have di\u000berent al-\nlowed wavevectors con\fned in the ydirection, as deter-\nmined by surface pinning. As such, the frequencies of the\nFEX modes include contributions from exchange, and are\nsensitive to surfaces and interfaces. The lower symmetry\nat \flm boundaries can give rise to local anisotropy \felds,\nand interfaces between di\u000berent magnetic layers can sup-\nport exchange coupling. In these cases, spin wave oscil-\nlations may be pinned at one or more boundaries of a\nferromagnetic \flm. Pinning of this type is accompanied\nby contributions through exchange energies, and can re-\nsult in substantial frequency shifts [12].\nA simple means of analysing frequencies obtained\nfor thickness modes was suggested long ago by Rado\nand Weertman[5, 13, 14]. In this approach, surface\nanisotropies are assumed, which then dictate the bound-\nary conditions for FEX modes in thin \flm geometries. It\nshould be noted that the FMR mode will also be a\u000bected\nand given a non-zero wavevector resulting from surface\npinning. If we associate a surface energy[15] of the fol-\nlowing form with the exchange biased interface:\nESA=p\u0001Ms (2)\nWe can then calculate allowed spin wave wavevectors as\na result of pinning. In this equation Msis the satura-\ntion magnetisaion and pis the pinning parameter which\nacts parallel to the applied \feld. As demonstrated in\n[13], if one starts with the Landau-Lifshitz equation and\nintegrates over an in\fnitesimal volume region across the\ninterface, the following is obtained:\n(2A\nMs)M\u0002@M\n@n+Tsurf = 0 (3)\nHereMrepresents the total magnetisation, nis the di-\nrection normal to the interface and Tsurf is the interface\ntorque. Using Eq.(2), we have:\nTsurf =\u0000M\u0002rMESA=\u0000M\u0002p (4)\nWe approximate the exchange biased interface by suppos-\ning the pinning to come entirely from one of the bound-\naries, hence introducing an asymmetry into the model.After solving Eq.(3) in combination with Eq.(4), the re-\nlationship between these surface anisotropies and kyfor\nHfapplied at an angle \u0012to the easy axis is:\np(\u0012) = (2A\nMs)(\u0000ky(\u0012)\ncot(ky(\u0012)teff)) (5)\nIt is important to note that teffis the magnetic thickness\nof the ferromagnet, as opposed to the structural thickness\n(which may be di\u000berent)[16, 17]. This di\u000berence may be\ncaused by deviations away from uniform ferromagnetic\norder near the interface due to local pinning \felds.\nThe remainder of the paper is organized as follows.\nFirst, preparation of, and magnetization measurements\nfrom, exchange biased Ni 80Fi20/Ir25Mn75are discussed.\nNext we present results from coplanar FMR studies of the\nfundamental and \frst thickness modes for these struc-\ntures, and discuss their interpretation in terms of the\npinning parameter pand e\u000bective thickness teff.\nII. SAMPLE PREPARATION AND\nCHARACTERIZATION\nMagnetic bi-layer specimens consisting of\nTa(50 \u0017A)/ Ni80Fe20(605 \u0017A)/ Ir25Mn75( tAF\u0017A)/ Ta(50 \u0017A)\nwere sequentially deposited onto Si(001) substrates by\ndc-magnetron sputtering at an argon working pressure of\n2.5 mTorr to minimise growth variations. A nanometer\nlayer of native oxide on the silicon surface created condi-\ntions for polycrystalline growth. Typical deposition rates\nwere 2\u00002.5\u0017As\u00001, which were determined by measuring\nthe thickness of calibration \flms by low-angle x-ray\nre\rectometry. The base pressure prior to the deposition\nwas of the order of 1 \u000210\u00008Torr and the samples were\ndeposited at ambient temperature. An in-plane forming\n\feld of 200 Oe was applied during the growth to induce\na macroscopic uniaxial anisotropy in the NiFe (Py) layer\nin a de\fned direction. The thickness of the IrMn layer,\ntAF, for this study was varied from 0 to 60 \u0017A which is\nalso the region where the onset of biasing appears at\nroom temperature for such systems[18]. The samples\nwere cut into 10mm \u000210mm squares.\nFilm thickness was accurately characterised with a\nSiemens two-circle di\u000bractometer, to within \u00066\u0017A. In-\nplane and out-of-plane FMR magnetometry was used to\nextract\u00160Ms, which could consistently be used in fur-\nther FMR data analysis. In-plane FMR magnetometry\nalong the easy axis of a Py sample with no IrMn re-\nvealed a saturation magnetisation \u00160Msof 0.80\u00060.05T,\na gyromagnetic ratio \rof 2.8\u00021010HzT\u00001and in plane\nbulk anisotropy \felds of 0.0002T \u00060.0005T. Further mag-\nnetometry was performed using the magnetoopical Kerr\ne\u000bect (MOKE). A 635nm diode laser, rated at 5mW,\nwas used to illuminate the sample. A di\u000berential ampli-\n\fer was used to analyse polarisation rotation. Example\nresults are shown in Fig. 1.\nAs demonstrated in Fig. 1, the samples saturate mag-\nnetically above 20Oe. The loops are non-symmetric3\nFIG. 1. Shown above is a sample of data taken\nwith a MOKE magnetometry setup focused onto the\nNiFe(60.5nm)/IrMn(6nm) sample. The vertical axis uses ar-\nbitrary units and represents the average magnetisation over\nthe laser spot focused onto the sample. The horizontal axis\ndisplays \feld applied across the sample in units of Oersteds.\nAlso the exchange bias shifting of the loop is shown by the\ndotted line and denoted by H EB.\nabout a non-zero \feld with a small coercivity, and com-\npare well with what has been found in similar studies[7,\n19]. The bias \feld as measured with FMR is de\fned\nasHEB=Hf+\u0000Hf\u0000\n2, shown in Fig.2, where Hf+corre-\nsponds toHf(0) in eq.1, and Hf\u0000corresponds to Hf(\u0019).\nThe coercivity increases with increasing IrMn thickness,\nwith a maximum in the region of thicknesses where there\nexists little exchange biasing.\nIII. RESONANCE MEASUREMENTS AND\nINTERPRETATION\nA 20GHz Vector Network Analyser was used to excite\nand detect FMR and FEX modes of the samples. The\ncoplanar stripline (0.3mm wide) which is coupled to 50\naxial cables, excites the sample with microwaves in the\n2-9 GHz regime. Example results are shown in Fig.2.\nWe choose a particular excitation frequency !and sweep\nthe applied magnetic \feld H(usually between 0 and 600\nOe), in a particular direction until microwave power is\nabsorbed strongly by the sample, indicating a standing\nspin wave is on resonance. This procedure is repeated\nfor the samples' easy axis aligned along di\u000berent direc-\ntions with respect to the applied \feld, denoted by \u0012. A\n\feld sweep was chosen rather then a frequency sweep, as\na \feld sweep avoids the problems of variable microwave\nfrequency attenuation in the waveguides with varying !\nand shows the magnetic response of the sample as op-\nposed to both magnetic and electric response.\nAn example of FMR and FEX resonances, at a driving\nfrequency of 7 GHz, is shown in Fig.2(b). A number of\nfactors determine the observed amplitudes of FMR and\nFIG. 2. Panel (a) shows the experimental geometry, with\nthe sample placed on top of the coplanar stripline. Hrefers\nto the applied \feld direction at some angle \u0012,Mrefers to\nthe magnetisation direction and HRFdemonstrates the mi-\ncrowave rf \feld generated by the waveguide. The sample is\nrotated in place in order to change the direction of Hwith\nrespect to the sample's easy axis. Panel (b) shows microwave\ntransmission as a function of static applied \feld for the 0nm\nIrMn sample. The values Hf\u0006correspond to applied reso-\nnant \felds in antiparallel directions for + and - respectively.\nMicrowave absorbtions are seen which correspond to the fun-\ndamental mode (FMR) and the \frst exchange mode (FEX).\nThe microwave excitation frequency !used was 7GHz.\nFEX modes in coplanar geometries [20{23], in particular\na combination of surface pinning and eddy current in-\nduced inhomogeneity in the driving microwave \feld. The\nFEX absorpton amplitude is approximately 23 times less\nthan that of the FMR mode as measured at 7GHz. The\nlinewidths of the modes at 7GHz are \u0001 fFMR =49Oe and\n\u0001fFEX=25Oe respectively. It should be noted that the\nFMR mode has a Lorentzian like absorption shape, but\nthe FEX mode does not, so the linewidths may not be\ndirectly comparable.\nThe bias determined from FMR and FEX are shown in\ncomparison to the bias determined from MOKE data in\nFig.3. Unidirectional exchange anisotropies are present\nat room temperature only for a certain critical thickness4\n>2.5nm of IrMn as shown in Fig.3.\nFIG. 3. Shown is the exchange bias as measured from the\nFMR mode (empty circle solid line), MOKE (empty diamond\nsolid line) and FEX mode (empty square solid line), as a\nfunction of IrMn \flm thickness. The NiFe layer thickness is\nalways 60.5nm. For comparison the cooercivity as measured\nwith MOKE is shown (hollow triangle dashed line)\nFor thicknesses above this value, the MOKE and FEX\nresults indicate a non-monotonic behavior of the bias\nwith respect to IrMn thickness begining at 4 nm. We\nhave at present no explanation for this, though this could\nbe due to sample to sample variation. It is possible that it\nmay have other origins, as such behaviour in similar sys-\ntems has been noted and explained via the domain-state\nmodel[24]. Most signi\fcantly, the pinning \feld is unidi-\nrectional. This is fully consistent with exchange bias as\nan interface e\u000bect. The bias acts as an e\u000bective volume\nunidirectional anisotropy when averaged by the FMR\nmode, and appears as a superposition with other volume\nanisotropies. This superposition can be seen most clearly\nby measuring bias at di\u000berent orientations of the applied\n\feld relative to the bias \feld direction. Example results\nfor the 2.5 and 6 nm thick IrMn samples are shown in Fig.\n4. Results for FMR and FEX peaks are shown as func-\ntion of angle, demonstrating that both modes contain\nequal contributions from a uniaxial anisotropy, whereas\nthe modes are a\u000bected di\u000berently by the exchange bias.\nThe results shown in Fig.4 illustrate the magnitude\nof exchange bias as measured by the FMR and FEX\nmodes. The di\u000berence in magnitude can be understood\nthrough pinning e\u000bects on the frequency of the FEX\nmodes. The FEX modes contain greater exchange energy\nthan the FMR because of their shorter wavelengths, and\npinning acts to e\u000bectively change the wavelength of an\nFEX mode. In this way, pinning by exchange bias is an\nampli\fcation of exchange anisotropy by a\u000becting directly\nthe exchange energy contribution to an FEX mode. This\nis demonstrated explicitly in Eq.(1), where the exchange-\nrelated e\u000bective anisotropy \feld Dk2\nyscales as the square\nof the wavenumber ky. Therefore one should expect dif-\nferent strengths of e\u000bective anisotropy from the FMR andFEX modes. Indeed, such di\u000berences are seen in Fig.4\nfor these two modes, con\frming the interface origins of\nthe anisotropy \felds in this exchange biased system.\nFIG. 4. Shown are the resonant \felds Hffor the FMR (empty\ncircle solid line)) and FEX (empty square dashed line) stand-\ning spin wave modes at di\u000berent applied \feld angles with\nrespect to the easy axis ( \u0012). The solid lines show \fts to the\ndata using cos(\u0012) andcos(2\u0012) components. Presented is the\nresonance data for di\u000berent IrMn thickness capping layers a)\nIrMn=0nm, b) IrMn=2.5nm, c)IrMn=6nm.\nPinning factors p, calculated according to Eq.(5) as a\nfunction of IrMn thickness are shown in Fig.5 for data\ntaken at room temperature. Interface anisotropy calcu-\nlated for the applied \feld along \u0012= 0 is denoted p+,\nand represents the situation there the applied \feld is an-\ntiparallel to the bias \feld direction. Conversely, p\u0000is5\nthe pinning calculated for the \feld applied along the bias\ndirection\u0012=\u0019. In these calculations, we have used ma-\nterial parameters determined experimentally as above.\nThe exchange coupling strength D=1 :3693\u000210\u000017JA\u00001\nwas chosen such that an e\u000bective thickness of 60.5nm\nwas extracted from the monolayer permalloy \flm. Er-\nror bars in Fig.5 were estimated by incorporating exper-\nimental \feld uncertainties. We consider pas the more\nfundamental quantity then exchange bias \feld. Pinning\nwill act with the same strength on both modes, but the\nwavelength of each mode will be distorted to a di\u000berent\ndegree. Importantly, in our \fttings we have the condition\nthatpshould have the same value for all observed modes.\nWe \fnd this condition cannot be satis\fed unless some\nvalue is modi\fed for one of the physical parameters in\nEq.5. The derivation of Eq.5 and previous works [16, 17]\nsuggest that the suitable parameter is the thickness of\nthe ferromagnetic layer. Therefore the second parameter\nextracted from the \fts is the e\u000bective thickness of the\nferromagnetic layer. As previously mentioned, the dif-\nference between teffand the structural thickness of the\nferromagnet might be related to deviation from uniform\nferromagnetic order close to the interface.\nThe dependence of pon IrMn thickness shows a cu-\nrious peak for the 4 nm thick \flm, but otherwise is a\nnearly linear function of tAFabove 2.5 nm. In addition\nto an interface pinning, we also simultaneously extract\nan e\u000bective magnetic thickness tefffrom the data. The\ngreatest change of teffwith in-plane \feld direction ap-\npears fortAFbetween 5 and 5.5nm, a range in which the\nlargest degree of exchange bias is observed with MOKE\nbut not FMR.\nLikep, the e\u000bective thickness varies as a function of\napplied \feld direction. The IrMn free permalloy layer\n(Fig.6a) does not show any signi\fcant variation of teff\nwith\u0012with the implication that no signi\fcant micromag-\nnetic con\fgurational changes take place when aligning\nthe magnetisation along di\u000berent anisotropy directions.\nThis is in sharp contrast to the 6nm IrMn \flm (Fig.6b),\nwhich does display a rougly 1 nm thickness variation of\nteffover the angular range 0 to 180o.\nAn interpretation of e\u000bective magnetic \flm thickness is\ndi\u000ecult as it does not allow identi\fcation of speci\fc mi-\ncromagnetic structures across the interface region. Nev-\nertheless, it does not seem unreasonable that teffpro-\nvides some measure of the size over which magnetization\nin the interface region contributes to pinning, perhaps\nthrough local modi\fcation of the magnetic order[17, 25].\nLastly, we discuss measured dependence of bias and\npinning on temperature for the 2.5 nm thick IrMn bi-\nlayer. This layer was most interesting because it does\nnot show signi\fcant bias at room temperature, but does\ndevelope bias at lower temperatures. A summary of re-\nsults is shown in Fig.7. A linear increase in exchange\nbias below 240K was found from the FMR mode data,\nand has been reported previously in literature[24, 26, 27].\nA linear increase in the magnitude of the pinning param-\neters was found over the same temperature region, with\nFIG. 5. a) The calculated strengths of pinning p along the\nbias direction (empty circle solid line) and against the bias\ndirection (empty square dashed line).\nb) The corresponding e\u000bective magnetic thickness teffof the\nNiFe along the bias direction (empty circle solid line) and\nagainst the bias direction (empty square dashed line).\ndi\u000berent slopes for pmeasured parallel and antiparallel to\nthe bias direction. The behaviour of teffhowever reveals\nsimilar behavior and slopes for the two \feld orientations.\nThe interfacial region involved in pinning is determined\nby the di\u000berence between values obtained from parallel\nand antiparallel orientations. This di\u000berence is about 0.5\nnm and independent of temperature.\nIV. DISCUSSION AND CONCLUSIONS\nIn this paper we have presented results for resonant\n\feld shifts due to exchange bias in NiFe/IrMn bilay-\ners. The unidirectional exchange anisotropy was de-\ntermined from angular resolved resonance experiments.\nWe observed \feld di\u000berences for the lowest order stand-\ning spin wave mode that are twice the magnitude of\nthe corresponding di\u000berence for the fundamental reso-\nnance. We show that interpretation of these results can\nbe made in terms of pinning e\u000bects due to an e\u000bective\nsurface exchange anisotropy. The distortion each spin\nwave mode experiences due to this pinning is not the\nsame for every mode. Experimentally this results in\ndi\u000berent exchange anisotropies observed for FMR and\nFEX resonances. The assumption of an e\u000bective sur-\nface anisotropy is possible because resonances of the IrMn6\nFIG. 6. The e\u000bective magnetic thickness of NiFe as a function\nof\u0012with respect to the easy axis for a) 0nm IrMn \flm, b)\n6nm IrMn \flm.\nare at much higher frequencies than those probed with\nour coplanar resonance technique, so that the NiFe spin\nwaves are driving the IrMn far o\u000b resonance. Because\nof this mismatch in frequencies, the e\u000bective \felds act-\ning on the NiFe spins near the interface are governed by\nanisotropies induced through exchange coupling to the\nIrMn, and other dynamics in the antiferromagnet can be\nsafely neglected[12, 28]. One can understand the pinning\nsimply as a unidirectional anisotropy whose magnitude\nvaries as cos( \u0012), where\u0012is the angle of the static \feld\nrelative to the bias direction.\nWhen calculating the wavevectors of the FMR and\nFEX modes, deviations from values expected assuming\nno pinning are found. Analysing the data this way re-\nturns a pinning parameter that charaterises the strength\nof interface coupling and gives an e\u000bective magnetic\nthickness over which the NiFe \flm acts as a saturated fer-\nromagnet. As the structural thickness of the NiFe \flms\nare well known, deviations from this value in teffmay\narise from the magnetisation close to the interface. Thus\none can also interpret the observed e\u000bective thickness as\nan exchange bias e\u000bect that involves a deformation of the\nmagnetization near the interface that reduces the mag-\nnetic thickness of the ferromagnet participating in the\nspin wave resonance. Such a deformation might be pos-\nsible through either pinning of ferromagnetic spins near\nFIG. 7. a) This \fgure illustrates the calculated strengths\nof pinning p along the bias direction (empty circle dashed)\nand against the bias direction (empty square dashed line) for\nthe IrMn 2.5nm \flm cooled to the temperature indicated on\nthe horizontal axis, in a 40Oe \feld. Also the complementary\ninformation on the exchange bias shift for the FMR mode\n(solid triangle solid line) and FEX mode (solid diamond solid\nline) is shown here.\nb) The corresponding e\u000bective magnetic thickness teffof the\nNiFe along the bias direction (empty circle solid line) and\nagainst the bias direction (empty square dashed line) for the\nsame range of \feld cooled temperatures.\nthe interface, or formation of a twist on the ferromagnet\nside of the interface. We note that this interpretation is\nanalogous to the e\u000bective boundary conditions derived\nby Guslienko and Slavin for dipolar contributions to res-\nonance in stripes [15].\nWe close with two \fnal remarks. First, there exists\na di\u000berence between exchange bias measurements be-\ntween FMR and MOKE of at most 30%. This is a well\nknown e\u000bect[29] and is due primarily to FMR being a\nperturbative measurement of local \felds whereas MOKE\nmeasurements of hysteresis necessarily involve magneti-\nzation processes. Though there has not previously been\nan FEX to MOKE comparison, we note that FEX fol-\nlows the same trend as the FMR data, but with di\u000berent\nmagnitude as both are perturbative measures of the ex-\nchange anisotropy. Secondly, possible e\u000bects associated\nwith \feld cooling were also sought. As shown above, the\n2.5nm IrMn sample has a blocking temperature below7\nroom temperature and that it does not experience signif-\nicant exchange biasing until below 240K.\nV. ACKNOWLEDGEMENTS\nSupport from the Australian Research Council under\nthe Discovery and Australian Postgraduate Award pro-grammes is acknowledged. Furthermore support from\nthe United Kingdom's Engineering and Physical Sciences\nResearch Council is also acknowledged.\n[1] R. Stamps, J. Phys. D: App. Phys. 33, R247 (2000).\n[2] J. Nogu\u0013 es and I. K. Schuller, Journal of Magnetism and\nMagnetic Materials 192, 203 (1999), ISSN 0304-8853.\n[3] W. Stoecklein, S. S. P. Parkin, and J. C. Scott, Phys.\nRev. B 38, 6847 (1988).\n[4] R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F.\nEgelho\u000b, Phys. Rev. B 58, 8605 (1998).\n[5] C. Kittel, Phys. Rev. 110, 1295 (1958).\n[6] B. K. Kuanr, S. Maat, S. Chandrashekariah, V. Veeraku-\nmar, R. E. Camley, and Z. Celinski, J. Appl. Phys. 103\n(2008).\n[7] J. McCord, R. Mattheis, and D. Elefant, Phys. Rev. B\n70, 094420 (2004).\n[8] C. Kittel, Introduction to Solid State Physics, 8th ed.\n(John Wiley & Sons, Inc, 2005).\n[9] A. E. Berkowitz and K. Takano, Journal of Magnetism\nand Magnetic Materials 200, 552 (1999), ISSN 0304-\n8853.\n[10] M. Kostylev, \\Strong asymmetry of microwave ab-\nsorption by bi-layer conducting ferromagnetic \flms in\nthe microstrip-line based broadband ferromagnetic res-\nonance,\" (2008).\n[11] K. J. Kennewell, M. Kostylev, M. Ali, A. A. Stashkevich,\nR. Magaraggia, D. Greig, B. J. Hickey, and R. L. Stamps,\nunpublished, arXiv: 1001.1837v1(2010).\n[12] R. L. Stamps, R. E. Camley, and R. J. Hicken, Phys.\nRev. B 54, 4159 (1996).\n[13] G. Rado and J. Weertman, J. Phys. Chem. Solids 11,\n315 (1959).\n[14] P. Yen, T. S. Stakelon, and P. E. Wigen, Phys. Rev. B\n19, 4575 (1979).\n[15] K. Y. Guslienko and A. N. Slavin, Phys. Rev. B. 72\n(2005).\n[16] S. Br uck, G. Sch utz, E. Goering, X. Ji, and K. M. Krish-\nnan, Physical Review Letters 101, 126402 (2008).[17] S. Roy, M. R. Fitzsimmons, S. Park, M. Dorn, O. Pe-\ntracic, I. V. Roshchin, Z.-P. Li, X. Batlle, R. Morales,\nA. Misra, X. Zhang, K. Chesnel, J. B. Kortright, S. K.\nSinha, and I. K. Schuller, Phys. Rev. Lett. 95, 047201\n(2005).\n[18] M. Ali, C. H. Marrows, and B. J. Hickey, Phys. Rev. B\n67, 172405 (2003).\n[19] C. Leighton and I. K. Schuller, Phys. Rev. B 63, 174419\n(2001).\n[20] M. Kostylev, K. J. Kennewell, R. Magaraggia, R. L.\nStamps, M. Ali, and B. J. Hickey(2009), http://arxiv.\norg/pdf/0908.4443 .\n[21] W. S. Ament and G. T. Rado, Phys. Rev. 97, 1558\n(1955).\n[22] R. D. McMichael, D. J. Twisselmann, and A. Kunz, Phys.\nRev. Lett. 90, 227601 (2003).\n[23] P. Pincus, Phys. Rev. 118, 658 (1960).\n[24] M. Ali, C. H. Marrows, M. Al-Jawad, B. J. Hickey,\nA. Misra, U. Nowak, and K. D. Usadel, Phys. Rev. B\n68, 214420 (2003).\n[25] H. Ohldag, A. Scholl, F. Nolting, E. Arenholz, S. Maat,\nA. T. Young, M. Carey, and J. St ohr, Phys. Rev. Lett.\n91, 017203 (2003).\n[26] C. Tsang and K. Lee, Journal of Applied Physics 53,\n2605 (1982).\n[27] S.-F. Cheng and P. Lubitz, J. Appl. Phys 87, 4927 (2000).\n[28] A. Ercole, W. S. Lew, G. Lauho\u000b, E. T. M. Kernohan,\nJ. Lee, and J. A. C. Bland, Phys. Rev. B 62, 6429 (Sep\n2000).\n[29] H. Xi, R. M. White, and S. M. Rezende, J. Appl. Phys.\n87, 4960 (2000)."    },    {        "title": "1603.05419v1.Proposal_of_a_micromagnetic_standard_problem_for_ferromagnetic_resonance_simulations.pdf",        "content": "Proposal of a micromagnetic standard problem for ferromagnetic\nresonance simulations\nAlexander Baker,1Marijan Beg,2Gregory Ashton,2Maximilian Albert,2Dmitri\nChernyshenko,2Weiwei Wang,3Shilei Zhang,1Marc-Antonio Bisotti,2Matteo\nFranchin,2Chun Lian Hu,4Robert Stamps,4Thorsten Hesjedal,1and Hans Fangohr2,\u0003\n1Department of Physics, Clarendon Laboratory,\nUniversity of Oxford, Oxford OX1 3PU, United Kingdom\n2Faculty of Engineering and the Environment,\nUniversity of Southampton, SO17 1BJ, Southampton, United Kingdom\n3Department of Physics, Ningbo University, Ningbo 315211, China\n4SUPA School of Physics and Astronomy,\nUniversity of Glasgow, Glasgow G12 8QQ, United Kingdom\nAbstract\nNowadays, micromagnetic simulations are a common tool for studying a wide range of di\u000berent\nmagnetic phenomena, including the ferromagnetic resonance. A technique for evaluating reliability\nand validity of di\u000berent micromagnetic simulation tools is the simulation of proposed standard\nproblems. We propose a new standard problem by providing a detailed speci\fcation and analysis\nof a su\u000eciently simple problem. By analyzing the magnetization dynamics in a thin permalloy\nsquare sample, triggered by a well de\fned excitation, we obtain the ferromagnetic resonance spec-\ntrum and identify the resonance modes via Fourier transform. Simulations are performed using\nboth \fnite di\u000berence and \fnite element numerical methods, with OOMMF andNmag simulators,\nrespectively. We report the e\u000bects of initial conditions and simulation parameters on the character\nof the observed resonance modes for this standard problem. We provide detailed instructions and\ncode to assist in using the results for evaluation of new simulator tools, and to help with numerical\ncalculation of ferromagnetic resonance spectra and modes in general.\nPACS numbers: 75.40.Mg; 76.50.+g; 75.70.-i\n1arXiv:1603.05419v1  [cond-mat.mes-hall]  17 Mar 2016I. INTRODUCTION\nComputational micromagnetics is a well developed \feld that sees widespread use in both\nmodern physics and magnetic device engineering communities.1{3With the advancement of\nmicromagnetic models, simulation techniques, and processing power, the list of phenomena\nthat can be studied has grown substantially and includes such diverse \felds as the spin\ntransfer torque4and spin wave dispersion in magnonic crystals.5An essential equation in\nmost of the micromagnetic system models6is the Landau-Lifshitz-Gilbert (LLG) equation\n{ a di\u000berential equation governing the magnetization dynamics. However, this equation\ncan be analytically solved only for a very limited number of systems and, because of that,\nthe complexity of common problems requires the use of micromagnetic simulation packages\nsuch as OOMMF7,LLG Micromagnetics ,8Micromagnum ,9andMumax10, which use the Finite\nDi\u000berence (FD) approach, and Nmag11and Magpar ,12employing the Finite Element (FE)\napproach to spatial discretization. To compare this range of numerical solvers, as well as\nto evaluate their validity and reliability, NIST's Micromagnetic Modelling Activity Group\n(\u0016Mag) publishes standard problems.13{15Recent additions have included the spin transfer\ntorque4and the spin wave dispersion16standard problems. In the light of this, it is natural\nto extend the coverage of standard problems in order to include the FerroMagnetic Reso-\nnance (FMR), a technique closely associated with many practical uses ranging from material\ncharacterization to the study of spin dynamics.17\nFMR probes the magnetization dynamics in samples using microwave \felds. The ab-\nsorption of the applied microwave \feld is at its maximum when the microwave's frequency\nmatches the frequency of the studied system's resonant modes. By analyzing the resonance\nmodes as a function of an applied magnetic \feld, some material parameters, such as the\nGilbert damping and magnetic anisotropy constants, can be determined.17This makes FMR\na powerful technique in the characterization of ferromagnetic nanostructures; including mea-\nsurements of spin pumping18and exchange coupling.19In a typical experiment, microwaves\nare directed across the sample using a coplanar waveguide, and their transmission is mea-\nsured as a function of both external bias \feld and excitation frequency.20\nIn terms of computational micromagnetics, there are at least three methods that can be\nused to simulate the FMR:\n1. Application of a time-dependent periodic sinusoidal magnetic microwave \feld of \fxed\n2frequencyfto determine the magnetization precession amplitude in response to the\nsystem. If the precession amplitude is small, the power absorption of the microwave\n\feld would be small as the excitation frequency does not couple well to the set of\nnatural frequencies of the system. This method is conceptually simple but compu-\ntationally very demanding as, for every frequency f, the micromagnetic simulation\nneeds to compute the time evolution of the system's magnetization after the transient\ndynamics has been damped and steady magnetization precession is reached. This will\nonly provide one point on the frequency-absorption curve and only a micromagnetic\nsimulation software that supports a time dependent external magnetic \feld can be\nused.\n2. Ringdown method:21the system is perturbed from its equilibrium state by applying\na short-lived and su\u000eciently weak excitation, followed by simulation and recording\nof the magnetization dynamics. Resonance frequencies and corresponding modes are\nextracted by performing the Fourier transform on the recorded data. This is an e\u000ecient\nway to determine the eigenmodes of the system.\n3. Eigenvalue method:22instead of simulating the time evolution of the system's magne-\ntization as in the methods above, the problem is represented as an eigenvalue problem,\nwhose solutions provide the frequencies (eigenvalues) and mode shapes (eigenvectors)\nof the system. This method requires specialist software that is not widely available.\nOur goal is to establish a standard problem to serve as a benchmark against which future\nsimulation tools and computational studies of the FMR can be compared and validated. In\nthis standard problem proposal, we will follow the second (ringdown) method, which is sup-\nported by most micromagnetic packages and compare its output with the third (eigenvalue)\nmethod. We provide a detailed standard problem description and speci\fcation as well as\nthe complete set of computational steps and code repository23in order to make it easily\nreproducible and accessible to a wide community. Parts of the code repository can also be\nused as an example to compute FMR data and modes from micromagnetic simulations. It\nis hoped that this work will aid the development of micromagnetic simulations of systems\nundergoing FMR and support and drive experimental e\u000borts.\nSec. II introduces and motivates the choice of the FMR standard problem, and introduces\nthe frequency spectrum computed in di\u000berent ways. Sec. III provides a more detailed dis-\n3cussion including computation of the normal mode shape, the eigenvalue problem approach\nas an alternative way of computing the frequency spectrum and normal modes, and a sys-\ntematic study of the dependence of the results on variations in the simulation parameters\nsuch as damping, relaxation of the initial state, nature of the perturbation and mesh dis-\ncretization. We close with a summary in Sec. IV. The Appendix provides more details on\nparameters used in the Nmag simulations, the eigenvalue approach and simulation results\nobtained in the absence of demagnetization e\u000bects.\nII. SELECTION AND DEFINITION OF STANDARD PROBLEM\nA. Problem de\fnition\nWe choose a cuboidal thin \flm permalloy sample measuring 120 \u0002120\u000210 nm3, as\nshown in Fig. 1. The choice of a cuboid is important as it ensures that the \fnite di\u000berence\nmethod employed by OOMMF does not introduce errors due to irregular boundaries that\ncannot be discretized well.24We choose the thin \flm geometry to be thin enough so that\nthe variation of magnetization dynamics along the out-of-\flm direction can be neglected.\nMaterial parameters based on permalloy are shown in Table I. An external magnetic bias\n\feldHextwith magnitude Hext= 80 kA/m is applied along the direction e= (1;0:715;0)\n(at 35:56\u000eto thex-axis), i.e. Hext=Hext\u0001e=jej\u0019(65:1;46:5;0) kA/m as shown in Fig. 1.\nWe choose the external magnetic \feld direction slightly o\u000b the sample diagonal in order to\nbreak the system's symmetry and thus avoid degenerate eigenmodes.\nFirst, we initialize the system with a uniform out-of-plane magnetization m0= (0;0;1).\nThe system is allowed to relax for 5 ns, which was found to be su\u000ecient time to obtain a\nwell-converged equilibrium magnetization con\fguration. We refer to this stage of simulation\nas the relaxation stage , and its \fnal relaxed magnetization con\fguration is saved to serve as\nthe initial con\fguration for the next dynamic stage . Conceptually, what is required to \fnd\nthe relaxed state is to minimize the system's energy in the presence of an external magnetic\nbias \feld, taking into account exchange and demagnetization energy contributions. We note\nthat there are other ways of obtaining this con\fguration, including energy minimization (as\nfor example supported by OOMMF ), or solution of the LLG without the precession term\n(as supported by Nmag ). Because we want to use a well de\fned method that is supported\n4by all simulation tools, we minimize the system's energy by integrating the LLG equation\nwith a large, quasistatic Gilbert damping \u000b= 1 for 5 ns. The use of any of these methods\nis expected to lead to the same relaxed equilibrium magnetization con\fguration.\nIn the next step ( dynamic stage ), a simulation is started using the equilibrium mag-\nnetization con\fguration from the relaxation stage as the initial con\fguration. Now, the\ndirection of an external magnetic \feld is altered to e= (1;0:7;0), i.e. Hext=Hext(e=jej)\u0019\n(65:5;45:9;0) kA/m. This corresponds to a rotation of the bias \feld to 35\u000ewith respect to\nthex-axis. Due to the change in xandycomponents of the external magnetic \feld, the ini-\ntial magnetization con\fguration is now out of equilibrium. Consequently, the system tends\nto relax towards the lowest energy con\fguration in the presence of a new external magnetic\n\feld. This simulation stage runs for T= 20 ns while the (average and spatially resolved)\nmagnetization M(t) is recorded every \u0001 t= 5 ps. The Gilbert damping in this dynamic\nsimulation stage is \u000b= 0:008. Using the recorded data, a Fourier transform is performed\nto produce the FMR spectrum and obtain eigenfrequencies (and the eigenmodes). Spa-\ntially resolved transformations allow examination of the shapes of the modes (see Sec. II C).\nSimulation parameters for both stages of the simulation are given in Tab. I.\nB. Problem Selection\nIn this section, we address the selection criteria for the standard problem, and give an\nexplanation of how each is met within the proposed framework:\n1.Initial magnetization con\fguration . This standard problem is de\fned in two stages:\n(i) relaxation stage and (ii) dynamic stage. The purpose of the relaxation stage is\nto bring the system into a well de\fned state. Starting from an initial uniform out-of-\nplane magnetization m0= (0;0;1) combined with the in-place bias \feld H0, the system\ntransitions into a \\relaxed\" state in an attempt to reach a (local) energy minimum.\nThe relaxed state is used as the initial con\fguration for the dynamic stage.\n2.Excitation of system . Apart from being reproducible, the perturbation or excitation\n\feld must be su\u000eciently large to excite magnetization dynamics, yet be small enough\nso that the system remains in the linear regime. This is achieved by altering the\ndirection of the bias \feld, as a simple practical approach that does not require time-\n5dependent applied \felds. The power spectrum obtained is speci\fc for the chosen\nexcitation, and thus the excitation is a key part of the problem de\fnition. All sim-\nulations tools, even the ones that do not support time-dependent external magnetic\n\felds, are expected to be able to excite the system in this manner.\n3.Computation time . Standard problems, apart from being simple and reproducible,\nrequire as short as possible computation time. In micromagnetic simulations, the\ncomputational time depends mostly on the number of degrees of freedom in the dis-\ncretized problem. Accordingly, the spatial discretization of 5 nm is chosen as a balance\nbetween computational time and accuracy. Although the second simulation stage is\nperformed with realistic Gilbert damping value \u000b= 0:008 over a limited simulation\ntime, in the \frst (relaxation) stage, we set \u000b= 1 to ensure the magnetization reaches\na well converged state within the allotted time.\n4.Veri\fcation of results . Ideally, results should be veri\fed against other methods of\nobtaining them. In this work, we use di\u000berent simulation packages (including \fnite\ndi\u000berence and \fnite element discretization schemes) that have been developed by dif-\nferent groups. Furthermore, we use a completely di\u000berent computational (eigenvalue\nbased) method to obtain the power density spectrum and excited normal modes sep-\narately.\nC. Data Analysis\nWe outline two di\u000berent ways to compute the power spectrum of the simulated system.\nMethod 1: Global power spectrum and Sy(f)\nIn this case, the observable we use is the spatially averaged magnetization hMir(t), as\nit is easily accessible in all known simulation tools. Using a discrete Fourier transform,25\nwe can obtain the power spectrum of the average magnetization in the frequency domain.\nAs the dynamic simulation progresses, at uniform time steps tk, we record the spatially\naveraged magnetization hMir(tk), wheretk=k\u0001twith \u0001t= 5 ps, and k= 1;2;:::;N , with\nN= 4000 being the number of time steps. However, we only consider the y-component of\n6spatially averaged magnetization hMyir(tk) to compute the power spectrum Sy(f) using\nSy(f) =jFy(f)j2with (1)\nFy(f) =NX\nk=1hMyir(tk)e\u0000i2\u0019ftk: (2)\nAccording to the chosen parameter values, the sampling frequency is fs= 1=\u0001t= 50 MHz,\nwhich implies that the maximum frequency that can be sampled (Nyquist frequency25) is\nfN= 2fs= 100 GHz. We term this approach \\method 1\". It requires that the discrete\nFourier transform is performed once (on the time series of the average magnetization) in\norder to compute the power spectrum Sy(f).\nMethod 2: Local power spectrum and ~Sy(f)\nEquation (2) uses the spatially averaged magnetization to compute its frequency spec-\ntrum. Following McMichael and Stiles' approach21to compute a collection of local power\nspectra over the extent of the sample we introduce a second method which allows to gain\nmore detailed information about the spectrum. In contrast to the \frst method, this requires\ncomputation of discrete Fourier transforms at all spatial sampling points.\nWe analyze n=nxnyscalar time-dependent signals: for every recording time tkwe\nsample the magnetization on a two-dimensional grid of positions rm;pwherenxandnyare\nthe number of sampling points in xandydirections, respectively. More precisely, rm;p=\n((m\u00001\n2)Lx\nnx;(p\u00001\n2)Ly\nny;2:5 nm) with m= 1;2;:::;nx,p= 1;2;:::;ny, andLx=Ly= 120 nm.\nIn the remainder of this work, we have used nx= 24 andny= 24. For simplicity and\ngenerality, we label the sampling points rm;pasrj, withj= 1;2;:::;nxny.\nWe term this approach \\method 2\", and compute the local power spectrum\nSy(rj;f) =jFy(rj;f)j2(3)\nfor each of the recorded signals (i.e. for each position rj), with\nFy(rj;f) =NX\nk=1My(rj;tk)e\u0000i2\u0019ftk: (4)\nBy averaging the local power spectra Sy(rj;f), we obtain\n~Sy(f) =1\nnxnynxnyX\nj=1Sy(rj;f): (5)\n7Both entities Sy(f) and ~Sy(f) are shown in Fig. 3 in a logarithmic scale, and strong resonance\npeaks are observed at f1= 8:1 GHz and f2= 11:1 GHz.\nThis method allows us to obtain spatially resolved information Sy(r;f) about the normal\nmodes of the system. See further discussion in Sec. III A and Figs. 4 and Fig. 5.\n1. Phase information\nIn order to understand the precession of a resonance mode qat a particular frequency fq\nacross the extent of a thin \flm, we need to extract the phase information from the spatially\nresolved Fourier transform. We start with the complex Fourier coe\u000ecient Fy(rj;fq) which\nrepresents the contribution of the frequency fqto the time series of the magnetization y-\ncomponent My(rj;t) of the magnetization dynamics at position rj. In our discrete Fourier\ntransform, we have a set of Ncomplex Fourier coe\u000ecients Fy(rj;f) at discrete frequencies\nfk. The modulus (= absolute value) of the Fourier coe\u000ecient contains the information\nabout the amplitude, whereas its argument (in the polar representation) contains the phase\ninformation. Consequently, the information about the resonance mode qphase can be\nextracted as the complex Fourier coe\u000ecient argument as a function of position rj, which\nallows us to identify the relative phases between di\u000berent spatial domains in a normal mode.\nIII. RESULTS AND DISCUSSION\nA. Standard Problem Simulation Results\nFigures 2 and 3 show the main results from the standard problem, as outlined in Sec. II A,\nobtained using the OOMMF simulation tool. Time evolution of the average magnetization\ny-component for the \frst 2 :5 ns of dynamic stage is shown in Fig. 2(a), and the associ-\nated ferromagnetic resonance spectrum (Fourier transform of hMyir(t) over the entire 20 ns\ndynamic simulation) is shown in Fig. 2(b). Performing the Fourier transform of spatially\naveraged magnetization (method 1) produces a slightly di\u000berent result in comparison to the\nspatially resolved (method 2) approach, which is shown in Fig. 3.\nUsing the spatially resolved approach, one can plot the power spectrum coe\u000ecients\nSy(rj;fq) as a function of position rjfor the normal mode frequency fqto represent both\n8the power amplitude and phase of the normal mode q, as described in Sec. II C 1. Figure 4\nshows the spatial resolution of the resonance mode at f1= 8:1 GHz with both the amplitude\njSj(rj;f1) and phase information arg( S)(rj;f1) forx,yandzmagnetization components\nthat were calculated from the OOMMF simulation using Eq. (5). The magnetization pre-\ncession is present in all three directions, with the highest amplitude in the y-direction as\nexpected since the largest external bias \feld perturbation is performed along the y-direction.\nFigure 3 shows that the frequency spectrum is dominated by two modes. The low frequency\nmode extends across the middle of the sample; this corresponds to the mode of uniform\nprecession observed in macroscopic samples.\nThe largest precession amplitude of the normal mode at 11 GHz (spatially resolved plot\nshown in Fig. 5) is located at the corners of the sample and is dominated by the demagneti-\nzation energy associated with magnetization canting at the sample boundaries. In terms of\nthe normal mode phase representation, an abrupt phase shift occurs as one moves away from\nthe sample corner to the sample center. This normal mode is associated with the particular\nshape and size of the sample. Note that the precession amplitude in Fig. 4 (top row) and\n5 (top row) is generally small where the phase changes: these oscillation nodes separate\ndomains that show out-of-phase precession relative to each other. Similar e\u000bects have been\nobserved, for example, in permalloy nanodisks: Guo et al.26used ferromagnetic resonance\nforce microscopy to spatially resolve the resonance modes. They observed the same mode\nshapes simulated here, and demonstrated a strong relationship between the size of the disk\nand the relative strength of the modes. Appendix C details the results of simulations per-\nformed without the demagnetization energy contribution (only one resonance is observed,\ncorresponding to a macrospin model of uniform coherent precession).\nA resonance mode also exists in the z-direction,Fz(rj;f). The precession of the moments\ndescribes an ellipse around the bias \feld, which has greatest amplitude in the x\u0000yplane,\nwith the component in zbeing smaller due to the demagnetization \feld.\nB. Eigenvalue method results\nAn alternative approach to calculating the normal modes is to linearize the LLG equation\nfor the studied system around its equilibrium state; the normal modes of the resulting linear\nsystem of equations can then be determined by solving an eigenvalue problem. This approach\n9does not require running and post-processing of a dynamic micromagnetic simulation, and\nis thus a good way to check the veracity of the results. A detailed description of this method\nproviding resonance frequencies and normal mode shapes can be found in Ref. 22.\nWe have extended the method to be able to also compute the FMR spectrum of the\nsimulated system, and report the new methodology in Appendix B not to distract from the\nresults obtained with the method.\nTable II shows the \frst \ffteen resonance frequencies, calculated with the eigenvalue ap-\nproach using a \fnite di\u000berence discretization with cell size 5 \u00025\u00025 nm, matching the\nsimulation parameters used by OOMMF . The spatial distribution of these modes are plotted\nin Fig. 6. The power density spectrum, and thus the amplitude of each mode excited during\nthe simulation, is dependent upon the perturbation of the system. Using the method de-\nscribed in Appendix B 4, we compute the coupling of the used excitation to each mode and\nreconstruct the spectrum shown in Fig. 7, demonstrating an excellent agreement between\nthe ringdown method and the eigenvalue method.\nFinally, we show the comparison of the spatial pro\fles generated by the ringdown and\neigenvalue methods. Figure 8 shows a comparison of the three lowest frequency modes,\ndemonstrating excellent agreement for the two modes visible in Fig. 3. This agreement gets\nworse as the frequency of the normal modes increases and their amplitude in the ringdown\nmethod decreases, increasing the signal-to-noise ratio; above 14 GHz the data quality is not\nsu\u000ecient to make a meaningful comparison. Nevertheless, the close agreement of results\ndemonstrates the equivalence of these two approaches.\nC. Falsi\fcation Properties\nIn de\fning a standard problem, it is useful to investigate how changing the parameters\nof the simulation will distort the results. This is intended to allow users to isolate incon-\nsistencies within their own simulations when attempting to reproduce the output of this\nproblem.\n101. Damping Parameter\nThe magnitude of the Gilbert damping parameter during ringdown method determines\nthe time taken for the system to reach its stable con\fguration. However, this did not a\u000bect\nthe resonance frequencies produced by the Fourier transform, except in the strongly damped\ncase where \u000b\u00150:1. Figure 9 shows the power spectrum produced by the simulation for \u000b\n= 10\u00001, 10\u00002, 10\u00003and 10\u00004. As the damping parameter is decreased, the peaks become\nnarrower and taller as expected. For the highest damping the spectrum is heavily suppressed,\nshowing only two broad features, with the 11 :25 GHz mode barely visible above the tails\nof the 8:25 GHz mode. In this case the system is approaching overdamping; if we choose\na large damping of \u000b= 1 then no precession occurs and the Fourier transform shows no\npeaks. As damping decreases extra peaks begin to form, for example at f\u001912 GHz and\n13.5 GHz. At lower dampings the intensity of these features increases, but never surpasses\nthat observed for the two dominant modes.\n2. Relaxation Time\nThis standard problem de\fnition (see Sec. II A) asks that the computation of the relaxed\ncon\fguration should be carried out by integrating the damped equation of motion for 5ns.\nIn this subsection, we explore how the obtained frequency spectrum changes if a shorter\nperiod is used.\nFigure 10 shows that starting the dynamic simulation stage from an improperly converged\ncon\fguration from the relaxation stage causes signi\fcant instability within the system. Al-\nthough there are still peaks at the resonance frequencies for relaxation times shorter than\n5ns, there are also many other peaks corresponding to domains aligned in other directions\nrelaxing back to align with the bias \feld. The frequency of the normal modes does not\nchange, but the strength of the contributions from spurious modes is too large to allow a\nmeaningful analysis. The importance of allowing the relaxation stage su\u000ecient time to reach\na converged state is clear; the di\u000berence occurs because the the system dynamics contains\nthe components that exist as a consequence of the system tending to reach the equilibrium\nstate during the dynamic stage of simulation. We can see that the curves for 500 ps and\n5000 ps produce very similar results.\n113. Relaxation Stage Perturbation Angle\nThe problem de\fnition (Sec. II A) suggests a change of 0 :56\u000ebetween the bias \feld in\nthe relaxation and dynamic stage. Figure 11 shows the e\u000bect of changing the perturbation\nangle between bias \feld in the relaxation and dynamic stages from 0 :1\u000eto 55\u000e.\nChanging the perturbation angle of the bias \feld changes the amount of energy supplied\nto the system in the initial excitation, which manifests as a greater area under the power\nspectrum curve (not shown). If the perturbation angle is too small ( <0:1\u000e), no peaks are\nobserved above the noise level of the power spectrum. Conversely, if the perturbation angle\nis too large the system deviates signi\fcantly from the equilibrium state, and additional\nmodes form, leading to a distorted power spectrum. While the resonance frequency does\nnot signi\fcantly change, the spectrum is eventually dominated by these other features and\nmodes. At such high perturbations both Nmag andOOMMF show a slight drop in resonant\nfrequency.\n4. Spatial Discretization\nIn micromagnetics, it is generally recommended to keep the cell size smaller than the\nexchange length, and for this standard problem, we use a cell with an edge length of 5 nm.\nThe e\u000bects of changing the cell size from 2 :5 nm to 120 nm are shown in Fig. 12. Decreas-\ning the resolution of the mesh (increasing the size of the tetrahedra in FE or cuboids in FD)\ncauses the divergence between FD and FE codes. This is to be expected, as the di\u000bering\napproach to calculation of demagnetization \feld is one of the key di\u000berences between the\ntwo approaches. In OOMMF the frequency of the low frequency mode decreases, while the\nfrequency of the main edge mode increases. This comes about due to changes in relative\nimportance of demagnetization e\u000bects from the edge of the sample, with fewer nodes near\nthe boundaries the sample becomes more like an idealized in\fnite thin \flm.\nIn both codes, high frequency features are suppressed with increasing element size. These\ncorrespond to higher order modes that cannot form if there are too few elements to support\ntheir spatial variation. It is well known that choice of an appropriate mesh discretization\nis crucial in computational micromagnetics, an aphorism that is well supported by these\nresults. The deviation of resonant frequency with mesh resolution therefore suggests that a\n12resolution comparable to the exchange length in permalloy ( \u00185 nm) is appropriate. We can\nalso see that OOMMF 's \fnite di\u000berence approach is more robust than Nmag 's \fnite element\nbased result here.\nD. Comparison of Simulation Methodologies\nFor the standard problem de\fned above with a cell size of 5 \u00025\u00025 nm3, the deviations\nbetween \fnite di\u000berence and \fnite element methods for both the resonance frequencies are\nnoticeable, as shown in Fig. 13, reaching 0.2 GHz for the low frequency mode, and 0.4 GHz\nat the higher frequency. Note that in the case of the tetrahedra used in the \fnite element\nmethod this means that space is divided into 6 tetrahedra that together form a cube of\ndimensions 5\u00025\u00025 nm3. A smaller cell size 2 \u00022\u00021 nm3for the \fnite element code will\nreduce the deviations signi\fcantly, bringing the two codes to within 0.05 GHz of agreement.\nThe corresponding comparison for the average magnetization ( y-component) evolution\nis shown in Fig. 14. It is obvious that the two micromagnetic package produce di\u000berent\nsimulation results when the cell size is 4 \u00025\u00025 nm3, but good agreement is found for the\nsmaller cell size of 2 \u00022\u00021 nm3.\nIn FD the computation takes place at the center of a series of cuboids used to build the\nsample, while in FEM it takes place at the nodes of the mesh tetrahedra. While tetrahedra\ngive signi\fcantly better approximations to irregular shapes than the cuboids, computing\nvalues on vertices is problematic when the values of the demagnetization tensor vary sharply.\nIf the mesh is not \fne enough to accurately resolve the change, the e\u000bective \felds will be\ncalculated less accurately, and spurious results will be produced. In this simulation, the\nerror arises from contributions from the top and bottom surfaces of the \flm, and a fourfold\nincrease in resolution in the z-direction brings the FDM simulations into agreement with\nthe FEM, at the cost of signi\fcantly increasing the runtime. This problem could also be\nalleviated through the use of a spatially varying mesh density, placing more mesh nodes in\nthe regions near the surfaces to accurately sample the demagnetization tensor.\n13IV. SUMMARY AND CONCLUSIONS\nA standard problem for micromagnetic simulations of ferromagnetic resonance in a thin\n\flm has been introduced. FMR is a technique that is widely used for material characteriza-\ntion and the study of spin transfer phenomena. While micromagnetic simulations are able\nto provide insightful analysis and prediction of FMR experiments, it is not trivial to con-\nduct those simulations. With this paper, we provide step by step instructions and speci\fc\nparameters and results that can be used to validate simulation tools before they are applied\nto new problems.\nWe provide performance data from two popular micromagnetics packages ( OOMMF and\nNmag ), thus providing data for the deviations that can be expected between di\u000berent dis-\ncretization and computation strategies. This standard problem may serve as an introduction\nto the procedures involved and allow benchmarking and testing of new simulation packages.\nExample scripts to run the simulations and analyse the data, as well as raw data for all\nthe \fgures, are available in the associated electronic supplementary material.23\nACKNOWLEDGEMENTS\nA.A.B. acknowledges support from Diamond Light Source, and the EPSRC through a\nDoctoral Training Grant. This work has been supported through the EPSRC Centre for\nDoctoral Training grant EP/G03690X/1.\nAppendix A: Nmag tolerances\nBy analyzing the time evolution of the average magnetization z-component, obtained\nby running the Nmag simulation with default time integration tolerances, we observe that\nnumerical noise is present after approximatelly 0 :8 ns. By simply performing the Fourier\ntransform on this data, this numerical noise can be interpreted as a particular eigenmode\nof certain frequency. Although this does not a\u000bect any results presented in this work, we\nprovide the following improved demagnetization \feld computation settings that suppress\nthis:\nksp_tols = {\"DBC.rtol\":1e-7,\n14\"DBC.atol\":1e-7,\n\"DBC.maxits\":1000000,\n\"NBC.rtol\":1e-7,\n\"NBC.atol\":1e-7,\n\"NBC.maxits\":1000000,\n\"PC.rtol\":1e-3,\n\"PC.atol\":1e-6,\n\"PC.maxits\":1000000}\nand time integration tolerances:\nsim.set_params(stopping_dm_dt=0.0, ts_abs_tol=1e-7, ts_rel_tol=1e-7)\nin the dynamic simulation stage. The improved tolerances remove the numerical noise from\nthe average magnetization time evolution, but increases the running time. The full scripts\nto run the simulations are available.23\nAppendix B: Eigenvalue approach\nIn this Appendix, we provide a brief summary of the eigenvalue method described in\nRef. 22, with modi\fcations required to compute the Gilbert damping and excitation depen-\ndent FMR spectrum of the system along with the resonance frequencies and corresponding\nnormal modes.\nThe dynamics of the micromagnetic system is governed by the Landau-Lifshitz-Gilbert\n(LLG) equation:\n_m=\u0000\r\n1 +\u000b2[m\u0002He\u000b+\u000bm\u0002m\u0002He\u000b] =L(m); (B1)\nwhere mis the normalized magnetization: m=M=Ms, withjMj=Msbeing the satura-\ntion magnetization. If the system is in its equilibrium state m0, thenL(m0) = 0. Small\nperturbations from the equilibrium (for example, those generated by the removal of the ex-\nternal magnetic \feld perturbation when moving from the relaxation to the dynamic stage of\nsimulation) can be described as m=m0+\"v, with v?m0since thejmj= 1 condition is\nimposed. For a small \", terms of theO(\"2) order and higher can be neglected, which results\n15in the linearized equation (for the general case):\n_v=@L\n@m\f\f\f\f\nm=m0v: (B2)\nIf the \fxed linear operator ^L=@L\n@m\f\f\nm=m0is de\fned, the linearized equation can be written\nas_v=^Lv. This is an ordinary di\u000berential equation, which can be solved by an ansatz of\nthe form v= Re( ~vei2\u0019ft). The normal modes (eigenvectors) ~vand oscillation frequencies\n(eigenvalues) fcan be found from the following eigenvalue problem\ni2\u0019f~v=^L~v: (B3)\n1. Linearized equation without damping\nFirst, we consider the case when the damping term in the LLG equation is neglected ( \u000b=\n0). Without damping, the magnetic moments precess inde\fnitely, and the LLG equation\npreserves energy. In this simplest case the calculation of the linearized operator ^Lis fairly\nstraightforward and results in the following linearized equation of motion22\n_v=\rm0\u0002^Av; ^A=jHe\u000b(m0)jId\u0000@He\u000b\n@v; (B4)\nwhere ^Ais a positive de\fnite Hermitian operator. The normal modes ~vand frequencies f\nof the linearized equation can be found from the eigenvalue problem\n\u0000i2\u0019fm0\u0002~v=\r^A~v: (B5)\nThe left-hand side of this eigenvalue problem also contains a Hermitian operator describing\nthe uniform precession ^Bv=\u0000im0\u0002~v, however it is not positive de\fnite (its eigenvalues\nare\u00061).\nBecause of the energy conservation, the oscillation frequencies fkthat satisfy this eigen-\nvalue problem will be real and the normal modes ~vkcorresponding to di\u000berent frequencies\nwill be orthogonal. These properties enable the e\u000ecient numerical solution of Eq. (B5); the\neigenvalues fare the resonant frequencies and the complex magnitudes of the eigenvectors ~v\nare the normal mode amplitude plots (the complex phase of ~vcorresponds to the phase of\nthe oscillations at the corresponding sites).\nHowever, in order to compute the FMR spectrum via the eigenvalue approach, we have\nto consider the more complicated case of non-zero damping.\n162. Linearized equation with damping | perturbative analysis\nFor the case of su\u000eciently small non-zero damping \u000b, a perturbative analysis can be\nperformed to determine the corrections to the eigenvalues. In this case, eigenvalues have the\nform\u0015=i2\u0019f\u00001=\u001c, where\u001cis the characteristic time for the mode to decay to 1 =eof its\nstarting amplitude value. It turns out that to the \frst order, the resonance frequencies are\nunchanged, and the damping times can be found using a relatively simple analytic calculation\nwithout solving the perturbed eigenvalue equation numerically.22Additionally, the coupling\nbetween the perturbed normal modes is small if their frequencies are su\u000eciently separated\n| this property will be useful for the calculation of the FMR spectrum. We have found\nthat for our test system, which has a low damping constant, the damping times computed\nusing both the perturbative method and the numerical method (from the next section) are\nvery close; up to the 4 digits shown in Table 6 the results are identical for both methods.\n3. Linearized equation with damping | numerical solution\nTo compute the actual FMR spectrum, we have to derive the linearized equation in the\npresence of damping, and solve the corresponding eigenvalue problem. The derivation of\nthe linearized equation with damping is straightforward but slightly tedious. We skip this\nderivation and instead compute the linearized equation using a numerical di\u000berentiation\ntrick.\nFor the linearized equation, we have to compute the directional derivative\n^Lv=\u0012@L\n@m\f\f\f\f\nm=m0\u0013\n[v] =d\nd\u000fL(m0+\u000fv)j\u000f=0: (B6)\nFor the test problem, the components of the e\u000bective \feld (demagnetization, exchange, bias)\nare all either constant, or linear functions of m; therefore as a function of \u000f,L(m0+\u000fv) is a\ndegree 3 polynomial (the highest degree coming from the damping term m\u0002m\u0002He\u000b). This\nmeans that a numerical di\u000berentiation rule of order 3 or higher will compute the derivative\nd\nd\u000fL(m0+\u000fv)j\u000f=0exactly.\n174. Linearized equation with damping | spectrum computation\nThe previous sections outlined the method used to compute the frequencies and normal\nmode shapes for the linearized equation, with or without damping. In this section we\ndescribe the subsequent computation of the FMR spectrum, which also depends on the\ninitial state of the system. To determine the contributions of each normal mode to the total\nspectrum, we have to compute the coupling between the initial state and the normal modes\nin the presence of damping. More precisely, let nbe the total number of the degrees of\nfreedom (for a mesh with Nnodes,n= 3N). Let ~vi,i= 1;2;:::;n be the set of eigenvalues\n(normal modes) without damping, ~ v(p)\ni,i= 1;2;:::;n the set of perturbed eigenvalues in\nthe presence of damping, and fiand\u001cithe corresponding mode frequencies and damping\ntimes. Letvinitial =minitial\u0000m0be the initial state of the system. Due to the orthogonality\nproperty described in Sec. B 1, we can assume that with the respect to the Hermitian inner\nproduct de\fned by the operator ^Avia (x;y) :=x\u0001^A\u0001y\u0003the non-perturbed eigenvectors\nform an orthonormal basis, i.e. ( vi;vj) =\u000eij.\nTo solve the linearized equation of motion Eq. (B2), we need to expand the initial state\nvinitial in the perturbed ~ v(p)\nibasis:\nvinitial =nX\ni=1Ci~v(p)\ni: (B7)\nOnce the coe\u000ecients Ciare known, the full solution of the linearized equation (B2) is\nm(t) =m0+nX\ni=1Cie(2\u0019i!i\u00001=\u001ci)t: (B8)\nGiven this full analytic solution, we can then calculate the spectrum using either of the\nmethods described in Sec. II C. Unfortunately, this expansion requires the knowledge of\nthe complete set of eigenvectors ~ v(p)\ni, which is numerically unfeasible to compute. Instead,\nwe will attempt to reconstruct the spectrum based on the \frst kperturbed modes with\nthe lowest frequencies (we used k= 40). We would like to do this by \fnding the \\best\"\napproximation\nvinitial =kX\ni=1ci~v(p)\ni+R: (B9)\nWe will look for this approximation in the subspace spanned by the \frst k0non-perturbed\nnormal modes ~ vi, withk0>k(we usedk0= 60). Due to the frequency separation property\n18mentioned earlier, we can expect that this restriction will not a\u000bect the residue (the high-\nfrequency modes will not measurably contribute to the low-frequency spectrum). When\nrestricted to this subspace, we arrive at a system of k0equations with kunknownsci\n(vinitial;vj) =kX\ni=1ci(~v(p)\ni;vj); i= 1:::k0: (B10)\nThis linear system is overspeci\fed but any residue will only contain the high-frequency modes\nwith frequencies above f0\nk, which don't contribute to the spectrum for frequencies below fk\nthat we are trying to compute. We solve this linear system with a standard linear least-\nsquares (linear regression) method, allowing the determination of the coupling coe\u000ecients ci\nand thus the full solution Eq. (B8).\nAppendix C: Simulations without Demagnetization\nIn this section, we show the results of this standard problem in a setup where only the\nexchange and the applied Zeeman e\u000bective \felds are considered. In particular, the demag-\nnetization energy has been ignored. Figure 15 shows the power spectrum of a simulation\ncarried out with demagnetization e\u000bects disabled in OOMMF andNmag . The data has been\nobtained using the ringdown method. It can be seen that the two packages are in excellent\nagreement, producing only one mode at 2 :8 GHz. In the absence of the demagnetization en-\nergy, we obtain this single mode corresponding to coherent precession of the magnetization\nas shown in Fig. 15. This matches the result from the Kittel equation for a material when\ndemagnetization energy contribution is neglected, for which:27\nf=\r\n2\u0019\u0002\u00160\u0002Happlied; (C1)\nyieldsf= 2:81 GHz.\nAs the simulation starts from a uniform, well-converged state only the lowest order,\nuniform, mode is observed. Modes located at the edge of the sample are suppressed due to\nthe absence of demagnetization.\nAs discussed in Sec. III D di\u000berences can arise between simulations performed using the\n\fnite di\u000berence and \fnite element approaches due to their handling of demagnetization\ne\u000bects at the \flm boundaries. The data above shows that both approaches produce very\nsimilar spectra in the absence of demagnetization e\u000bects. We stress that this information\n19is presented for comparative purposes only - it does not have physical meaning. 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Dmytriiev, M. Mruczkiewicz, H. Fangohr, A. Barman,\nM. Krawczyk, and A. Prabhakar, IEEE Trans. Magn. 49, 524 (2013).\n17M. Farle, Rep. Prog. Phys. 61, 755 (1998).\n18B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer,\nPhys. Rev. Lett. 90, 187601 (2003).\n19B. Heinrich, Z. Celinski, J. F. Cochran, W. B. Muir, J. Rudd, Q. M. Zhong, A. S. Arrott,\nK. Myrtle, and J. Kirschner, Phys. Rev. Lett. 64, 673 (1990).\n20H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider, M. J. Carey, S. Maat, and J. R.\nChildress, Phys. Rev. B 84, 054424 (2011).\n21R. D. McMichael and M. D. Stiles, J. Appl. Phys. 97, 10J901 (2005).\n22M. d'Aquino, C. Serpico, G. Miano, and C. Forestiere, J. Comput. Phys. 228, 6130 (2009).\n23A. Baker, M. Beg, G. Ashton, W. Wang, M. Albert, D. Chernyshenko, S. Zhang, M.-\nA. Bisotti, M. Franchin, C. L. Hu, R. Stamps, T. Hesjedal, and H. Fangohr, \\Supple-\nmentary information for proposal of a micromagnetic standard problem for ferromagnetic\nresonance simulations,\" Github (2015), https://github.com/fangohr/micromagnetic-standard-\nproblem-ferromagnetic-resonance.\n24M. J. Donahue and R. D. McMichael, IEEE Trans. Magn. 43, 2878 (2007).\n25W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,\nNumerical Recipes 3rd Edition: The Art of Scienti\fc Computing (Cambridge University\nPress, 2007).\n26F. Guo, L. M. Belova, and R. D. McMichael, Phys. Rev. Lett. 110, 017601 (2013).\n27C. Kittel, Introduction to solid state physics (Wiley, 2005).\n21FIGURES\nFIG. 1. Geometry of the thin \flm sample, showing the static bias \feld Hext. The \feld is slightly\no\u000b-diagonal to break the symmetry of the system and thus avoid degenerate eigenmodes.\n0.0 0.5 1.0 1.5 2.0 2.5\nTime (ns)0.5800.5820.5840.5860.5880.5900.5920.594Magnetisation in y-direction(a)\n5 10 15 20\nFrequency (GHz)10-510-410-310-210-1100Spectral density(b)\nFIG. 2. (a) Spatially averaged (method 1) y-componenthMyir(t) of the magnetization My(t),\nas determined by the ringdown method in by OOMMF . (b) Power spectrum Sy(f) obtained from\nFourier transform of the spatially averaged y-component of the magnetization hMyir(t) data, cal-\nculated using Eq. 2.\n225 10 15 20\nFrequency (GHz)10-510-410-310-210-1100Spectral densityMethod 1\nMethod 2FIG. 3. Power spectrum Sy(f) from Eq. (2) (method 1, solid red line) and ~Sy(f) from Eq. (5)\n(method 2, dashed blue line) from ringdown method in OOMMF .\n23x\n y\n z\n0.0000.5811.1621.7432.324\nAmplitude\nx\n y\n z\n3.14\n0.003.14\nPhase8.25 GHzFIG. 4. Spatially resolved resonance modes in all three Cartesian directions plotted over the\nextent of the sample at f1= 8:25 GHz obtained from ringdown method in OOMMF . Top row:\nbase 10 logarithmic scale of power spectra for x-,y- andz-component, respectively. Bottom row:\ncorresponding phase distributions for three components.\n24x\n y\n z\n0.00000.21070.42140.63220.8429\nAmplitude\nx\n y\n z\n3.14\n0.003.14\nPhase11.25 GHzFIG. 5. Spatially resolved resonance modes in all three Cartesian directions plotted over the\nextent of the sample at f2= 11:25 GHz obtained from ringdown method in OOMMF . Top row:\nbase 10 logarithmic scale of power spectra for x-,y- andz-component, respectively. Bottom row:\ncorresponding phase distributions for three components.\n258.271 GHz\n 9.403 GHz\n 10.840 GHz\n 11.234 GHz\n 11.993 GHz\n13.046 GHz\n 13.817 GHz\n 14.277 GHz\n 15.317 GHz\n 15.908 GHz\n16.719 GHz\n 17.235 GHz\n 17.458 GHz\n 18.410 GHz\n 19.808 GHzFIG. 6. The spatial power spectrum of y-component of magnetization for the 15 lowest frequency\nmodes. The squares measure 120 nm of each side.\n260 4 8 12 16 20\nFrequency (GHz)10-510-410-310-210-1|Fy(f)|(arb.unit)\n(a)\n(b)\n(c)Eigenvalues\nSimulationFIG. 7. Comparison of the resonance spectra calculated using method 1 [ jFy(f)j, from Eq.\n(2)] obtained by simulation using OOMMF (dashed blue line) and from the eigenvalue problem\nformulation (solid red line). Excellent agreement is observed over the whole frequency range,\nalthough the peak heights are slightly di\u000berent. Arrows denote the positions of the modes plotted\nin Fig. 8.\n278.25 GHz - Eigenmodes\n012345\n8.25 GHz - Ringdown\n012345\n8.25 GHz - Difference\n0.008\n0.006\n0.004\n0.002\n0.0000.0020.0040.0060.008\n11.25 GHz - Eigenmodes\n012345\n11.25 GHz - Ringdown\n012345\n11.25 GHz - Difference\n0.18\n0.12\n0.06\n0.000.060.120.180.240.30\n13.9 GHz - Eigenmodes\n012345\n13.9 GHz - Ringdown\n012345\n13.9 GHz - Difference\n1.6\n1.2\n0.8\n0.4\n0.00.40.81.21.6(a)\n(b)\n(c)FIG. 8. Comparison of the spatially resolved power spectrum given by the ringdown method 2\nfrom OOMMF (middle column) and the eigenvalue problem (left column) for the y-component of\nthe 3 lowest frequency modes ( top row: 8:25 GHz, middle row: 11:25 GHz, bottom row: 13:9 GHz).\nExcellent agreement is observed for 8 :25 GHz and 11 :25 GHz. The agreement gets worse as the\namplitude of the mode generated by the ringdown method decreases, leading to a larger signal-to-\nnoise ratio and a less well de\fned spatial plot.\n285 10 15 20\nFrequency (GHz)10-1110-910-710-510-310-1101103|Fy(f)|(arb.unit)α=0.1\nα=0.01\nα=0.001\nα=0.0001FIG. 9. Normalized FMR spectrum for systems in the dynamic stage with a range of damping\nconstants. At \u000b\u00150:5 the system is over-damped, not producing resonance modes. As the damping\ndecreases the peaks become taller and sharper.\n295 10 15 20\nFrequency (GHz)10-1110-1010-910-810-710-610-510-410-310-210-1100101|Fy(f)|(arb.unit)200 ps\n400 ps\n500 ps\n5000 psFIG. 10. Normalized FMR spectrum as calculated for systems entering the dynamic stage after\nvarying the time in the relaxation stage. Allowing more time to relax leads to a lower amplitude,\nless noise and more well-de\fned peaks.\n10-1100101102\nPerturbation angle (Degrees)56789101112Resonance frequency (GHz)\nOOMMF LF Mode\nNmag LF Mode\nOOMMF HF Mode\nNmag HF Mode\n30FIG. 11. Changes to the resonance frequency of the main modes in the FMR spectrum as the\nmagnitude of the initial perturbation is altered. The FD method used by OOMMF is relatively\nuna\u000bected by this change for small angles, at high angles the spectrum becomes noisy and resonance\nfrequency drops. LF stands for Low Frequency peak at \u00198 GHz and HF for High Frequency at\n\u001911 GHz\n100101102103\nCell size (nm)3456789101112Resonance frequency (GHz)\nOOMMF LF Mode\nNmag LF Mode\nOOMMF HF Mode\nNmag HF Mode\nFIG. 12. Location of the main resonance modes in the FMR spectrum as a function of resolution\nof the mesh. The FE method shows greater deviation from standard results with changes to the\nparameters, due to its more sensitive handling of demagnetization e\u000bects.\n0 5 10 15 20\nFrequency (GHz)0.00.10.20.30.40.50.60.70.80.9|Fy(f)|(arb.units)OOMMF, 5×5×5 nm3\nnmag, 5×5×5 nm3\nnmag, 2×2×1 nm3\nFIG. 13. Comparison of the resonance spectra jFy(f)jobtained using OOMMF andNmag with\ndi\u000berent resolutions of the mesh. A cell size of 5 \u00025\u00025 nm3is used for OOMMF . Data from\nOOMMF andNmag agree well when the cell size for Nmag is reduced.\n310.0 0.5 1.0 1.5 2.0 2.5\nTime (ns)0.5780.5820.5860.5900.594My/MsOOMMF, 5×5×5 nm3\nnmag, 5×5×5 nm3\nnmag, 2×2×1 nm3FIG. 14. Comparison of average magnetization ( y-component) evolution between OOMMF and\nNmag with di\u000berent resolutions of the mesh. Note the phase shift that develops between di\u000berent\nspatial resolutions in Nmag , corresponding to a di\u000berent mode frequency in Fig. 14. Data from\nOOMMF andNmag agree well when the cell size for Nmag is reduced.\n2 4 6 8 10\nFrequency (GHz)10-810-710-610-510-410-310-210-1100101102103|Fy(f)|(arb.unit)OOMMF\nNmag\n32FIG. 15. Power spectrum for the proposed standard problem with the demagnetization \feld\ndisabled. This removes all but one peak, wherein the entire sample is in resonance together.\nFinite element and \fnite di\u000berent codes produce the same result under these conditions.\nTABLES\nParameter Value Unit\nsaturation magnetization ( Ms) 800 kA/m\nexchange constant ( A) 1 :3\u000210\u000011J/m\nanisotropy constant ( K) 0 J/m3\ngyromagnetic ratio ( \r\u0003) 2:210173\u0002105m/(As)\nGilbert damping ( \u000b), relaxation 1 :0\nGilbert damping ( \u000b), dynamic 0 :008\nDC bias \feld magnitude ( jH0j) 80 kA/m\nDC bias \feld ( e), relaxation [1, 0.715, 0]\nDC bias \feld ( e), dynamic [1, 0.7, 0]\nTABLE I. External magnetic \felds and material (permalloy) parameters used. Where these\nchange between the initial relaxation stage of the simulation, and the subsequent dynamic stage,\nboth values are shown.\nMode Frequency (GHz) Damping Time (ns)\n8.270 1.549\n9.402 1.639\n10.839 1.437\n11.233 1.452\n11.992 1.401\n13.045 1.345\n13.816 1.292\n14.276 1.253\n3315.316 1.191\n15.907 1.156\n16.718 1.126\n17.234 1.094\n17.457 1.094\n18.409 1.030\n19.806 0.963\nTABLE II. Frequency and damping time of the 15 lowest frequency modes, calculated using an\neigenvalue problem approach.\n34"    },    {        "title": "1909.03924v4.Ferromagnetic_resonance_assisted_optomechanical_magnetometer.pdf",        "content": "Ferromagnetic resonance assisted optomechanical magnetometer\nM. F. Colombano,1, 2G. Arregui,1, 2F. Bonell,1N. E. Capuj,3, 4E. Chavez-Angel,1A. Pitanti,5\nS.O. Valenzuela,1, 6C. M. Sotomayor-Torres,1, 6D. Navarro-Urrios,7,\u0003and M. V. Costache1\n1Catalan Institute of Nanoscience and Nanotechnology (ICN2),\nCSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain\n2Depto. de F\u0012 \u0010sica, Universitat Aut\u0012 onoma de Barcelona, 08193 Bellaterra, Spain\n3Depto. F\u0013 \u0010sica, Universidad de La Laguna, 38200 San Crist\u0013 obal de La Laguna, Spain\n4Instituto Universitario de Materiales y Nanotecnolog\u0013 \u0010a,\nUniversidad de La Laguna, 38071 Santa Cruz de Tenerife, Spain.\n5NEST, CNR - Istituto Nanoscienze and Scuola Normale Superiore, Piazza San Silvestro 12, 56127 Pisa, Italy\n6ICREA - Instituci\u0012 o Catalana de Recerca i Estudis Avan\u0018 cats, 08010 Barcelona, Spain\n7MIND-IN2UB, Departament d'Enginyer\u0012 \u0010a Electr\u0012 onica i Biom\u0012 edica, Facultat de F\u0012 \u0010sica,\nUniversitat de Barcelona, Mart\u0012 \u0010 i Franqu\u0012 es 1, 08028 Barcelona, Spain\n(Dated: September 4, 2020)\nThe resonant enhancement of mechanical and optical interaction in optomechanical cavities en-\nables their use as extremely sensitive displacement and force detectors. In this work we demonstrate\na hybrid magnetometer that exploits the coupling between the resonant excitation of spin waves in\na ferromagnetic insulator and the resonant excitation of the breathing mechanical modes of a glass\nmicrosphere deposited on top. The interaction is mediated by magnetostriction in the ferromagnetic\nmaterial and the consequent mechanical driving of the microsphere. The magnetometer response\nthus relies on the spectral overlap between the ferromagnetic resonance and the mechanical modes\nof the sphere, leading to a peak sensitivity of 850 pT Hz\u00001=2at 206 MHz when the overlap is maxi-\nmized. By externally tuning the ferromagnetic resonance frequency with a static magnetic \feld we\ndemonstrate sensitivity values at resonance around a few nT Hz\u00001=2up to the GHz range. Our\nresults show that our hybrid system can be used to build a high-speed sensor of oscillating magnetic\n\felds.\nCavity optomechanics (OM) focuses on the low-energy\ninteraction between photons and micro and nanomechan-\nical systems embedded in an optical cavity with main ap-\nplications as high performance detectors and as interfaces\nfor quantum information processing [1{5]. OM cavities\nenable ultra-sensitive optical transduction of mechanical\nmotion in km-scale systems such as LIGO [2] down to\nnano-scale quantum resonators [3]. Stimuli read-out ex-\nperiments based on such platforms have already reached\nthe state-of-the-art for force sensors [4] and accelerome-\nters [5]. In addition, the interaction of mechanical ele-\nments with magnetic \felds also makes OM devices high\nperformance magnetometers, i.e, room temperature OM\nmagnetometer (OMM) of small size [6, 7], high sensitivity\n[8] and large dynamic range [9]. The ability to measure\nsmall magnetic \felds over a broad frequency range is im-\nportant for numerous applications playing a key role in\nareas such as geology [10], space exploration [11], biology\n[12] and medical imaging [13].\nIn this Letter, we report a mechanical and magnetic\nhybrid resonator based on a thin \flm of magnetic insu-\nlator yttrium iron garnet (YIG) coupled to a glass mi-\ncrosphere optical cavity. We show that the combined\ncoupling of optical whispering gallery modes (WGM)\nto mechanical breathing modes in the microsphere and\nthe presence of a ferromagnetic resonance (FMR) in the\nYIG \flm enables a sensitive RF magnetic-\feld detector.\nWhen the magnetic \feld frequency is able to excite the\nYIG FMR and further coincides with a mechanical modeof the sphere, the detection sensitivity is maximized. The\nbasic transduction principle involves the conversion of an\nRF magnetic \feld, that resonantly excites magnons, into\nmechanical vibrations via magnetostriction in the YIG\n\flm [14{17]. Although the frequencies of the FMR mode\nand mechanical breathing mode in the microsphere may\ndi\u000ber, the FMR frequency can be tuned by a static mag-\nnetic \feld until both resonances are aligned, hence giv-\ning rise to a modulation of the microsphere WGM due\nto the OM interaction. We show that, by following that\nprocedure, it is possible to maximize the peak sensitiv-\nity at the mechanical mode frequencies. This allows the\nmagnetometer to operate in multiple windows where me-\nchanical modes are found, from 50 MHz to 1.1GHz. The\nsensitivities obtained for those frequencies are in the nT\nHz\u00001=2range. We attain a maximum sensitivity of \u0018850\npT Hz\u00001=2at 206 MHz operating at room temperature.\nAn illustration of the main part of the hybrid system is\nshown in Fig. 1a. It consists of a 0.5 \u00020.5 mm2and 1\u0016m\nthick \flm of YIG grown over a Gadolinium Garnet sub-\nstrate. Glass microspheres of Barium-Titanium-Silicate\n(BTS) with a diameter between 40 \u0016m and 70\u0016m [18, 19]\nwere deposited on the YIG thin \flm. These microspheres\nare used as high-quality OM cavities supporting both op-\ntical WGM and mechanical breathing modes with large\nOM coupling (G) values, de\fned as the optical frequency\nshift by a unit mechanical displacement [20]. A schematic\npicture of the OMM principle is shown in Fig. 1b. The\nsensor can be modeled as a Fabry-Perot optical interfer-arXiv:1909.03924v4  [cond-mat.mes-hall]  3 Sep 20202\nometer in which one of the mirrors responds mechani-\ncally to an applied magnetic \feld. This response is due\nto magnetostriction in the bulk of the material, i.e, the\ngeneration of an oscillating stress when a magnetic \feld is\napplied. This stress acts as a source force for mechanical\nmotion of the mirror, greatly ampli\fed when the initial\ndrive is resonant with a mechanical eigenfrequency.\nIn Fig. 1b, we also illustrate that the two main\nforces actuating the mechanical modes are the thermal\nLangevin force ( Fth), and the one associated with the\nRF magnetic \feld ( Fmag). These forces cause a variation\nof the cavity length, x, shifting the optical resonance by\n\u000e!optical =Gx.\nThe experimental setup is shown in Fig. 1c. An in-\nfrared tunable laser source is used to couple light into the\nWGM of the microsphere using a micro-looped tapered\n\fber placed close enough to ensure an overlap between\nthe evanescent \feld of the \fber fundamental mode and\nthe WGM. The mechanical motion is detected by mea-\nsuring the RF modulation of the transmitted light, which\nis collected and sent to a photo-detector with an opera-\ntional bandwidth of 12 GHz. The output signal can be\nanalyzed by a spectrum analyzer (SA) and a vector net-\nwork analyzer (VNA). The latter is also used to inject RF\nsignals into a shorted end microstrip waveguide (MSW),\nwhich we use to generate RF magnetic \felds (see Fig.\n1c, and Supplementary Information), and to character-\nize the FMR modes of the YIG \flm. A static magnetic\n\feld B DCis generated by two coils connected to a current\nsource. All experiments were carried out under ambient\nconditions of temperature and pressure.\nIn Fig. 1d we show the typical optical transmission\nspectrum from a microsphere of about 40 \u0016m in diameter\nwith multiple WGMs resonances. The mode used here\nnear 1509 nm has a quality factor of 108and couples\nto several mechanical radial breathing modes (Fig. 1e),\nat 109, 206, 292, 338, and 465 MHz. The displacement\npro\fles of the \frst three modes obtained using COMSOL\nMultiphysics software are shown in Fig. 1e (inset). Due\nto the disadvantageous refractive index contrast between\nthe YIG \flm ( n= 2.19) and the BTS microsphere sphere\n(n= 1.9), we use an intermediate 100 nm thick layer of\nPolymethyl methacrylate (PMMA), which has a smaller\nrefractive index ( n= 1.49), to preserve the high Q-factors\nof the optical modes.\nThe method used to excite and measure uniform\nmagnon modes, from MHz to GHz range, in the YIG\n\flm, is the broadband FMR method. The MSW cre-\nates a RF \feld perpendicular to B DCthat excites the\nprecessional motion of the magnetization around B DC.\nWhen the excitation frequency matches the FMR con-\ndition, energy is absorbed. Consequently, a dip is ob-\nserved in the re\rection spectrum (S 11). With the mag-\nnetic \feld generated in our set up (B DC\u001410 mT), the\nFMR frequency can be tuned from 0.2 to 1 GHz (see\nFig. 1f). In the case of a thin \flm with a static mag-netic \feld applied in-plane and the RF \feld perpendic-\nular to the direction of magnetization M, the resonance\nfrequency is !0\u0011\rp\nBint(Bint+\u0016M), with the internal\n\feldBint=Ban+BDC,Banbeing the anisotropy \feld\nand\rthe gyromagnetic ratio (See Supplementary Infor-\nmation). We note that the linewidth of the FMR mode\n(Fig. 1f) decreases as a function of B DCfrom about 46\nMHz to 24 MHz (at 2.5 mT). At low B DCvalues, the\nmagnetization is not uniform, leading to inhomogeneous\nspectral broadening of the resonance.\nThe spectral response of the system to an applied mag-\nnetic excitation at a calibration frequency, !cal, is shown\nin Fig. 2a. In Fig. 2b, we plot a zoom area with and\nwithout excitation. The thermal spectrum (black line)\nshows a double-peak response between 205 and 210 MHz.\nWhen the system is excited at a calibration frequency\n!cal= 206 MHz, with an RF power level of -10 dBm (2.3\n\u0016T), a sharp peak emerges (red line). Such a peak dis-\nappears when the sphere is lifted from the YIG and the\nmechanical contact is lost, which demonstrates the me-\nchanical origin of the signal. The force induced by vibra-\ntions in the YIG modi\fes the mechanical spectrum of the\nmicrosphere with a corresponding Signal-to-Noise-Ratio\n(SNR) of 8 dB. As shown in Fig. 2c, we observe a linear\ndependence of square root of SNR at !calon the applied\nRF magnetic \feld magnitude ( B). HereBis estimated\nusing the characteristic impedance of the of circuit and\nthe applied RF power (see Supplementary Information).\nTheBsensitivity ( Bmin) is given by the \feld strength\nat which the spectral peak height is equal to the noise\n(SNR=1) for a 1 Hz measurement resolution bandwidth\n(RBW) [21]. The corresponding magnetic \feld at !cal\nisBmin(!cal) = 0.5\u0016T for RBW = 30 kHz. Then, the\nsensitivity at !calis given by\u000eBmin(!cal) =Bmin(!cal)p\nRBW\u0018\n3 nT Hz\u00001=2. The dynamic response of the sensor, N( !),\nover a wide frequency range is obtained by varying the\ninput frequency from port 1 of the VNA and by looking\nat S 21, where port 2 is directly connected to the detector.\nAs shown in Fig. 2d, we observe a peak in signal N( !)\nwherever!is resonant with a mechanical mode (labelled\nin Fig. 2d from I to IV) with a high OM coupling (see\nFig. 1e). Due to the enhanced noise rejection of the\nVNA, we can detect modes at 374 MHz and 456 MHz\nthat in the thermally activated spectrum (Fig. 2c) were\nbelow the noise level. By following a similar procedure\nas in Ref.[6], the frequency dependence of the sensitivity\n\u000eBmin(!) is obtained by combining the spectral calibra-\ntion at a single frequency !cal, the noise power spectrum\nin absence of a magnetic \feld S(!), andN(!) on the\nmechanical modes.\n\u000eBmin(!) =s\nS(!)N(!cal)\nN(!)S(!cal)\u000eBmin(!cal) (1)\nFig. 2e plots the sensitivity within the frequency range3\nWhispering\nGallery Mode Fiber\nFmagFth\nBBDCMicrosphere\nWaveguide\nYIG\n1.4 - 1.6 µm\nFIG. 1: ( a) Schematic representation of the magnetometer, including a microstrip waveguide, a ferromagnetic YIG \flm and\na barium-titanium-silicate microsphere. ( b) Conceptual schematic of the device. The optomechanical system is modeled as a\nFabry-P\u0013 erot cavity with a moving mirror. FthandFmagdenote the thermal force and the magnetostrictive forces, respectively.\n(c) A simpli\fed schematic of the full experimental setup. A tapered \fber is used to probe the optical modes of the microsphere.\nTE polarization is set with a \fber polarization controller (FPC) and the transmitted signal is sent to a fast photodetector. Two\nelectromagnetic coils are used to generate static magnetic \felds that tune ferromagnetic resonance modes on the YIG \flm. ( d)\nOptical whispering gallery modes spectrum measured on the microsphere. The inset shows a schematic of the coupling scheme\nbetween the tapered and the sphere and a simulation of the optical WGMs. ( e) Mechanical mode spectrum of the microsphere.\nOnly thermally driven motion is observed corresponding to radial breathing modes. The inset shows the displacement pro\fle\nof the \frst three modes. ( f) FMR resonances of the YIG \flm applying di\u000berent static magnetic \felds measured by detecting\nthe re\rected signal Re(S11).\nassociated to the \frst \fve mechanical modes of the micro-\nsphere observed in Fig. 1c. The lowest sensitivity value\nobtained is\u0018850 pT Hz\u00001=2close to the mechanical\nmode at 206 MHz. This particular mode presents a large\noverlap with the optical WGM, since its displacement\n\feld pro\fle is concentrated along the edge of the sphere\n(see Fig. 1a and 1e). It is also worth noting that the\nfrequency of the mode is still several tens of MHz away\nfrom the center of the FMR resonance (see Fig. 2d) so\nthat the reported value of minimum sensitivity could be\nimproved by \fne tuning the FMR position. The sensi-\ntivity remains around 1 nT Hz\u00001=2within the linewidth\nof the mechanical resonances (about 10 MHz), for \fve\nmechanical modes.\nIn Fig. 3a, we report the system response N(!) as a\nfunction of frequency for di\u000berent values of BDC. We\nuse a second sphere with a similar radius, mechanical\nspectrum and optical quality factors. As evidenced by\nthe FMR spectra in Fig. 3a (grey curves), the FMR fre-quency increases increasing BDC, and shifts the magneto-\nmoeter spectral response N(!) (red curves). In addition,\nthis shift is acompanied by a spectral narrowing of the\nOMM bandwith, clearly following a spectral narrowing of\nthe FMR dip (Fig. 1f). This behavior con\frms that the\nmagnetic signal appears only where the linewidth of the\nFMR resonance overlaps with the mechanical resonances\nof the microsphere. It also evidences the presence of me-\nchanical modes that are hidden below the noise level of\nthe SA. In Fig. 3b we obtain the peak sensitivity for\nthe di\u000berent measured positions of the FMR mode. We\nnote that the OMM detects magnetic \felds even above\nan operational frequency of 1 GHz (see Supplementary\nInformation). The measured sensitivities are comparable\nwith the one reported in Fig. 3d ( s1 nT Hz\u00001=2). This\nvalue of the operational frequency is a lower bound lim-\nited by the minimum BDCreachable in our experimental\nset up.\nAs noted above, the FMR resonances obtained are4\nN( ) (dB)ωS( ) (dBm)ωS( ) (dBm)ω\nFIG. 2: ( a) Spectral response of the magnetometer S(!) mea-\nsured with a spectrum analyzer. ( b) Mechanical spectrum of\nthe microsphere excited by applying an RF magnetic \feld of\n2.3\u0016T at 206 MHz, in red. The black curve corresponds to\nthe mechanical mode without excitation. ( c) Square root of\nsignal to noise ratio (SNR) of the system as a function of the\napplied RF magnetic \feld. ( d) System response N(!) as a\nfunction of the frequency of the RF \feld. The FMR is shown\n(inverted) on the back of the graph to illustrate the mechan-\nical modes a\u000bected by the resonant e\u000bect. ( e) Magnetic \feld\nsensitivity\u000eBmin(!) as a function of frequency de\fned where\nthere is overlapped with mechanical modes. Five excited me-\nchanical modes (labelled from I to V) allow to calculate the\nsensitivity in a frequency window of s10 MHz around the\nfrequencies of the mechanical resonances. A peak sensitivity\nof\u000eBmin(!)\u0018850 pT Hz\u00001=2is achieved.\nrather broad given that the electromagnets were placed\nseveral centimetres away from the sample, resulting in a\nnot fully uniform B DC. The linewidths of the FMR res-\nonances shown in Fig. 2c are a factor of two larger than\nthe one shown in Fig. 1f, which were obtained measur-\ning the YIG \flm in a set up with a highly homogeneous\nBDC. The origin of this broadening is due to the excita-\ntion of non-uniform modes and low-\feld losses [22, 23].\nOn the positive hand, under these conditions, the opera-\ntional frequency range is increased and several mechan-\nical modes can be covered without changing BDC. On\nN(ω) (dB)FIG. 3: ( a) System response for di\u000berent FMR modes excited\non the YIG \flm. Gray curves show magnon resonances for\ndi\u000berent values of static magnetic \feld. ( b) Peak magnetic\n\feld sensitivity obtained by moving the FMR mode.\nthe other hand, a much improved sensitivity could be at-\ntained with a narrower and deeper FMR resonant with a\nmechanical mode.\nA further evidence that the mechanical modes of the\nspheres are excited by mechanical vibrations within the\nYIG layer generated by magnetostriction is given by ad-\nditional experiments performed with a high-frequency\nDoppler Vibrometer. This setup implements an optical\ntechnique for non-contact measurements of displacement\nin the vertical direction with picometer accuracy. We use\nthis technique to measure the YIG surface displacement\nwithout the microsphere. The measured displacement as\na function of frequency results in non zero amplitudes\nonly in a frequency range that is coincident with a given\nFMR (See the Supplementary Information). The defor-\nmation spatial pro\fle is in phase throughout the YIG\nlayer surface, i.e., there is a spatially homogenous out-of-\nplane displacement. We cannot rule out the excitation of\nphonon modes with in-plane deformation within the YIG\nlayer, but those are not playing an active role activating\nthe mechanical modes of the sphere, which is also veri\fed\nwith Finite Element Method (FEM) simulations.\nThe \feld sensitivity presented in Fig. 2e, is similar\nto the best sensitivity obtained in previous cavity OMM\nstudies [6{9]. In those references, a magnetostrictive ma-\nterial (Terfenol-D) was used due to its high magnetostric-\ntive coe\u000ecient [24]. Despite the fact that single crystal\nYIG was found to be around a factor of two less mag-\nnetostrictive than Terfenol-D [25], the high performance\nof the OMM reported here is due to the use of YIG to5\ndisplay a high quality FMR and a high Q glass resonator.\nCompared with room temperature devices like dia-\nmond NV-centers magnetometers, the device reported\nhere shows a factor of two smaller peak sensitivity than\nthe sub-pico Tesla NV magnetometer reported in Refs.\n[26, 27], with the advantage of having a \fber-based op-\ntical detection. The sensitivity values reported here out-\nperform electrical Lorentz-Force magnetometers [28] of\ncomparable size by three orders of magnitude. SQUID\nmagnetometry can detect magnetic \felds that are \fve\norders of magnitude smaller than our scheme [29, 30],\nreaching sensitivities of 1 fT Hz\u00001=2ats100 Hz, but it\nrequires cryogenic environments to operate.\nIn summary, we have demonstrated a hybrid system\ncomposed of a magnetic resonator coupled by mechanical\ninteraction to a whispering gallery mode optomechanical\ncavity to detect weak oscillating magnetic \felds. A peak\nmagnetic \feld sensitivity of \u0018850 pT Hz\u00001=2is achieved\nby exciting a mechanical mode at 206 MHz. This value\ncan be further improved by optimizing the overlap be-\ntween the FMR resonance and the mechanical resonance\nof the optomechanical cavity. Besides the excellent \fg-\nures of merit, the tuneability of the frequency response\nup to 1 GHz, room temperature operation and simplicity\nin fabrication o\u000ber the opportunity of designing a high-\nperformance magnetometer. Large bandwidths are nec-\nessary for applications such as high-speed detection, me-\nchanical signal processing and for high-resolution imag-\ning methods [11]. In this regard, the frequency response\nof our device could be further extended to higher fre-\nquencies by increasing the static magnetic \feld. The\nmagnetometer's sensitivity can be further improved fol-\nlowing di\u000berent strategies. For example, measuring at\nlow temperatures or high vacuum conditions would result\nin better sensitivity values, since the sensitivity behaves\nas\u000eBmin(!)\u0018pTQm. Moreover, using a harder mate-\nrial than PMMA would reduce the mechanical impedance\nmismatch between PMMA and YIG, avoiding mechani-\ncal energy to be dissipated at the interface before reach-\ning the sphere. In addition to the technological possi-\nbilities of designing a new magnetometer our hybrid de-\nvice also opens a path towards studying phenomena re-\nlated to phonon-magnon coupling. Currently, magnons\nare gathering increasing attention in spintronics exper-\niments (e.g. magnonics [34] and spin caloritronics ar-\neas [35{37]) as means of processing spin information and\nmanaging heat in nanoscale structures. Even though\nits superior properties make YIG a common choice for\nspintronic applications, the underlying physical mecha-\nnisms involved in phonon-magnon coupling are only an-\nalyzed by controlling the magnonic system. In contrast,\nour hybrid-resonator can be used as a novel approach to\nstudy phonon-magnon coupling, controlling the phonon\ncontribution using optical techniques.\nThis work was supported by the Spanish Severo Ochoa\nExcellence program. D. N. U. gratefully acknowledgesthe support of a Ramn y Cajal postdoctoral fellowship\n(RYC-2014-15392) and the Ministry of Science, Innova-\ntion and Universities (PGC2018-094490-B-C22). M. V.\nC. acknowledges support from the Spanish Ministry of\nEconomy and Competitiveness, MINECO (under Con-\ntract FIS2015-62641-ERC). M. V. C. thanks H. Tang and\nG. E. W. Bauer for discussions. M. F. C. acknowledges\nsupport of a Severo Ochoa studenship. G. A. aknowl-\nedges support of a BIST studentship. The authors thank\nG. Jakob for the YIG samples and M. Cecchini for ex-\nperimental support.\n\u0003Corresponding author; Electronic address:\ndnavarro@ub.edu\n[1] J.-M. Pirkkalainen, S.U. Cho, F. Massel, J. Tuorila, T.T.\nHeikkil a, P.J. Hakonen, and M.A. Sillanp a a, Nat. Com-\nmun., 6, 6981 (2015).\n[2] J. Abadie, B.P. Abbott, R. Abbott, T.D. Abbott[...], and\nJ. Zweizig, Nat. Phys. 7, 962 (2011).\n[3] T.J. Kippenberg, and K.J. Vahala, Science 321, 1172{\n1176 (2008).\n[4] E. Gavartin, P. Verlot, and T.J. Kippenberg, Nat. Nan-\notechnol. 7, 509{514 (2012).\n[5] A. Krause, G.M. Winger, T.D. Blasius, Q. Lin, and O.\nPainter, Nat. Photonics 6, 768 (2012).\n[6] S. Forstner, S. 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Valen-\nzuela, Nat. Mater. 11, 199{202 (2012)."    },    {        "title": "1610.04715v1.Spin_excitations_in_an_all_organic_double_quantum_dot_molecule.pdf",        "content": "Spin excitations in an all-organic double quantum dot molecule\nMax Koole,1Jan C. Hummelen,2and Herre S.J. van der Zant1,\u0003\n1Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, Netherlands\n2Stratingh Institute for Chemistry and Zernike Institute for Advanced Materials,\nUniversity of Groningen, Nijenborgh 4, 9747 AG, Groningen, Netherlands\n(Dated: April 20, 2022)\nWe realize a strongly coupled double quantum dot in a single all-organic molecule by introduc-\ning a non-conjugated bridge in between two identical conjugated moieties. Spin-1/2 Kondo and\nKondo enhanced low-energy excitations for respectively the odd and even electron occupation are\nobserved in o\u000b-resonant transport. The ground state in the even occupation can be the singlet or\nthe triplet state varying between samples. This observation suggests that both anti-ferromagnetic\nand ferromagnetic interactions between spins are of the same order of magnitude.\nPACS numbers: 85.65.+h, 73.63.-b, 73.21.La\nI. INTRODUCTION\nElectron transport through double quantum dots made\nit possible to investigate new avenues of physics, such as\nthe study of Kondo e\u000bects beyond the single-impurity\nAnderson model1and the realization of qubits2. Ex-\nperimental systems encompass coupled semiconductor3,\ngraphene4and carbon nanotube5quantum dots and\nquantum dots built from atom pairs6. For single\nmolecules, electron transport through an organic double\nquantum dot (DQD) molecule has not been studied to the\nsame extend despite the fact that the energy scales of its\nelectronic states are generally larger than in its counter-\nparts, therefore relaxing the requirements for millikelvin\ntemperatures for operation. Furthermore, the use of or-\nganic atoms like carbon and oxygen is expected to result\nin small spin-orbit coupling. This suggest that an all or-\nganic DQD molecule can potentially be a model system\nfor the study of spin interactions that can be chemically\ntuned.7\nTo study transport in an organic DQD molecule, we\nuse a 9,10-dihydroanthracene core with acetyl-protected\nsulfur anchoring groups connected by spacers8(see \fg-\nure 1a; named AH from now on). The conjugation of\nAH is broken due to the sp3hybridization of the two car-\nbon atoms in the center of the 9,10-dihydroanthracene\ncore. This results in AH being made up of two conju-\ngated halves (dark green in \fgure 1a) connected by a\nnon-conjugated bridge (red in \fgure 1a). This peculiar\nelectronic structure results in a decreased conductance9\nand a higher energy UV-VIS absorption threshold10com-\npared to the fully conjugated version of the molecule.\nThe electronic structure of AH also results in negative\ndi\u000berential conductance in charge transport11, which is\nexplained by a two-level model formed by the broken con-\njugation. Past research has thus shown that the broken\nconjugation in\ruences the electronic structure and trans-\nport, suggesting that it may be regarded as a molecu-\nlar DQD. In this paper, we explore this possibility by\nusing low-temperature transport spectroscopy measure-\nments in a three-terminal con\fguration.II. MEASUREMENTS\nTo form a gold-molecule-gold junction, we employ\nthree-terminal electromigration junctions (\fgure 1b). A\nsilicon/silicon oxide substrate is used, on top of which\nan aluminum gate is deposited by e-beam evaporation.\nUnder a pure oxygen atmosphere, the aluminum is oxi-\ndized to create an electrically insulating layer surround-\ning the gate terminal. On top of the gate a 12 nm thick,\n100 nm wide gold nano-wire is subsequently deposited,\nwhich is connected with 100 nm thick Au leads to bond\nFIG. 1. Molecule and setup. a) 9,10-dihydroanthracene\n(AH) with thiolated spacers. The dark green parts are the\nconjugated halves and the red colored part in the middle is\nthe non-conjugated bridge. The yellowish colored parts at\nthe end of the molecule are the anchoring groups. b) False-\ncolored scanning electron microscopy image of a junction be-\nfore electromigration. The purple area is the aluminum oxide\nsurrounding the aluminum gate electrode underneath it. The\ndarker green area is the 10 nm thick gold nano-wire. Lighter\ngreen areas correspond to the thicker leads, which are con-\nnected to wire bond pads. The white bar at the bottom has\na length of 250 nm.arXiv:1610.04715v1  [cond-mat.mes-hall]  15 Oct 20162\npads. To create the nano-gap the thin wire is controllably\nelectromigrated12at room-temperature in a solution of\ndichloromethane with 0.5 mM AH to a resistance of the\norder of 5 K\n. After this the wire is let to self-brake13to\navoid the formation of spurious gold nano-particles when\nopening a gap in the wire to form the junction. When\njunction resistances are of the order of 1 M\n, the sam-\nple chamber is pumped and cooled to cryogenic temper-\natures. The three-terminal geometry of these junctions\nmakes it possible to measure the current ( I) as a function\nof bias (V) and gate ( Vg) voltages. Furthermore, with the\nhelp of a 1K-pot, heater resistor and a superconducting\nmagnet, temperature and magnetic-\feld dependent mea-\nsurements can be performed.\nFigure 2a shows a di\u000berential conductance map of a\njunction (S1) prepared as described above. A single-\nelectron tunneling (SET) regime is present around Vg=\n1 V, marked by the dashed lines. The SET edges of\nthe left charge state are visible; the edges of the right\ncharge state are, however, suppressed. We \frst focus\non the left charge state in \fgure 2a. It shows a pro-\nnounced gate-dependent zero-bias peak. The peak splits\nas a function of magnetic \feld and its height decays ex-\nponentially as a function of temperature (see supporting\ninformation)14. This behavior suggests spin-1/2 Kondo\nphysics to be the origin of the zero-bias peak; the g-factor\ndetermined from the magnetic \feld dependent data is\ng\u00182:6, which matches with values observed for single\nmolecules15. The presence of a spin-1/2 Kondo peak in-\ndicates an uneven electron occupation for the left charge\nstate.\nWhen the gate voltage in \fgure 2a is increased past\nVg= 1 V, an electron is added to the molecule. Instead\nof a zero-bias peak, a broader peak appears with a sup-\npression around zero bias. Figure 3a shows the temper-\nature dependence of this feature. With increasing tem-\nperature the suppression is lifted and a single broad peak\nremains above T= 5 K. Increasing the temperature even\nfurther results in a decrease in the height of the broad\npeak. This is seen more clearly in \fgure 3b, which shows\nthe zero-bias conductance at Vg= 3 V as a function\nof temperature. This temperature dependence resembles\na two-stage Kondo process16, which can be caused by\nthe full screening of a triplet17or a nearly degenerate\nsinglet/triplet state18. These scenario's are further sup-\nported by the fact that in this charge state there should\nbe an even number of electrons on the molecule, consid-\nering the odd occupation of the adjacent charge state to\nthe left.\nThe magnetic \feld dependence in \fgure 3c shows that\nthe suppression is lifted as a function of increasing mag-\nnetic \feld. This observation matches a singlet ground\nstate (S= 0), because in case of a triplet ground state\nthe excited states would move up in energy when increas-\ning the magnetic \feld. Taking all information together,\nwe assign the feature in the right charge state to two spins\nthat interact anti-ferro magnetically; the ground state is\na singlet and the peak at \fnite bias in dI=dV is caused\nFIG. 2. Di\u000berential conductance map as a function of bias\n(V) and gate ( Vg) voltage of electromigration junction S1 (a)\n(T= 2:0 K, B= 0 T) and junction S2 (b) ( T= 2:0 K,\nB= 0 T). Dashed lines show the SET edges. In a) a zero-\nbias peak is present in the left charge state and an excitation\nat a few meV in the right charge state. In b) a weak zero-bias\npeak (white line, see \fgure P7 in the supporting information\nfor a linecut)14is present in the right charge state and an\nexcitation of a few meV in the left one.\nby the singlet to triplet excitation. Using a Heisenberg\nhamiltonian H=\u0000JS1\u0001S2, the exchange coupling ( J)\ncan be determined from the \fgure, J=\u00000:7 meV. It is\nalso interesting to note that when the suppression is lifted\n(when the singlet and triplet state are brought into de-\ngeneracy), the peak full width half maximum (FWHM)\nis larger than the spin-1/2 Kondo FWHM in the adja-\ncent charge state at the same temperature. This matches\nother observations where a Kondo resonance caused by a\nsinglet-triplet degeneracy has a larger FWHM than the\none of the spin-1/2 Kondo19.\nFigure 2b shows another sample (S2) which has two\ncharge states separated by a SET regime around Vg=\n\u00001:3V(marked by the dashed lines). However, in this\nsample a zero-bias peak is present in the right charge\nstate. Temperature and magnetic \feld dependent data\npoint again to a spin-1/2 Kondo origin and thus suggest\nan uneven occupation of the molecule in this charge state\n(see supporting information)14. Going to the left charge\nstate by removing an electron from the molecule results3\nFIG. 3. Temperature and magnetic \feld dependence of low-bias features in the even charge state for sample S1 (a,b,c) and\nS2 (d,e,f). a) Temperature dependence of the di\u000berential conductance at Vg= 3 V. The individual curves are taken at a\ntemperature of 1.9, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 7.0, 8.0 and 9.0 K and are o\u000b-set from each other. The red dashed\nline is the zero-bias conductance plotted in panel b. b) Zero-bias conductance as a function of temperature. c) Magnetic-\feld\ndependence of the low-bias feature in the right charge state ( Vg= 3 V, T= 2 K). d,e,f) show the same measurements as a,b,c)\nwith curves taken at 1.95, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 7.0 and 9.0 K for sample S2. In d,e) Vg=\u00001:9 V and in f)\nT= 2 K, Vg=\u00001:9 V.\nin the appearance of a low-bias excitation. Figure 3d and\ne show the temperature dependence and \fgure 3f shows\nthe magnetic-\feld dependence of this low-bias feature.\nA strong dependence on temperature is seen, suggesting\nKondo correlations to play a role. Furthermore, the mag-\nnetic \feld dependence shows that the excitation energy\nshifts outwards to higher energies with increasing mag-\nnetic \feld. The presence of an even number of electrons\non the molecule and the magnetic \feld and temperature\ndependences suggest a triplet ( S= 1) ground state with\na Kondo enhanced excitation to the singlet state (the\nexchange coupling Jis +0:5 meV and ferro-magnetic)19.\nSample S2 therefore has two di\u000berences compared to sam-\nple S1; the ground state in the even charge state is a\ntriplet instead of a singlet state and the spin-1/2 Kondo\ne\u000bect is found to the right of an even charge state with\nlow-bias features instead of to the left of such a state.\nThe pattern of spin-1/2 Kondo physics in one charge\nstate and a low-bias feature due to the excitation be-\ntween the singlet and triplet in a neighboring charge\nstate has been seen in eight samples (see supporting\ninformation)14. Table I shows the addition energy ( Uadd),\nthe level broadening of the SET edge (\u0000), the magni-\ntude and sign of the exchange coupling ( J) of the eight\nsamples (for an overview of all parameters used in this\ntext see supporting information)14. It can be seen that\nthe addition energies are relatively small compared to\nthose of molecules in gas phase. This may in part bedue to the small energy di\u000berence between the HOMO\nand HOMO-1 (see discussion), but re-normalization of\nthe energy levels (image charge e\u000bects associated with\nthe gate and electrodes) may also play a role20{22. The\nlevel broadening is smaller than the addition energy, but\nstill of the same order explaining why Kondo correla-\ntions can be seen in the measurements. The magni-\ntude of the coupling between the spins ( jJj) ranges from\napproximately degenerate up to 5 meV. The sign of J\nhas been determined for 2 samples (S1 and S2, anti-\nferromagnetic and ferro-magnetic respectively). Further-\nmore for sample S6 and S7 the singlet and triplet states\nare approximately degenerate (see \fgure P7 in the sup-\nporting information)14. For 4 samples magnetic \feld\ndata have not been recorded so the sign of Jcould not\nbe determined.\nIII. DISCUSSION\nThe observation of low-bias features in a charge state\nwith an even occupation suggests that the singlet to\ntriplet energy di\u000berence for the two highest lying elec-\ntrons is small. For molecules this may seem counter in-\ntuitive due to their small size, however, when inspecting\nthe density functional calculations (DFT)11of the or-\nbitals of AH (\fgure 4a) it can be seen that the HOMO\nand HOMO-1 are nearly degenerate in energy. Analo-4\ngous to the prototypical hydrogen molecule, these nearly\ndegenerate orbitals are formed by the hybridization of\nboth conjugated halves of AH and thus show symmet-\nric and anti-symmetric orbital wave-functions. The level\nsplitting (\u0001 E) of the HOMO and HOMO-1 is of the or-\nder of 10 to 50 meV, depending on the geometry of the\nmolecule (see the supplementary information of Perrin\net al.11), which is signi\fcantly smaller then in fully con-\njugated molecules. The level splitting gives an estimate\nfor the internal coupling ( \u001c) between the two conjugated\nhalves of the molecule by using \u001c= \u0001E=2. This results\nin the coupling of the conjugated moieties to be between\n5 to 25 meV.\nWe attribute the features in the measurements to the\nsequential oxidation of AH from a N = -1 occupation\ndown to N = -3 occupation (where N is the di\u000berence in\nelectrons with the neutrally charged molecule). Oxida-\ntion of the AH is supported by DFT calculations which\npredict the Fermi energy to lie close to the HOMO or-\nbital. Furthermore, multiple reduction of AH is unlikely\nas the LUMO and LUMO+1 are not close in energy. This\nleads us to the following features in transport (\fgure 4b)\nand the electronic states (\fgure 4c) as a function of elec-\ntron occupation of the HOMO and HOMO-1 in AH. In\nthe N = -3 state, a single unpaired electron is present in\nthe HOMO-1, which results in spin-1/2 Kondo in trans-\nport. Adding an electron results in the formation of a\nsinglet or triplet state. Depending on the ground state,\nthe excitation visible in the N = -2 charge state is the\ntriplet or singlet respectively (this will be discussed fur-\nther on). Addition of a second electron results in a pair\nof electrons in the HOMO-1 and a free unpaired spin in\nthe HOMO; spin-1/2 Kondo reappears. No samples were\nmeasured that showed the three consecutive charge states\nshown in \fgure 4b, however, all eight samples show the\nN = -2 charge state. Next to the N = -2 charge state\nTABLE I. Parameters extracted from the di\u000berential con-\nductance maps (see also supporting information)14. Some\nvalues could not be determined; in those cases a upper or\nlower limit is given. Uaddis the addition energy of the spec-\ni\fed charge state with an estimated error of \u00065 meV, \u0000 is\nthe level broadening ( \u00062 meV) andjJjthe magnitude of the\nexchange coupling ( \u00060:2 meV). The last column displays the\ncharacter of the coupling between the two spins, which can be\nanti-ferromagnetic (AF), ferromagnetic (F), degenerate (de-\ngen.) or not determined (not det.) due to the absence of\nmagnetic \feld dependent data.\nUadd(meV) \u0000 (meV) jJj(meV) coupling\nS1 49 (N=-1) 11 0.7 AF\nS2 >63 (N=-2) 5.5 0.5 F\nS3 50 (N=-2) 9 1.4 not det.\nS4 73 (N=-2) 5 3 not det.\nS5 37 (N=-2) 8 1.2 not det.\nS6 58 (N=-2) 10 <0.2 degen.\nS7 >53 (N=-1) 18 <0.5 degen.\nS8 >50 (N=-2) 40 3.5 not det.\nFIG. 4. a) Energy spectrum of the AH molecular orbitals\n(not to scale); L stand for lowest unoccupied molecular or-\nbital (LUMO) and H for highest occupied molecular orbital\n(HOMO). The isosurface of the nearly degenerate HOMO and\nHOMO-1 are shown next to it. It can be seen that the HOMO-\n1 is symmetric whereas the HOMO is anti-symmetric with re-\nspect to the molecule. b) Schematic of the transport features\nin AH at low temperature and B= 0 T. Green dashed lines\nare Coulomb edges; red lines represent higher-order transport\nfeatures in the Coulomb blockade region, i.e, zero-bias peaks\nassociated with S= 1=2 Kondo or inelastic tunneling excita-\ntions at \fnite bias. c) Orbital \flling of the nearly degenerate\nHOMO and HOMO-1. In the N = -2 charge state two ground\nstates are possible depending on the exchange coupling.\nsamples S1, S3, S7 and S8 show the N = -3 state and\nsamples S2, S5 and S6 show the N = -1 charge state.\nThe fact that both the singlet and triplet ground states\nare observed at B= 0 T, indicates that the ferromagnetic\nJFand anti-ferromagnetic JAFterms contribute almost\nequally to the total exchange coupling J=JF\u0000JAF. To\n\frst order23,JFis governed by the Coulomb exchange\nandJAFby the kinetic exchange. The kinetic exchange\ncan be approximated by the Hubbard model, for t<U ,\nJAF= 4t2=U, whereUis the on-site Coulomb interaction\nandtis the inter-site hopping. Using the addition energy\nas an approximation for the on-site Coulomb interaction\nU\u0019\u0001E= 40\u000080 meV and using t\u0019\u001c= 10 meV as cal-\nculated from DFT, one \fnds JAF= 5\u000010 meV. A singlet\nground state can thus be explained by considering only\nthis term. However, the appearance of the triplet ground\nstate suggests that the Coulomb exchange should also be5\ntaken into account. This can be intuitively understood\nfrom the fact that AH is approximately 2 nm long, much\nsmaller than for other double quantum dot implemen-\ntations. Therefore, electrons are closer together thereby\nincreasing Coulomb e\u000bects and promoting ferromagnetic\ncoupling (Hund's rule).\nFurther indications for a nearly degenerate HOMO and\nHOMO-1 can be seen in the o\u000b-resonant transport in the\nN = -1 and N = -3 charge states of S2 and S7 respectively.\nHere, in addition to the spin-1/2 Kondo resonance, low-\nbias excitations are visible. These can be related to\nthe excitation of an electron from the HOMO-1 to the\nHOMO24. In both samples the excitation energy changes\nas a function of gate voltage; a possible mechanism for\nthis could be di\u000berential gating25of the two conjugated\nhalves of AH that form the HOMO and HOMO-1. If\nAH is situated asymmetrically in between the electrodes,\nthe left conjugated half of AH can exhibit a slightly dif-\nferent gate coupling than the right conjugated half. It\nwould thus modify the energy splitting (\u0001 E) between\nthe HOMO and HOMO-1, leading to low-lying gate de-\npendent excitations as observed in the experiment.\nThe delocalization and splitting of the HOMO and\nHOMO-1 and the observation of Kondo e\u000bects, suggest\nthat the measurements show a DQD that has a strong\ninternal coupling ( \u001c) and is well coupled to the leads (\u0000).\nTo further probe the spin states in AH and study spin re-\nlaxation mechanisms, it is bene\fcial to signi\fcantly lower\n\u0000 and\u001cso that the spins on the two halves are local-\nized more. This has been done in semiconductor26andcarbon nanotube27DQD's, where with Pauli spin block-\nade the spin dynamics and relaxation mechanisms have\nbeen studied. This localization can be achieved by using\nchemical synthesis to decrease the coupling to the leads28\nor the intramolecular coupling of the molecular DQD by\nusing for example a larger non-conjugated bridge7or a\nrotation of the two halves29.\nIn conclusion, we have shown transport spectroscopy of\na strongly coupled double quantum dot molecule formed\nin a symmetric molecule with a non-conjugated bridge\nin the middle. A pattern of consecutive charge states\nhas been identi\fed which shows spin-1/2 Kondo, sin-\nglet/triplet states, spin-1/2 Kondo as a function of in-\ncreasing electron occupation. This pattern can be ex-\nplained with the formation of a pair of nearly degener-\nate orbitals due to the broken conjugation. In the even\ncharge state transport features are observed, which indi-\ncate the presence of singlet and triplet states. 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Hedeg\u0017 ard,\nN. Stuhr-Hansen, K. Moth-Poulsen, and T. Bj\u001crnholm,\nNano Lett. 8, 1 (2008).\n29L. Venkataraman, J. E. Klare, C. Nuckolls, M. S. Hybert-\nsen, and M. L. Steigerwald, Nature 442, 904 (2006)."    },    {        "title": "1806.04111v3.Competing_magnetic_correlations_across_the_ferromagnetic_quantum_critical_point_in_the_Kondo_system_CeTi___1_x__V__x_Ge__3______51__V_NMR_as_a_local_probe.pdf",        "content": "arXiv:1806.04111v3  [cond-mat.str-el]  8 Oct 2019Competing magnetic correlations across the ferromagnetic quantum critical point in\nthe Kondo system CeTi 1−xVxGe3:51V NMR as a local probe\nM. Majumder1∗,†W. Kittler2, V. Fritsch2,3, H. v. L¨ ohneysen2,4,‡H. Yasuoka1, and M. Baenitz1§\n1Max Plank Institute for Chemical Physics of Solids, 01187 Dr esden, Germany\n2Physics Institute, Karlsruhe Institute of Technology, 761 31-Karlsruhe, Germany\n3Experimental Physics VI, Center for Electronic Correlatio ns and Magnetism,\nInstitute of Physics, University of Augsburg, 86135 Augsbu rg, Germany and\n4Institute for Solid State Physics, Karlsruhe Institute of T echnology, 76012 Karlsruhe, Germany\n(Dated: October 9, 2019)\n51V nuclear magnetic resonance (NMR) and magnetization studi es on CeTi 1−xVxGe3have been\nperformed to explore the evolution from the ferromagnetic ( x= 0.113) to the antiferromagnetic\nKondo lattice state ( x= 1), with focus on the emergence of a possible ferromagnetic quantum\ncritical point (FMQCP) at xc≈0.4. From the temperature dependence of the nuclear spin-latt ice\nrelaxation rate, 1 /T1T, and the Knight shift, K, forx= 0.113 and x= 1 a considerable competi-\ntion between ferro- and antiferromagnetic correlations is found. Around the critical concentration\n(x= 0.35,0.405) quantum-critical spin fluctuations entail weak antife rromagnetic spin fluctuations\nadmixed with ferromagnetic spin fluctuations. The FMQCP in C eTi1−xVxGe3therefore is not\npurely ferromagnetic in nature.\nPACS numbers: 76.60.-k, 75.50.Gg, 75.30.Et, 75.25.Dk\nINTRODUCTION\nThe generic Doniach diagram of heavy-fermion mate-\nrials is based on the competition of Ruderman-Kittel-\nKasuya-Yosida (RKKY) interaction between localized\nmagnetic moments, and Kondo interaction between mo-\nments and conduction electrons. It has been derived for\nantiferromagnetic (AF) and isotropic RKKY-type of ex-\nchange interaction [1]. For 4 fsystems this competition\nis a prime scenariofor the emergence of a quantum phase\ntransition with a critical point (QCP) at zero tempera-\nture. The tuning of AF correlated 4 fsystems by an ex-\nternal control parameter (e.g., chemical composition, hy-\ndrostatic or uniaxial pressure, magnetic field) towards an\nAFQCP has been demonstrated for many materials[2–4].\nThe more recent search for ferromagnetic (FM) quantum\ncritical points (FMQCP) among 4 fand 3dsystems [5]\nhas provided a number of examples. However, due to\nthe interplay between FM and AF correlations and, even\nmore importantly, the low-lying quasiparticle excitations\nin metallic systems, these systems often undergo tran-\nsitions to other phases and thus appear to ”avoid” a\nbona-fide FMQCP. For correlated Yb systems FM or-\nder is rare and so far YbNi 4P2is the only FM 4 fsys-\ntem (with a Curie temperature TCof 170 mK) which\ncould be tuned towards the FMQCP [6]. Among the Ce\nsystems, CeRuPO [7], CeRu 2Ge2[8], CeAgSb 2[9], and\nCeNiSb 2[10] exhibit FM order but application of hydro-\nstatic pressure induces long-range AF order thus avoid-\ning an FMQCP. In CeFePO [11] and CePd 1−xRhx[12]\n∗Present address: Experimental Physics VI, Center for Elect ronic\nCorrelations and Magnetism, University of Augsburg, Germa nya glass-like ”Kondo cluster” state forms which likewise\nimpedes tuning towards an FMQCP.\nNMR measurements were carried out on a number of\nthose systems to probe locally the magnetic fluctuations\nandunveiltheirnature(AF vs. FMfluctuations) bytem-\nperature and field scaling of the spin-lattice relaxation\nrate, 1/T1T, and the NMR shift, K. In particular, NMR\nisabletoqualitativelydisentanglefinite- q(AF)and q= 0\n(FM) excitations [13, 14]. At the same time, NMR pro-\nvides information about the degree of disorder in alloy\nsystems which might affect the nature of the quantum\ncritical point. For example, in the heavy-fermion com-\npound CeFePO where long-range order is absent,31P\nNMR gave clear evidence for FM correlations admixed\nto AF correlations [15] whereas in the structural homo-\nlogue CeRuPO the existence of stable long range FM\norder was demonstrated by31P NMR measurements [7].\nRecently, Kittler et al.succeeded to tune the Curie\ntemperature of the FM Kondo-lattice system CeTiGe 3\n(TC≈14 K) monotonically down to T= 0 indicating a\npossible FMQCP upon V substitution for Ti [16, 17]. TC\ndecreases linearly with increasing V concentration, the\nstructure type (hexagonal perovskite P63/mmc, see Fig.\n6) is preserved and no crossover to other phases occurs.\nUpon V substitution the c/aratio is reduced [16]. The\nsingle-ionanisotropyof the Ce3+ions arising from the ef-\nfectofthecrystalelectricfield(CEF)onthe4 fmoments,\nchanges drastically with increasing V content: while the\neasy direction is parallel to the caxis for CeTiGe 3, it is\nperpendicular to cfor CeVGe 3[17, 18]. The end mem-\nber on the V-rich side, CeVGe 3, is an antiferromagnetic\nKondo system with a N´ eel temperature TNof about 6 K\nandaKondotemperatureof TK≈10K[18]. Itshouldbe\nnoted that in contrast to most systems discussed above\nthe V substitution of Ti in CeTiGe 3is not isoelectronic2\nbut differs by one delectron. Recent studies on CeTiGe 3\nsuggest an avoided FMQCP and a splitting of the TC(p)\nline into a wing-like structure [19] as observed in other\nclean ferromagnets [5], with the complication of several\nintermediate AF phases occurring in this system.\nEXPERIMENTAL DETAILS\nCeTi1−xVxGe3alloys crystallize in the hexago-\nnal perovskite (BaNiO 3-type) structure (space group\nP63/mmc) without any structural phase transitions\nacross the entire V concentration range. The samples\n(x= 0.113, 0.35, 0.405 and 1) used in this experiment\nwere cut from single crystals [17] and ground to pow-\nder. The51V NMR spectra were taken by the transient\n(pulsed) NMR technique with a commercial NMR spec-\ntrometer (TecMag Apollo). The NMR line profiles were\nobtained by integration of the spin-echo amplitude in the\ntime domain with sweeping an external field at constant\nfrequency. By simulating the experimental line profile,\nthe isotropic and anisotropic (from the V local symme-\ntry an axial distribution of shift is expected, see Fig. 6)\nKnight shifts ( KisoandKax) have been extracted. The\nspin-lattice relaxation rate 1 /T1was measured by the\nsaturation recovery method where the recovery of nu-\nclear magnetization was fitted to a stretched exponential\nfunction (see Fig. 11).\nIn general, NMR probes the hyperfine field at the nu-\nclear site originating from the electron magnetic moment\nwhich has static and dynamic components. The51V nu-\ncleus (natural abundance 100 %) has a nuclear spin of\nI= 7/2, hence a quadrupolar splitting is expected under\nan electric-field gradient (EFG). However, the absence of\nsuchsatellitesindicatesasmallEFGattheVnuclearsite.\nWe have also checked by simulation that even if there is\na quadrupolar effect, that is less than 100 kHz, does not\ncontribute to the temperature dependence of the shift\nparameters. This is originated by the fact that51V nu-\ncleus has a small quadrupole moment (0.051 barns) and\nthat the EFG usually is screened by conduction electrons\nin a metal, so the51VνQeffect has been ignored in our\nstudy. The temperature dependent static part of the hy-\nperfine field can be measuredby the NMR (Knight) shift,\nK(T), which is related to the electron spin susceptibil-\nity,χ(T), byK(T) = (Ahf/NµB)χ(T)+K0, whereAhf\nis the hyperfine coupling constant between the nuclear\nand the electron spins, and K0is a temperature inde-\npendent contribution. In metals, K0arises from both\nthe orbital interaction associated with non- selectrons\nandthe Fermicontactinteractionfrom sconductionelec-\ntrons (K0=Korb+Kce). For non-cubic materials like\nCeTi1−xVxGe3, the Knight shift is anisotropic reflecting\nthe anisotropy of the spin susceptibility. In this case, one\nobserves a powder pattern of the NMR spectrum arising\nfrom the different orientation of the crystallites with re-spect to applied field direction. For the present uniaxial\ncase, we can extract values of K/bardbl(B0is parallel to the\nsymmetry axis) and K⊥(B0perpendicular to the sym-\nmetry axis) from the singularities of the spectrum. Then\nisotropic and axial Knight shifts, KisoandKax, respec-\ntively, are given by\nKiso= (K/bardbl+2K⊥)/3,Kax= (K/bardbl−K⊥)/3.(1)\nNote that in the present case K/bardblcorresponds to Kpar-\nallel to the caxis and K⊥toKperpendicular to the c\naxis.\nIt is to be noted that we have found a considerable\nchangeinthemagnitudeof Ahfwith doping(listed inthe\nTable I). Such a variationcan be due to the changeof lat-\ntice constant, along with the change of electronic struc-\nture (due to electron doing in the present case) as well.\nReplacing Ti4+by V5+introduces carriers (doping) into\nthe system while the structure type is preserved. It re-\nduces the c/aratio and changes the single-ion anisotropy\nof the Ce3+ion. The competition between these two ef-\nfects might result in the observed considerable change in\nAhf.\nThe fluctuating part of the hyperfine field is related to\nthe spin-lattice relaxation rate 1 /T1given by,\n1/T1∝T/summationdisplay\nq|Ahf(q)|2χ′′\n⊥(q,ωn)/ωn.(2)\nwhere the sum extends over the wave vectors qwithin\nthe first Brillouin zone, χ′′\n⊥(q,ωn) is the imaginary part\nof the transverse dynamical electron spin susceptibility\nandωnis the Larmor frequency for NMR. Ahf(q) is the\nq-dependent hyperfine coupling constant.\nThe recovery of the nuclear magnetization, M(t), in\nthese inhomogeneoussly broadened cases generally fol-\nlowsastretchedexponentialform, then the recoverydata\nhave been fitted to M(t) = (1−exp(−t/T1)β), where t is\nthe time after a saturation of nuclear magnetization and\nβis the stretch exponent related to the distribution of\ntheT1value across the spectrum.\nRESULTS AND DISCUSSIONS\nBefore discussing the compounds close to the FMQCP\n(x= 0.35and 0.405)we presentthe results for the parent\ncompounds (AF, x= 1.0 and FM, x= 0.113).\nCeVGe 3(x= 1) is a heavy-fermion itinerant antiferro-\nmagnet with TK≈10 K and TN≈6 K. The51V NMR\nspectrahavebeentakenatafixed frequencyof31.0MHz.\nThe field-swept51V NMR spectra at 8 K, the tempera-\nture dependence of Kisoand the relaxation rate divided\nby temperature, 1 /T1T, are depicted in Fig. 1(a), (b)\nand (d), respectively. In order to obtain the isotropic\nhyperfine coupling constant Aiso\nhf, the observed Kisodata\nare plotted as a function of the magnetic susceptibility3\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s52/s54/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50/s48/s50/s52/s54\n/s50/s46/s52 /s50/s46/s54 /s50/s46/s56 /s51/s46/s48\n/s49/s48/s48/s50/s48/s52/s48/s54/s48/s32/s32\n/s32/s75 /s32/s40/s37/s41/s32\n/s32/s75\n/s124/s124/s32/s40/s37/s41/s32\n/s32/s75\n/s105/s115/s111\n/s32/s75\n/s105/s115/s111/s32/s61/s32/s75/s39\n/s48/s43/s67/s39/s47/s40/s84/s45 /s41\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s75/s39\n/s48/s32/s61/s32/s48/s46/s51/s56/s32/s37/s44/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s67/s39/s61/s49/s48/s57/s32/s75/s44/s32\n/s32/s61/s32/s45/s49/s55/s46/s53/s32/s75/s32/s75/s32/s40/s37/s41\n/s84/s32/s40/s75/s41\n/s75 \n/s105/s115/s111 /s32/s40/s37 /s41\n/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41/s32/s97/s116/s32/s50/s46/s56/s32/s84/s65/s105/s115/s111\n/s104/s102/s61/s32/s49/s46/s49 /s48/s46/s48/s51/s32/s84/s47\n/s66\n/s75\n/s48/s32/s61/s32/s49/s46/s51/s32/s37/s32\n/s40/s100/s41/s40/s99/s41\n/s40/s98/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s48/s72/s32/s40/s84/s41/s51/s49/s32/s77/s72/s122\n/s84/s32/s61/s32/s56/s32/s75/s40/s97/s41\n/s67/s101/s86/s71/s101\n/s51\n/s49/s47/s84\n/s49/s84/s32/s40/s115/s45/s49\n/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s40/s84/s45/s84\n/s78/s41/s45/s48/s46/s53\nFIG. 1: NMR data for x=1 sample. (a)51V NMR spectra\nmeasured at 31 MHz at 8 K, solid line is a fit of a Gaussian\nline profile to the data; (b) temperature dependence of K⊥\n(open triangles), K/bardbl(closed circles) and the resultant Kiso,\nsolid line is a fit of a Curie-Weiss law to the Kisodata; (c)\nKisoversusχmeasured at 2.8 T, solid line is a linear fit of the\ndata; (d) temperature dependence of 1 /T1T, dashed line is an\nexpected curve of the SCR theory for itinerant AF magnet.\n(χ) with the temperatureasanimplicit parameterin Fig.\n1(c). From the slope and the intercept Aiso\nhfandKorbare\nobtained as 1.1 T/Ce- µBand 1.3%, respectively. Here,\nAiso\nhfindicates that if a Ce 4 fmoment is polarized by one\nµB, the V nucleus is subjected to an isotropic hyperfine\nfield of 1.1 T along the field direction.\nKiso, hence the static susceptibility χ, has a strong\ntemperature dependence, implying that the wave-vector-\ndependent susceptibility χ(q)hasnotonlyapeakatwave\nvectorq=Qbut also must have a large FM q= 0\ncomponent. As to the 1 /T1TvsTplot in Fig.1(d), the\nself-consistent renormalization (SCR) theory for a weak\nitinerant AF predicts a 1 /T1T∝1/√T−TNdivergence\natTN[20]. Althoughthereisananomalyat TNin1/T1T,\nthe experimental data cannot be described by the above\nrelation. This is also an evidence for the existence of\nFM fluctuations which is suppressed by an external field\naroundTN.\nFor the other end member of the series we have taken\nthex= 0.113 sample which undergoes FM ordering at\nTC≈9 K in zero field (as determined from the sharp\nmaximum in the specific heat [17]). The51V NMR\nspectrum shown in Fig.2(a) exhibits a large anisotropy\n(K⊥/K/bardbl≃0.26 at 20 K) which is consistent with the\nanisotropyinthemagneticsusceptibilityreportedforsin-\ngle crystals [17, 21]. As will be described below, the\nanisotropy tends to vanish towards the FM QCP but in-\ncreasesagain in CeVGe 3where the anisotropyis reversed\nto yieldK⊥/K/bardbl≃2.37 at 8 K, again in accord with the\nsusceptibility measurements [16, 17].\nThe temperature dependence of Kisoextracted from\nK/bardblandK⊥is shown in Fig. 2(b), and that of 1 /T1Tin/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s53/s49/s48/s49/s53/s50/s48/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51/s48/s50/s52/s54\n/s53/s46/s54 /s53/s46/s56 /s54/s46/s48 /s54/s46/s50 /s54/s46/s52\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s51/s48/s54/s48/s57/s48/s49/s50/s48/s49/s53/s48\n/s48 /s49 /s50 /s51/s48/s52/s48/s56/s48\n/s32/s75 /s32/s40/s37/s41\n/s32/s75\n/s124 /s124 /s32/s40/s37/s41\n/s32/s75\n/s105/s115/s111/s32/s40/s37/s41\n/s32/s75\n/s105/s115/s111/s32/s61/s32/s75/s39\n/s48/s43/s67/s39/s47/s40/s84/s45 /s41\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s75/s39\n/s48/s61/s49/s46/s51/s57/s32/s37/s44/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s67/s39/s61/s32/s52/s57/s46/s50/s32/s75/s44/s32\n/s32/s61/s32/s50/s46/s55/s56/s32/s75/s75/s40/s37/s41\n/s84/s32/s40/s75/s41\n/s75\n/s105/s115/s111/s32/s40/s37/s41\n/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41/s32/s97/s116/s32/s54/s46/s52/s32/s84/s65/s105/s115/s111\n/s104/s102/s32/s61/s32/s48/s46/s57 /s48/s46/s48/s50/s32/s84/s47\n/s66\n/s75\n/s48/s32/s61/s32/s49/s46/s54/s32/s37\n/s40/s100/s41/s40/s99/s41\n/s40/s98/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s48/s72/s40/s84/s41/s55/s48/s32/s77/s72/s122\n/s84/s32/s61/s32/s50/s48/s32/s75/s40/s97/s41\n/s49/s47/s84\n/s49/s84/s32/s40/s115/s45/s49\n/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41\n/s49/s47/s84\n/s49/s84/s32/s40/s115/s45/s49\n/s75/s45/s49\n/s41\n/s75\n/s105/s115/s111/s32/s40/s37 /s41\nFIG. 2: NMR data for the x=0.113 sample. (a)51V NMR\nspectra measured at 70 MHz at 20 K, (b) temperature depen-\ndence of K⊥,K/bardbl, and the calculated Kiso, (c)Kiso(%) versus\nχmeasured at 2.8 T, (d) temperature dependence of 1 /T1T.\nThe solid line is a fit of Eq. (3) to the data (for details see\nmain text), the inset shows 1 /T1TvsKiso.\nFig. 2(d). Although 1 /T1Tscales with K2\niso(Korringa\nlaw) in simple metals, 1 /T1Tforx= 0.113 depends lin-\nearly on Kisoas shown in the insert of Fig. 2(d). This\nbehavior can be described by the SCR theory for a weak\nitinerant ferromagnet in the frame of SCR theory. In\nthe SCR theory, 1 /T1Tunder an external field is given\nby[20, 22, 23]\n(1/T1T) =κχ/(1+χ3B2\n0P), (3)\nwhich reproduces very well the experimental 1 /T1T(T)\nshown in Fig. 2(d) for B0= 6.25 T. Here, Pis a con-\nstant related to the area of the Fermi surface of the itin-\nerant electrons, and κis related to the spin-fluctuation\nparameter, T0. In the paramagnetic state\n1/T1T≃3/planckover2pi1γ2\nnAhfKiso/16πµBT0. (4)\nThe slope of 1 /T1Tvs.Kisoin the inset of Fig. 2(d)\nyields the value of T0= 21 K which is rather close to TC.\nThe deviation from 1 of the ratio TC/T0is a measure\nof the degree of itinerancy of the 4 felectrons [20, 22–\n24].TC≈T0indicates that the enhancement of χ(q)\nis not confined to q= 0 but rather extends to larger q\nvalues. This situation is reminiscent of but opposite to\nthat ofx= 1 described above, where χ(q) extends from\nthe AF wave vector Qtowards q= 0. Therefore, we\nconclude that there is a considerable mixture of FM and\nAF correlations in both end materials.\nWe now discuss51V NMR data for the samples with\nx= 0.35 and 0.405 which are close to the proposed\nFMQCP. Here the NMR measurements were performed\nunder different applied magnetic fields to elucidate the\nproperties in low fields. The spectra shown in Fig. 3\n(a, b) for x= 0.35 andx= 0.405, are less anisotropic4\n/s53/s46/s54 /s53/s46/s56 /s54/s46/s48 /s54/s46/s50 /s54/s46/s52/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56/s57\n/s40/s99/s41/s40/s98/s41/s32/s32 /s32/s120/s32/s61/s32/s48/s46/s51/s53/s40/s97/s41\n/s55/s48/s32/s77/s72/s122\n/s50/s48/s32/s75\n/s55/s48/s32/s77/s72/s122\n/s50/s48/s32/s75\n/s32\n/s48/s72/s32/s40/s84/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s32/s32/s120/s32/s61/s32/s48/s46/s52/s48/s53/s32/s51/s49/s46/s48/s77/s72/s122/s40/s50/s46/s56/s84/s41/s32/s120/s32/s61/s32/s48/s46/s51/s53\n/s32/s55/s48/s46/s48/s77/s72/s122/s40/s54/s46/s50/s53/s84/s41/s32/s120/s32/s61/s32/s48/s46/s51/s53\n/s32/s51/s49/s46/s48/s77/s72/s122/s40/s50/s46/s56/s84/s41/s32/s120/s32/s61/s32/s48/s46/s52/s48/s53\n/s32/s55/s48/s46/s48/s77/s72/s122/s40/s54/s46/s50/s53/s84/s41/s32/s120/s32/s61/s32/s48/s46/s52/s48/s53\n/s32/s75\n/s105/s115/s111/s32/s40/s37/s41\n/s84/s32/s40/s75/s41\nFIG. 3:51V NMR spectra measured at 70 MHz and 20 K for\n(a)x= 0.35 and (b) 0.405, (c) Kisoforx= 0.35 and 0.405\nsample as a function of temperature in two different fields.\ncompared to those for x= 1.0 and 0.113. The tem-\nperature dependence of Kisoat 2.76 T and 6.25 T is\nshown in Fig. 3(c). The change of characteristics in\nthe spin-fluctuations can easily be seen from Fig. 12,\nwhere 1/T1Thave been plotted as a fucntion of tem-\npetaure for all the different doping concertrations. The\ntemperature dependences of the relaxation rate 1 /T1in\ndifferent fields are shown in Fig.4(a) and (b) for x= 0.35\nandx= 0.405, respectively. 1 /T1shows a crossover\nfromalow-temperatureKorringa-likeprocess(1 /T1∝T)\nto a temperature-independent process above a temper-\nature,T∗which depends on field. The SCR theory\nhas been successfully adopted to understand thermo-\ndynamic properties of weakly or nearly itinerant FM\nand AF systems. This model has been extended to\ncover cases where the spin fluctuations have large am-\nplitudes and their significant qcomponents extend over\nthe entire qspace of the first Brillouin zone. This ex-\ntension of the SCR theory, ”temperature-induced-local-\nmoment” (TILM) model, predicts that the amplitude of\nlocal spin fluctuations, < S2\nL>, increases rapidly at\nlow temperatures and saturates at a certain tempera-\ntureT∗[25]. Above T∗, the thermodynamic properties\naregovernedbythe transversecomponent(local-moment\ntype) of the fluctuations and the susceptibility shows\nCurie-Weiss behavior. As described above, the 1 /T1data\nof CeTi 1−xVxGe3aroundx≈0.4 are in accord with\nTILM model. It has been shown that exchange-coupled\nlocal-moment fluctuations bring about a temperature in-\ndependent 1 /T1a magnitude given by the correlation\ntimeτcassociated with the local spin fluctuations [26]:\n(1/T1)TILM= (Ahf//planckover2pi1)2(2π)1/2< S2\nL>\n3·τc(5)\nwhere (1/T1)TILMis the temperature independent value\nobserved above T∗, and< S2\nL>is the amplitude ofthe local spin density. Using experimental values of\n(1/T1)TILMand effective moments (obtained from the\nCurie-Weiss fit) assumed to be the same as < S2\nL>,τcis\nevaluated and shown in Fig. 4(c) as a function of mag-\nnetic field. τcfor both samples around the FMQCP ( x\n= 0.35 and 0.405) increases linearly with field, indicating\nthat the characteristic frequency of the spin fluctuations\ndecreases with external field, i.e., the samples approach\nthe field-polarized state.\nAn possible microscopic reason for a constant 1 /T1\nvalue may be the suppression of Kondo interaction with\nincreasing temperature, with a crossover from T-linear\nFermi-liquid-like relaxation rate. Because TKwhich is\naround 10 K in CeTi 1−xVxGe3, the high-temperature\nrelaxation process may be due to fluctuations of purely\nlocal moments associated with the thermal quenching of\nthe Kondointeraction. The relaxationrateofthis typeof\nprocess can easily be calculated using the hyperfine cou-\nplingconstantandanexchangefrequencyestimatedfrom\nthe Weiss constant[22]. The calculated 1 /T1is around\n2×103sec−1, which is about one order of magnitude\nlarger than the experimental values. Also, if we assume\nobserved T∗is associated with TKone would expect that\ntheT∗should be decreased with increasing filed that is\nopposite to the experimental findings. Therefore, we be-\nlieve TILM is more plausible scenario to explain the ex-\nperimental results for the samples close to the quantum\ncritical point.\nFor a detailed discussion, we plot T1Twhich is a mea-\nsure of the inverse χ′′(q,ωn) (cf. Eq. 2) as a function\nof temperature in Fig. 5(a) and (b) for x= 0.35 and\nx= 0.405. For both samples 1 /T1TaboveT∗follows\nthe CW law with the Weiss constant θincreasing with\napplied magnetic field (figure 5(c)). The enhancement\nofθsimply illustrates the tendency to a saturated para-\nmagnet with increasing field. The fact that θfor the\nlowest field for both samples is very close to zero signals\nquantum critical spin-fluctuations. Note that the field\ndependent Curie constants correspond to the inverse of\nτc(cf. Eq. 5).\nAsshowninFig. 4(a)and(b), attemperatureswellbe-\nlowT∗the nuclearrelaxationisgovernedbythe Korringa\nprocess where interacting electron-hole excitations are\nthe primary source of the magnetic excitations. Treating\nthe interaction of the quasiparticles in the frame of the\nrandom phase approximation (RPA) the modified Kor-\nringa relation can be expressed as S0/T1TK2\nspin=K(α),\nwithS0= (/planckover2pi1/4πkB)(γe/γn)2.K(α) =<(1−α0)2/(1−\nαq)2)>FSwithαq=α0[χ0(0,q)/χ0(0,0)]where χ0(ω,q)\nisthe magneticsusceptibility, and < ... > FSindicatesthe\nqaverage over the Fermi surface. K(α) is a modification\nfactor of the Korringarelation, which depends on the ex-\nchange enhancement factor α. If the spin fluctuations\nare enhanced around q= 0 as for dominant FM corre-\nlations, then K(α)<1. On the other hand, K(α)>1\nindicatesthat finite- q(typicallyAF) spin fluctuationsare5\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s40/s99/s41/s40/s98/s41/s120/s61/s48/s46/s51/s53\n/s32/s49/s48/s46/s55/s77/s72/s122/s32/s40/s48/s46/s57/s55/s84/s41\n/s32/s51/s49/s46/s48/s77/s72/s122/s32/s40/s50/s46/s56/s84/s41\n/s32/s55/s48/s46/s48/s77/s72/s122/s32/s40/s54/s46/s50/s53/s84/s41/s54/s46/s50/s53/s32/s84\n/s48/s46/s57/s55/s32/s84/s50/s46/s56/s32/s84/s49/s47/s84\n/s49/s32/s40/s115/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s40/s97/s41\n/s84/s32/s40/s75/s41/s49/s47/s84\n/s49/s32/s40/s115/s45/s49\n/s41/s120/s61/s48/s46/s52/s48/s53\n/s32/s49/s48/s46/s55/s77/s72/s122/s32/s40/s48/s46/s57/s55/s84/s41\n/s32/s51/s49/s46/s48/s77/s72/s122/s32/s40/s50/s46/s56/s84/s41\n/s32/s55/s48/s46/s48/s77/s72/s122/s32/s40/s54/s46/s50/s53/s84/s41/s54/s46/s50/s53/s32/s84\n/s48/s46/s57/s55/s32/s84/s50/s46/s56/s32/s84/s49/s47/s84\n/s49/s32/s126/s32\n/s99/s32/s40/s115/s45/s49\n/s41\n/s120/s61/s48/s46/s52/s48/s53\n/s120/s61/s48/s46/s51/s53\n/s48/s72/s32/s40/s84/s41\nFIG. 4: (a) and (b): 1 /T1vs. temperature T for x= 0.35\nand 0.405 at different fields, (c): τc(correlation time of the\nspin fluctuations) as a function of applied field. The dotted\nand solid lines are for guide to the eye.\ndominant. The estimated K(α) values for x= 0.35 and\n0.405at 2 K (a temperature where the modified Korringa\nlaw is valid) are shown in Table I which indicates that\nthedominantferromagneticcorrelationsarereducedwith\nincreasing xor AF correlations are becoming dominant\nwith increasing xtowards CeVGe 3.\nEvidence for the presence of weak AF spin fluctua-\ntions on top of dominant FM spin fluctuations close to\nan FMQCP has been seen recently in Ru-doped Ce-\nFePO [28]. The appearance of AF spin fluctuations was\natttributed to the Fermi-surface instability which might\nappear in case of a local QCP. The presence of consid-\nerable AF correlations for x= 0.113 even far away from\nthe possible QCP indicates that the QCP in this sys-\ntem is not driven solely by FM fluctuations. Further-\nmore, the Knight-shift anisotropy reduces considerably\nin CeTi 1−xVxGe3upon approaching QCP which indi-\ncates the isotropic nature of local fields at the V site.\nThis behavior is very different from that of the layered/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s120/s32/s61/s32/s48/s46/s52/s48/s53\n/s84/s32/s40/s75/s41/s84/s32/s40/s75/s41\n/s32/s84\n/s49/s84/s32/s40/s115/s75/s41/s32 /s32 /s34/s40/s113/s44 /s41/s40/s97/s41\n/s48/s46/s57/s55/s32/s84/s50/s46/s56/s32/s84\n/s54/s46/s50/s53/s32/s84/s120/s32/s61/s32/s48/s46/s51/s53\n/s40/s99/s41/s84\n/s49/s84/s32/s40/s115/s75/s41/s32 /s32 /s34/s40/s113/s44 /s41\n/s54/s46/s50/s53/s32/s84/s50/s46/s56/s32/s84\n/s48/s46/s57/s55/s32/s84/s40/s98/s41/s32\n/s32\n/s32/s32/s32/s40/s75/s41\n/s48/s72/s40/s84/s41/s32/s120/s32/s61/s32/s48/s46/s51/s53\n/s32/s120/s32/s61/s32/s48/s46/s52/s48/s53\nFIG. 5: (a) and (b): Tempereature dependecne of T1Tat dif-\nferent fields for x= 0.35 and 0.405, (c): θobtained from high\ntemperatiure Curie-Weiss fits (straight lines) as a functio n of\nfield for x= 0.35 and 0.405. The curved lines are for guide\nto the eye.\nFM CeRuPO, which is driven to a QCP by substitu-\ntion of Ru by Fe, where the fluctuations become strongly\nanisotropic upon approaching the QCP [27, 28].\nTABLE I: Summary of estimated parameters of\nCeTi1−xVxGe3.\nCeTi1−xVxGe3x=0.113 x=0.35 x=0.405 x=1\nT0(K) 22.9 21 11 -\n1/T1T(1/sK) at 4K 18.6 50 45 33\nAiso\nhf(T/µB) 0.9 0.6 0.5 1.1\nK(α) - 0.28 ±0.02 0.46 ±0.05 -\nOn more general grounds, the issue of an FMQCP re-\nmains very challenging despite the considerable amount\nof work in this field [5]. It is generally believed that a\nQCP in clean systems is intrinsically unstable because of\nthe dynamics of low-lying fermionic excitations. There-\nfore, alloysystemswithvaryingdegreesofdisorderarean\nimportantsubject. ThesystemCeTi 1−xVxGe3isunique,\nin that FM for x= 0 gives way to AF for x= 1 and, at\nthe same time, the single-ion anisotropy changes from6\nuniaxial (Ising) to planar (XY). Close to the QCP, the\nsystem becomes isotropic. It should be mentioned that\nunusually slowly fluctuating glass-like electronic phases\nnear FM quantum criticality due to the competing in-\nteractions have been proposed by theory [29]. In our\ncase, the competition between dominant FM and weak\nAF correlationsmight induce such phases which prevents\nthe formation of a pure FMQCP. However, our NMR\ndata do not indicate pronounced line broadening which\nwould signify a glassy-like inhomogeneous state. Numer-\nous experiments on clean three-dimensional FM metals\nhave shown that, in contrast to the original Hertz-Millis-\nMoriya model of quantum criticality, the FMQCP is un-\nstable, and FM metals undergo a first-orderphase transi-\ntion to the paramagnetic or to an incommensurate phase\nas predicted by theory [30–32]. As a matter of fact, pure\nCeTiGe 3under highhydrostaticpressurefollowsthissce-\nnario, with several intervening magnetic phases until the\nparamagnetic state is finally reached around 6 GPa. Un-\nder magnetic field a wing-like structure appears at high\npressures, again as predicted by theory[33] and observed\nin othercleansystems[34]. Foranalloyedsystem asstud-\nied in the present work, with a considerable degreeof dis-\norder at the QCP at x≈0.4, one might expect a genuine\nsecond-order transition. Indeed, it was suggested on the-\noretical grounds that a first-order FM transition might\nbe ”tuned” continuously to a FMQCP by disorder [35].\nFurther work is necessary to elucidate how these features\ncompete or cooperate at the QCP. In this respect it is\nworth to mention that, our study indicates the presence\nof anisotropy in shift and which in principle should also\ninduce anisotropy in the relaxation processes. The effect\nof anisotropy in general have not been included in our\npresent study, especially in case of interpreting the na-\nture of spin-fluctuations as it is not possible to estimate\nthe relaxation in different directions without having sin-\ngle crystals. So we have mainly used the isotropic part\nof the Knight shift and the ”average” T1by using the\nstretched exponential function which includes the distri-\nbution of T1due to the anisotropyand alsodisorder. The\nissues related to such anisotropy will be of interest for fu-\nture studies by employing single crystals.\nCONCLUSION\nSystematic51V NMR measurements have been per-\nformedonCeTi 1−xVxGe3withtheendmembersshowing\nferromagneticand antiferromagneticorder, for x= 0 and\nx= 1 respectively. NMR as a local probe provides in-\nformations about magnetic fluctuations across the phase\ndiagram. The temperature dependence of Kand 1/T1T\nin CeVGe 3shows strong admixture of FM fluctuations\nto the dominant AF fluctuations. The temperature de-\npendence of 1 /T1Tforx= 0.113 at 6.4 T can be well\nexplained by self-consistent renormalization theory foritinerant ferromagnets. Around the critical concentra-\ntion (x= 0.35,0.405), quantum-critical spin fluctuations\ncompriseweakbut finite AF spin fluctuationsadmixedto\nFM spin fluctuations. The spin-fluctuation parameters\nT0andK(α) (the latter probing the relative strength\nof AF vs. FM spin fluctuations) have been estimated\nforx= 0.35 and 0.405. K(α) shows a considerable en-\nhancement with xindicating the growing importance of\nAF fluctuations towards the QCP. The critical samples\nlack the NMR finger print of a pure FMQCP, i.e., the\n1/T1T∼T−4/3NMR power law [36]. Hence, the gen-\neral presence of both FM and AF fluctuations across the\nwhole CeTi 1−xVxGe3is a constituting trait of this sys-\ntem. Furtherworkshouldelucidateifandhowthe chang-\ning single-ion anisotropies affect the quantum criticality\nin this system.\nACKNOWLEDGMENT\nWe thank M. Brando, C. Geibel and B. Pilawa for\nfruitful discussions. Furthermore, we thank H. Rave, C.\nKlausnitzer and R. Hempel-Weber for technical support.\nAPPENDIX A: CRYSTAL STRUCTURE\nThe crystal structure of Ce(V,Ti)Ge 3has been shown\nin Fig. 6. It crystallizes in the hexagonal perovskite\n(BaNiO 3-type) structure P63/ mmcwith a=b=6.306\n(6.2744)˚A, c=5.6732 (5.882) ˚A,α=β= 90◦,γ= 120◦\nfor CeVGe 3(CeTiGe 3). The crystal structure has only\none crystallographic V (Ti), Ce, and Ge sites. Fig. 6(a)\nis the full view and Fig. 6(b) is the view along the c axis,\nshowing a uniaxial symmetry of the V site transferred\nhyperfine interaction from the Ce sites.\nFIG. 6: Crystal structure of Ce(Ti,V)Ge 3.7\nAPPENDIX B: MAGNETIZATION\nThe dc-magnetization was measured in various mag-\nnetic fields and temperatures between 5 K and 100 K in\ntemperature using commercially available SQUID mag-\nnetometer (Quantum Design MPMS).\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52\n/s32/s32/s32/s40/s101/s109 /s117/s47/s109 /s111/s108/s101/s41\n/s84/s32/s40/s75/s41/s32/s54/s46/s52/s32/s84/s32/s40 /s78/s77/s82/s41\n/s120/s32/s61/s32/s48/s46/s49/s49/s51\nFIG. 7: Temperature dependence of M/Hatµ0H = 6.4 T for\nthex=0.113 sample.\nFig. 7 shows the temperature dependence of ( χ=\nM/H) for x=0.113 sample at 6.4 T close to which the\n51VNMR has been conducted.\nFig. 8 alsoshowsthe temperature dependence of M/H\nandχac(=δM/δH) at different magnetic fields in a log-\nlog plot. We could not find any signature of long-range\nmagnetic ordering. With increasing field the critical be-\nhaviorof χhas been suppressed for the two samples close\nto the quantum-critical V concentration which is also\nconsistent with the results of51VNMR described in the\nmain text.\nAPPENDIX C: NMR\nThe51VNMR spectra have been obtained using a\ncommerciallyavailableTecmagNMRspectrometerinthe\nfield sweep mode down to 1.8 K at various resonance\nfrequencies (Fig. 9). NMR measurements have been\nperformed on the51Vnucleus with a spin (I=7/2) with\n99.75% natural abundance and a quadrupole moment of\nQ= -5.2×10−30m2.\nThe crystal structure of Ce(Ti,V)Ge 3exhibits one V-\nlattice site. The51VNMR spectra shows only a single\nNMR line. There are no further lines (satellites) related\nto quadrupolar interaction. The spin-echo intensity was\nobtained by integrating over the spin echo in the time\ndomain at a given magnetic field. The final spectrum is\ngiven by plotting the spin echo intensity as a function of\nthe applied field. Due to the small quadrupolar moment\nand also small EFG at the V site the satellites are not/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s48/s46/s48/s49/s48/s46/s48/s49/s53/s48/s46/s48/s50/s48/s46/s48/s50/s53/s48/s46/s48/s51/s48/s46/s48/s51/s53\n/s53 /s54 /s55 /s56 /s57 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s49/s48\n/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s46/s48/s49/s48/s46/s48/s49/s53/s48/s46/s48/s50/s48/s46/s48/s50/s53/s48/s46/s48/s51/s48/s46/s48/s51/s53\n/s53 /s54 /s55 /s56 /s57 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s49/s48/s84/s32/s40/s75/s41\n/s32/s32/s48/s46/s57/s55/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s50/s46/s56/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s54/s46/s52/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s56/s46/s56/s32/s84/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s120/s32/s61/s32/s48/s46/s51/s53\n/s32/s32/s49/s48/s48/s48/s32/s79/s101\n/s32/s49/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s50/s46/s56/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s54/s46/s52/s32/s84/s32/s40 /s78/s77/s82/s41/s97/s99/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s84/s32/s40/s75/s41/s120/s32/s61/s32/s48/s46/s51/s53/s32/s32\n/s32/s48/s46/s57/s55/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s50/s46/s56/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s54/s46/s52/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s56/s46/s56/s32/s84/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41/s120/s32/s61/s32/s48/s46/s52/s48/s53/s120/s32/s61/s32/s48/s46/s52/s48/s53\n/s32/s32/s97/s99/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s84/s32/s40/s75/s41/s32/s49/s48/s48/s48/s32/s79/s101\n/s32/s49/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s50/s46/s56/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s54/s46/s52/s32/s84/s32/s40 /s78/s77/s82/s41\n/s32/s56/s46/s56/s32/s84\nFIG. 8: Temperature dependence of M/Handχac\n(=δM/δH) in different magnetic fields (mostly at which51V\nhas been conducted) for the x=0.35 and 0.405 sample. The\nfields where51VNMR was performed are indicated.\n/s50/s46/s52/s48 /s50/s46/s52/s53 /s50/s46/s53/s48 /s50/s46/s53/s53 /s50/s46/s54/s48 /s50/s46/s54/s53 /s50/s46/s55/s48 /s50/s46/s55/s53 /s50/s46/s56/s48 /s50/s46/s56/s53 /s50/s46/s57/s48/s32/s51/s32/s75\n/s32/s52/s46/s51/s32/s75\n/s32/s54/s32/s75\n/s32/s56/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s49/s53/s32/s75\n/s32/s50/s48/s32/s75\n/s32/s50/s53/s32/s75\n/s32/s51/s48/s32/s75\n/s32\n/s48/s72/s40/s84/s41\n/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41/s120/s32/s61/s32/s48/s46/s51/s53/s51/s49/s32/s77/s72/s122\n/s72/s42\n/s32/s50/s32/s75\n/s32/s51/s32/s75\n/s32/s52/s46/s51/s32/s75\n/s32/s54/s32/s75\n/s32/s55/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s49/s50/s46/s53/s32/s75\n/s32/s49/s53/s32/s75/s51/s49/s32/s77/s72/s122\n/s120/s32/s61/s32/s48/s46/s52/s48/s53\n/s32/s32/s72\n/s76\n/s32/s52/s46/s51/s32/s75\n/s32/s54/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s49/s53/s32/s75\n/s32/s50/s48/s32/s75\n/s32/s50/s53/s32/s75\n/s32/s51/s48/s32/s75\n/s32/s52/s48/s32/s75\n/s32/s53/s48/s32/s75/s51/s49/s32/s77/s72/s122\n/s120/s32/s61/s32/s49\n/s32/s32\nFIG. 9: Field sweep51Vspectra at 31 MHz. Vertical dotted\nlines atHLandH∗(see text) are attributed to non-magnetic\nimpurity phases.\nvisible,ortheyaresuperimposedwiththebroadenedcen-\ntral line. The spectra for the x= 0.35 and 0.405 samples\nare quite isotropic in nature, whereas for x=0.113, the\nspectra are axially symmetric and for x=1 exhibit planar\nanisotropy. Interestingly the nature of anisotropy is op-\nposite for both the samples ( x=0.113 and 1), which we\nsuggest is due to the opposite nature of anisotropy seen\nfrom magnetization measurements in single crystals [21].\nIn addition, we have found two extra peaks in almost\nall spectra. One peak is at the51V-Larmor field ( HL)\nwhich indicates the presence of unreacted V 2O5. An-\nother peak (at H∗) has a negative (but also temperature\nindependent) shift which might originate from a binary\nV5+-containing phase (probably V-Ge binary phase).\nTo estimate the hyperfine coupling constants Aiso\nhf,8\n/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51\n/s48/s50/s52/s54/s56\n/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50/s48/s49/s50/s51/s52\n/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50/s48/s49/s50/s51/s52/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50\n/s48/s49/s50/s51/s52/s53/s54/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41/s75 \n/s105/s115/s111 /s32/s40/s37 /s41/s32\n/s32\n/s75\n/s48/s32/s61/s32/s49/s46/s53/s32/s37 /s75\n/s48/s32/s61/s32/s49/s46/s53/s53/s32/s37 /s75\n/s48/s32/s61/s32/s49/s46/s54/s32/s37 /s72\n/s105/s115/s111/s32/s61/s32/s48/s46/s57 /s48/s46/s48/s50/s32/s84/s47\n/s66/s32/s120/s32/s61/s32/s48/s46/s49/s49/s51/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41/s75 \n/s105/s115/s111 /s32/s40/s37 /s41/s32/s32\n/s72\n/s105/s115/s111/s32/s61/s32/s48/s46/s54 /s48/s46/s48/s49/s53/s32/s84/s47\n/s66/s32/s120/s32/s61/s32/s48/s46/s51/s53/s75 \n/s105/s115/s111 /s32/s40/s37 /s41\n/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41/s32\n/s32\n/s72\n/s105/s115/s111/s32/s61/s32/s48/s46/s53 /s48/s46/s48/s50/s32/s84/s47\n/s66/s32/s120/s32/s61/s32/s48/s46/s52/s48/s53\n/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41/s32\n/s75\n/s48/s32/s61/s32/s49/s46/s51/s32/s37 /s72\n/s105/s115/s111/s32/s61/s32/s49/s46/s49 /s48/s46/s48/s51/s32/s84/s47\n/s66/s32/s32/s120/s32/s61/s32/s49/s75 \n/s105/s115/s111 /s32/s40/s37 /s41\nFIG. 10: Kisoversusχfor all the samples indicating a linear\nrelation.\nKiso(%) is plotted versus χ(Fig. 10) and found to follow\na linear relation as expected. From the slope we estimate\nAiso\nhfwhich is plotted in Figure 4 in the main manuscript\nfor all samples investigated.\n/s49/s48/s45/s53\n/s49/s48/s45/s52\n/s49/s48/s45/s51\n/s49/s48/s45/s50\n/s49/s48/s45/s49/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s120/s32/s61/s32/s48/s46/s49/s49/s51\n/s64/s32/s55/s48/s32/s77/s72/s122\n/s84/s32/s61/s32/s56/s46/s53/s32/s75\n/s84\n/s49/s32/s61/s32/s48/s46/s48/s48/s51/s53/s32/s115/s101/s99\n/s32/s61/s32/s48/s46/s55/s77/s40/s116/s41/s47/s77/s40 /s41\n/s116/s32/s40/s115/s101/s99/s41\nFIG. 11: Magnetization recovery curve at 70 MHz and at 8.5\nK forx=0.113 sample.\nThe spin-lattice relaxation rate was obtained by the\nstandard saturation-recovery method. The exponent\nβ≈0.7-0.75 was kept constant for all temperatures for\nx=0.35, 0.405 and 0.113 but β=1 for the pure end mem-\nber with x=1. A typical recovery curve has been shown\nin Fig. 11.\n†Electronic address: mayukh.cu@gmail.com\n‡Electronic address: Hilbert.loehneysen@kit.edu\n§Electronic address: Michael.Baenitz@cpfs.mpg.de\n[1]S. Doniach in ”Valence instabilities and related narrow\nband phenomena” edited by R. D. Parks, page 1669/s49/s48/s49/s48/s49/s48/s48\n/s32/s120/s61/s49/s32/s40/s50/s46/s56/s32/s84/s41\n/s32/s120/s61/s48/s46/s49/s49/s51/s32/s40/s54/s46/s52/s32/s84/s41\n/s32/s120/s61/s48/s46/s51/s53/s32/s40/s48/s46/s57/s55/s32/s84/s41\n/s32/s120/s61/s48/s46/s52/s48/s53/s32/s40/s48/s46/s57/s55/s32/s84/s41/s49/s47/s84\n/s49/s84/s32/s40/s115/s45/s49\n/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41\nFIG. 12: Temperature depedence of 1 /T1Tat the respec-\ntive lowest fields for each of the compounds. The figure\nindicates the doping evolution of electron spin-dynamics i n\nCeTi1−xVxGe3.\n(Plenum New York 1977).\n[2]Hilbert v. 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Raikh\nDepartment of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA\nMotivated by recent experiments on spin pumping from a ferromagnet into organic materials in\nwhich the charge transport is due to hopping, we study theoretically the generation and propagation\nof spin current in a hopping insulator. Unlike metals, the spin polarization at the boundary with\nferromagnet is created as a result of magnon absorption within pairs of localized states and it spreads\nfollowing the current-currying resistor network (although the charge current is absent). We consider\na classic resonant mechanism of the ac absorption in insulators and adapt it to the absorption of\nmagnons. A strong enhancement of pumping e\u000eciency is predicted when the Zeeman splitting of\nthe localized states in external magnetic \feld is equal to the frequency of ferromagnetic resonance.\nUnder this condition the absorption of a magnon takes place within individual sites.\nPACS numbers: 85.75.-d,72.25.Rb, 78.47.-p\nI. INTRODUCTION\nThe phenomenon of spin pumping from a ferromagnet\n(F) into a normal (N) layer is one of the most promi-\nnent approaches to the generation of pure spin currents.\nA prime manifestation that pumping indeed takes place\nin realistic F-N structures is the additional broadening1\nof the ferromagnetic resonance (FMR) in F, caused by\na contact with N-layer. This additional broadening\nwas \frst observed experimentally in Ref. [2]. Another,\nmore delicate, manifestation of pumping was reported\nshortly after. Namely, the injected spin current, entering\nthe nonmagnetic material with spin-orbit coupling (like\nPt) causes a voltage drop across the current direction.\nThis voltage drop is due to the inverse spin-Hall e\u000bect3\n(ISHE), and has a maximum when the frequency of the\nmicrowave radiation driving the ferromagnet, !, is equal\nto the FMR frequency, !FMR. Pioneering observations of\npumping via ISHE in Refs. [4{6] utilized Pt as the nor-\nmal layer.7{9They were followed by reports on similar\nobservations of pumping into di\u000berent materials10{13, in-\ncluding prominent semiconductors GaAs14, Si15,16, Ge17,\nand, most recently, graphene.18Experimental results on\nthe electric \feld generated due to ISHE , EISHE, are an-\nalyzed using the relation EISHE/J(s)\u0002\u001b, whereJ(s)\ndetermines the spatial direction of the spin current \row\nand its magnitude, while \u001bis its polarization. The mag-\nnitude of the spin current is given by\nJ(s)=g\"#Ch\nm(t)\u0002dm(t)\ndti\nz; (1)\nwherezaxis is taken along the static part of the mag-\nnetization. In Eq. (1) the constant Ccharacterizes the\nproperties of the normal layer (like ratio of thickness to\nthe spin-di\u000busion length), while m(t) describes the mag-\nnetization dynamics in the ferromagnet. The expression\nforJ(s)has the same form as the damping term in the\nequation that governs m(t). It was a remarkable experi-\nmental \fnding5that ISHE voltage exhibits essentially the\nsame behavior as a function of microwave power and thedeviation of !from!FMRas the additional FMR damp-\ning.\nMicroscopic physics of pumping is encoded in the mix-\ning constant1,19,20g\"#in Eq. (1). A fundamental process\nunderlying the pumping is the inelastic electron-magnon\nscattering at the F-N interface. Microscopic treatment\nof this scattering21,22assumes that electrons of the nor-\nmal layer impinging on the interface with ferromagnet are\nplane waves . On the other hand, in a number of recent\npapers23{26spin pumping into organic materials sand-\nwiched between ferromagnet and Pt has been reported.\nStrong temperature dependence of the resistance in these\nmaterials27suggests that the charge transport is due to\nhopping of polarons24,26, so that the description of pump-\ning based on plane waves does not apply. This raises the\nquestion about the microscopics of spin pumping in the\nlocalized regime.\nIn the present paper we consider theoretically the spin\npumping into a hopping insulator using the minimal\nmodel of coupling of localized states to a ferromagnet.\nWe demonstrate that, unlike metals, the underlying pro-\ncess responsible for pumping is the resonant magnon ab-\nsorption accompanied by transitions between localized\nstates, see Fig. 1. A distinctive feature of pumping into\nan insulator is that that the pumping e\u000eciency, com-\nmonly described by a constant, g\"#, depends strongly on\nthe external dc magnetic \feld. This is because, in addi-\ntion to causing the spin precession in ferromagnet, this\n\feld modi\fes the spin structure of the localized states\nbetween which the magnon is absorbed, see Fig. 1. The\ne\u000bect of external \feld is most pronounced when the wait-\ning time for a hop is longer than the period of the ac \feld\nwhich drives the FMR. Since the resonance frequency,\n!FMR, depends on the orientation of the external \feld28,\nfor certain orientations29this frequency coincides with\nthe Zeeman splitting of the localized states, Fig. 2. Spin\npumping is most e\u000ecient for such orientations, since the\nabsorption of magnon takes place within individual sites.\nWe also show that, with no charge current, the spin po-\nlarization generated at the F-N boundary, spreads in thearXiv:1505.01211v1  [cond-mat.mes-hall]  5 May 20152\nFIG. 1. (Color online) Elementary processes underlying the\nspin pumping into a metal (a), and into an insulator (b). In\nthe metal, an\"electron, impinging on the N-F boundary, is\nprimarily re\rected elastically with amplitude r\". Spin pre-\ncession in F gives rise to inelastic re\rection with amplitude\n~r\"#associated with the emission of a magnon. A #electron\nis either re\rected elastically with amplitude r#, or inelasti-\ncally, after absorbing of a magnon, with amplitude ~ r#\". The\ninjected spin current is proportional to j~r\"#j2!@f\n@\". In the in-\nsulator, only inelastic processes are at work. Emission and\nabsorption of magnons take place within pairs of localized\nstates.\ninsulator along the same percolation network30,31that\ndetermines the electrical resistance.\nII. ABSORPTION OF MAGNONS AT F-N\nBOUNDARY\nA. General considerations\nFigure 2 illustrates the di\u000berence between pumping\ninto a metal, and into an insulator in an applied mag-\nnetic \feld,H. WhileHis responsible for the magne-\ntization precession precession in the ferromagnet, it also\ncauses a spin splitting, \u0001 z, of the spectrum in the metal-\nlic normal layer, Fig. 2a. This splitting, however, does\nnot a\u000bect the absorption of magnons. The reason is that\nthe absorption at a boundary does not require momen-\ntum conservation, i.e. the matrix element is constant,\nand thus there is no dependence of the spin current, I(s),\non the dc \feld in the normal layer.\nThe situation is di\u000berent for an insulator, where the\nFIG. 2. (Color online) Illustration of pumping in metal (a)\nand in insulator (b) in the presence of a Zeeman splitting,\n\u0001z. In metal, the absorption (emission) of a magnon, ~!,\nnear the F-N boundary does not conserve momentum, and\nthus is insensitive to the ratio \u0001 z=~!. By contrast, in and\ninsulator, and near the condition ~!= \u0001z, the absorption\n(emission) of a magnon is resonant .\nmagnon absorption takes place between the discrete lev-\nels, Fig. 2(b). In this case, and for a general orien-\ntation ofH, the Zeeman levels are the linear combina-\ntions of\"and#spin states. As a result, transitions from\neach of the initial states on site itoboth \fnal states on\nsitejare allowed. This fact distinguishes absorption of\nmagnons from the conventional absorption of an ac elec-\ntric\feld32{34, and, as we will see below, gives rise to\nH-dependence of the spin current. Another origin of H-\ndependence is the possibility of intrasite absorption of\nmagnons at the boundary. We will see that the intrasite\ntransitions dominate the absorption near the resonant\ncondition ~!= \u0001z. Away from this condition, the inter-\nsite transitions dominate.\nB. The model\nConsider a pair of localized states, iandj, Fig. 2(b).\nAssume for simplicity that the ferromagnet is an insu-\nlator, i.e. it is a barrier for electrons in N. Precession,\nm(t), of magnetization in ferromagnet can be modeled\nas a time-dependent correction /m(t)^\u001bto the barrier\npotential. The pumping takes place since the wave func-\ntion, \ti, can penetrate under the barrier. As a result,\nthe Hamiltonian of site ihas a correction\n\u000e^Hi=Jh\n^\u001bxmxsin!t+ ^\u001bymycos!ti\n; (2)\nwhereJaccounts for tunneling. Projections mx(t) and\nmy(t) are proportional to the magnitude of the mi-\ncrowave \feld and depend in a resonant way on the prox-\nimity of!to!FMR. Analytical expressions for these pro-\njections can be found e.g. in Ref. [9].\nThe Hamiltonian \u000e^Hiof Eq. (2) causes transitions of\nelectrons between the sites iandj. Absorption of energy\nin course of these transitions is quite similar to the ab-\nsorption of the ac electric \feld by pairs of the localized\nstates. However, the transitions caused by \u000e^Hiare ac-\ncompanied by spin \rips, both from \"to#, and from#to\n\". With regard to absorption of energy, one should add\nup the contributions of the both types of transitions, i.e.\nI(e)=I#!\"+I\"!# (3)3\nHowever, the spin current results from the fact that these\ncontributions are not equal to each other, so that\nI(s)=I#!\"\u0000I\"!#: (4)\nThus, for calculation of the spin current into hopping\ninsulator, one can use the standard \\resonant\" phonon-\nless absorption theory32and substitute the corresponding\nrates into Eq. (4).\nC. Resonant absorption at H= 0\nWe \frst neglect the Zeeman splitting in the normal\nlayer. In this case resonant transitions happen within\npairs of localized states, Fig. 1b. The correction \u000e^Hi\ncauses such transitions between the sites iandjbecause\nthe corresponding wave functions jiiandjjihave a \fnite\noverlap integral, tij.33Due to this overlap, the eigenfunc-\ntions of the pair get modi\fed as\njii=r\n\u0000 +\u000e\"\n2\u0000jii+r\n\u0000\u0000\u000e\"\n2\u0000jji;\njji=\u0000r\n\u0000\u0000\u000e\"\n2\u0000jii+r\n\u0000 +\u000e\"\n2\u0000jji; (5)\nfor\u000e\"=\"j\u0000\"i>0. The corresponding energies are\n~\"i;j=\"i+\"j\n2\u0007\u0000\n2;\u0000 =h\n\u000e\"2+ 4t2\niji1=2\n: (6)\nSince both modi\fed eigenfunctions contain jii, the ma-\ntrix element of \u000e^Hibetween them is \fnite, and the\nGolden-rule expression for the spin-\rip part of the i!j\ntransition rate for \"j>\"ireads\nI(s)\ni!j=\u0000mxmyJ2F(~\"i;~\"j;!); (7)\nwhere the function F is de\fned as\nF(~\"i;~\"j;!) =2t2\nij\n(~\"j\u0000~\"i)21\n\u001c[f(~\"i)\u0000f(~\"j)]\n(~\"j\u0000~\"i\u0000~!)2+\u0000~\n\u001c\u00012\n=2t2\nij\n\u000021\n\u001c[f(~\"i)\u0000f(~\"j)]\n(\u0000\u0000~!)2+\u0000~\n\u001c\u00012: (8)\nHere we have introduced the phonon broadening of the\nlevels,\u001c\u00001.\nIt is easy to see that the transition rate to states with\n\"j<\"iis given by Eq. (7) with function F from Eq. (8),\nbut withf(~\"i)$f(~\"j), and thus the rate has the same\nsign as Eq. (7). Physically, this can be seen from the fol-\nlowing argument: Consider the simple case of mx=my.\nThe Hamiltonian of Eq. (2) implies that for a given site\nat the interface, spins \"are transferred to states of higher\nenergy (and there is a back\row of spins \"converted from\n#from those states), while spins #are pushed to stateswith lower energy (and there is a back\row of spins #con-\nverted from\"). Since the occupation of the state at the\ninterface is larger than of those at higher energy, there is\na negative#!\" conversion rate because of transitions up\nthe energy. This is exactly what Eq. (7) suggests. Fur-\nther, since the occupation of the state at the interface\nis lower than of those at lower energy, there is a positive\n\"!# conversion rate, or, again, negative #!\" one. Hence\na simple permutation f(~\"i)$f(~\"j) su\u000eces to describe\ntransitions to states with \"j<\"i.\nThe product mxmyin Eq. (7) is speci\fc for spin pump-\ning, see Eq. (1). The expression for the net absorption\nrate contains1\n2(m2\nx+m2\ny) instead. Another di\u000berence\nfrom the conventional resonance absorption31,32is the\nstructure of the matrix element in Eq. (7). This, how-\never, modi\fes the result of averaging over the sites, j,\nonly by a numerical factor. A crucial observation in the\naveraging procedure32is that the relevant sites, j, are\nlocated within a narrow spherical layer with a radius r!\nwhich is found from the condition 2 jtij(r!)j=~!. As-\nsuming the exponential decay of the overlap integral with\ndistance,jtij(r)j=t0exp(\u0000rij=a), we have\nr!=aln2t0\n~!: (9)\nThe result of averaging and summing over sites far away\nfrom the boundary reads\nI(s)(!) = 2\u00192mxmyJ2\u0000\ng!ar2\n!\u0001@f\n@\"; (10)\nwheregis the density of states. The transition rate of\nEq. (10) should be interpreted as the spin current gener-\nated per a localized state coupled to the ferromagnet.\nD. Resonant absorption at \fnite H\nTo generalize Eq. (7) to a \fnite magnetic \feld in the\nnormal layer, one must take into account the modi\fcation\nof the spin eigenstates, as well as the Zeeman splitting in\nenergies of the latter. The spin structure of the spin-split\nlevels depends on the orientation of Has follows\nj\u001fH+i= cos\u0010\u0012H\u0000\u0012M\n2\u0011\nj\u001fM+i+isin\u0010\u0012H\u0000\u0012M\n2\u0011\nj\u001fM\u0000i;\n(11)\nj\u001fH\u0000i= cos\u0010\u0012H\u0000\u0012M\n2\u0011\nj\u001fM\u0000i+isin\u0010\u0012H\u0000\u0012M\n2\u0011\nj\u001fM+i:\n(12)\nHere the quantization axes for j\u001fM\u0006iandj\u001fH\u0006ispinors\nare chosen along the static part of the magnetization, and\nthe external magnetic \feld, respectively, see Fig. 3(a).\nThe statesj\u001fM\u0006iat sitesiandjare split by \u0001 z.\nAll four transitions between states with j\u001fM\u0006ispin\nwave functions, Fig. 2(b), are allowed for a general ori-\nentation of the magnetic \feld. For spin-conserving tran-\nsitions (+!+ and\u0000!\u0000 ), the frequency dependence4\nofI(s)remains!r2\n!, i.e. the same as in Eq. (10). Ori-\nentation ofHenters into the prefactor: The product\nmxmyshould be replaced with1\n4sin2(\u0012H\u0000\u0012M)m2\nxfor\nboth transitions.\nWhile the spin-conserving transitions do a\u000bect the spin\ncurrent density distribution in the sample, they are non-\nresonant, and it is the spin-\ripping ones (+ $\u0000 ) that\nare responsible for the spin current generation at the in-\nterface. In other words, no spin current is possible in a\nstationary state without the latter processes. Therefore,\nin what follows we concentrate on the frequency and mag-\nnetic \feld dependence of the corresponding rates.\nAs far as +! \u0000 and\u0000 ! + transitions are con-\ncerned, only the + !\u0000 with absorption of a magnon,\nand\u0000! + with emission of a magnon become important\nin the vicinity of the resonance ~!= \u0001z. The other two\ntransitions are non-resonant, and therefore disregarded\nhere. For the +$\u0000 transitions, the prefactor !in the\nspin current remains intact, since it comes from the dif-\nference in the populations of levels involved. However,\ndespite the upper and lower Zeeman levels being sepa-\nrated in energy, the overlap of the spatial wave functions\nis determined by \"i,\"jinzero magnetic \feld. Thus,\nthe +! \u0000 transitions take place between pairs with\n(\"j\u0000\"i)\u0018j~!\u0000\u0001zj. These pairs have the \\shoulder\"\nr~!\u0000\u0001z=aln2t0\nj~!\u0000\u0001zj: (13)\nLogarithmic divergence of Eq. (13), which is cut o\u000b at\nj~!\u0000\u0001zj\u0018~=\u001c, ensures the resonant character of spin-\n\ripping transitions that we took into account.\nIn addition to the replacement of r!byr!\u0000\u0001zin the\nspin current, the prefactor mxmyshould be modi\fed as\nmxmy!G(mx;my), where the function G is de\fned as\nG(mx;my) =1\n4(mx+mycos(\u0012H\u0000\u0012M))2;(14)\nso that the absorption, and thus the FMR damping, do\nnot have the usual form /mxmy.\nThe most spectacular manifestation of the resonance\n~!= \u0001zis that the intrasite transitions become pos-\nsible, as illustrated in Fig. 2(b). For these transitions\nthe overlap of the spatial parts of the on-site wave func-\ntions is equal to 1, and the magnetic-\feld dependence of\nabsorption is a pure Lorentzian. Orientation-dependent\nprefactor, which is the matrix element of \u000e^Hibetween\nthe spinorsj\u001fH+iandj\u001fH\u0000iis the same as in Eq. (14).\nSummarizing, we present the expression for spin current\nclose to the resonance ~!= \u0001zin the form\nI(s)(!) = 2G(mx;my)J2!@f\n@\"\n\u0002h ~\n\u001c\n(\u0001z\u0000~!)2+\u0000~\n\u001c\u00012+\u00192ga3ln2 2t0\nj~!\u0000\u0001zji\n;(15)\nwhere the \frst term comes from intrasite and the second\nterm from intersite transitions. Directly at the resonance,\nthe \frst term dominates. This is ensured by the condition\n0.51.01.52.02.53.00.51.01.52.02.53.0\n0.51.01.52.02.53.00.51.01.52.00.51.01.52.02.53.00.20.40.60.81.01.21.4\n(b)\n✓H\n✓H\n(c)✓H\n(d)\n˜H\n˜H˜HFIG. 3. (Color online) (a) The geometry of FMR; For a \fxed\ndimensionless frequency, ~ !=!FMR=4\u0019Ms, the dimensionless\nmagnitude, ~H=H=4\u0019Ms, and orientation, \u0012H, of dc mag-\nnetic \feld are related via Eq. (17). This dependencies are\nshown for the values of ~ !=\r: (b) 0:5, (c) 1:2, and (d) 2.\nRed dots indicate the values of H, for which the condition\n\rH=!FMRis satis\fed.\n(a)\nFIG. 4. (Color online) (a) the resonant condition \rH=!FMR\nis satis\fed along the solid lines on the\u0010\nH\n4\u0019Ms\u0011\n\u0000\u0012Hplane. The\ncuto\u000b values of \u0012Hare cos\u00001\u0010\n1p\n3\u0011\n\u001955\u000eand\u0019\u0000cos\u00001\u0010\n1p\n3\u0011\n.\n(b) the behavior of the spin current calculated from Eq. (15)\nforga3~=\u001c= 4\u000110\u00003andt0\u001c=~= 15.\nga3~=\u001c\u001c1. Since the combination 1 =ga3is the minimal\nenergy spacing between two sites in the insulator located\nwithin\u0018afrom each other, the above condition implies\nthat this spacing is much bigger than the phonon broad-\nening of individual levels, which is the de\fnition of the\nAnderson insulator. As the deviation from the resonance\nincreases, the behavior of I(s)(!) is dominated by the sec-\nond term. Neglecting the logarithm, the crossover takes\nplace atj\u0001z\u0000~!j\u001c=~&\u0000\n\u001c=~ga3\u00011=2\u001d1. The behavior\nof spin current near the resonance is shown in Fig. 4(b),\nwhere the logarithm was cut o\u000b at j~!\u0000\u0001zj=t0=15.\nIII. RESONANT ORIENTATIONS OF\nEXTERNAL FIELD\nEquation (15) is our main result. To make connec-\ntion to the experimental papers Refs. [23{26], below we\ncalculate the magnitude and orientation of the dc \feld\nwhere the anomalous behavior of ISHE voltage takes5\nplace. Such behavior takes place when two conditions\nare met: The Zeeman splitting of the localized states is\nequal to ~!, and!=!FMR.\nWe specify the orientation of Hand magnetization,\nM, using the notations common in the literature, see\ne.g. Refs. [9, 15, and 24], and Fig. 3. We will also intro-\nduce dimensionless variables ~H,~Mand ~!, which stand\nforH,Mand!FMRin the units of 4 \u0019Ms, whereMsis\nthe saturation magnetization. Then the angle \u0012M, corre-\nsponding the equilibrium orientation of M, is found from\nthe condition that Mis parallel to the e\u000bective magnetic\n\feld, with the demagnetizing term taken into account9\n2~Hsin(\u0012H\u0000\u0012M) + sin 2\u0012M= 0; (16)\nwhile the expression for the resonant frequency, ~ !,\nreads28\n\u0010~!\n\r\u00112\n=h\n~Hcos(\u0012H\u0000\u0012M)\u0000cos 2\u0012Mi\n\u0002h\n~Hcos(\u0012H\u0000\u0012M)\u0000cos2\u0012Mi\n:(17)\nFrom these two equations we exclude \u0012Mand plot the di-\nmensionless \feld ~Hversus\u0012H, for a given FMR frequency\n~!. Examples of these curves are shown in Fig. 3. Res-\nonant orientation is obtained by crossing a curve ~H(\u0012H)\nby the line ~!=\r~H. Two intersections determine the ori-\nentations for which !FMRis equal to the Zeeman splitting\nof the localized states. Upon changing !FMR, we get two\nlines of resonances, Fig. 4(a). They occupy two domains:\n0< \u0012H<cos\u000011p\n3and (\u0019\u0000cos\u000011p\n3)< \u0012H< \u0019. At\nthe boundaries of the domains ~Hgoes to in\fnity. Then\nit follows from Eqs. (16) and (17) that at these bound-\naries sin(\u0012M\u0000\u0012H) = 0, and \u0012Hsatis\fes the equation\ncos(2\u0012H)+cos2\u0012H= 0, yielding \u0012H= cos\u00001\u0010\n1p\n3\u0011\n\u001955\u000e.\nIn Refs. [23 and 24] on pumping into organics the mi-\ncrowave frequency driving the resonance was 9 :45 Ghz,\nwhile the values of 4 \u0019Mswere very di\u000berent, namely,\n4\u0019Ms= 0:175 T in Ref. [23] and 4 \u0019Ms= 0:805 T in\nRef. [24]. Then from Fig. 4(a) we \fnd that the resonant\nangle\u0012Hshould be close to 45\u000efor Ref. [23] and 23\u000efor\nRef. [24].\nIV. SPIN-RESISTOR NETWORK\nAfter the spin polarization is generated at the bound-\nary, it should spread into the bulk of the insulator to\navoid the back\row.35In a metal, where Pis a contin-\nuous function of coordinates, this spreading is by spin\ndi\u000busion accompanied by the Larmor precession. In a\nhopping insulator Ptakes discrete values, Pi, which are\nthe polarizations on the sites, i. The Larmor precession\nis accounted for by the on-site Zeeman splitting, \u0001 z, of\nthe levels, see Fig. 2. The frequencies of electron hops\nFIG. 5. (Color online) \\Spin-resistor\" network. Polarizations\nPiandPjon the sites i,jdetermine the spin current be-\ntween these sites. The coe\u000ecients, Rij, are proportional to\nthe electric hopping resistances.\nbetween two sites, iandj, depend strongly on their en-\nergies,\"i,\"j, and their spatial separation, rij. Then the\nissue of spreading of the spin polarization reduces to the\nquestion: what is the spin current I(s)\ni!jbetween the sites\nwith polarizations PiandPj, provided that, on average,\nthere is no charge current between these sites?\nIf bias were applied between the two sites, then the\naverage charge current, proportional to this bias, could\nbe found by ascribing an e\u000bective resistance, Rij, to the\npair of sites30,31. It is easy to see that the same Rij\ndetermines the proportionality coe\u000ecient between I(s)\ni!j\nandPi\u0000Pj, namely\nI(s)\ni!j= 2Pi\u0000Pj\nRij@f\n@\": (18)\nIn Eq. (18) we have assumed that the di\u000berence ( \"i\u0000\"j)\nis much smaller than the temperature, so that@f\n@\"is the\nsame for both sites. Equation (18) follows from the fact\nthat the on-site chemical potentials of the local majority\nand minority electrons are shifted by \u0007jPij=@f\n@\", respec-\ntively. The spinors that correspond to these local spin\neigenstates are de\fned by the directions of Pi,Pj. Im-\nportantly, the fact that the chemical potential splitting is\nsymmetric around the chemical potential of the unpolar-\nized system ensures the absence of the charge current, i.e.\nthe net current \row i!jis compensated by the net cur-\nrent \rowj!i. With di\u000berent spin polarizations of the\nsites, the compensation of the charge \rows leads to the\nimbalance of the spin \rows, and thus to Eq. (18). Note\nthat Eq. (18) remains valid in external magnetic \feld,\nwhich enters only via the magnitudes of polarizations.\nOverall, Eq. (18) suggests that polarization built up at\nthe F-N boundary spreads along the current-carrying re-\nsistor network, as illustrated in Fig. 5.\nV. CONCLUDING REMARKS\n(i) Our result Eq. (15) applies when the phonon-induces\nbroadening of the levels is smaller than !. In the oppo-\nsite case,!\u001c\u001c1, the mechanism of absorption is the\nPollak-Geballe relaxation mechanism, Ref. [36]; no sharp6\ndependence of pumping near the resonance is expected\nin this regime. Unlike pumping into metals, the pumping\nrate Eq. (15) is not simply proportional to m\u0002dm\ndt. The\nprefactor in Eq. (14) depends on the relative orientation\nofmand the external magnetic \feld, H.\n(ii) Suppose that we are at resonance ~!= \u0001z. The\nmicrowave \feld acts both by driving the FMR but also\ndirectly, by causing transitions between the Zeeman lev-\nels. If the amplitude of the \feld in frequency units (Rabi\nfrequency) exceeds the inverse spin relaxation time, these\ntransitions will be saturated in the bulk. Then the pump-\ning becomes ine\u000ecient.\n(iii) In conventional theory of hopping transport the\napplied voltage drops not on all the resistors consti-\ntuting the network, but on the highest, critical, resis-\ntors representing the \\hardest\" hops.31The spin relax-\nation rate will be dominated by hyper\fne or spin-orbitenvironment37,38of this hop.\n(iv) We did not consider e\u000bects of electron-electron in-\nteraction, and did not describe in detail how \fnite resis-\ntance of the spin-current network, Section IV, a\u000bects the\nmeasured value of the spin current. In brief, Coulomb\ncorrelations enhance the absorption of magnons by in-\ncreasing the number of singly occupied pairs33, while the\nmeasured spin current is given by Eq. (15) only in the\nlimit of vanishingly small bulk resistance. These consid-\nerations are, however, completely standard, and do not\nchange the qualitative picture of spin current generation\nby resonant magnon absorption in hopping insulators,\ndeveloped in this paper.\nAcknowledgements. We are grateful to C. Boehme and\nZ. V. Vardeny for piquing our interest in the subject.\nThe work was supported by NSF grants MRSEC DMR-\n1121252 (ZY and MER), and DMR-1409089 (DAP).\n1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n2S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66,\n104413 (2002).\n3M. I. Dyakonov and A. V. Khaetskii, in Spin Physics\nin Semiconductors , ed. by M. I. Dyakonov, Chapter 8\n(Springer, Berlin, 2008).\n4E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n5H. Y. Inoue, K. Harii, K. Ando, K. Sasage, and E. Saitoh,\nJ. Appl. Phys. 102, 083915 (2007).\n6K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S.\nMaekawa, and E. Saitoh, Phys. Rev. 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Sirringhaus, Nat. Phys.\n10, 308 (2014).\n25Z. Qiu, M. Uruichi, D. Hou, K. Uchida, H. M. Yamamoto,\nE. Saitoh, arXiv:1502.05244.\n26M. Kimata, D. Nozaki, Y. Niimi, H. Tajima, and Y. Otani,\narXiv:1411.2740.\n27For example, the bulk resistance, \u001a(T), in Ref. [26] in-\ncreased by two orders of magnitude between the room tem-\nperature and 20 K following the Mott's law ln \u001a/T\u00001=4.\nThe localization length inferred by the authors from the\ntransport measurements is quite small: a\u001911 nm.\n28C. Kittel, Phys. Rev. 73, 155 (1948).\n29R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,\n4382 (1979). The authors demonstrate theoretically and\nexperimentally that paramagnetic resonance in copper foil\nis greatly enhanced when the Zeeman frequency coincides\nwith the frequency of FMR in ferromagnet coupled to the\nfoil.\n30V. Ambegaokar, B. I. Halperin, and J. S. Langer, Phys.\nRev. B 4, 2612 (1971).\n31B. I. Shklovskii and A. L. Efros, \\Electronic Properties of\nDoped Semiconductors\" (Springer-Verlag, Berlin, 1984).\n32N. F. Mott, Philos. Mag. 22, 7 (1970).\n33A. L. Efros and B. I. Shklovskii, in Electron-Electron Inter-\nactions in Disordered Systems , edited by A. L. Efros and\nM. Pollak North-Holland, Amsterdam, 1985.\n34B. I. Shklovskii and A. L. Efros, Zh. Eksp. Teor. Fiz. 81,7\n406 (1981) [Sov. Phys. JETP 54, 218 (1981)].\n35A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404(R) (2002).\n36M. Pollak and T. H. Geballe, Phys. Rev. 122, 1742 (1961).37N. J. Harmon and M. E. Flatt\u0013 e Phys. Rev. Lett. 110,\n176602 (2013).\n38R. C. Roundy and M. E. Raikh Phys. Rev. B 88, 205206\n(2013)."    },    {        "title": "1411.5339v2.Dynamical_spin_injection_at_a_quasi_one_dimensional_ferromagnet_graphene_interface.pdf",        "content": "1 \n Dynamical  spin injection at a quasi -one-dimensional ferromagnet -graphene \ninterface  \nS. Singh1, A. Ahmadi1, C.T. Cherian2, 4, E. R. Mucciolo1, E. del Barco1, and \nB. Özyilmaz2,3,4,5  \n1Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542  \n3NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n4Graphene Research Center, National University of Singapore, Singapore 117542  \n5NUS Graduate School for Integrative Sciences and Engineering (NGS), National University of \nSingapore, Singapore 117456  \nAbstract:  We present a s tudy of dynamical spin injection  from a three -dimensional ferromagnet \ninto two-dimensional single -layer graphene.  Comparative f erromagnetic resonance (F MR) \nstudies of f erromagnet/g raphene strips buried underneath the central line of a coplanar \nwaveguide show that t he FMR line width  broadening is the largest  when the graphene layer \nprotrudes laterally  away from  the ferromagnet ic strip, indicating that the spin current is injected \ninto the graphene areas away from the area directly underneath the ferromagnet being excited. \nOur results confirm that the observed damping is indeed a signature of dynamical spin injec tion, \nwherein a pure spin current  is pumped into the single -layer graphene from the precessing \nmagnetization of the ferromagnet . The observed spin pumping efficiency is difficult to reconcile \nwith the expected backflow of spins according to  the standard sp in pumping theory and the \ncharacteristics of graphene, and constitutes a n enigma for spin pumping in two-dimensional \nstructures.  2 \n The efficient generation  of pure spin currents holds a great deal of prom ise for spintronics \napplications , with several existin g methods  already demonstrated, such as  the spin H all effect  1, \nelectrical spin injection  2, voltage -based spin pumping3 dynamical spin pumping  4,5, and optical \ngenerat ion of  spin packets6. Particularly,  the dynamical generation of spin current s carries special  \ninterest because no net charge current is involved in the process . In this method, spin a ngular \nmomentum is transferred from the precessional magnetization in a ferromagnet  (FM)  to an \nadjacent  non-magnetic  (NM)  system. This approach has already been experimentally \ndemonstrated in several FM/NM interfaces, including NM systems such as metals7, \nsemiconductors  5,8 or organic based materials9. A few advantages of this method are  that it does \nnot suffer from the impedance mismatch problem, it is scalable to large samples, it provides a \nhigh spin injection efficiency , and it is not based on the spin -orbit coupling to function, allowing \nits employment in sy stems without this interaction .  \nWithin the exciting cur rent trend to explore novel low -dimensional systems, the possibility to \ninject pure spin currents in graphene and other two -dimensional (2D) crystals has attracted \nconsiderable  attention in the past f ew years . In particular, graphene rises as  a prototype system to \nexplore this physics  due to its crystalline nature, excellent  electronic  properties10, tunable spin -\norbit co upling (i.e., via adatom engineering )11,12 and long spin relaxation lengths13. In addition, \ngraphene can act as a  high-fidelity channel for spin information  transfer  (due to its small intrinsic \nspin-orbit coupling and absence  of nuclear spins) , as well as provide the platform  for electrical \nmanipulat ion of the  spin polarization  (i.e., on-demand enhancement of spin -orbit coupling11,12). \nDynamical spin injection in FM/g raphene (FM/Gr) interfaces has been recently demonstrated  \n14,15. In the original work, we associated  the observed enhan cement in dynamical damping of \nextended FM/Gr films  to the generation of pure spin current in graphene  result ing from loss es of 3 \n spin angular mome ntum  in the ferro magnet  (i.e., spin pumping14). Subsequently, the dynamical \ninjection of spin currents in graphene was demonstrated by spin -charge conversio n \nmeasurements in a P d strip  placed laterally and in close proximity to a FM/Gr interface  \nundergoing FMR15. Although providing evidence for spin injection, none of these w orks shed \nlight into the real nature of spin pumping at the FM/Gr interface.  Estimates of the spin -mixing \nconductance obtained from the broadening of the FMR peaks resulted in surprisingly high values \n(e.g., \ng 5.261019 m-2 from Py/Gr  = Py/Gr - Py = 110-2), comparable to systems with high \nspin-orbit coupling7 (such as heavy metals  Pt and Pd). In addition, the direct deposition of the \nferromagnetic Permalloy (Py) film on top of the graphene layer could cause magneto -structural16 \nchanges in the Py surface  and a subsequent increase in straight fields altering the spin dynamics \nand accumulation in the semiconductor17, responsible for the observed change in damping when \ncomparing with films without graphene, i.e., deposited on the bare wafer.  A direct measuremen t \nof the spin Hall angle in FM/Gr interfaces has not been reported yet. A  recent work by R. \nOhshima et al.  18 published during the preparation of this manuscript , claims the observation of \nthe ISHE signal in single -layer graphene upon conversion of a spin current pumped from an \nextended insulating ferromagnet ( i.e., YIG) into a charge current. According to th at report, the \nspin current is injected perpendicularly to the YIG/G r interface , and the ISHE electric field \nmeasured within the graphene plane, as is the norm in other ferromagnet/conductor \nheterostructures. However, we find this interpretation rather questionable, since  conduction \nperpendicular to a single -layer graphene in not possible  simply due to the lack of a third \ndimension. Since conduction perpendicular to the graphene sheet is not possible, the geometry \nproposed by the authors, where the ISHE is measured along the graphene  plane, is impracticable.  4 \n The observations in that work  could be explained in terms of alternative physics, such as an \ninverse Rashba -Edelstein effect19 or similar, but never in terms of the ISHE.  \nIn this letter we present FMR experiments performed on different FM/Gr interfaces designed \nto systematically identify and eliminate  damping enhancement arising  from process es other than  \nspin pumping. In particular, a substantial enhancement of the Gilbert  damping observed in Py/Gr \nstrips when the graphene layer protrudes a few micrometers away from the edges of a narrow Py \nstrip is univocally a ssociated to spin pumping at the quasi -one-dimensional  interface between the \nPy edge and graphene , which shows lower spin -mixing conductance values (\ng 6.891018 \nm-2) than in extended films but still comparable to those obtained in Py/Pt interfaces  (e.g., \ng\n~1-41019 m-2). We also provide a theoretical analysis which  shows  the observed  spin \ninjection efficiency to lie well beyond that expected from the spin conduction channels provided \nby single -layer graphene, open ing fundamental  question s about the nature of spin injection in to \nthis 2D crystal . \nThe graphene  layers used in our experiments are  grown by the standard chemical vapor \ndeposition ( CVD ) method on thin Cu foil s20. Graphene is subsequently transferred onto the \nsubstrate  using a wet chemistry  process  and characterized by Raman spectroscopy . We use \n14 nm-thick films of Ni20Fe80 Permalloy  (deposited by e -beam evaporation in high vacuum \nconditions) as the ferromagnet for  all our studies. For FMR on extended films, we place  the \nsample  upside -down  on the central part of our broad -band -CPW  FMR sensor , which is  coated \nwith an insulating polymer  to prevent electrical contact with the sample14. For FMR on patterned \nfilms  in the shape of long and narrow  Py/Gr strips , the sample is buried directly underneath the \ncentral line of the CPW , isolated  from it by a thick  (~100 nm) insulating layer of oxide.    5 \n Before getting into the detailed discussion of the main results of this work , we want to briefly \ndiscuss a set of experiments designed to address the effect of magneto -structural changes in the \nsurface of the Py due to the immediate presence of graphene underneath, which  could  cause a \nnon-dynamical  broadening of the FMR 21. In the first experiment, the stacking order  of the Py \nand graphene layers has been reversed with respect to the original experiments14, where the Py \nwas deposited directly on top of the graphene layer  (i.e., a Py/Gr stacking) . In the present case, \nthe Py film is depo sited on a bare Si wafer coated with 300nm of thermally grown SiO 2 and \ngraphene transferred on top afterwards  (i.e., a Gr/Py stacking) , with the objective of maintaining \nthe Py film unaltered by the presence of graphene. A clear enhancement of the Gilbert damping \nis obtained when graphene is present  (i.e.,  Gr/Py   = Gr/Py  - Py = 3.410-3), resulting in a spin -\nmixing conductance of \ng 1.951019 m-2, i.e., three times lower than in previous Py/Gr \nsamples. In the second experiment, a 20  nm-thick Cu spacing l ayer was inserted in between the \nPy and graphene ( i.e., a Py/Cu/Gr stacking), and FMR results compared to those in Py and Py/Cu \nsamples. The rationale is to use the Cu layer as a structural spacer between the Py and the single -\nlayer graphene in order to ma intain the Py film unchanged. Note that a thin Cu film does not \ncontribute to the absorption/diffusion of the spin pumped away from the Py film, since Cu has a \nsubstantially larger spin diffusion length than the Cu  layer  thickness used in these experiments22. \nAgain, a clear enhancement of the G ilbert damping is observed when graphene is present ( i.e., \nPy/Cu/Gr   =  Py/Cu/Gr  -  Py/Cu = 4.210-3), resulting in a spin -mixing conductance of \ng\n2.381019 m-2, i.e., comparable to the values obtained in Gr/Py samples.   6 \n \n \nFigure 1: Sketches illustrating the strips used in the experiments. The Py strips are all the same \ndimensions, with a length of 3 mm and a width w  = 25m. a) Py strip. b) Py/Gr strip. c)  Py/Gr -prt strip, \nwith graphene protruding from the sides of the Py strip by d  = 20 m. \n \nThe set of experiments described above eliminate structu ral changes in the Py as a possible \ncause for the observed damping enhancement. However, the presence of the ferromagnet i n close \nproximity to the single -layer graphene , even in areas away from the FMR excitation , may \ninfluence the diffusion of the spins p umped away from the Py film, which can  still act as a spin \nsink since electrons can flow back into it . To avoid this situation , we patterned the Py/Gr film \ninto long  (l = 3mm)  and narrow  (w = 25 m) strips that  are placed directly underneath the central \nline of the -CPW, as shown in Fig.  1 and in the inset s to Fig. 2. Essentially, we prepare d three \ndifferent samples for this study : a) a Py strip  (Fig.  1a); b) a Py/Gr  strip (Fig.  1b); and,  c) a Py/Gr \nstrip with  the single -layer  graphene protruding  away  from  the Py strip  on both sides , which we \nshall call Py/Gr -prt henceforth (Fig.1c) . The upper inset to Fig. 2 shows a  scanning e lectron  \nmicroscope  image of a Py/Gr -prt strip, where  one can clearly see the cont inuous sheet of \ngraphene extending  away from the central ferromagnet  strip. Note that t he length by which \ngraphene protrudes  on each side of the ferromagnet strip , i.e., d ~12 m, is larger than the  spin \ndiffusion length of CVD graphene ( s ~2 m)23, in order to allow for a total relaxation of the spin \npumped away from the ferromagnet.  The devices are prepared by  transferr ing a single -layer 7 \n graphene  onto a GaAs (undoped) /SiO 2(100nm)  substrate , after which  unwanted graphene areas  \nare etch ed away using a photo resist mask  and standard optical lithography . Following etching  \nand e-beam  evaporation  of the Py strip , a 100  nm-thick layer of silicon oxide is grown atop to \ninsulate the device from the  central line of the -CPW , which ultimately cov ers the sample  (as \ndepicted in  the lower inset to Fig. 2). This geometry guarantees a homogeneous FMR excitation \nof the whole Py strip .  \n \nFigure 2: Field -derivative of the  FMR response for the three strips measured  (Py: Permalloy Gr: \ngraphene and Gr -prt: graphene protruding from the sides of the Py strip).  The upper inset shows an \nelectronic microscope image of the Py/Gr -prt stripe before being placed underneath the central line of \nthe -CPW sensor  (lower inset) . \n \nStandard  broadban d FMR measurements are performed on the samples described above to \nextract the FMR line width s. The corresponding field-derivative s, dM/dH, obtained at  an \nirradiation frequency of  12 GHz with the dc magnetic field applied in the plane of the Py strips \nare shown in F ig. 2. The FMR linewidth, defined as the peak -to-peak distance in the dM/dH data, \nis the largest for the sample with graphene protruding away from the Py strip ( i.e., the Py/Gr -prt 8 \n strip) and the smallest for the sample with Py only (i.e., the Py strip).  The frequency dependence \nof the in-plane excited FMR linewidth for the se samples is shown in Fig. 3a. The observed linear \nfrequency dependence of the linewidth can be explained by  means of the dynamical Gilbert \ndamping model , using the following ex pression:  \n                  \nf H H\n\n34\n0  ,                                (1)                                                                                        \nwhere \nh gB/  is the gyromagnetic ratio and α is the damping parameter, which is related to \nthe Gilbert damp ing through the expression \nSM G , with \nSM  being the saturation \nmagnetization . The two different contributions to the damping in Eqn. 1 are: a) a sample \ndependent i nhomogeneous part  (first term in the equation) , which can be calculated from the \nintercept at zero  extrapolated frequency and it does not depend on frequency ; and, b) dynamical \ndamping  (second term in the equation) , which scales  linearly with frequency  and from whose \nslope the Gilbert damping can be calculated . These extracted damping parameters  for in -plane \nFMR excitation (Fig.  3a) are as follows:   Py = 9.110-3 & GPy = 0.239  GHz; Py/Gr = 11.310-3 \n& GPy/Gr = 0.299  GHz ; Py/Gr-prt = 13.010-3 & GPy/Gr-prt = 0.333 GHz . There is a considerable \nenhancement in damping when going  from the Py -only strip t o the Py/Gr strip, where graphene \nis only present underneath the ferromagnet . This damping can not be attributed to spin pumping \ngiven the 2D nature of graphene, which  is located only underneath the Py strip and  does not \nprovide any conduction channel perpe ndicular to the interface.  Indeed , it has been shown that \ngraphene can act as an effective tunnel barrier for electrical spin injection into silicon due to the \nvery large resistivity of carriers across the graphene sheet24. Most likely, t he observed FMR \nbroadening is due to changes in the magnetic response of the Py due to surface changes induced \nby the graphene, such as an enhancement  of two-magnon scattering  processes . It has been 9 \n recently shown that the deposition of Co films  on graphene results  in magnetic variations a nd \nenhanced ma gnetic coercivity16. In our case, for example, we observe a slight  change (<10%) in \nthe magnetization saturation when graphene is present  (not shown here) . Out-of-plane excited \nFMR measurements on these strips seem to support this hypothesis . Two-magnon processes are \nsubstantially weaker when the precession of the magnet ization is excited wit h the dc field out of \nthe plane, which would explain the similar  H vs. f slopes for Py and Py/Gr strips in the out -of-\nplane excited FMR data of Fig.  3b (black and red data) .  \n \nFigure 3: a) In -plane and b) ou t-of-plane frequency dependences of the FMR linewidth of the three strips \nmeasured  (Py: Permalloy Gr: graphene and Gr -prt: graphene protruding from the sides of the Py strip) . \nThe insets show the FMR field excitation situations and the corresponding direct ions of propagation and \npolarization of the pumped spin currents  (Js) into the graphene area protruding away from the edges of \nthe Py strip.  \n \nThe central result of this work is the clear enhancement of the dynamical damping observed \nin Py/Gr -prt strips und er both in -plane and out -of-plane excited FMR (blue data in Figs.  3a and \n3b). Importantly, t his additional damping can only result from the relaxation of spins  in the area \nof graphene away from the Py strip, where an enhanced relaxation due to proximity ef fects as 10 \n discussed in the introduction for extended films is not an option . Comparing the FMR \nbroadening in the Py/Gr and Py/Gr -prt strips, and using Eqn.  (1) we extract a change in the \ndamping parameter Py/Gr -prt  = Py/Gr -prt - Py/Gr = 1.310-3, resul ting in a spin -mixing \nconductance of \ng 6.891018 m-2. These are a factor 2 -3 smaller than the values found in \nexperiments performed on extended films but still comparable to those found in Py/Pt or Py/Pd \nsamples.  \nWe now discuss the fundamental implications of the experiments described above. The \nobserved additional damping enhancement  provided by the protruding graphene  support s our \nassertion  that spin  pumping must occur across the quasi -one-dimensional  Py/Gr -prt interface at \nthe very edge of the Py strip. The picture we propose is the following. The proximity  of Py to \ngraphene induces a wea k equilibrium ferromagnetization in the latter25. The precessing \nmagnetization in the Py film pumps a spin current into the graphene layer  underneath the Py , \nthus creating an additional non-equilibrium spin accumulation in that layer. Part of the excess \nspin polarization is rel axed by local defects and impurities present on graphene (through  local -\nmoment scattering or spin -orbit coupling). When the graphene layer does not protrude away \nfrom the Py, t he remainder non -equilibrium spin accumulation creates a coherent backflow spin \ncurrent into the Py.  Thus, in steady state, there is no net spin current and the enhanced damping \nof the FMR is mainly due to spin relaxation in the graphene layer underneath the Py. However, \nwhen the graphene layer extends beyond the Py, the non -equilibri um spin accumulation causes a \nspin current to flow into the protruded graphene regions, reducing the amount of coherent \nbackflow into the Py and thus increasing the damping of the FMR due to losses of angular \nmomentum . In this case, it is standard to obtai n the spin -mixing conductance associated to the \npumping of spin into the extended graphene regions through  the expression26 11 \n  \n .                                                      (2) \nYet, this is only justified if the spin current relaxes much faster than the charge diffuses ( i.e., \nwhen the electronic motion in the extended graphene regions is ballistic or when very strong spin \nscattering is present, which are unlikely in our samples). We rather expect the charge to diffus e \nwith a relaxation time \n , where \n  is the spin relaxation time. In this case, a non -\nequilibrium spin population  builds up  on the protruding graphene  near the Py edge. This causes \nthe spin current  to partially diffuse  back into the graphene underneath the Py.  Thus, t he resulting  \nspin-mixing conductance is smaller than that obtained from  measuring the excess damping26 \n                                                               (3) \nwhere \n  is the backflow (dimensionless) parameter, with \n  denot ing the \ndensity of states of graphene at the Fermi level , \n being the length of the Py strip,  A being the \narea of the Py/graphene interface, and \n  representing the spin diffusion length (notice \nthat \n , where \n  = 20 m is the length of the protruding  graphene region). Assuming \ns, \n m2/s, and \n , with \n  meV and \nm/s (see Ref.13 ) we find \n  m2, which i s a much large r value (by several \norders of magnitude) than  the experimental value \n  m2. This implies  a 12 \n negative  value for \n  and therefore indicates that Eqn. (3) may not be directly applicable to \nour setup . We note that this analysi s may change in experiments  performed  in devices using h -\nBN substrates, for which the spin relaxation parameters would drastically change. H -BN has also \nbeen shown to decrease the conductance mismatch in electrical spin injection  and therefore may \naffect s pin reflection and relaxation at FM/graphene interfaces  27. We believe that the main \nproblem  in our analysis  is not in the est imate of the backflow parameter  β, since this follows \nstraightforwardly from reasonable estimates for the graphene parameters l, D, τs, and NF. Instead, \nwe believe that the problem lies on the assumption that spin currents pumped by the Py are fully \ninjected into the protruding graphene sheets. It may be possible that very close to the edge where \nthe protruding graphene meets the Py, there is a strongly enhanced spin relaxation, effectively \nmaking λs a much shorter length scale, rendering the backflow negligible. This relaxation could \nbe due to the Py -Gr edge acting approximately as a semiconducting p -n junction, regaining the \nconducta nce mismatch that were supposed to be eliminated by the spin pumping method, with \ngraphene underneath the Py being heavily p -doped, while protruding areas  are almost undoped . \nHowever, this interpretation requires further experimental verification and a mor e detailed \ntheoretical modeling.   \nIn conclusion, we have presented experimental evidence of a substantial increase in damping \nin Py/Gr strips when graphene is left to protrude from the sides of the ferromagnet. The \nsurprisingly high spin mixing conductance  obtained from the observations raises question s about \nthe physics of dynamical spin injection into two -dimensional structures such as graphene.  Our \nimmediate future objective is the direct measurement of the ISHE voltage generated in the \nprotruding graph ene region as a result of the  pumped spins from the ferromagnet, for which \nelectrodes will be placed at the opposite ends of the protruding graphene lines. However, YIG-13 \n based insulating ferromagnetic strips will be used for this purpose, since the low -resistivity Py \nstrip in the present configuration acts as an electrical shunt and prevents the observation of the \neffect.   \n \nAknowledgements  \nWe thank A. Brataas for valuable discussions. S.S. E.M. and E.d.B. acknowledge support from \nthe National Science Foundat ion (ECCS #1402990).  A.A., C.T.C. and B.O. acknowledge support \nfrom the Singapore National Research Foundation (Grant Nos. NRFRF2008 -07 and NUS -YIA \nR144 -000-283-101). \n \n \n \n \n \n \n \n \n \n 14 \n Supplemental Material  \nDynamical spin injection at a quasi -one-dimensional ferrom agnet -graphene \ninterface  \nS. Singh1, A. Ahmadi1, C.T. Cherian2, 4, E. R. Mucciolo1, E. del Barco1, and \nB. Özyilmaz2,3,4,5  \n1Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n2Department of Physics, National University of Singa pore, 2 Science Drive 3, Singapore 117542  \n3NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n4Graphene Research Center, National University of Singapore, Singapore 117542  \n5NUS Graduate School for Integrative Sciences and E ngineering (NGS), National University of \nSingapore, Singapore 117456  \nIn this document we present a set of experiments designed to study and prevent changes of the \nmagnetic properties of the Py films due to structural changes associate to the adjacent singl e-\nlayer graphene and their possible effect in broadening the FMR linewidth of the ferromagnet.  \n1. Reversing the FM/Gr stacking order  \nWe first turn to experiments in which the graphene layer is transferred onto pre -deposited Py \nfilms, in contrast with our prev ious studies14, where the graphene was transferred onto the \nsubstrate ( i.e., GaAs) and subsequently Py was deposited atop. Since graphene is not atomically \nflat when deposited on most su bstrates28 (with exception of boron nitride), the growth of the \nferromagnet on top of the graphene surface can result in a rough interface and damping 15 \n contributions due to processes other than the spin pumping, such as two magnon scattering. In \naddition, the saturation magnetization of the film can also be modified. We have performed \nnanoscale tunneling microscopy on our graph ene films on SiO 2 substrates and observed the \nnanometer sized ripples of our graphene surfaces. To eliminate an enhancement of damping due \nto structural changes at the interface, we transfer single -layer graphene onto of Py films which \nhave been previously  deposited on SiO 2 wafers, so that the Py is unaffected, or weakly affected \n(see Fig.  SM1a).  \n \nFigure SM4: a) Sketch illustrating a Py film in where graphene is wet transferred on the top of \nthe film. b) Frequency dependence of th e FMR linewidth of Py and Gr/Py films.  \n \nWe performed FMR measurements in both Py and Gr/Py by applying the dc magnetic field \nin the plane of the films. The width of the resonance, in units of external dc field, is extracted by \ntaking the peak -to-peak dista nce of the FMR field -derivatives. The frequency dependences of \nFMR linewidth for both samples are shown in figure SM1b . The frequency -dependent width of a \nFMR resonance represents the effective dynamical damping of the precessing magnetization of \nthe ferro magnet. Any loss of spin angular momentum  (in this case due to the presence of 16 \n graphene)  will correspondingly result in an increase in linewidth. Since pure spin currents carry \naway spin angular momentum, we can theoretically associate t he damping enhancem ent to be a \ndirect consequence of dynamical spin pumping from the Py into the graphene layer .  \nThe damping parameters extracted for these samples are: Py = 9.610-3 & GPy = 0.257  GHz; \nPy/Gr = 13.010-3 & GPy/Gr = 0.339  GHz. This corresponds to a change i n the damping \nparameter ( Δα) of 35%, as compared to 88% change observed in our previous studies of films \nwith the Py deposited onto of the graphene layer14. The difference in the relative damping change \njust by reversing the stacking order of the Py and graphene layers is indicative that a good \nfraction of the FMR broadening in Py/Gr samples was likely due to structural changes in the \nsurface of the Py film, which has been deposited on top of the rough graphene s urface. It is \npossible that in our experiment the wet transfer of graphene into only one of the Py films causes \nsome degree of contamination ( e.g., oxidation of the Py surface) altering the magnetic properties \nof the ferromagnetic film or the spin polariza tion at the interface. Recent experiments have \nshown how contamination during the wet transfer process of two -dimensional crystals can \ndecrease spin polarization at the interface29. We do not observe any appreciable change in the \nsaturation magnetization, so such contamination is not drastically changing the \nmagnetostructural properties of the Py film.  Nevertheless, the substantial amount of damping \nobserved in the Gr/Py samples o f Fig.  SM1 indicates that spin pumping at the Gr/Py interface is \nlikely the main cause of FMR broadening.  \n2. Inserting a Cu spacing layer in between the Py film and graphene  \nIt was brought to our attention that while graphene is transferred onto evaporated Py  films \nfrom an aqueous solution there is a possibility that the Py is oxidized, causing an enhancement of 17 \n damping of structural origin. Although we find this possibility unlikely to explain the observed \nresults, we made controlled extended films by inserti ng a thin layer of Copper (Cu) in between \nthe Py film and graphene, and compared the FMR results with Py/Cu films without graphene \nunderneath. The actual samples that we developed are: a) bare Py (14nm); b) Py(14nm)/Cu \n(20nm); and, Py(14nm)/Cu(20nm)/Gr. Th e cartoons representing samples are shown in the inset \nto Fig. SM2a. We use 20nm -thick Cu because it is substantially less than the spin diffusion \nlength and one expects to see negligible spin relaxation and hence no enhanced damping due to \nthe presence of  Cu itself in the FMR experiments22. In addition, the effect of a graphene layer \n20nm away from the Py sample will be negligible and the Py/Cu interface will remain unaltered, \nwhether or not graphene is underneath the heterostructures.  \n \nFigure SM5: a) Illustrations of the Py, Cu and Gr heterostructures. b) Frequency dependence of \nthe FMR linewidth of Py , Py/Cu and Py/Cu/Gr  films . \n \nThe in -plane excited FMR linewidths for the above mentioned samples are presented as a \nfunction of frequency in Fig.  SM2b. As exp ected, there is no change in damping due to presence \nof the Cu film underneath the Py, since the pumped spins reflect coherently back into the 18 \n ferromagnet (no diffusion in the Cu film). One would need to increase the thickness of the Cu \nfilm beyond the spi n diffusion length for pumping -induced damping to be observable22. Now, in \nthe case of the Py/Cu/Gr stacking we do observe a clear change in the damping (i.e. the slope of \nthe frequency dependence of FMR linewidth changes). The extracted damping parameters for \nthese samples are: Py = 1210-3 & GPy = 0.315  GHz; Cu/Py = 11.910-3 & GCu/Py = 0.312  GHz ; \nand, Gr/Cu/Py = 16.210-3 & GGr/Cu/Py = 0.422  GHz . This is about 35 % change in damping due to \nthe presence of graphene underneath the Cu film, which is similar to what we observe in \nexperiments with graphene transferred atop of the Py films.  \n3. Comparison between different experiments  \nA first inclination is to attribute the observed damping enhancements to spin pumping into \nthe whole graphene area underneath the ferromagnet. However, one  needs to keep in mind that \ngraphene is a two dimensional material with no conduction channels perpendicular to the plane. \nFor standard 3D FM/NM heterostructures where spin pumping has been studied7, the spin \ngradient is perpendicular to the FM/NM interface, i.e. the spin polarization decays \nperpendicularly to the plane. In the case of graphene, the only d irection spins can decay is along \nthe plane of the graphene sheet, and this needs to be away from the microwave excitation area. In \nour above explained experiments on extended thin films only a small part (~20  um  1000  µm) \nof the total film (~5  mm  5 mm)  is excited by to the central line of the µ -CPW. Therefore, the \nspin density pumped from the excited part of the ferromagnet into the graphene directly \nunderneath can only decay in the horizontal graphene plane which is not under microwave \nexcitation. Unfo rtunately, in experiments with extended films (as those presented in this \nSupplemental Information document), the areas of graphene away from excitation are in close 19 \n proximity to the ferromagnet, which is covering the whole film. Given that graphene is hig hly \nsensitive to its environment30, the presence of the ferromagnet is expected to change the intrinsic \nelectronic and spin relaxation properties and result in an enhanced diffusion of the spin currents. \nThis asks for a device geometry in which the damping assoc iated with spin pumping can be \nsystematically differentiated, which is the purpose of the experiments discussed in the main text \nof this article.  \nFor the sake of clarity, we have compared the change in damping observed for different \nconfiguration samples i n Table SM1, including the results in Py/Gr -prt strips discussed in the \nmain text, where the enhanced spin diffusion due to the proximity of the ferromagnet in areas of \ngraphene away from the excitation has been eliminated. Although being the smallest, t he 11% \ndamping enhancement observed in the latter case can be univocally associated to spin pumping \ninto single -layer graphene and still stands comparable to values observed in heavy metals. It has  \nbeen shown that a small density of  adatoms ( e.g., atomic hyd rogen) on the graphene surface can \ninduce local deformations resulting in huge enhancement s of spin orbit coupling11. Recently, a \nlarge spin Hall Effect  has also been reported for CVD grown graphene (similar to what is used \nfor this work) du e to presence of residual Cu adatoms at the graphene surface. Although Cu itself \nhas low spin-orbit  interaction , it can induce local deformation in the graphene lattice which \nresults in an enhance ment of the spin -orbit interaction .  \n \n \n \n 20 \n  \nTable  SM1 : Comparison between damping enhancements for all the samples discussed in this \narticle.  \n \nReferences:  \n1 A. K. Patra, S. Singh, B. Barin, Y. Lee, J. -H. 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V enkata Kamalakar, André Dankert, Johan Bergsten, Tommy Ive, and Saroj P. Dash,  Sci. Rep. \n4 (2014).  \n28 Masa Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D. Williams,  Nano Letters 7 (6), \n1643 (2007).  \n29 André Dankert, M. Venkata Kamalakar, Abdu l Wajid, R. S. Patel, and SarojP Dash,  Nano Res., 1 \n(2014).  \n30 F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov,  \nNat Mater 6 (9), 652 (2007).  \n "    },    {        "title": "1206.3041v1.Low_field_microwave_absorption_in_epitaxial_La_Sr_Mn_O_films_resulting_from_the_angle_tuned_ferromagnetic_resonance_in_the_multidomain_state.pdf",        "content": "arXiv:1206.3041v1  [cond-mat.mtrl-sci]  14 Jun 2012Low-field microwave absorption in epitaxial La 0.7Sr0.3MnO 3films resulting from the\nangle-tuned ferromagnetic resonance in the multidomain st ate\nM. Golosovsky∗, P. Monod,\nLaboratoire de Physique du Solide, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France\nP.K. Muduli and R.C. Budhani\nDepartment of Physics,Indian Institute of Technology,\nKanpur, 208016, India\n(Dated: June 20, 2018)\nWe studied magnetic-field induced microwave absorption in 1 00-200 nm thick La 0.7Sr0.3MnO3\nfilms on SrTiO 3substrate and found a low-field absorption with a very peculi ar angular dependence:\nit appears only in the oblique field and is absent both in the pa rallel and in the perpendicular\norientations. Wedemonstrate thatthis low-field absorptio n results from the ferromagnetic resonance\nin the multidomain state (domain-mode resonance). Its unus ual angular dependence arises from the\ninterplay between the parallel component of the magnetic fie ld that drives the film into multidomain\nstate and the perpendicular field component that controls th e domain width through its effect on\ndomain wall energy. The low-field microwave absorption in th e multidomain state can be a tool to\nprobe domain structure in magnetic films with in-plane magne tization.\nPACS numbers: 76.50+g, 75.60.Ch, 75.78.Fg, 75.47.Gk\nI. INTRODUCTION\nLow-field microwave absorption is a useful tool of de-\ntection ofmagnetic transitions in bulk materialsand thin\nfilms [1–3]. It is also known as nonresonant, zero-field,\nor magnetically-modulated microwave absorption and it\nwas used to find the traces of ferromagnetic and super-\nconducting phases in bulk materials [1, 4]. The low-field\nabsorption in magnetic materials has multiple sources\nthat operate in different frequency ranges [5]. The dom-\ninant contribution at radiofrequencies arises from mag-\nnetoimpedance [6–8] (the field dependence of the skin-\ndepth) and domain wall resonances [9–11], while at mi-\ncrowave frequencies it is commonly attributed to var-\nious ferromagnetic resonances in the magnetically un-\nsaturated state. These include non-aligned FMR mode\n[12, 13], the natural FMR [14], off-resonant absorption\nassociated with the FMR tail [15, 16], and the domain\nmode resonances[17–25]. Magnetoresistiveabsorption in\nconducting ferromagnetics contributes at all frequencies\nand is most spectacular at the field when magnetization\nchanges abruptly, namely at zero field [26, 27] and upon\ntransition from the unsaturated to saturated state [28–\n32].\nManganites are conducting ferromagnetics with excep-\ntionally high magnetoresistance [33]. Their microwave\nproperties in magnetic field have been extensively stud-\nied and the low-field microwave absorption was found in\nmany compounds. While the dominant contribution in\n∗e-mail michael.golosovsky@mail.huji.ac.il, Permanent a ddress:\nthe Racah Institute ofPhysics, the Hebrew Universityof Jer usalem,\n91904 Jerusalem, Israelbulk manganites arises from magnetoimpedance [14, 34]\nthere are many observations of the low-field microwave\nabsorption in thin films that can’t be explained in this\nway. Some of these observations were attributed to\nmagnetoresistance in the magnetically-unsaturated state\n[8, 35, 36] while the origin of others [14, 32, 37, 38] is\nunclear.\nHere, we studied microwave absorption in high-quality\nepitaxial La 0.7Sr0.3MnO3(LSMO) films in the presence\nof magnetic field and observed the low-field microwave\nabsorption peak with a very peculiar angular depen-\ndence: it appears only in the oblique field. We show\nthat this peak is not related to magnetoresistance but\narises from the ferromagnetic resonance in the multido-\nmain state.\nII. DOMAIN MODE RESONANCE IN A THIN\nFILM WITH ”EASY-PLANE” ANISOTROPY IN\nTHE OBLIQUE MAGNETIC FIELD\nThe theory of the ferromagnetic resonance in the mul-\ntidomain state was originally developed in relation to\nbubble orstripe domainsin ferromagneticfilms with out-\nof-plane anisotropy [17, 19–21]. The few studies of films\nwith in-plane magnetizationwererestrictedto the caseof\nuniaxial anisotropy and parallel field orientation [18, 19].\nHere, we consider a ferromagnetic film with biaxial easy-\nplane anisotropy in oblique field and calculate its high-\nfrequency susceptibility following the approach of Refs.\n[18–20].2\nA. Magnetostatics\nConsider a thin ferromagnetic film in oblique magnetic\nfieldH(Fig.1). Its free energy is\nW=W/bardbl+W⊥+Wdomain+WZeeman+W(M) (1)\nwhereW/bardblandW⊥are the in-plane and out-of-plane\nanisotropy energies, Wdomainis the energy associated\nwith domainwalls andstrayfield ofdomains, WZeemanis\nthe Zeeman energy, and W(M) absorbs all contributions\nthat do not depend on field orientation. We assume a\nstrong easy-plane anisotropy, W⊥=NzM2\nz\n2, and a weak\nin-plane anisotropy, W/bardbl=βM4\nx+M4\ny\n4M2. Here,Mis the sat-\nuration magnetization, Nzis the effective demagnetizing\nfactor that includes both the shape and the crystalline\nanisotropies, x,yare hard axes and βis the (positive)\nin-plane anisotropy constant, whereas β << N z. We\nassume that the external field has only y−andz−com-\nponents, in such a way that the field projection on the\nfilm plane is parallel to one of the in-plane hard axes.\nΨHn\nMΨHn\nMΨH\nnM(a) Saturated state (b) Unsaturated state \n(c) Multidomainstate \nΨ> Ψcr \nHz\nyx\nn(d) Multidomainstate \nΨ< Ψcr \nhmw hmw hmw hmw \nFIG. 1: Schematic drawing of a thin ferromagnetic film with\nbiaxial easy-plane anisotropy in magnetic field Hthat makes\nangle Ψ with the normal to the film n.Mis magnetization,\nH⊥,H/bardblare the perpendicular and parallel anisotropy fields,\nhard axes are along the x−andy−directions. (a) Saturated\nstate. Magnetization is collinear with the field. (b) Unsatu -\nrated state. Magnetization is not collinear with the field an d\ntendstolie in thefilm plane. (c) Multidomain state, Ψ >Ψcr.\nThe film is split into parallel domains with the preferential\norientation of the walls in x-direction (i.e. perpendicular to\nHy). In the presence of a small microwave magnetic field\nhmw, which is perpendicular to the Hfield and has a com-\nponent parallel to domain walls, the domain mode resonance\ncan be excited. (d) Multidomain state, Ψ <Ψcr. The result-\ning domain structure is not clear and we plot only one of the\npossibilities. The film is split into irregular domains with out\npreferential orientation of the walls, in such a way that the\ndomain mode resonance is impeded.\nFigure 1 schematicallyshowsthe magnetic structureof\nthe film when the external field slowly decreases to zero.At high field the film is in the single domain state, the\nmagnetizationliesinthe y−zplane(Fig.1a,b), and Mx=\n0. At lower field the magnetization deviates towards the\nin-plane easy axes (which lie at 450tox−andy-axes)\nandMx/negationslash= 0. The film splits onto parallel domains with\ndomainwallsinthe x−zplane(Fig.1c). Whenthefieldis\nnearly perpendicular to the film, the resulting magnetic\nstructure is not clear. We believe that the film exhibits\nan irregular domain pattern (Fig.1d).\nConsider the structure shown in Fig.1c. Magnetiza-\ntions of the adjacent domains, M1,M2, have the same\nmagnitude but different orientation. The free energy of\na domain is:\nW1=βM4\n1x+M4\n1y\n4M2−(M1yHy+M1zHz)+\nNzM2\n1z\n2+Ny(M1y−M2y)2\n4+λM2\n2(2)\nwhereNy,Nzare the demagnetizing factors in the y−\nandz−directions, M2=M2\n1x+M2\n1y+M2\n1zis the mag-\nnetization, Hy=HsinΨ,Hz=HcosΨ are projections\nof the field onto y−andz−axes, and λis the Lagrange\nmultiplier. The contribution of the domain wall energy\nand stray field of domains was neglected. The effective\nfield is\nHeff\n1=−∂W1\n∂M1=−iM1x/parenleftbigg\nλ+βM2\n1x\nM2/parenrightbigg\n+\nj/bracketleftBigg\nHy−λM1y+βM3\n1y\nM2−Ny\n2(M1y−M2y)/bracketrightBigg\n+k[Hz−(λ+Nz)M1z] (3)\nA similar expression holds for Heff\n2. The equilibrium\ncondition, Heff\n1=Heff\n2= 0, yields magnetization com-\nponentsMeq\n1y=Meq\n2y,Meq\n1x=−Meq\n2xand\nMeq\nx/bracketleftBigg\nλ+β/parenleftbiggMeq\nx\nM/parenrightbigg2/bracketrightBigg\n= 0;\nMeq\ny/bracketleftBigg\nλ+β/parenleftbiggMeq\ny\nM/parenrightbigg2/bracketrightBigg\n=Hy;\nMeq\nz(λ+Nz) =Hz. (4)\nwhere indices 1,2 were dropped. The single domain state\ncorresponds to Meq\nx= 0,λ/negationslash= 0, while the multidomain\nstate corresponds to finite Mxandλ=−β(Meq\nx/M)2.\nTheonset ofthe multidomainstate occurswhen Meq\nx= 0\nandλ= 0. Equation 4 yields the field H0that corre-\nsponds to the onset of the domain state,\n/parenleftbiggH0sinΨ\nH/bardbl/parenrightbigg2\n3\n+/parenleftbiggH0cosΨ\nH⊥/parenrightbigg2\n= 1 (5)\nwhereH/bardbl=βMandH⊥=NzMare the parallel and\nthe perpendicular anisotropy fields. For weak in-plane\nanisotropy, H/bardbl<< H ⊥, Eq.5 yields H0≈H/bardbl/sinΨ for3\nΨ>>Ψcr, andH0≈H⊥for Ψ<<Ψcrwhere Ψ cr=\nH/bardbl/H⊥is the critical angle that delineates between two\nregimes that exhibit different domain patterns.\n•Oblique field, Ψ >Ψcr. Upon decreasing field the\nfilm goes from the magnetically-saturated to the\nunsaturated state when the following condition is\nmet:HcosΨ = Hz≈H⊥. At lower field when\nHsinΨ =Hy=H/bardblthe film goes into multidomain\nstate and splits into parallel domains. The prefer-\nential direction of the domain walls is determined\nby the field projection onto the film plane (Fig.1,c).\n•Nearly perpendicular orientation, Ψ <Ψcr. When\nthe film goes from the magnetically-saturated to\nunsaturated state at HcosΨ =Hz≈H⊥, the sec-\nond condition, Hy=HsinΨ≈H/bardbl, is already met.\nThe in-plane component of the external field at the\nonset of the domain state is too small to impose\npreferential orientation of the domain walls, hence\nthe film most probably splits into irregular domain\npattern (Fig.1d).\nB. Domain mode resonance\nToderivethedynamicresponseofthefilmwithparallel\ndomains to the microwave magnetic field h=heiwtori-entedalongthedomainwalls(Fig.1c)weusetheLandau-\nLifshitz equation without damping\n∂M\n∂t=−γ[M×Heff] (6)\nwhereγis the gyromagnetic ratio. We consider the mi-\ncrowave field as a small perturbation that induces rota-\ntion of magnetization of each domain, in other words we\nassumeM=Meq+mwhereMeqis the static equi-\nlibrium magnetization and mis a small time-dependent\ncontribution. The effective field also acquires small dy-\nnamic contribution, δHeff=∂Heff\n∂Mm+hmw, where\nHeffis given by Eq.3. We linearize Eqs.3,6 and find\nthe dynamic effective field:\nδHeff\n1=i(h−λm1x)−j[(β+λ)m1y+\nNy\n2(m1y−m2y)]−k(λ+Nz)m1z(7)\nA similar expression holds for domain 2. Solution of Eqs.\n6,7 yields two eigenfrequencies\nω2\nl=γ2M2\nxβ/parenleftBigg\nNz5M2\ny−M2\nx\nM2+β6M2\nzM2\ny−2M2\nzM2\nx−5M2\nyM2\nx+M4\nx\nM4/parenrightBigg\n(8)\nω2\nh=γ2M2\nx/bracketleftBigg\nNyNz+β/parenleftBigg\nNy4M2\nz−M2\nx\nM2+Nz5M2\ny−M2\nx\nM2+β6M2\nzM2\ny−2M2\nzM2\nx−5M2\nyM2\nx+M4\nx\nM4/parenrightBigg/bracketrightBigg\n(9)\nSince magnetization in the unsaturated state is almost\nparallel to the film, we can simplify these cumbersome\nexpressions by omitting the terms with Mz. Equations\n8, 9 reduce to:\nω2\nl≈γ2M2\nxβ/parenleftbigg\nNz−βM2\nx\nM2/parenrightbigg/parenleftBigg\n5M2\ny−M2\nx\nM2/parenrightBigg\n(10)\nω2\nh≈γ2M2\nxNy/parenleftbigg\nNz−βM2\nx\nM2/parenrightbigg\n(11)\nBoth eigenfrequencies depend on external field (through\nMx)andgotozeroattheonsetofthedomainstatewhere\nMx= 0. Thelowerfrequency, ωl,almostdoesnotdepend\nonthedomainstructuresinceinthis”acoustic”modethe\nmagnetizations of neighboring domains precess almost inphase. On the other hand, the higher frequency, ωh, de-\npends on the domain shape (through the demagnetizing\nfactorNy) since in this ”optical” mode the magnetiza-\ntionsofneighboringdomainsprecessinantiphaseandthe\nresulting magnetization has poles on the domain walls.\nThe complex susceptibility\nχxx=m1x+m2x\n2h≈γ2M2\ny/parenleftBig\nNz−βM2\nx\nM2/parenrightBig\nω2\nh−ω2(12)\nexhibits resonance at ω=ωh. The resonant frequency\ncan be crudely estimated from Eq.11,\nωh∼γMx/radicalbig\nNyNz. (13)\nSince the domain demagnetizing factors are Nz≈\nw\nd+w,Ny≈d\nd+wwheredis the film thickness and w4\nis the domain width [18], Eq.13 yields ωh≈γMx√\nwd\nw+d\n(we neglected here the contribution of stress and crys-\ntallographic anisotropy to Nz). When the field is de-\ncreased to zero the magnetization in domains rotate (af-\nfectingMx), domain width walso changes, in such a way\nthatωhand, correspondingly, the microwave susceptibil-\nity (Eq.12) can pass through the resonance. For nearly\nperpendicular field orientation, when the film splits into\nirregular domains (Fig.1d) such resonance is impeded.\nAlthough the above model assumes that the field pro-\njection on the film plane is exactly along the hard mag-\nnetization axis (Fig.1) this assumption is probably not\ntoo restrictive. To observe the domain mode resonance\nit is necessary to have a system of parallel domains with\nalternating magnetizationsand the component ofthe mi-\ncrowavemagnetic field parallel to the domain walls. This\nrequirement can be satisfied when the field projection on\nthe film considerably deviates from the hard magnetiza-\ntion axis, especially for the films with biaxial in-plane\nanisotropy.\nTE 102 cavity sample holder \nhmw \nnΨ\nsample \nelectromagnet H\nFIG. 2: Schematic drawing of the sample in magnetic field. A\nsample holder with attached film on substrate is mounted in\nthe center of the 9.4 GHz TE102resonant cavity. The polar\nangle of the magnetic field His varied by sample rotation.\nThe microwave magnetic field hmwis always perpendicular\ntoHand parallel to the film plane.\nIII. EXPERIMENT AND COMPARISON TO\nMODEL\nA. Experimental procedure\nTo measure the low-field microwave absorption associ-\nated with the domain mode resonance we used a bipo-\nlarX-band Bruker ESR spectrometer, a TE102resonant\ncavity, and an Oxford helium flow cryostat (Fig.2). We\nstudied La 0.7Sr0.3MnO3films with thicknesses d=50,\n100, 150, and 200 nm. The films were grown by the\npulsed laser deposition technique on the ∼5×5×1\nmm3(001) SrTiO 3substrates in two different labora-\ntories [39, 40] and have TCof 330 K. The FMR mea-\nsurements in the parallel field showed biaxial (four-fold)in-plane anisotropy of all these epitaxial films. We mea-\nsuredmagnetically-modulatedmicrowaveabsorptionand\ndispersion in 1 ×1 mm2pieces of these films when the\nmagnetic field was swept from large negative to large\npositive values (typically, from -10 kOe to 10 kOe). To\nchange the field orientation with respect to the film we\nrotated the sample whereas the microwavemagnetic field\nwas always parallel to the film plane and perpendicular\nto the dc magnetic field. To find the exact perpendicular\norientation we slightly tilted the sample holder in differ-\nent directions, measured the field of the ferromagnetic\nresonance (FMR), and found the orientation in which\nthe resonant field attained its maximum value. In what\nfollows we focus on the results for the different pieces of\nthe 200 nm thick film. We got similar results for the 100\nnm and 150 nm thick films but not for the 50 nm thick\nfilm. According to our interpretation this difference can\nbe related to the different domain structure in thin films\nwhose thickness is comparable to the domain wall width.\n-2 10 4-1.5 10 4-1 10 4-5 10 30 10 05 10 31 10 41.5 10 42 10 4\n-1 10 4-5000 0 5000 1 10 4\nH(Oe) dχ\"/dH (arb.units) domain mode \nresonance ψ =2 0\nmagnetoresistance FMR SWR \nFIG. 3: Derivative microwave absorption for a 200 nm thick\nLSMOfilmon(001)STOsubstrateinmagneticfielddeviating\nfrom the perpendicular orientation by Ψ = 20. The ferromag-\nnetic (FMR) and the spin-wave resonances (SWR) appear at\n±7.6 kOe, the absorption baseline results from magnetore-\nsistance. Note strong zero-field feature between -2 kOe and\n2 kOe that shows considerable hysteresis. We attribute it to\nthe domain mode resonance.\nB. Low-field microwave absorption at 295 K\nFigure 3 shows derivative microwave absorption in\noblique field for the 200 nm thick film and for two op-\nposite directions of the field sweep. The narrow peaks\nat±7.6 kOe arise from the ferromagnetic and spin-wave\nresonances, the baseline arises from magnetoresistance,\nthe pronounced zero-field feature exhibiting hysteresis is\nattributed to domain mode resonance. This zero-field\nfeature does not depend on the magnitude (1-10 Oe) or\nfrequency of the modulation field (1-100 kHz).5\n-1 10 701 10 72 10 73 10 74 10 7\n-1 10 4-5000 0 5000 1 10 4\nH(Oe) χ  (arb.units) \nχ'χ\" \nx2 FMR magnetoresistance \ndomain mode \nresonance ψ =2 0\nHonset Hending \nFIG. 4: Integrated microwave absorption χ′′and dispersion\nχ′for the film shown in Fig.3. The linear baseline in the ab-\nsorption spectrum results from magnetoresistance while th e\nzero-field peak and the corresponding zero-field dip in the\ndispersion spectrum are attributed to the domain mode reso-\nnance.HonsetandHendingindicate the field range where the\nlow-field absorption appears.\nFigure4showscorrespondingintegratedabsorption χ′′\nanddispersion χ′(detailsoftheintegrationprocedureare\nin Ref. [36]). The absorption baseline linearly increas-\ning with field arises from magnetoresistance (rigorously\nspeaking,frommagnetoconductance)andisabsentinthe\ndispersion spectrum, as expected. The focus of our study\nis a low-field peak in the absorption spectrum and a cor-\nresponding dip in the dispersion spectrum. We attribute\nboth these features to the tail of the domain mode res-\nonance. Indeed, we substitute into Eq.13 w=0.5µm\n[48, 49], d= 200 nm, 4 πMx= 4πM= 4 kG and find\nωh/2π=5.1 GHz. This value is below but not far away\nfrom our operating frequency of 9.4 GHz, hence the tail\nof the resonance should introduce positive contribution\ntoχ′′and negative contribution to χ′(Eq.12,ωh< ω)\nthat conforms to our observations (Fig.4).\nNote that the magnitude of the zero-field absorption\npeak is close to the magnitude of the FMR peak. Since\nthe FMR absorption peak arises from the microwave ab-\nsorption in the whole film, the zero-field absorption peak\nis an intrinsic effect (if it were related to the microwave\nabsorption associated with some defect, its magnitude\nshould be much smaller). Moreover, when we took a film\nwiththesize5 ×5mm2whereandcutittoseveralsmaller\npieces we observed the zero-field absorption peak in each\npiece.02 10 74 10 76 10 78 10 71 10 8\n-1 10 4-5000 0 5000 1 10 4\nH(Oe) χ\"  (arb.units) \n10\n0090\n20305070domain mode \nresonance \nHonsetHending \nFIG. 5: Integrated microwave absorption at different devia-\ntions of the field from the perpendicular orientation. The lo w-\nfield absorption peak appears in the narrow angular range of\n80>|Ψ|>10close to the perpendicular orientation. The\nred dashed line shows the field interval where this low-field\nabsorption appears. This interval narrows with increasing\nΨ. The field corresponding to the maximum absorption also\nshifts with the angle and depends on the direction of the field\nsweep. The curves are vertically shifted for clarity.\n1. Angular dependence\nFigure 5 shows that the low-field absorption peak ap-\npears in the oblique field and disappears when the field\ndeviation from the perpendicular orientation is too large\nor too small. Since the conventional protocol of the\nangular-dependent microwave absorption measurements\nwith the ESR spectrometer starts from Ψ = 00and goes\nwith 100steps, it is not a surprise that the low-field ab-\nsorption peak with such a peculiar angular dependence\nwas overlooked in previous studies.\nFigure 6 shows angular dependence of the maximum\nlow-field absorption and dispersion, χ′\nmax,χ′′\nmax, as well\nas the magnitude of the FMR peak, χ′\nFMR,χ′′\nFMR. In\nthe narrow angular range around perpendicular orienta-\ntion the latter are nearly constant, while the former ap-\npear at|Ψ|<80, grow towards perpendicular orientation\nand suddenly disappear in the even more narrow angular\nrange of |Ψ|<10.\n2. Field dependence\nThe low-field absorption achieves its maximum value\nχ′\nmaxat small but non-zero field, Hcenter, that depends\non the direction of the field sweep (Fig.3) and on the6\nfield orientation (Fig.7). The angular dependence of this\nfield is satisfactorily accounted by the following function\nHcenter≈H0/sinΨ where H0= 7.8 Oe which is close\nto the coercive field. This means that the maximum ab-\nsorption is achieved when the in-plane component of the\nmagnetic field is close to the coercive field that is con-\nsistent with Eq.13 (since Mx≈MatHy=Hcthenωh\nmost closely approaches ω).\n0246810 12 \n-5 0 5 10 Integrated signal (arb.units) \nPolar angle, Ψ (deg.) −χ 'H=0 χ\"FMR \nχ'FMR \nχ\"H=0 parallel domains \nparallel domains irregular domains Ψcr Ψcr \nFIG. 6: Angular dependence of the microwave susceptibil-\nity at zero field, χ′′\nH=0,χ′\nH=0; and at the ferromagnetic res-\nonance, χ′′\nFMR,χ′\nFMR(peak-to-peak). The filled circles stay\nfor absorption, the open circles stay for dispersion. The lo w-\nfield absorption peak appears only at some deviation from\nthe perpendicular orientation when the parallel domains ar e\nexpected (gray area). The absorption peak disappears when\nthe deviation from the perpendicular orientation is less th an\n10(yellow area) when the domain pattern is, presumably, ir-\nregular.\nFigure 8 shows the field range where the low-field ab-\nsorption peak is observed at 295 K. This range has a\nfour-lobed shape with the lobes oriented along the di-\nrections |Ψ|= 2.60. Remarkably, this value coincides\nwith the critical angle that was defined in relation to\nEq.5. Indeed, by substituting into Eq.5 the perpendic-\nular anisotropy field, H⊥= 4 kOe (measured from the\nFMR in the perpendicular geometry), and the parallel\nanisotropy field, H/bardbl= 180 Oe (found from magnetiza-\ntion measurements) we find Ψ cr=H/bardbl/H⊥= 2.60. The\nlobes are within the rectangle defined by the conditions:\nHy< H/bardbl,Hz<0.7H⊥. The former defines the onset of\nthe multidomain state for the nearly parallel field orien-\ntation, while the latter means that the magnetization lies\npredominantly in-plane (more precisely, the deviation of\nthe magnetization from the in-plane orientation does not\nexceed 450).0100 200 300 400 500 600 \n-4 -3 -2 -1 0 1 2 3 4\nPolar angle, Ψ (deg.) Hcenter (Oe) Hcenter =H 0/sin Ψ\nHc=15 Oe \nT=295 K \nFIG. 7: The field at which the maximum zero-field absorption\nappears at T= 295 K. The solid lines show approximation\nby the following dependence, Hcenter=H0/sinΨ, where Ψ is\nthe polar angle and H0= 7.8 Oe.\n3. Comparison to the model\nThe unusual angular dependence of the low-field ab-\nsorption peak (Fig.6) and its hysteretic behavior (Fig.3)\nimply that the peak magnitude strongly depends on the\nfield orientation during magnetic field sweep. How such\ndependence on magnetic history can be qualitatively ex-\nplained by Eqs.11,12? The film has out-of-plane hard\naxis in the z-direction and the in-plane hard axes in the\nx−andy−directions. Figure 8 shows that the low-field\nabsorption appears when the film is in the unsaturated\nstate both with respect to the out-of-plane and in-plane\nhard axes. This state is presumably the multidomain\nstate. Indeed, there are quite a few observations of mag-\nnetic domains in the LSMO films on STO [43–48]. Pho-\ntoelectron emission spectroscopy [46] revealed 3-30 µm\nwideparalleldomains, magneticforcemicroscopyshowed\nmuch smaller, 1 µm wide domains [47], or the checker-\nboard domain pattern with 0.5-0.75 µm wide domains\n[48], and even smaller 0.3 µm wide domains (in LCMO\nthin films) [49]. To explain our results we assume paral-\nlel domains whose width and orientation depend on the\nvalues of HzandHyat the onset of the domain state,\nhence the dependence on magnetic history.\nWe attribute the absorption peaks shown in Figs.4, 5\nto the tail of the domain mode resonance. The peak\nmagnitude is set by the proximity of the resonant fre-\nquencyωhto the operating frequency ω. Dependence\nofωhon magnetic history is captured by the parameter\nNy(Eq.13) which is determined by the domain width w.\nThe latter is found by minimizing the sum of the energy\nassociated with the stray field of domains and the to-\ntal energy of domain walls WDW[41]. The onset of the\nmultidomain state is set by Hywhile the domain wall7\nHy(Oe) H||-4000 -2000 02000 4000 \n-200 -100 0 100 200 Hz(Oe) \nH||Hperp \nHperp field sweep \nΨcr \nHonset Hending irregular domains \nwide \ndomains wide \ndomains irregular domains \nFIG. 8: The range of magnetic fields where the low-field mi-\ncrowave absorption peak appears. The red and purple circles\nstand, correspondingly, for the onset and the ending fields\nat which the low-field absorption peak is observed upon the\nfield sweep from -10 kOe to 10 kOe (see Fig.5). Note different\nhorizontal and vertical scales. H⊥andH/bardblstand for the per-\npendicularandparallel anisotropy fields. Thepale yellowa rea\nindicates the field range where domain structure can appear.\nThe bright yellow area enclosed by the data points shows the\nfield range where the low-field microwave absorption appears .\nenergy and, correspondingly, the domain width are set\nbyHz. The latter can strongly affect the energy and\nstructure of the domain walls and even drive the tran-\nsition from the Bloch to N´ eel wall [41, 42]. When the\nfield is lowered, the domain width changes but due to\npinning some memory of the magnetic conditions at the\nonset of the multidomain state is conserved. Since Hy\nandHzplay very different roles in establishing the do-\nmain state, the domain pattern and the frequency of the\ndomain mode resonance (Eq.13) depend on field orien-\ntation during field sweep, hence the resulting absorption\npeak is strongly angular- and field-dependent. We can\nspeak on ”angle-tuned” domain mode resonance.\nWhen the field is parallel to the film, domain width is\nlargeand ωhis toolowtoproduceappreciableabsorption\natω/2π=9.4 GHz. When the field is close to the per-\npendicular orientation, the domain width is small, ωhap-\nproaches ωand absorption strongly increases. When the\nfield is too close to the perpendicular orientation there\nis no preferential domain wall orientation, domain pat-\ntern is irregular, and the resonance is not excited. Theseconsiderations can be succinctly summarized as follows:\nHz→WDW→w→Ny→ωh→χmax.\nC. Low-field microwave absorption at low\ntemperatures\n05 10 71 10 81.5 10 82 10 8\n-1.5 10 4-1 10 4-5000 0 5000 1 10 41.5 10 400\n10 0\n20 0\n30 0\n40 0\n50 0\n60 0\n70 0\n80 0\n90 0\nH(Oe) χ\"  (arb.units) perpendicular \nparallelT=4.5 K \nFMR \nLFA \nFIG. 9: Integrated absorption for the sample of the Fig.5 at\n4.5 K at different field orientations. The purple arrows indi-\ncate FMR peaks, the red arrows indicate low-field absorption\npeaks associated with the domain mode resonance. The FMR\npeaks appear in all orientations while the domain mode res-\nonance appears only in oblique orientation and disappears\nboth in the parallel and perpendicular orientations. Unlik e\nFig.5 where at each field orientation there is a single low-fie ld\nabsorption peak, here there are two overlapping peaks.\nTable I demonstrates that the maximum low-field ab-\nsorption χ′′\nmax(achieved at different angles for each tem-\nperature) does not disappear at low temperatures and\neven increases. This stays in sharp contrast with the\nmicrowave losses associated with magnetoconductance\n(magnetoresistance), whose magnitude for our highly\nconducting films dramatically decreases at low temper-\natures. This proves once again that the low-field mi-\ncrowave absorption is unrelated to magnetoresistance.\nWe observe also that χ′′\nmaxis on the same order as the\nFMR absorption peak in the parallel geometry. This is\nnot surprising since according to our model the zero-field\nabsorption peak is nothing else but the FMR absorption\npeakinthe parallelgeometrywhichisshifted tozerofield\ndue to dynamic demagnetization factor associated with\nmagnetic domains.8\nTABLE I: Comparison of differentcontributions tomicrowave\nabsorption. (The units are arbitrary but the same for all\nentries).\nT(K) χ′′\nH=0aχ′′\nFMRbχ′′\nMRc\n295 3.3 2.5 31\n270 3 1 12\n220 2.5 3.4 6.2\n110 4.3 4 2.8\n4.5 8.8 2.6 <0.3\naThe maximum low-field microwave absorption (at correspondi ng\nangle).\nbThe magnitude of the FMR absorption peak in parallel field.\ncNonresonant microwave lossesassociated with magnetoresi stance\nin parallel field, χ′′\nMR=χ′′\nH=10kOe−χ′′\nH=0.\nThe temperature dependence of the low-field absorp-\ntion is dictated mostly by ωhand its proximity to the\noperating frequency ω(Eq.13). The low-field absorption\nat 295 K arises from the tail of the domain mode reso-\nnance since ωh< ω. To estimate ωhat 4.5 K we note\nthatat4.5K4 πM=7.4kG(saturationmagnetizationof\nLSMO). We substitute this value and d= 200 nm, and\nw=0.5µm into Eq.13 and find ωh/2π=12.2 GHz, in\nsuch a way that ωh> ω. This means that the maximum\nabsorption is achieved at resonant field that satisfies the\ncondition ωh=ω. Upon field sweep across the zero this\ncondition is met twice- once for negative and once for\npositive field.\nFigure9showsthelow-temperaturemicrowaveabsorp-\ntion spectraat different angles. In contrastto Fig.5 thereare two low-field absorption peaks: one at negative and\none at positive field. There are other differences with\nrespect to Fig.5 as well. The baseline in Fig.9 is flat indi-\ncating negligible contribution from magnetoresistance, in\nsuch a way that in the parallel orientation the absorption\nspectrum exhibits only the FMR peak. When the field is\nrotated away from the parallel orientation there appears\na low-field absorptionpeak that becomes a dominant fea-\nture at Ψ = 50−200. This peak disappears again when\nthe deviation from the perpendicular orientation is less\nthan 10.\nIV. CONCLUSIONS\nIn thin La 0.7Sr0.3MnO3films with easy-plane\nanisotropy there is a special kind of the domain mode\nresonance that is absent in the parallel and perpendic-\nular field orientation and appears only in the oblique\nmagnetic field. It can be ”angle-tuned” by varying the\nfield orientation with respect to the film. The resonance\nin oblique field should not be unique to manganites and\nit can be a source of pronounced low-field microwave\nabsorption in other magnetic materials as well.\nV. ACKNOWLEDGMENTS\nWe are grateful to S. Mercone, S.V. Kapel’nitskii, and\nK.N. Rozanov for stimulating discussions and construc-\ntive comments.\n[1] R. C. Haddon, S. H. Glarum, S. V. Chichester, A. P.\nRamirez, and N. M. 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B 64, 172407 (2001)."    },    {        "title": "1810.01606v1.Selective_Activation_of_an_Isolated_Magnetic_Skyrmion_in_a_Ferromagnet_with_Microwave_Electric_Fields.pdf",        "content": "Selective Activation of an Isolated Magnetic Skyrmion in a Ferromagnet with\nMicrowave Electric Fields\nAkihito Takeuchi1,a)and Masahito Mochizuki2, 3,b)\n1)Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 252-5258,\nJapan\n2)Department of Applied Physics, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555,\nJapan\n3)PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012,\nJapan\nWe theoretically reveal that pure eigenmodes of an isolated magnetic skyrmion embedded in a ferromagnetic\nenvironment can be selectively activated using microwave electric \felds without exciting gigantic ferromagnetic\nresonances, in contrast to conventional methods using microwave magnetic \felds. We also demonstrate that\nthis selective activation of a skyrmion can e\u000eciently drive its translational motion in a ferromagnetic nanotrack\nunder application of an external magnetic \feld inclined from the normal direction. We \fnd that a mode with\ncombined breathing and rotational oscillations induces much faster skyrmion propagation than the breathing\nmode studied previously by Wang et al: [Phys. Rev. B 92, 020403(R) (2015)].\nPACS numbers: 76.50.+g,78.20.Ls,78.20.Bh,78.70.Gq\nA skyrmion crystal, a hexagonally crystalized state\nof magnetic skyrmions1{6, has characteristic resonance\nmodes at microwave frequencies7{10, which give rise to\nintriguing physical phenomena11such as microwave di-\nrectional dichroism12{15, spin-voltage induction16,17, and\nspin-current generation18. When a static magnetic \feld\nHexis applied perpendicular to a thin-plate specimen\nof the skyrmion-hosting magnet, several types of spin-\nwave modes emerge depending on the microwave po-\nlarization7. A microwave magnetic \feld H!normal to\nHz\nHxHex\n/g84\nxyz\nD0+/g39D(t)~\nmiz-1         0         1Csi\n-0.01     0      0.01 -0.700.7\n-0.010-0.005-0.002(a) (b)\n(c) (e) (d) (f)24 sites24 sites/g84=0˃ /g84=30˃ /g84=0˃ /g84=30˃\nFIG. 1. (color online). (a) Schematics of a magnetic bi-\nlayer system hosting skyrmions stabilized by the interfacial\nDzyaloshinskii-Moriya interaction. (b) External magnetic\n\feldHexwhere\u0012is an inclination angle from the normal\ndirection. (c), (d) Color maps of the normal component of\nmagnetizations mz(c) and scalar spin chiralities cs(d) of a\nskyrmion under a perpendicular Hex\feld. In-plane compo-\nnents of the magnetizations ( mx;my) are displayed by arrows.\n(e), (f) Those under an inclined Hex\feld with\u0012=30\u000e.\na)Electronic mail: akihito@phys.aoyama.ac.jp\nb)Electronic mail: masa mochizuki@waseda.jpthe skyrmion plane ( H!\n?) activates the so-called breath-\ning mode in which all the skyrmions constituting the\nskyrmion crystal uniformly expand and shrink in an os-\ncillatory manner. On the other hand, the H!\feld within\nthe skyrmion plane ( H!\nk) activates two types of rotation\nmodes with opposite rotational senses, in which cores of\nall the skyrmions circulate uniformly in counterclockwise\nand clockwise ways.\nIn addition to the crystallized form, skyrmions can\nappear as individual defects in a ferromagnetic state;\nsuch skyrmions are also expected to have peculiar col-\nlective modes19. Isolated skyrmions con\fned in a nano-\nferromagnet are potentially useful for applications20to\nmemory devices21, magnonics devices22{24, spin-torque\noscillators25,26, and microwave sensing devices27. As\nsuch, it is necessary to clarify the microwave-active eigen-\nmodes of a single skyrmion in a ferromagnetic specimen.\nIn addition, it is important to establish a way to ma-\nnipulate isolated skyrmions using microwaves for their\ndevice application. As the microwave \feld H!\n?cannot\nactivate precessions of the magnetizations when the mi-\ncrowave \feld is applied parallel to them, we can activate\npure breathing-type skyrmion oscillations with H!\n?un-\nder a perpendicular Hex\feld without exciting the back-\nground ferromagnetic state. However, once the Hex\feld\nis inclined, the microwave magnetic \feld H!\n?excites huge\nferromagnetic resonances, which drown out the weaker\nskyrmion resonances. Moreover, the microwave energy is\nabsorbed by the sample when exciting the gigantic fer-\nromagnetic resonances, which would inevitably result in\nhigh energy consumption and considerable temperature\nrise. Therefore, a technique to selectively activate an iso-\nlated skyrmion is urgently required.\nIn this Letter, we \frst theoretically show that the\neigenmodes of an isolated skyrmion embedded in a ferro-\nmagnetic environment can be selectively activated with a\nmicrowave electric \feld E!via oscillatory modulation of\nthe Dzyaloshinskii-Moriya interaction (DMI) without ac-arXiv:1810.01606v1  [cond-mat.mes-hall]  3 Oct 20182\ntivating ferromagnetic resonances. We then demonstrate\nthat translational motion of the skyrmion can be driven\nthrough activating its resonance modes using this electric\ntechnique in an inclined Hex\feld. The latter part of the\nresearch was motivated by the recent theoretical work\nby Wangetal: that demonstrated skyrmion propagation\nvia activating the breathing mode with a microwave H!\n?\n\feld28. Our study reveals that skyrmion motion can be\ndriven not only by the breathing mode but also by other\nE!-active modes. Furthermore, we \fnd that a mode with\ncombined clockwise and breathing oscillations can induce\nmuch faster propagation of the skyrmion than the previ-\nously studied breathing mode. Our \fndings pave a new\nway toward e\u000ecient manipulation of isolated skyrmions\nin nano-devices via the application of microwaves.\nWe consider a magnetic bilayer system composed of a\nferromagnetic layer and a heavy-metal layer with strong\nspin-orbit interactions [see Fig. 1(a)], where the spa-\ntial inversion symmetry is broken at their interface, and\nthereby the DMI is active. An inclined magnetic \feld\nHex= (Hx;0;Hz) withHx=Hztan\u0012is applied where \u0012\nis the inclination angle [see Fig. 1(b)]. For 0\u000e<\u0012\u001490\u000e,\ntheHex\feld is inclined toward the positive xdirection.\nThe DMI favors a rotating alignment of the magneti-\nzations, which results in the formation of a Neel-type\nskyrmion. The skyrmion has a circular symmetry un-\nder a perpendicular Hex\feld (\u0012= 0\u000e), but has dis-\nproportionate weight in distributions of the magnetiza-\ntions and scalar spin chiralities [see Fig. 1(c) and (d)].\nTo describe the magnetism in this magnetic bilayer sys-\ntem, we employ a classical Heisenberg model on a square\nlattice with magnetization vectors miwhose norm mis\nunity29,30. The Hamiltonian contains the ferromagnetic-\nexchange interaction, the Zeeman coupling to the mag-\nnetic \felds, and the interfacial DMI:\nH=\u0000JX\n<i;j>mi\u0001mj\u0000[Hex+H(t)]\u0001X\nimi\n+D(t)X\ni[(mi\u0002mi+^x)\u0001^y\u0000(mi\u0002mi+^y)\u0001^x]:(1)\nHereH(t) = (0;0;Hz(t)) and E(t) = (0;0;Ez(t)) rep-\nresent time-dependent magnetic and electric \felds ap-\nplied perpendicular to the sample plane, respectively.\nWe neglect magnetic anisotropies because they do not\nalter the results qualitatively although stability of the\nskyrmions and resonant frequencies of the eigenmodes\nmay be slightly changed. The strength of the interfacial\nDMI can be tuned by applying a gate electric \feld normal\nto the plane via varying the extent of the spatial inversion\nasymmetry31,32. The DMI coe\u000ecient D(t) =D0+\u0001D(t)\nhas two components, namely, a steady component D0and\naE(t)-dependent component \u0001 D(t) =\u0014Ez(t) with\u0014be-\ning the coupling constant. We take J=1 for the energy\nunits and take D0=J=0.27. For the inclined magnetic\n\feld, we take Hz=0.057 with \u0012being a variable. The\nunit conversions when J=1 meV are summarized in Ta-\nble I.Exchange int. J= 1 1 meV\nTime t= 1 0.66 ps\nFrequencyf=!=2\u0019!= 0:01 2.41 GHz\nMagnetic \feld H= 1 8.64 T\nTABLE I. Unit conversion table when J=1 meV.\nWe simulate the magnetization dynamics by numeri-\ncally solving the Landau-Lifshitz-Gilbert equation using\nthe fourth-order Runge-Kutta method. The equation is\ngiven by,\ndmi\ndt=\u0000\rmmi\u0002He\u000b\ni+\u000bG\nmmi\u0002dmi\ndt: (2)\nHere\u000bG(=0.04) and \rmare the Gilbert-damping con-\nstant and the gyrotropic ratio, respectively. The e\u000bective\n\feldHe\u000b\niis calculated as He\u000b\ni=\u0000(1=\rm\u0016h)@H=@mi.\nWe \frst calculate the dynamical electromagnetic and\nmagnetic susceptibilities \u001femand\u001fmm:\n\u001fem(!) =r\u00160\n\u000f0\u0001M!\nz\nEpulse; \u001fmm(!) =\u0001M!\nz\n\u00160Hpulse:(3)\nAfter applying a short pulse Hz(t) orEz(t) with dura-\ntion of \u0001t=1 in the units of J=\u0016h, we trace time pro\fles\nof the net magnetization Mz(t) = (1=N)P\nimzi(t) and\n\u0001Mz(t) =Mz(t)\u0000Mz(0) and obtain the Fourier trans-\nform \u0001M!\nz. Dividing this quantity by an amplitude of\nthe pulseHpulse orEpulse, we obtain these susceptibili-\nties. The calculations are performed using a system of\nN=160\u0002160 sites with periodic boundary conditions.\nFigure 2(a) displays the imaginary parts of the electro-\nmagnetic susceptibilities Im \u001fem(!) for several values of\n\u0012, which describe the response of the magnetizations to\ntheE!\feld. When \u0012= 0\u000e, the spectrum exhibits a sin-\ngle peak corresponding to the breathing mode activated\nby the oscillating DMI under the application of an AC\nelectric \feld. When the Hex\feld is inclined with \u00126= 0\u000e,\ntwo novel modes emerge, one with a higher and one with\na lower frequency than the breathing mode. The inten-\nsities of the novel modes increase, whereas the intensity\nof the original breathing mode is increasingly suppressed\nas\u0012increases.\nThe imaginary parts of the magnetic susceptibilities\nIm\u001fmm(!) in Fig. 2(b) describe the response of the mag-\nnetizations to the H!\feld. We \fnd that only a breath-\ning mode (m-Mode 2) appears when the Hex\feld is\nperpendicular ( \u0012= 0\u000e); however, its intensity decreases\nas\u0012increases. Remarkably, a large ferromagnetic reso-\nnance from the surrouding ferromagnetic magnetizations\nemerges under an inclined Hex\feld, whereas it is totally\nsilent under the perpendicular Hex\feld. We also \fnd a\nnovel mode (m-Mode 1) at lower frequencies.\nIn reality, the skyrmion has another H!-active mode\n(m-Mode 3) at higher frequencies, but it is hidden be-\nhind the gigantic ferromagnetic resonance and thus can-\nnot be seen in the spectra of Im \u001fmm(!). We can see this3\n0 0.04 0.0800.0050.010.015\n00.010.02\n0 0.04 0.080 0.04 0.08/g84=0˃\n/g90/g90Im/g70mmIm/g70em(arb. units)\nIm/g70mc(arb. units)\n00.010.02 Im/g70mm(arb. units)\n/g84=30˃/g84=25˃/g84=15˃/g84=10˃\n/g84=20˃/g84=0˃\n/g84=30˃/g84=25˃/g84=15˃/g84=10˃\n/g84=20˃/g84=0˃\n/g84=30˃/g84=25˃/g84=15˃/g84=10˃\n/g84=20˃\n/g90(a) (b) (c)\nIm/g70emIm/g70mc\ne-Mode 1e-Mode 2\ne-Mode 3m-Mode 3m-Mode2\nm-Mode1m-Mode 2\nm-Mode 1\nFMR/g68/g68=x\n/g68=y\ntotal\nE/g90||zHz\nHxH\n/g84\nxyz\nH/g90||zHxH\n/g84\nxyzHz\nFIG. 2. (color online). Imaginary parts of (a) the electromagnetic susceptibility Im \u001fem(!), (b) the magnetic susceptibility\nIm\u001fmm(!), and (c) the chirality susceptibility Im \u001fmc\n\u000b(!) for several values of \u0012. Here, an inclined magnetic \feld Hex=\n(Hztan\u0012;0;Hz) withHz=0.057 is applied. Three E!-active modes are labeled as e-Modes 1-3 in (a), whereas the three H!-\nactive modes are labeled as m-Modes 1-3 in (b) and (c). The extremely intense mode around !\u00180:05\u00000:06 in (b) is the\nferromagnetic resonance (FMR).\nweak response of the skyrmion to the H!\feld by fo-\ncusing on the vector spin chiralities, si=P\n\rmi\u0002mi+\r\n(\r= ^x;^y). The calculated imaginary parts of the dynam-\nical susceptibilities Im \u001fmc\n\u000b(!) for the\u000b-component of the\nvector spin chirality S\u000b= (1=N)P\nis\u000bi(\u000b=x;y) are\nshown in Fig. 2(c). We \fnd that they coincide with the\nspectra of Im \u001fem(!) in Fig. 2(a). These results indicate\nthat the magnetic method using H!cannot selectively\nactivate the eigenmodes of an isolated skyrmion in the\nferromagnetic specimen; however, the results show that\nthe electrical method using E!can achieve this. This\nelectrical technique is anticipated to play a crucial role\nfor developing future skyrmion-based devices.\nBased on the susceptibility data, we \fnd that an iso-\nlated skyrmion in a ferromagnetic specimen has three\nlow-lying eigenmodes. In Fig. 3, we show simulation re-\nsults of snapshots for \u0012= 30\u000e. It is found that all of\nthese modes have the breathing component, i.e., they\nshow oscillatory expansion and shrinkage. Among these\nthree modes, the higher-frequency mode ( !=0.0664) can\nbe regarded as a pure breathing mode, whereas the other\ntwo modes show distinct behaviors. The lower-frequency\nmode (!= 0:0453) is accompanied by a clockwise rota-\ntion of skyrmion in an elliptical orbit oriented horizon-\ntally against the inclined direction of Hexas shown in\nFig. 3(a). The moderate-frequency mode ( != 0:0513)\nis also accompanied by the clockwise rotation of the\nskyrmion in an elliptical orbit, but its trajectory is ori-\nented vertically against the inclined direction of Hexas\nshown in Fig. 3(b). The higher-lying pure breathing\nmode at!=0.0664 does not show any rotational compo-\nnent [Fig. 3(c)]. The three E!-active modes are referred\n123\n4\n1\n234\n1234\nxy25 sites25 sitest=2030 t=2064 t=2098 t=2132(a) e-Mode 1: CW + Breathing 1 ( /g90=0.0453)\n(b) e-Mode 2: CW + Breathing 2 ( /g90=0.0513)\n(c) e-Mode 3: Breathing ( /g90=0.0664)t=2053 t=2083 t=2112 t=2142\nt=2057 t=2081 t=2105 t=2129\n-1 1 mizCW\nrotation+\nrotation+\nCW\nbreathingbreathingbreathingt - t0Mz2/g83//g90\n12\n34FIG. 3. (color online). Simulated snapshots of the magnetiza-\ntion dynamics for three E!-active eigenmodes (e-Modes 1-3)\nof an isolated skyrmion in the ferromagnetic specimen under\nan inclined Hex\feld where Hz=0.057 and \u0012= 30\u000e.\nto as e-Modes 1, 2, and, 3 [see Fig. 2(a) and Fig 3].\nNext we numerically investigate the translational mo-\ntion of a skyrmion driven by the electrically activated res-\nonance modes under an inclined Hex\feld [see Fig. 4(a)].\nHere the inclination angle of Hexis \fxed at \u0012= 30\u000e.\nThe amplitude of the AC electric \feld Ez(t) =E!\nzsin!t\nis \fxed at \u0014E!\nz= 0:05D0= 0:0135. A recent ex-\nperiment for Ta/FeCoB/TaO xreported a huge E-\feld-\ninduced variation of the interfacial DMI that reaches\n140% when the applied voltage is 10 V. This observation\nsupports the experimental feasibility of the 5% modula-4\n-4 -3 -2 -1 0(/g90=0.0453)-8-6-4-20\n0 0.004 0.008 0.012/g90=0.0513/g90=0.0664\n/g90=0.0453/g90=0.0664\n/g90=0.0513-50\n-11 0 0.04 0.0800.005\n/g84=30°y coordinate\nx coordinate/g90\n0\n-11y coordinate-4 -3 -2 -1 0 -5\n0 0.04 0.0800.01\n/g84=30°\n/g90E/g90 activation\n(trajectory)\n/g78Ez/g90Displacement in x\n0 0.001 0.002 0.003-202Disp. in y\n-8-6-4-20Displacement in x\n-202Disp. in y\nHz/g90Im/g70em\nH/g90 activation\n(trajectory)E/g90 activation\n(drift velocity)\nH/g90 activation\n(drift velocity)(c)\nIm/g70mm(d)\n(e)\n(f)(g)\n(h)vx (m/s)-0.2\n-0.4\n-0.6\n-0.80vx (m/s)-0.2\n-0.4\n-0.6\n-0.80-0.200.2vy (m/s)\n-0.200.2vy (m/s)\n36 sites\n36 sitest=0 t=8ʷ103t=24ʷ103\nxy\ne-Mode 1\ne-Mode 3\ne-Mode 2e-Mode 3e-Mode 2e-Mode 1\nm-Mode 1\nm-Mode 3m-Mode 2m-Mode 3 + FMR\nm-Mode 2m-Mode 1\n~\nxyzHz\nHxHex\n/g84(b)\nmz\n-1         0         1(a)\nFIG. 4. (color online). (a) Illustration of skyrmion propa-\ngation driven by the E!\feld through activating a resonant\nmode of the skyrmion under an inclined Hex\feld. (b) Simu-\nlated snapshots of the skyrmion propagation for the E!-active\nmode with !=0.0513 (e-Mode 2). (c)-(e) Trajectories (c) and\ndrift velocities vx(d) andvy(e) of the propagating skyrmion\ndriven by the E!\feld for three di\u000berent resonant modes.\n(f)-(h) Those of the propagating skyrmion driven by the H!\n\feld. All the simulations were performed with an inclined\nHex\feld where Hz=0.057 and \u0012= 30\u000e.\ntion assumed here. Figure 4(b) shows simulated snap-\nshots of the skyrmion motion when the E!\feld with\n!=0.0513 is applied, which indeed displays propagation\nin the negative xdirection.\nThe trajectories of the propagating skyrmion during a\ntime period from t=0 tot=5000 are shown in Fig. 4(c) for\nthree di\u000berent E!-active modes at !=0.0453, 0.0513, and\n0.0664. They were obtained by tracing the center-of-mass\ncoordinate ( jx;jy) of the topological-charge distribution:\nj\r=X\ni=(ix;iy)i\rcs(ix;iy)=X\ni=(ix;iy)cs(ix;iy) (4)with\ncs=1\n8\u0019[mi\u0001(mi+^x\u0002mi+^y) +mi\u0001(mi\u0000^x\u0002mi\u0000^y)]:\n(5)\nWe \fnd that the direction and velocity of the motion\nvary depending on the skyrmion resonance mode. For\nall the modes, the skyrmion moves mainly in the nega-\ntivexdirection. However, the trajectories for e-Modes\n2 and 3 are slightly slanted toward the negative ydirec-\ntion; meanwhile, the trajectory for e-Mode 1 is perfectly\nparallel to the xaxis. Interestingly, despite the slanted\ntrajectory for e-Mode 2, its traveling distance along the\nxaxis is identical to that of the trajectory for e-Mode 1,\nwhich is directed perfectly along the xaxis. It can also\nbe seen that the directions of the skyrmion movement for\ne-Modes 2 and 3 are the same, even though their trav-\neling distances are di\u000berent. The traveling distance for\ne-Mode 3 is much shorter than that for e-Mode 2 because\nof the smaller intensity of the latter mode, as can be seen\nin the inset of Fig. 4(c).\nIn Fig. 4(d) and (e), we plot the velocities v=(vx;vy)\nof the skyrmion for three di\u000berent E!-active modes (see\nthe right vertical axes) as functions of the strength of the\ntime-dependent DMI, i.e., \u0014E!\nz. The values are calcu-\nlated from the simulated displacements of the skyrmion\nin thexandydirections for a time period from t=2000\ntot=10000 (see the left vertical axes) by assuming J= 1\nmeV anda=5\u0017A withabeing the lattice constant. It can\nbe seen that the velocities are proportional to the square\nof\u0014E!\nz, and they are on the order of 10\u00001m/s. In fact,\nthe traveling speed of the skyrmions achieved using this\ntechnique turns out to be relatively slow compared with\nthe speed achieved in techniques based on electric-current\ninjection33{40. However, the present technique has an ad-\nvantage: it is free from the Joule heating, and thus the\nenergy consumption and the temperature rise could be\nsigni\fcantly suppressed.\nWe \fnally study the skyrmion motion driven by AC\nmagnetic \felds H!under an inclined Hex\feld with\nHz=0.057 and \u0012= 30\u000e. The H!is applied perpendicular\nto the skyrmion plane, which is given by H!\nzsin!twith\nH!\nz= 0:05Hz= 0:00285. The simulated trajectories\nand velocities v=(vx;vy) are shown in Fig. 4(f) and (g),\nrespectively. We \fnd that the trajectories are again ori-\nented almost in the negative xdirection; however, for the\nferromagnetic resonance mode with !=0.0664 the trajec-\ntory is slanted toward the positive ydirection, which con-\ntrasts with the case of the E!-active mode. Noticeably,\nthe higher-frequency mode with !=0.0664 has a much\nfaster propagation of the skyrmion than the other two\nmodes. However, the usage of this mode is not ener-\ngetically e\u000ecient because this mode is not an eigenmode\nof the isolated skyrmion but a very intense resonance of\nthe vast ferromagnetic magnetizations, which necessarily\nleads to large energy consumption and considerable rise\nof device temperatures.\nIn summary, we have theoretically found that reso-\nnance modes of an isolated sykrmion in a ferromagnet can5\nbe activated by application of AC electric \felds through\noscillatory variation of the interfacial DMI. The advan-\ntage of this method compared with conventional methods\nusing AC magnetic \felds is that we can selectively excite\nskyrmions without activating gigantic ferromagnetic res-\nonances; this results in a signi\fcant suppression of both\nenergy consumption and temperature rise. Our result re-\nvealed that among the three E!-active modes, the mode\nwith combined clockwise and breathing oscillations in-\nduces much faster skyrmion propagation than the pre-\nviously studied breathing mode. Our \fndings will pave\na new way toward the e\u000ecient manipulation of isolated\nskyrmions and thus will be useful for future skyrmion-\nbased devices.\nThis work was supported by JSPS KAKENHI (Grant\nNo. 17H02924), Waseda University Grant for Special Re-\nsearch Projects (Project Nos. 2017S-101, 2018K-257),\nand JST PRESTO (Grant No. JPMJPR132A).\n1A. N. Bogdanov and D.A. Yablonskii, Sov. Phys. JETP 68, 101\n(1989).\n2A. Bogdanov and A. Hubert, J. Mag. Mag. Mat. 138, 255 (1994).\n3S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer, A. Rosch, A.\nNeubauer, R. Georgii, and P. B oni, Science 323, 915 (2009).\n4X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y.\nMatsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010).\n5N. Nagaosa and Y. Tokura, Nat. 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Phys. 13,\n162 (2017)."    },    {        "title": "1906.11688v2.Magnetic_proximity_effect_induced_FMR_frequency_enhancement_in__Py_FeMn__bilayers.pdf",        "content": "arXiv:1906.11688v2  [cond-mat.mes-hall]  27 Sep 2020Magnetic proximity effect induced FMR frequency enhancemen t in Py/FeMn bilayers\nD. M. Polishchuk,1,2,∗T. I. Polek,2V. Yu. Borynskyi,2A. F. Kravets,2A. I. Tovstolytkin,2and V. Korenivski1\n1Nanostructure Physics, Royal Institute of Technology, 106 91 Stockholm, Sweden\n2Institute of Magnetism, NASU and MESU, 03142 Kyiv, Ukraine\n(Dated: September 29, 2020)\nFerromagnetic resonance (FMR) in exchange-coupled ferrom agnet-antiferromagnet (FM/AFM)\nbilayers commonly shows a moderate increase in the resonanc e frequency owing to the induced\nunidirectional anisotropy. Here we report a large FMR frequ ency enhancement toward the sub-THz\nrange observed in Py/FeMn with ultrathin AFM FeMn. The effect is connected with a sizable\ninduced magnetic moment in FeMn caused by the magnetic proxi mity effect from the Py layer. The\nobserved FMR properties are explained as due to the competin g intrinsic antiferromagnetic order\nand the ferromagnetic proximity effect in nanometer thin FeM n. Our results show that combining\nmaterials with strong and weak anti/ferromagnetic orderin g can potentially close the notoriously\ndifficult GHz-THz gap important for high-speed spintronic ap plications.\nThe significant difference in the magnetization dynam-\nics of ferromagnetic (FM) and antiferromagnetic (AFM)\nmaterials leads to a wide gap between the respective res-\nonance frequencies, reachingseveralordersofmagnitude.\nThis frequency gap is due to drastically different intrin-\nsic fields acting on the spins in FMs and AFMs under\nFMR1. The spins in FMs are in the effective field of the\nmagnetic anisotropy, which is usually not larger than a\nkG (100 mT) for ferromagnetic 3d-metals and their al-\nloys2. On the contrary, the spins belonging to one of\nthe FM spin sublattices in an AFM material experience\na strong exchange field ( /greaterorsimilar100 T) from the other (an-\ntiparallel) sublattice. This 1000-fold factor between the\nintrinsic effective fields governing the FMR in FMs and\nAFMs leads to a corresponding GHz-THz frequency gap.\nThe idea to enhance the ferromagnetic-resonance\n(FMR) frequencies by combining FM and AFM mate-\nrials, e.g. in thin-film multilayers3–11, meets difficul-\nties in practice, which arise from typically weak ex-\nchange coupling at FM/AFM interfaces2. This problem\nis well known and well-studied in relation to exchange\nbias12,13widely used in spintronic applications for cre-\nating the so-called exchange pinning of FM layers. The\nlatter manifests as relatively weak unidirectional mag-\nnetic anisotropy (typically <1 kG), which can offer only\na weak enhancement of the FMR frequency.\nIn this work, we report a several-fold FMR frequency\nenhancement due to coupled magnetization dynamics in\nthin-film bilayersof Ni 80Fe20/Fe50Mn50(Py/FeMn). De-\nspite the fact that Py/FeMn is one of the most studied\nexchange-bias systems14–17, the magnetization dynam-\nics of this and similar bilayers, specifically near the N´ eel\ntemperature of the AFM layer, TN, had received little\nattention, since the focus with exchange-pinning is on\nthe relevant magnetic properties at T≪TN. The main\nthesis of this work is that the magnetization dynamics\nof Py/FeMn should be strongly modified near TNdue\nto the pronounced magnetic proximity effect of Py on\nFeMn. When approaching TNfrom bellow, AFM order-\ning in FeMn becomes weaker, while the exchange-field\nfrom Py penetrates into FeMn and re-aligns the AFM\nspins ferromagnetically. Such magnetic proximity effecthas been observed for Ni/FeMn/Co trilayers and shown\nto decrease TNof the FeMn layer18. The strength of the\nproximity-exchange is known to drop exponentially with\ndepth into adjacent weakly magnetic layers19,20, making\nthe effect most pronounced for nanometer thin layers.\nOn the other hand, FeMn has an fcc lattice with random\nchemical occupation of the sites and a noncollinear spin\nstructure21, which together can result in an uncompen-\nsated magnetic moment in a thin film22–24. Since the\nspin structure of FeMn in Py/FeMn is aligned with the\nFM moment (of Py), M,25,26one can assume that the\nproximity effect can induce and stabilize a non-zero net\nmagneticmoment, Mb, in thin FeMn layers(Fig. 1). The\ndynamic interplay between MandMbcan then lead to\ninteresting magnetization dynamics. We indeed observe\na 3-fold enhancement of the FMR frequency in Py/FeMn\nbilayers with an ultrathin FeMn near its effective TN.\nSamples and Experiment.— A series of multilayers\nTa(5)/Py(5)/FeMn( t= 0, 3, 5 and 7 nm)/Al(4) were\ndeposited on thermally oxidized Si substrates using mag-\nnetron sputtering. To induce exchange pinning, a mag-\nnetic field of /lessorsimilar1 kG was applied in the film plane dur-\ning deposition. The FMR measurements were carried\nout in a temperature interval of 200–320 K using an X-\nband ELEXSYS E500 spectrometer (Bruker) at a con-\nstant operating frequency of 9.46 GHz. Additionally,\nbroad-band FMR was performed at room temperature\nin the frequency range of 9–20 GHz. The FMR spectra\nwere measured for varying in-plane orientation of the ex-\nternal field and using reverse field sweeping from 5 kG to\nzero. The hysteresis loops were obtained using vibrating-\nsample (VSM) and longitudinal magneto-optical Kerr-\neffect (MOKE) magnetometry. Temperature-dependent\nmeasurementswereperformedaftercoolinginamagnetic\nfield applied along the pinning direction.\nRoom-Temperature Magnetometry.— The magneto-\nstatic measurements indicate pronounced changes in the\nmagnetic state of the Py/FeMn( t) bilayers as a function\noftheFeMn thickness, t, atroomtemperature; Fig. 1(b)–\n(d). The enhanced coercivity field, Hc, manifests the on-\nset of antiferromagnetism in FeMn already for t= 3 nm,\nwhereas the loop offset observed for t≥5 nm (non-zero2\n/s48 /s49 /s48 /s48 /s50 /s48 /s48 \n/s40 /s98 /s41 \n/s40 /s99/s41 \n/s72\n/s99 /s72\n/s98 /s44 /s32 /s72\n/s99 /s32 /s40 /s71/s41 /s72\n/s98 \n/s40 /s97 /s41 \n/s48 /s50 /s52 /s54 /s56 /s48 /s46 /s56 /s53 /s49 /s46 /s48 /s48 /s49 /s46 /s49 /s53 \n/s116 /s32 /s40 /s110 /s109 /s41 /s40 /s100 /s41 /s77\n/s115 /s47 /s77/s32 /s114 /s101 /s102 /s80 /s121 /s115 FIG. 1. (a) Schematic represen-\ntation of FMR-induced coupled mag-\nnetization dynamics in an exchange-\ncoupled system “strong ferromagnet\n/ weak antiferromagnet” (F/AF) and\n(b)-(d) magnetostatic properties of\nthe system implemented experimen-\ntally, Py/FeMn( t), at room tempera-\nture. Rising exchange bias and coerciv-\nity fields, HbandHc, reflect the onset\nof antiferromagnetic ordering in thicker\nFeMn layers. Saturation magnetiza-\ntion,Ms, normalized by that of the\nreference Py film, MrefPy\ns, indicates a\nmagnetic-proximity induced magnetic\nmoment in FeMn, Mb.\nexchange bias field, Hb) reflects relatively strong AF-\ninduced exchange pinning; Fig. 1(b)–(c). Such thickness\ndependence is conventionally related to the reduction of\neffective TNand explained by the finite-size effects aris-\ning in thin films27. In Py/FeMn bilayers, however, the\nmagnetic proximity effect from Py should additionally\ncontribute to the reduction of TN18,28.\nThe magnetometry data, shown in Fig. 1(d), indicate\na remarkable increase in the total magnetization, Ms, for\nthe bilayers with t≈2–4 nm when compared to that of\nthe reference Py(5 nm) film. Mswas obtained as the\nmeasured magnetic moment normalized by the nominal\nvolume of the Py layer. This increase in Ms(about 15%)\nis a strong evidence for a magnetic moment in FeMn in-\nduced by the magnetic proximity effect from Py. This\nadditional magnetic moment vanishes with thickness in-\ncrease,t >4 nm, which can be explained as due to sup-\npression of the magnetic proximity effect by the stronger\nAFM ordering of the thicker FeMn. We find this induced\nmagnetic moment in thin FeMn to be important for ex-\nplaining the unusual FMR properties described below.\nIn-Plane FMR.— Figure2(a)–(c) show select FMR\nspectra for Py/FeMn bilayers with different thickness t\nof FeMn, measured at room temperature and with vary-\ning in-plane orientation of the applied field (angle ϕH).\nDepending on the thickness of FeMn, the spectra can be\ngrouped into two categories; panels (b) and (c) in Fig. 2,\nrespectively. First, the bilayers with the thicker FeMn\n(t≥5 nm) exhibit two resonance lines, both with the\nresonance field, Hres, dependent on ϕH. This angular\ndependence reflects the exchange pinning in the system,\nas detailed above, and indicates that the thicker FeMn\nlayershavea significant antiferromagneticcharacterwith\nTNhigher than room temperature. In contrast, the bi-\nlayer with the thinner, 3-nm FeMn shows only one res-\nonance line. This line is independent of ϕHand has a\nlargeoffset ( ∼600G) from the position ofthe free-Pyline\n(LPy); Fig.2(a). This difference between the two types\nof FMR behavior is explained below in terms of a trans-\nformation of the interlayer exchange coupling near TN.\nThe observed two resonance lines for the structureswitht≥5 nm are attributed to the uniform (line L B,\nHres≈1 kG) and low-field non-uniform (L NU,Hres≤\n0.4 kG) FMR modes. Both L Band LNUlines exhibit a\npronounced angular dependence in-plane, reflecting the\nunidirectional in-plane magnetic anisotropy in Py aris-\ning from the exchange bias with FeMn. The correspond-\ning exchange bias fields Hb= 300–400 G, determined as\nHb= 0.5[Hres(180◦)−Hres(0◦)] for the main L Bline,\nagree well with the values obtained from the magnetom-\netry, Fig. 1(c). Onthe otherhand, the non-uniformFMR\nline LNUis observed at fields at which the Py magnetiza-\ntion switches back to the direction of exchange pinning.\nThis switching is usually accompanied by a non-uniform\nmagnetic state in the films, e.g. due to domain forma-\ntion, that can lead to non-uniform spin excitation and\nassociated resonance modes. The position of L NUthere-\nfore reflects the asymmetry of the hysteresis loop, which\nnaturally is angle dependent. The non-uniform nature of\nLNUwas confirmed by the broad-band FMR experiment\n(data not shown): the position of L NUwas essentially in-\ndependent of the excitation frequency, f, in contrast to\nLBshowing the expected f-dependence.\nThe standard FMR formalism29can yield parame-\nters of the magnetic anisotropy and magnetization. In\nthe case of an exchange-coupled bilayer for the para-\n/antiparallel orientations of the external magnetic field,\nH, with respect to the exchange pinning direction ( ϕH=\n0◦/180◦), the FMR frequency can be expressed as\nfFMR=γ\n2π/radicalbig\n(H±Hb)(H±Hb+4πMeff),(1)\nwhereγisthegyromagneticconstant; exchange-biasfield\nHbenters (1) with “+”/“ −” sign for ϕH= 0◦/180◦;\n4πMeffis the effective demagnetizing field, which can\nalso include an out-of-plane anisotropy, indistinguishable\nin this form from the thin-film demagnetization, 4 πMs.\nThat is why this term is commonly presented as 4 πMeff,\ncomprising the effective magnetization ,Meff.\nThe gap between the L Bbranches in Fig. 2(d), accord-\ning to (1), reflects an increase in Meffon decreasing the3\n/s114 /s101 /s102/s46 /s32 /s80 /s121 /s32 /s102/s105 /s108 /s109 \n/s40 /s97 /s41 \n/s48 /s111/s76 \n/s80 /s121 \n/s106\n/s32 /s72 /s32 /s61 /s32 /s49 /s56 /s48 /s111\n/s40 /s98 /s41 \n/s76 \n/s66 \n/s48 /s111/s49 /s56 /s48 /s111/s80 /s121 /s47 /s70 /s101 /s77 /s110 /s40 /s51 /s32 /s110 /s109 /s41 \n/s48 /s46 /s48 /s48 /s46 /s53 /s49 /s46 /s48 /s49 /s46 /s53 /s50 /s46 /s48 /s40 /s99/s41 \n/s76 \n/s78 /s85 \n/s48 /s111/s57 /s48 /s111/s49 /s56 /s48 /s111\n/s76 \n/s66 /s80 /s121 /s47 /s70 /s101 /s77 /s110 /s40 /s53 /s32 /s110 /s109 /s41 /s100 /s80 /s47 /s100 /s72 /s32 /s40 /s97 /s114 /s98 /s46 /s32 /s117 /s110 /s105 /s116 /s115/s41 \n/s72 /s32 /s40 /s107 /s71/s41 /s48 /s57 /s48 /s49 /s56 /s48 /s48 /s46 /s48 /s48 /s46 /s52 /s48 /s46 /s56 /s49 /s46 /s50 /s72\n/s114 /s101 /s115 /s32 /s40 /s107/s71/s41 \n/s106\n/s32 /s72 /s32 /s40 /s176 /s41 /s76 \n/s80 /s121 \n/s76 \n/s78 /s85 /s76 \n/s66 \n/s55 /s32 /s110 /s109 \n/s53 /s32 /s110 /s109 /s55 /s32 /s110 /s109 /s53 /s32 /s110 /s109 /s40 /s100 /s41 \n/s51 /s32 /s110 /s109 FIG. 2. In-plane FMR spectra\nfor the reference 5-nm Py film (a)\nand Py(5)/FeMn( t) bilayers (b)–(c)\nmeasured at room temperature and\nshown for select orientations of the\nexternal field with respect to the\nexchange pinning direction ( ϕH=\n0◦). Arrows in panel (c) track the\noffset of the resonance lines with\nvaryingϕH. (d) Corresponding res-\nonance field vs field angle ϕHfor\nlines L Band L UNof Py/FeMn bi-\nlayers with t= 3, 5, and 7 nm, and\nline L Pyof the reference Py film.\nthickness of FeMn from t= 7 nm to 5 nm. Interestingly\nbut not surprisingly, Mefffort= 5 nm is even higher\nthan the saturation magnetization of the reference Py\nfilm (details below). This increase in Meffwith the de-\ncrease in tfrom 7 nm to 5 nm perfectly agrees with the\nmagnetometry data shown in Fig. 1(d) and is due to the\nFM-proximity-induced magnetic moment in FeMn.\nThe FMR behavior of the Py/FeMn bilayer with t=\n3 nm is distinctly different from that of the structures\nwitht= 5 nm and 7 nm. It exhibits only one resonance\nline, L B, the uniformFMR asconfirmedinthe roomtem-\nperature broad-band FMR measurements: L Bhas the\nsamef-dependence as the resonance line of the reference\nPy film, L Py; Fig.3(a). However, line L Bis observed at\nmuch lower resonance fields than L Py, which can not be\nexplained by an in-plane magnetic anisotropy since L Bis\nangle-independent; cf. Fig. 2(d). Then, according to ( 1),\nsuch decrease in Hresshould be attributed to an increase\nin4πMeff. Thislargeofachangecannotbecausedbythe\nenhanced magnetization of the resonating FM layer, Ms.\nOn the contrary, the actual magnetization of the bilayer,\ncomprising Msof Py and the small, proximity-induced\nmagnetic moment in the adjacent quasi-paramagnetic\nFeMn, should be even lower. The remaining explanation\nis additional magnetic anisotropy that favors in-plane\norientation of the magnetization – the so-called “easy-\nplane” anisotropy. In fact, a similar isotropic resonance\nfield shift was reported for similar Py/FeMn bilayers30\nand was associated with the rotatable in-plane magnetic\nanisotropy due to thermally activated states of the AFM\nspins in ultra-thin FeMn layers. However, as shown be-\nlow, this consideration is not sufficient to explain the\nlow-temperature FMR properties we observe since the\nrequired anisotropy would be unrealistically high.\nFMR vs Temperature.— Temperature-dependent FMR\nmeasurements reveal the key properties of L Bin the\nPy/FeMn structure with t= 3 nm; Fig. 3(b)–(e). Be-ing essentially angle-independent at room temperature,\nLBshows unidirectional anisotropy at lower tempera-\ntures, as seen in Hres-vs-ϕH; Fig.3(c). This unidirec-\ntional anisotropy indicates the presence of exchange pin-\nningin thesystem andthereforesufficiently strongAFM-\nordering in the thin FeMn layer. As seen from both the\nFMR and MOKE data shown in Fig. 3(e), the exchange\npinning vanishes completely at T/greaterorsimilar300 K and is signif-\nicantly suppressed (with a relatively low TN) compared\nwith the thicker-FeMn structures, where FMR reveals\nexchange pinning in the whole temperature interval31.\nThe reduction of effective TNcan be explained by the\ncompetitionofthemagneticproximityeffectandintrinsic\nAFM ordering in the AF layers18. Since the magnetic\nproximity effect is relatively short range (a few nm), the\nreductionof TNismorepronouncedforthinnerAFlayers,\nfor which the finite-size effect can also take place18,28.\nThis explains the relatively large difference in TNfor the\nstructures in our series.\nThe temperature dependence of Meffobtained from\nthe FMR data using ( 1) and shown in Fig. 4(a) helps to\nexplain the pronounced difference in dynamic properties\nbetween the structures with different t. With increasing\ntemperature, Meffdecreases for the structure with the\n7-nm FeMn, which is typical for ferromagnetic materi-\nals. In contrast, Mefffor the structure with the 5-nm\nFeMn has an unconventional upturn at high tempera-\ntures. This can be explained by an increase in the net\nmagnetic moment of the 5-nm FeMn, since its AFM or-\nder weakens and the magnetic proximity effect becomes\nmore pronounced at higher temperatures.\nThe temperature behavior of Mefffor the structure\nwith the 3-nm FeMn cannot be explained in the same\nfashion as for the other structures. The reason is that\nthe obtained values for Meffare larger than the maxi-\nmum possible magnetization of the bilayer or even that\nof a corresponding Fe film; Fig. 4(a). This large en-4\n/s48 /s46 /s48 /s48 /s46 /s52 /s48 /s46 /s56 /s49 /s46 /s50 \n/s76 \n/s66 /s40 /s98 /s41 \n/s50 /s52 /s48 /s32 /s75 /s32 /s32 /s48 /s176 \n/s32 /s32 /s49 /s56 /s48 /s176 \n/s50 /s54 /s48 /s32 /s75 /s50 /s56 /s48 /s32 /s75 /s51 /s50 /s48 /s32 /s75 /s100 /s80 /s47 /s100 /s72 /s32 /s40 /s97 /s114 /s98 /s46 /s32 /s117 /s110 /s105 /s116 /s115/s41 \n/s72 /s32 /s40 /s107 /s71/s41 /s40 /s97 /s41 \n/s48 /s46 /s48 /s48 /s46 /s53 /s49 /s46 /s48 /s40 /s100 /s41 \n/s49 /s56 /s48 /s176 \n/s48 /s176 /s72\n/s114 /s101 /s115 /s32 /s40 /s107 /s71/s41 \n/s48 /s57 /s48 /s49 /s56 /s48 /s48 /s46 /s48 /s48 /s46 /s52 /s48 /s46 /s56 /s49 /s46 /s50 /s72\n/s114 /s101 /s115 /s32 /s40 /s107/s71/s41 \n/s50 /s52 /s48 /s32 /s75 /s50 /s56 /s48 /s32 /s75 /s51 /s50 /s48 /s32 /s75 /s76 \n/s66 /s114 /s101 /s102/s46 /s32 /s80 /s121 /s44 /s32 /s50 /s57 /s52 /s32 /s75 \n/s106\n/s32 /s72 /s32 /s40 /s176 /s41 /s40 /s99/s41 \n/s50 /s52 /s48 /s51 /s48 /s48 /s48 /s49 /s48 /s48 \n/s40 /s101 /s41 /s72\n/s98 /s32 /s40 /s71/s41 \n/s84 /s32 /s40 /s75 /s41 /s32 /s32 /s70/s77/s82\n/s32 /s32 /s77/s79 /s75/s69\nFIG. 3. (a) Frequency dependence of the resonance field, meas ured at room temperature for the Py/FeMn bilayer with 3-nm\nthin FeMn and compared to that for the reference Py film. (b) Te mperature induced changes in the FMR spectra measured\nalong (ϕH= 0◦) and against ( ϕH= 180◦) the pinning direction set by cooling the samples in a magnet ic field of ≈1 kG. (c)\nCorresponding angular profiles of the resonance field at sele ct temperatures. (d) Temperature dependence of the resonan ce\nfield along (0◦) and against (180◦) the exchange-pinning direction. (e) Exchange-pinning fie ld vs temperature, derived from\nthe FMR and MOKE data.\n/s50 /s48 /s48 /s50 /s52 /s48 /s50 /s56 /s48 /s51 /s50 /s48 /s48 /s46/s49 /s49 \n/s114 /s101 /s102/s46 /s32 /s70 /s101 /s40 /s97 /s41 \n/s114 /s101 /s102/s46 /s32 /s80 /s121 /s52 /s112 /s77\n/s101 /s102 /s102 /s32 /s180 /s49 /s48 /s53 \n/s32 /s40 /s71/s41 \n/s84 /s32 /s40 /s75 /s41 /s116 /s32 /s61 /s32 /s51 /s32 /s110 /s109 \n/s50 /s48 /s48 /s50 /s52 /s48 /s50 /s56 /s48 /s51 /s50 /s48 /s49 /s48 /s50 /s48 /s51 /s48 /s40 /s98 /s41 \n/s102 \n/s49 /s107 /s71 /s32 /s40 /s71/s72/s122/s41 \n/s84 /s32 /s40 /s75 /s41 /s53 /s32 /s110 /s109 \n/s55 /s32 /s110 /s109 \nFIG. 4. (a) Effective demagnetizing field 4 πMeffobtained\nfrom the FMR data as a function of temperature. The de-\nmagnetizing fields, obtained experimentally for the contro l\nPy film and expected for a Fe film, are shown for compari-\nson. (b) Resonance frequency corresponding to the measured\nresonance field for the single Py film ( ∼1 kG) for all samples.\nhancement in Meffis unlikely, however, to arise from\nthe presence of conventional magneto-crystalline or in-\nduced “easy-plane” magnetic anisotropy that would con-\ntribute to the 4 πMeffterm in ( 1). The reason is that\nthe required effective anisotropy-field strength would be\n∼100 kG – too high for any 3d transition-metal system.\nOntheotherhand, suchahighvaluecanbeanindication\nof its magnetic-exchangeorigin and thus be attributed to\nsome dynamic effective field, HAF, arisingfrom the inter-\naction of the FM subsystem with the AFM component in\nFeMn. The characteristicfrequencies of the AFM excita-tions near TNare close to the ferromagnetic excitations\nrange(lowGHz), whichshouldenablethe energytransfer\nbetween the FM and AFM subsystems. With decreasing\ntemperature, the AFM order becomes stronger, verified\nby a strengthening exchange-pinning. At the same time,\nthe resonance frequency of the bilayer increases signifi-\ncantly, trending toward the sub-terahertz range charac-\nteristic of the AFM resonance, as shown in Fig. 4(b).\nThe observed bilayer FMR-frequency enhancement and\nthe induced magnetic moment in nanometer thin FeMn\nindicate a complex interplay between the FM- and AFM-\ntype interfacial exchange in the system, which deserves a\nseparate comprehensive discussion32.\nConclusion.— The considerable frequency enhance-\nment we observe demonstrates an alternative way for\ndesigningmagnetic nanostructuresoperatingin the high-\nGHz frequency range. In contrasttothe conventionalap-\nproach to enhancing the FMR frequency by tailoring the\nmagnetic anisotropy, e.g. by using exchange bias3–11, we\nshow that the frequency can be significantly increased by\nemploying the proximity-magnetized regime of the AFM\nnear its N´ eel temperature, where the frequency gap be-\ntween the spin excitations in the two materials becomes\nsufficiently narrow. This approach can potentially result\nin a new class of ferromagnetic-like materials operating\nat sub-THz frequencies, important for a variety of high-\nspeed applications.\nSupport from the Swedish Research Council (VR:\n2018-03526), the Swedish Stiftelsen Olle Engkvist\nByggm¨ astare (2017-185-589), Volkswagen Foundation\n(90418) and the National Academy of Sciences of\nUkraine (0118U003265, 0119U100469 and 0120U100457)\nare gratefully acknowledged.5\n∗dpol@kth.se.\n1A. G. Gurevich and G. A. Melkov, Magnetization Oscilla-\ntions and Waves (CRC Press, 1996).\n2R. C. 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D.\nBar´ o, and J. Nogu´ es, Magnetic proximity effect features in\nantiferromagnetic/ferrimagnetic core-shell nanopartic les,\nPhysical Review Letters 102, 247201 (2009) .\n29C. Kittel, On the theory of ferromagnetic resonance ab-\nsorption, Physical Review 73, 155 (1948) .\n30W. Fan, X. Qiu, Z. Shi, S. Zhou, and Z. Cheng, Correla-\ntion between isotropic ferromagnetic resonance field shift\nand rotatable anisotropy in polycrystalline NiFe/FeMn bi-\nlayers,Thin Solid Films 518, 2175 (2010) .\n31See Supplemental Material at [link] for the details on\nsample fabrication and methods as well as additional\ntemperature-dependent FMR data.\n32Additional results on temperature-dependent magnetom-\netry and broad-band FMR measurements, supported by\nmicromagnetic simulations and focused on the physical\nmechanisms behind the observed f-enhancement, will be\ndiscussed in a separate publication.6\nAppendix: SUPPLEMENTAL MATERIAL\n1. Sample Fabrication and Methods\nA series of multilayers Ta(5)/Py(5)/FeMn( t= 0, 3, 5 and 7 nm)/Al(4) were deposited on oxidized Si substrates\nat room temperature using a dc magnetron sputtering system (AJ A Inc.). The thicknesses of individual layers\nwere controlled by setting the deposition time using the respective r ate calibrations. In order to induce exchange\npinning, all multilayers were deposited in a magnetic field of /lessorsimilar1 kG applied in the film plane. For temperature-\ndependent measurements, the samples were cooled down in a magne tic field applied in the same direction as during\nthe deposition. The magnetic properties were initially characterized at room temperature using a vibrating-sample\nmagnetometer (Lakeshore Cryogenics). The FMR measurements were carried out in a temperature interval of 200–\n320 K using an X-band ELEXSYS E500 spectrometer (Bruker) at a c onstant operating frequency of 9.46 GHz and\nsweeping the external magnetic field in the film plane. The field sweepin g was performed in the reversedirection, from\n5 kG to zero, in order to prevent FMR signals due to possible domain fo rmation at low fields (below ∼100–200 G).\n2. Temperature-dependent FMR behavior of bilayers with thi cker FeMn\nThe structures with the thicker FeMn layers ( t= 5 and 7 nm) have much higher TNthan the maximum temperature\navailable experimentally in our FMR measurements ( T≤320 K). With decreasing temperature, the unidirectional\nanisotropyobservedin Hres-vs-ϕHforline L Bincreases[Fig. A1(a)]. The correspondingeffective exchangefiled Hbcan\nbe derived from the temperature dependence of Hres(0◦) andHres(180◦) [Figs. A1(b),(c)]. A pronounced temperature\ndependence of Hexis typical for exchange-biased systems [ J. Magn. Magn. Mater. 192, 203–232 (1999)]. The\nhighly unusual result is that the derived effective magnetization is 10 –20 % larger than that for the reference Py film\n[Fig.4(a)], which is discussed in the main text.\n/s48 /s57 /s48 /s49 /s56 /s48 /s48 /s46/s48 /s48 /s46/s52 /s48 /s46/s56 /s49 /s46/s50 /s40/s97 /s41\n/s51/s50/s48/s32 /s75/s50/s56/s48/s32 /s75/s84 /s32 /s61/s32 /s50/s48/s48/s32 /s75\n/s32 /s32 /s70/s73 /s84\n/s32 /s32 /s69/s88/s80/s80 /s121/s40/s53 /s41 /s47/s70 /s101 /s77 /s110 /s40/s53/s41/s72\n/s114/s101 /s115/s32/s40/s107/s71/s41\n/s106\n/s32/s72 /s32/s40 /s176 /s41 /s50 /s48 /s48 /s51 /s48 /s48 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s49 /s46/s53 \n/s49/s56/s48 /s176 \n/s48 /s176 /s72 \n/s114/s101 /s115 /s32/s40/s107/s71/s41\n/s40/s98 /s41\n/s50 /s48 /s48 /s51 /s48 /s48 /s48 /s51 /s48 /s48 /s54 /s48 /s48 \n/s40/s99/s41/s72 \n/s98 /s32/s40/s71/s41\n/s84 /s32/s40/s75 /s41\nFigure A1. Temperature-dependent resonance field for the Py/FeMn bila yer with a thicker FeMn ( t= 5 nm; TNwell above\nroom temperature). (a) Angle profiles of the resonance field o f the L Bline versus temperature. (b) Temperature dependence\nof the resonance field along (0◦) and against (180◦) the exchange pinning direction. (c) Exchange field vs tempe rature derived\nfrom the data in panel (b).7\n/s45/s56 /s48 /s48 /s45/s54 /s48 /s48 /s45/s52 /s48 /s48 /s45/s50 /s48 /s48 /s48 /s50 /s48 /s48 /s52 /s48 /s48 \n/s77 /s79/s75/s69/s32/s109 /s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40 /s97/s114 /s98/s46/s32/s117/s110/s105/s116/s115/s41 \n/s72 /s32/s40 /s71/s41 /s80/s121/s40 /s53/s41 /s47/s70/s101/s77 /s110/s40 /s51/s41 \n/s32/s84/s32/s61/s32/s50 /s48 /s48 /s32/s75\n/s50 /s50 /s48 /s32/s75\n/s50 /s51 /s50 /s32/s75\n/s50 /s52 /s53 /s32/s75\n/s50 /s53 /s56 /s32/s75\n/s50 /s55 /s48 /s32/s75\n/s50 /s56 /s51 /s32/s75\n/s50 /s57 /s53 /s32/s75\n/s51 /s49 /s48 /s32/s75\n/s51 /s50 /s53 /s32/s75/s51 /s48 /s51 /s32/s75\nFigure A2. Magnetization loops for the Py(5)/FeMn(3) bilayer measure d at different temperatures using longitudinal\nMOKE. The measurements were performed after cooling in a mag netic field of ∼800 G and every hysteresis loop was\nobtained with an incremental increase in temperature. The d ata were used for obtaining the temperature dependence of th e\nexchange bias field Hb, shown in Fig. 3(e) in the main text.8\n/s48 /s49 /s48 /s48 /s48 /s50 /s48 /s48 /s48 /s51 /s48 /s48 /s48 /s100 /s80/s47/s100 /s72 /s32/s40/s97 /s114/s98 /s46/s32/s117 /s110 /s105 /s116/s115/s41\n/s72 /s32/s40/s71/s41/s32/s50 /s48 /s32 /s71/s72/s122 \n/s32/s49 /s57 /s32 /s71/s72/s122 \n/s32/s49 /s56 /s32 /s71/s72/s122 \n/s32/s49 /s55 /s32 /s71/s72/s122 \n/s32/s49 /s54 /s32 /s71/s72/s122 \n/s32/s49 /s53 /s32 /s71/s72/s122 \n/s32/s49 /s52 /s32 /s71/s72/s122 \n/s32/s49 /s51 /s32 /s71/s72/s122 \n/s32/s49 /s50 /s32 /s71/s72/s122 \n/s32/s49 /s49 /s32 /s71/s72/s122 \n/s32/s49 /s48 /s32 /s71/s72/s122 \n/s32/s57 /s32 /s71/s72/s122 /s80/s121/s40/s53 /s41/s32/s114/s101 /s102/s32/s108/s97 /s121/s101 /s114/s32/s124/s32/s82 /s84/s40/s97 /s41\n/s48 /s49 /s48 /s48 /s48 /s50 /s48 /s48 /s48 /s51 /s48 /s48 /s48 /s100 /s80/s47/s100 /s72 /s32/s40/s97 /s114/s98 /s46/s32/s117 /s110 /s105 /s116/s115/s41\n/s72 /s32/s40/s71/s41/s32/s50 /s48 /s32 /s71/s72/s122 \n/s32/s49 /s57 /s32 /s71/s72/s122 \n/s32/s49 /s56 /s32 /s71/s72/s122 \n/s32/s49 /s55 /s32 /s71/s72/s122 \n/s32/s49 /s54 /s32 /s71/s72/s122 \n/s32/s49 /s53 /s32 /s71/s72/s122 \n/s32/s49 /s52 /s32 /s71/s72/s122 \n/s32/s49 /s51 /s32 /s71/s72/s122 \n/s32/s49 /s50 /s32 /s71/s72/s122 \n/s32/s49 /s49 /s32 /s71/s72/s122 \n/s32/s49 /s48 /s32 /s71/s72/s122 \n/s32/s57 /s32 /s71/s72/s122 /s80/s121/s40/s53 /s41/s47/s70/s101 /s77/s110 /s40/s51 /s41/s32/s124/s32/s82 /s84/s40/s98 /s41\nFigure A3. Broadband FMR for the reference Py(5) structure (a) and for t he Py(5)/FeMn(3) bilayer (b), measured at room\ntemperature with the external field applied in-plane. The da ta were used for Fig. 3(a) in the main text."    },    {        "title": "2107.07939v2.Influence_of_inter_sublattice_coupling_on_the_terahertz_nutation_spin_dynamics_in_antiferromagnets.pdf",        "content": "Influence of inter-sublattice coupling on the terahertz nutation spin dynamics in\nantiferromagnets\nRitwik Mondal1;2\u0003and Peter M. Oppeneer1\n1Department of Physics and Astronomy, Uppsala University, Box 516, Uppsala, SE-75120, Sweden and\n2Department of Spintronics and Nanoelectronics, Institute of Physics of the Czech Academy of Sciences,\nCukrovarnická 10, CZ - 162 00 Praha 6, Czech Republic\n(Dated: July 20, 2021)\nSpin-nutation resonance has been well-explored in one-sublattice ferromagnets. Here, we investi-\ngate the spin nutation in two-sublattice antiferromagnets as well as, for comparison, ferrimagnets\nwith inter- and intra-sublattice nutation coupling. In particular, we derive the susceptibility of the\ntwo-sublattice magnetic system in response to an applied external magnetic field. To this end, the\nantiferromagnetic and ferrimagnetic (sub-THz) precession and THz nutation resonance frequencies\nare calculated. Our results show that the precession resonance frequencies and effective damping\ndecrease with intra-sublattice nutation coupling, while they increase with inter-sublattice nutation\nin an antiferromagnet. However, we find that the THz nutation resonance frequencies decrease with\nboth the intraandinter-sublattice nutation couplings. For ferrimagnets, conversely, we calculate\ntwo nutation modes with distinct frequencies, unlike antiferromagnets. The exchange-like precession\nresonance frequency of ferrimagnets decreases with intra-sublattice nutation coupling and increases\nwith inter-sublattice nutation coupling, like antiferromagnets, but the ferromagnetic-like precession\nfrequency of ferrimagnets is practically invariant to the intraandinter-sublattice nutation couplings.\nI. INTRODUCTION\nEfficientspinmanipulationatultrashorttimescalesholds\npromise for applications in future magnetic memory tech-\nnology [1–5]. Introduced by Landau and Lifshitz, the time\nevolutionofmagnetization M(r;t),canbedescribedbythe\nphenomenological Landau-Lifshitz-Gilbert (LLG) equation\nof motion, which reads [6–9]\n_M=\u0000\r(M\u0002H) +\u000b\nM0\u0010\nM\u0002_M\u0011\n;(1)\nwith the gyromagnetic ratio \r, constant magnetization am-\nplitudeM0, and Gilbert damping parameter \u000b. The LLG\nequation consists of the precession of spins around a field\nHand transverse damping that aligns the spins towards\nthe field direction. While the spin precessional motion can\nbe explained by Zeeman-like field-spin coupling, there are\nseveral fundamental and microscopic mechanisms leading\nto Gilbert damping [10–22].\nWhen one approaches the femtosecond regime, however,\nthe spin dynamics can not only be described by the tra-\nditional LLG dynamical equation of motion [23, 24], but\nit has to be supplemented by a fast dynamics term due\nto magnetic inertia [25–27]. Essentially, the inclusion of\nmagnetic inertia leads to a spin nutation at ultrashort\ntimescalesandcanbedescribedbyatorqueduetoadouble\ntime-derivative of the magnetization i.e., M\u0002M[26, 28].\nThe inertial LLG (ILLG) equation of motion has the form\n_M=\u0000\r(M\u0002H) +\u000b\nM0\u0010\nM\u0002_M\u0011\n+\u0011\nM0\u0010\nM\u0002M\u0011\n;\n(2)\nwith the inertial relaxation time \u0011. In general, the Gilbert\ndamping\u000band the inertial relaxation time \u0011are ten-\nsors [29], however, for an isotropic system, these param-\neters can be considered as scalars. The emergence of spin\n\u0003mondal@fzu.cznutation has been attributed to an extension of Kamberský\nbreathing Fermi surface model [30, 31], namely, an s\u0000d-\nlike interaction spin model between local magnetization\nand itinerant electrons [32, 33]. Moreover, the ILLG equa-\ntion has been derived from the fundamental Dirac equa-\ntion [20, 29]. Note that the Gilbert damping and inertial\nrelaxation time are related to each other as the Gilbert\ndamping is associated with the imaginary part of the sus-\nceptibility, while the inertial dynamics are associated with\nthe real part of the susceptibility [20, 34]. The characteris-\ntic timescales of the nutation have been predicted to be in a\nrangeof 1\u0000100fs[25,32,35,36]and 1\u000010ps[36,37]. More\nrecently, it has been demonstrated that simple classical me-\nchanical considerations superimposed with Gilbert dynam-\nics naturally lead to magnetic inertial dynamics [38, 39].\nTheoretically, the spin nutation has recently been exten-\nsively discussed for one-sublattice ferromagnets [25, 36, 40–\n44]. The nutation resonance has also been observed in ex-\nperiments, however for two-sublattice ferromagnets [37]. A\nrecent theoretical investigation predicts that the precession\nand nutation resonance frequencies may overlap in two-\nsublattice ferromagnets [45]. The spin nutation resonance\nhas been observed at a higher frequency than ferromagnetic\nresonance, e.g., while the ferromagnetic resonance occurs in\nthe GHz regime, the nutation resonance occurs in the THz\nregime [37, 46]. Moreover, the spin nutation shifts the fer-\nromagnetic resonance frequency to a lower value. Although\nthis shift is very small, the line-width of the resonance de-\ncreases, however, and thus the effective damping decreases,\ntoo.\nSpin nutation effects have not yet comprehensively been\ndiscussed in two-antiparallel aligned sublattice magnetic\nsystems (e.g., antiferromagnets, ferrimagnets). In a recent\ninvestigation, it has been predicted that the spin nutation\nin antiferromagnets may have much significance [46]. Due\nto sublattice exchange interaction, the antiferromagnetic\nresonance frequency lies in the THz regime, while the nu-\ntation resonance frequency has similar order of magnitude.\nThis helps to detect the antiferromagnetic precession andarXiv:2107.07939v2  [cond-mat.mtrl-sci]  19 Jul 20212\nnutation resonances experimentally as they fall in the same\nfrequency range. Moreover, the calculated shift of the anti-\nferromagnetic resonance frequency is stronger than that of\na ferromagnet. Additionally, the nutation resonance peak\nis exchange enhanced [46], which is beneficial for detection\nin experiments. However, the previous investigation only\nconsiders the intra-sublattice inertial dynamics, while the\neffect of inter-sublattice inertial dynamics is unknown.\nIn a previous work, the LLG equation of motion with\ninter-sublattice Gilbert damping has been explored by\nKamra et al.[47]. It was found that the introduction\nof inter-sublattice Gilbert damping enhances the damp-\ning [47–49]. In this study, we formulate the spin dynamical\nequations in a two-sublattice magnetic system with both\nintraandinter-sublattice inertial dynamics as well as in-\nterandintra-sublattice Gilbert damping, extending thus\nprevious work [46]. First, we derive the magnetic suscepti-\nbility with the inter-sublattice effects and compute the pre-\ncession and nutation resonance frequencies. We find that\ntheprecessionresonancefrequencyandtheeffectiveGilbert\ndamping decrease with the intra-sublattice nutation cou-\npling in antiferromagnets, however, they increase with the\ninter-sublattice nutation. Unlike antiferromagnets, we find\nfor ferrimagnets that the change of precession resonance\nfrequencies is more pronounced with both intra and inter-\nsublattice nutation coupling constants in the exchange-like\nmode, but nearly negligible for the ferromagnetic mode.\nThe article is organized as follows. First, in Sec. II, we\ndiscuss the linear-response theory of spin dynamics to cal-\nculate the magnetic susceptibility with the intra and inter-\nsublattice nutation effects. In Sec. III, the precession reso-\nnance frequencies have been calculated with analytical and\nnumerical tools for antiferromagnets (Sec. IIIA) and ferri-\nmagnets (Sec. IIIB). We summarize the obtained results in\nSec. IV.\nII. LINEAR-RESPONSE SUSCEPTIBILITY IN\nTWO-SUBLATTICE MAGNETS\nFor two-sublattice magnetic systems, namely AandB\nrepresentingthetwosublattices, theILLGequationsofmo-\ntion read\n_MA=\u0000\rA(MA\u0002HA) +\u000bAA\nMA0\u0010\nMA\u0002_MA\u0011\n+\u000bAB\nMB0\u0010\nMA\u0002_MB\u0011\n+\u0011AA\nMA0\u0010\nMA\u0002MA\u0011\n+\u0011AB\nMB0\u0010\nMA\u0002MB\u0011\n; (3)\n_MB=\u0000\rB(MB\u0002HB) +\u000bBB\nMB0\u0010\nMB\u0002_MB\u0011\n+\u000bBA\nMA0\u0010\nMB\u0002_MA\u0011\n+\u0011BB\nMB0\u0010\nMB\u0002MB\u0011\n+\u0011BA\nMA0\u0010\nMB\u0002MA\u0011\n: (4)\nIn the above dynamical equations, the first terms relate\nto the spin precession, the second and third terms repre-sent the intraandinter-sublattice Gilbert damping, and\nthe last two terms classify the intraandinter-sublattice\ninertial dynamics. The intra-sublattice magnetization dy-\nnamics has been characterized with the Gilbert damping\nconstants\u000bAA,\u000bBBand inertial relaxation time \u0011AAor\n\u0011BB, while the inter-sublattice dynamics is characterized\nby Gilbert damping \u000bABor\u000bBAand inertial relaxation\ntime\u0011ABor\u0011BA. Note that the Gilbert damping parame-\ntersaredimensionless, however, inertialrelaxationtimehas\na dimension of time [25, 26, 29]. The extended equations\nof motions in Eqs. (3) and (4) represent general magneti-\nzation dynamics for two-sublattice magnets (e.g., antifer-\nromagnets, ferrimagnets, two-sublattice ferromagnets, and\nso on).\nThe free energy of the considered two-sublattice system\nreads\nF(MA;MB) =\u0000H0(MAz+MBz)\u0000KA\nM2\nA0M2\nAz\n\u0000KB\nM2\nB0M2\nBz+J\nMA0MB0MA\u0001MB:(5)\nHere, the first term defines the Zeeman coupling of two\nsublattice spins with an external field H0=H0^z. The\nsecondandthirdtermsrepresenttheanisotropyenergiesfor\nthe sublattice AandB, respectively. The last term can be\nidentified as the Heisenberg exchange energy between the\ntwo sublattices. Note that the Heisenberg coupling energy,\nJ >0for antiferromagnets and ferrimagnets, however J <\n0for ferromagnetic-like coupling.\nWe calculate the effective field in the ILLG equation as\nthe derivative of free energy in Eq. (5) with respect to the\ncorresponding magnetization\nHA=\u0000@F(MA;MB)\n@MA\n=\u0012\nH0+2KA\nM2\nA0MAz\u0013\n^z\u0000J\nMA0MB0MB;(6)\nHB=\u0000@F(MA;MB)\n@MB\n=\u0012\nH0+2KB\nM2\nB0MBz\u0013\n^z\u0000J\nMA0MB0MA:(7)\nFirst, in the ground state, we consider that the Asub-\nlattice magnetization is MA=MA0^z, while the Bsub-\nlattice magnetization is antiparallel MB=\u0000MB0^z, such\nthat we can describe the antiferromagnets ( MA0=MB0)\nand ferrimagnets ( MA0> MB0). We then expand the\nmagnetization around the ground state in small deviations,\nMA=MA0^z+mA(t)andMB=\u0000MB0^z+mB(t). The\nsmall deviations mA=Bare induced by the transverse ex-\nternal field hA=B(t).\nFor convenience, we work in the circularly polar-\nized basis, i.e., mA=B\u0006=mA=Bx\u0006imA=By; hA=B\u0006=\nhA=Bx\u0006ihA=By, and define \nA=\rA=MA0(J+ 2KA+\nH0MA0);\nB=\rB=MB0(J+ 2KB\u0000H0MB0). With\nthe time-dependent harmonic fields and magnetizations\nhA=B\u0006; mA=B\u0006/e\u0006i!t, we obtain the magnetic suscep-\ntibility tensor [46]3\n\u0012\nmA\u0006\nmB\u0006\u0013\n=1\n\u0001\u00060\nBB@1\n\rBMB0\u0000\n\nB\u0006i!\u000bBB\u0000!2\u0011BB+!\u0001\n\u00001\n\rBMA0\u0012\rB\nMB0J\u0006i!\u000bBA\u0000!2\u0011BA\u0013\n\u00001\n\rAMB0\u0012\rA\nMA0J\u0006i!\u000bAB\u0000!2\u0011AB\u00131\n\rAMA0\u0000\n\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!\u00011\nCCA\u0012\nhA\u0006\nhB\u0006\u0013\n=\u001fAB\n\u0006\u0012\nhA\u0006\nhB\u0006\u0013\n; (8)\nwiththedefinitionofthedeterminant \u0001\u0006= (\rA\rBMA0MB0)\u00001\u0000\n\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!\u0001\u0000\n\nB\u0006i!\u000bBB\u0000!2\u0011BB+!\u0001\n\u0000\n(\rA\rBMA0MB0)\u00001\u0010\n\rA\nMA0J\u0006i!\u000bAB\u0000!2\u0011AB\u0011\u0010\n\rB\nMB0J\u0006i!\u000bBA\u0000!2\u0011BA\u0011\n.\nAs one expects, the inter-sublattice Gilbert damping and\ninertial dynamical contributions arise in the off-diagonal\ncomponents of the susceptibility tensor, while the intra-\nsublattice contributions are in the diagonal component of\nthe susceptibility [46]. Note that without inertial dynamics\nterms, the expression for the susceptibility is in accordance\nwith the one derived by Kamra et al.[47].\nTo find the resonance frequencies, the determinant \u0001\u0006\nmust go to zero, thus one has to solve the following fourth-\norder equation in frequency\nA\u0006!4+B\u0006!3+C\u0006!2+D\u0006!+E\u0006= 0;(9)\nwhere the coefficients have the following forms\nA\u0006=\u0011AA\u0011BB\u0000\u0011AB\u0011BA; (10)\nB\u0006=\u0007i (\u000bAA\u0011BB+\u000bBB\u0011AA)\u0000(\u0011AA\u0000\u0011BB)\n\u0006i (\u000bAB\u0011BA+\u000bBA\u0011AB); (11)\nC\u0006=\u00001\u0006i (\u000bAA\u0000\u000bBB)\u0000(\nA\u0011BB+ \nB\u0011AA)\n\u0000\u000bAA\u000bBB+\u0012\rA\nMA0\u0011BA+\rB\nMB0\u0011AB\u0013\nJ\n+\u000bAB\u000bBA; (12)\nD\u0006= (\nA\u0000\nB)\u0006i (\nA\u000bBB+ \nB\u000bAA)\n\u0007i\u0012\rA\nMA0\u000bBA+\rB\nMB0\u000bAB\u0013\nJ; (13)\nE\u0006= \nA\nB\u0000\rA\rB\nMA0MB0J2: (14)\nThe solutions of the above equation (9) result in four dif-\nferent frequencies in the presence of a finite external field.\nTwo of those frequencies can be associated with the mag-\nnetization precession resonance, !p\u0006(positive and negative\nmodes) that exists even without nutation. The other two\nfrequencies dictate the nutation resonance frequencies, !n\u0006\n(positive and negative modes).\nIII. RESULTS AND DISCUSSION\nTheintrinsicintra-sublatticeinertialdynamicshavebeen\ndiscussedextensivelyinRef.[46]. Essentially, theresonance\nfrequencies and effective damping decrease with increasing\nintra-sublattice inertial relaxation time for antiferromag-\nnets and ferrimagnets. Therefore, we consider a constant\nintra-sublatticeinertialrelaxationtimeinthiswork. Inthis\nsection, we specifically discuss the effects of inter-sublattice\nnutation in both antiferromagnets and ferrimagnets.\n-3-2-10123!/2º(THz)0.00.20.40.60.81.01.21.4PAB£ÆAA∞AMA0|hA|2¥= 0,¥0=0¥= 100 fs,¥0=0¥= 100 fs,¥0= 50 fs\n-0.6-0.4-0.200.20.40.600.010.02Figure 1. The calculated dissipated power vs. frequency for an\nantiferromagnet with MA0=MB0= 2\u0016B, and various values\nof the intra- and inter-sublattice nutations parameters, \u0011and\n\u00110. The inset shows the dissipated power close to the precession\nresonance frequencies. The other used parameters are \rA=\n\rB= 1:76\u00021011T\u00001s\u00001,J= 10\u000021J,KA=KB= 10\u000023J,\nH0= 1T,\u000bAA=\u000bBB= 0:05,\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\n\u0011and\u0011AB=\u0011BA=\u00110.\nA. Antiferromagnets\nTo start with, we calculate the frequency-dependent\ndissipated power of an antiferromagnet. Using the\nexpressions for the susceptibility in Eq. (8), we cal-\nculate the dissipated power in the inertial dynamics\nwith the following definition PAB=_mA\u0001hA+_mB\u0001\nhB=1\n2( _mA+hA\u0000+ _mA\u0000hA++ _mB+hB\u0000+ _mB\u0000hB+)\nwhich leads to a complicated expression (not given). For\nconvenience, we define \u000bAA=\u000bBB=\u000b,\u0011AA=\u0011BB=\u0011.\nTo focus on the inter-sublattice nutation \u0011AB=\u0011BA=\u00110,\nwe set the inter-sublattice Gilbert damping to zero, i.e.,\n\u000bAB=\u000bBA= 0, and choose MA0=MB0= 2\u0016B. The\nexchange and anisotropy energies, magnetic moments used\nin the here-presented computations are comparable to a\ntypical antiferromagnetic NiO [23, 50, 51] or CoO [52, 53]\nsystem. However, we mention that NiO or CoO bulk crys-\ntals have biaxial anisotropy. Also, the Gilbert damping of\nNiO is very small \u000b\u001810\u00004, i.e., less than the here-used\nvalue. In contrast, a large spin-orbit coupling in antifer-\nromagnetic CrPt (that has \u00182\u0016BCr moments) leads to a4\n10−1100101102\nη/prime(fs)0.160.20.240.280.32ωp±/2π(THz)\nη= 100 fs,η/prime= 0\nη= 100 fs,η/prime= 0η=η/prime= 0\nη=η/prime= 0(a)\nRe(ωp+)\nRe(ωp−)\nEq. (17)\n10−1100101102\nη/prime(fs)0.160.20.24Im(ωp±)/Re(ωp±)\nη=η/prime= 0 η=η/prime= 0\nη= 100 fs,η/prime= 0 η= 100 fs,η/prime= 0(b)\nIm(ωp+)/Re(ωp+)\nIm(ωp−)/Re(ωp−)\nEq. (18)\nFigure 2. The calculated precession frequencies as a function of inter-sublattice nutation \u00110for an antiferromagnet, setting MA0=\nMB0= 2\u0016B. The data points denote the numerical solution of Eq. (9) and the black lines correspond to the analytical solution\nin Eq. (16). (a) The real part of the resonance frequency, and (b) the ratio of imaginary and real part of the frequency have been\nplotted. The other used parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,J= 10\u000021J,KA=KB=K= 10\u000023J,H0= 1T,\n\u000bAA=\u000bBB=\u000b= 0:05,\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\u0011= 100fs and\u0011AB=\u0011BA=\u00110. The horizontal lines correspond to\nsolutions with zero inter-sublattice nutation ( \u00110= 0). Note that we show Re(!p\u0000)as\u0000Re(!p\u0000).\nhigherGilbertdamping \u000b\u001810\u00002[54,55]. Importantly, the\ninertialrelaxationtimes \u0011and\u00110arenotknowninthesean-\ntiferromagnetic systems. Our simulations pertain therefore\nto typical, selected model systems. We show the evaluated\ndissipated power with and without inertial dynamics for\nsuch antiferromagnet in Fig. 1. Note that the dissipated\npower has already been calculated in Ref. [46], however,\nwithout the inter-sublattice inertial dynamics. We can ob-\nserve that while the intra-sublattice inertial dynamics de-\ncreases the precessional resonance frequencies (see the cyan\nlines in Fig. 1), the inter-sublattice inertial dynamics works\noppositely. Note that the nutation resonance frequencies\ndecrease with the introduction of inter-sublattice inertial\ndynamics.\nTo understand the effect of the inter-sublattice nuta-\ntion terms, first, we solve the Eq. (9), considering again\n\u000bAA=\u000bBB=\u000b,\u0011AA=\u0011BB=\u0011,\u0011AB=\u0011BA=\u00110,\nand\u000bAB=\u000bBA= 0. As the nutation in antiferromagnets\nis exchange enhanced [46], we calculate the effect of inter-\nsublatticetermsontheprecessionandnutationfrequencies,\nsetting\rA=\rB=\randMA0=MB0=M0for antifer-\nromagnets. Therefore, the fourth-order equation in Eq. (9)\nreduces to an equation with AAFM\n\u0006 =\u00112\u0000\u001102,BAFM\n\u0006 =\n\u0007i2\u000b\u0011,CAFM\n\u0006 =\u00001\u0000(\nA+ \nB)\u0011+ 2\r\nM0\u00110J,DAFM\n\u0006 =\n(\nA\u0000\nB)\u0006i (\nA+ \nB)\u000b, andEAFM\n\u0006 = \nA\nB\u0000\u0010\n\r\nM0J\u00112\n.\nThesolutionoftheaboveequationresultsinprecessionand\nnutation resonance frequencies for the two modes (positive\nandnegative). Insertingtherealandimaginarypartsofthe\nsolutions!\u0006=Re(!\u0006)+iIm(!\u0006), we numerically calculate\nthe precession resonance frequencies and effective damping\n(the ratio of imaginary and real frequencies) for an anti-\nferromagnet as a function of inter-sublattice nutation. The\nresults are shown in Fig. 2, where the data points corre-\nspond to the numerical solutions.\nOntheotherhand, thefourth-orderequation, AAFM\n+!4+\nBAFM\n+!3+CAFM\n+!2+DAFM\n+!+EAFM\n+ = 0can analytically\nbe solved using the considerations that KA=KB=K,\nJ\u001dK,M0H0and\u000b\u001c1. Therefore, one has \nA=\nB\u0019\r(J+2K)=M0. Essentially,thefourth-orderequation\nreduces to\n\u0000\n\u00112\u0000\u001102\u0001\n!4\u0000\u0014\n1 + 2\r\u0011(J+ 2K)\nM0\u00002\r\u00110J\nM0\u0015\n!2\n\u00002i\u000b\u0011!3\n(0)+ 2\rH0!(0)+2i\r\u000b\nM0(J+ 2K)!(0)\n+\r2\nM2\n0(J+ 2K)2\u0000\r2J2\nM2\n0\u0000\r2H2\n0= 0; (15)\nwith!(0)being the solution of the above equation for \u000b= 0\nandH0= 0. The solutions of the above equation are rather\nsimple and provide the two precession frequency modes\n(positive and negative) for antiferromagnets. Expanding\nthe solutions of Eq. (15) up to the first order in \u000band\nH0, and also in first order in K=J\u001c1, the precession reso-\nnance frequencies are obtained (neglecting the higher-order\nin!(0)-terms) as\n!p\u0006\u0019\u0006\r\nM0p\n4K(K+J)r\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\n+\rH0+ i\r\u000b\nM0(J+ 2K)\nr\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)j!(0)j\n\r\nM0p\n4K(K+J):\n(16)\nNowsubstitutingthe j!(0)jfromtheleadingterminthefre-\nquency expression into the perturbative terms in Eq. (16),\nthe approximate precession frequencies are obtained as\n!p\u0006\u0019\u0006\r\nM0p\n4K(K+J)r\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\n+\rH0+ i\r\u000b\nM0(J+ 2K)\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110): (17)5\nThis equation has been plotted in Fig. 2 as black lines.\nNote that, for !p\u0000we show for convenience \u0000Re(!p\u0000)in\nFig. 2(a) and in the following. Due to the presence of \u0011\u0000\u00110\nin the denominator of the frequency expressions, the pre-\ncession resonance frequency increases when inter-sublattice\nnutation is taken into account ( \u00110<\u0011), which explains the\nincrease in frequency in Fig. 2(a). At the limit \u0011!\u00110, the\nnutation (intra and inter-sublattice) does not play a signifi-\ncant role as the precession resonance frequency is decreased\nby a factorq\n1 +4\r\u0011K\nM0which is very small due to K\u001cJ.\nNote that the two resonance frequencies are approximately\n0.332 THz and 0.276 THz with \u000b= 0and\u0011=\u00110= 0, while\nthese two frequencies are 0.322 THz and 0.266 THz with\n\u000b= 0:05and\u0011=\u00110= 0. The latter has been shown in Fig.\n2(a) as dashed lines. Therefore, the Gilbert damping has\nalready the effect that it reduces the resonance frequencies.\nThe effective Gilbert damping can be calculated using\nthe ratio between the imaginary and real parts of the fre-\nquencies, i.e., the line width. From Eq. (17) one arrives\nat\nIm(!p\u0006)\nRe(!p\u0006)\u0019\u000b(J+ 2K)p\n4K(K+J)1r\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110):\n(18)\nNote that the two resonance modes have the same effec-\ntive damping. For ferromagnets, the exchange energies do\nnot contribute and thus the effective damping remains the\nsame as\u000b, in the absence of magnetic inertial terms (see\n[45]). However, in antiferromagnets the effective damping\nis enhanced due to the exchange interaction by a factor\n(J+2K)p\n4K(K+J), even without any inertial terms. As investi-\ngated earlier [46], the effective damping decreases with the\nintra-sublattice relaxation time. However, similar to the\nincrease in frequency, the effective damping also increases\nwith the inter-sublattice inertial relaxation time, as seen in\nFig. 2(b). The analytical solution in Eq. (18) agrees ex-\ncellently with the numerical solutions. Close to the limit\n\u00110!\u0011, the effective Gilbert damping in Eq. (18) one ex-\npects the effective damping to be increased by a factor\u0010\n1 +4\r\u0011K\nM0\u0011\u00001=2\n, as can be seen in Fig. 2(b).\nNext, we discuss the field dependence of the reso-\nnance frequencies. The precession resonance frequencies\nand effective damping have been plotted as a function\nof the applied field H0for several inter-sublattice relax-\nation times in Fig. 3. As can be observed, at zero ap-\nplied field, the two modes (positive and negative) coin-\ncide in antiferromagnets, a fact that can be seen from\nEq. (17). However, the applied field induces the splitting\nof these two modes. The frequency splitting scales with\u0014\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\u0015\u00001\n\rH0, meaningthatthesplit-\nting is linear in the applied field, H0. On the other hand,\nat a constant field, the splitting also depends on the inter-\nand intra-sublattice nutation. From Eq. (17), it is clear\nthat the splitting is reduced with intra-sublattice nutation,\nwhile it is enhanced with inter-sublattice nutation. Such a\nconclusion can also be drawn from the numerical solutions\nin Fig. 3(a). The effective damping of the antiferromagnet\n0.160.20.240.280.32Re(ωp±)/2π(THz)\n+\n–+\n–+\n–+\n–(a)\n0.0 0.2 0.4 0.6 0.8 1.0\nH0(T)0.150.20.25Im(ωp±)/Re(ωp±)(b)\nη=η/prime= 0\nη= 100 fs,η/prime= 0\nη= 100 fs,η/prime= 10 fs\nη= 100 fs,η/prime= 50 fsFigure 3. The calculated precession frequencies at several inter-\nsublattice relaxation times as a function of applied field for anti-\nferromagnets using MA0=MB0= 2\u0016B. The solid and dashed\nlinesrepresentthepositiveandnegativemodes, respectively. (a)\nThe real part of the resonance frequencies and (b) the ratio of\nimaginary and real part of the frequency have been plotted. The\nother used parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,J=\n10\u000021J,KA=KB=K= 10\u000023J,\u000bAA=\u000bBB=\u000b= 0:05,\n\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\u0011= 100fs and\u0011AB=\u0011BA=\u00110.\nremains field independent which can be observed in Fig.\n3(b).\nProceeding as previously, we obtain the following nuta-\ntion frequencies\n!n\u0006\u0019\u00061\n\u0011vuuuuut1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\n1\u0000\u001102\n\u00112 \n1\u0000\n(\u00112\u0000\u001102)\r2\nM0\u00024K(J+K)\n2\u0014\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110)\u00152!\n\u0000\rH0\u0000i\u000b\u0014\u0011\n\u00112\u0000\u001102+\r\nM0(J+ 2K)\u0015\n1 +4\r\u0011K\nM0+2\rJ\nM0(\u0011\u0000\u00110):(19)\nNote that at the limit \u00110!0, the nutation frequen-\ncieswithouttheinter-sublatticecouplingarerecovered[46].\nThe dominant term in the calculated frequency is the first\nterm in Eq. (19). With the introduction of inter-sublattice\ncoupling\u00110, both the numerator and denominator of the\ndominant frequency term decrease and therefore, the nuta-\ntion frequencies approximately stay constant (with a slow6\n2.533.54ωn±/2π(THz)(a)\nη= 100 fs,η/prime= 0\nη= 100 fs,η/prime= 0\nRe(ωn+)\nRe(ωn−)\n10−1100101102\nη/prime(fs)0.020.040.060.08Im(ωn±)/Re(ωn±)(b)\nη= 100 fs,η/prime= 0\nη= 100 fs,η/prime= 0\nIm(ωn+)/Re(ωn+)\nIm(ωn−)/Re(ωn−)\nFigure 4. The calculated nutation frequencies as a function\nof inter-sublattice nutation for antiferromagnets using MA0=\nMB0= 2\u0016B. (a) The real part of the nutation resonance fre-\nquencies and (b) the ratio of imaginary and real part of the nu-\ntation resonance frequency have been plotted. The other used\nparameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,J= 10\u000021J,\nKA=KB=K= 10\u000023J,H0= 1T,\u000bAA=\u000bBB=\u000b= 0:05,\n\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\u0011= 100fs and\u0011AB=\u0011BA=\u00110.\ndecrease) with inter-sublattice nutation when \u0011 > \u00110as\nplotted in Fig. 4. However, in the limit \u00110!\u0011, the denom-\ninator vanishes, and thus the nutation frequencies diverge\nas can be seen in Fig. 4. It is interesting to note that\nthe inter-sublattice inertial dynamics increase the preces-\nsion resonance frequencies, however, decrease the nutation\nfrequency. Such observation is also consistent with the dis-\nsipatedpowerinFig.1. Thedampingoftheinertialdynam-\nics also shows a similar behavior: it stays nearly constant\nwith a divergence at the limit \u00110!\u0011.\nAs mentioned before, the inertial relaxation times \u0011and\n\u00110are not known in for typical antiferromagnetic systems.\nNotwithstanding, we obtain the general result that the pre-\ncession resonance frequencies decrease with intra-sublattice\ninertial dynamics, however, increase with inter-sublattice\ninertial dynamics. Thus, to experimentally realize the sig-\nnature of inertial dynamics, an antiferromagnet with a\nhigher ratio of intra to inter-sublattice inertial relaxation\ntime (\u0011=\u00110\u001d1) is better suited.\nB. Ferrimagnets\nNext, we consider a ferrimagnetic system where the mag-\nnetic moments in the two sublattices are different, i.e.,\nMA06=MB0. In this case, the analytical solution of\nEq. (9) becomes cumbersome. The main reason is that\n0.00.20.40.60.81.01.2ωp±/2π(THz)\nη= 0,η/prime= 0η= 0,η/prime= 0\nη= 100 fs,η/prime= 0η= 100 fs,η/prime= 0(a)\nRe(ωp+)\nRe(ωp−)\n10−1100101102\nη/prime(fs)2345ωn±/2π(THz)\nη= 100 fs,η/prime= 0η= 100 fs,η/prime= 0(b)\nRe(ωn+)\nRe(ωn−)Figure 5. The calculated precession and nutation frequencies\nas a function of inter-sublattice nutation for ferrimagnets using\nMA0= 5MB0= 10\u0016B. The real part of the (a) precession reso-\nnance and (b) nutation resonance frequencies have been plotted.\nThe other used parameters are \rA=\rB= 1:76\u00021011T\u00001s\u00001,\nJ= 10\u000021J,KA=KB=K= 10\u000023J,H0= 1T,\u000bAA=\n\u000bBB=\u000b= 0:05,\u000bAB=\u000bBA= 0,\u0011AA=\u0011BB=\u0011= 100fs\nand\u0011AB=\u0011BA=\u00110.\n\nA6= \nBfor ferrimagnets, in fact, we calculate \nA\u0000\nB=\n\r(J+2K)(MA0\u0000MB0)\nMA0MB0+2\rH0. For antiferromagnets, the mag-\nnetic moments in the two sublattices are exactly the same,\ni.e.,MA0=MB0and thus, within the approximation of\nJ\u001dM0H0, we find \nA= \nBwhich simplifies the ana-\nlytical solution of Eq. (9). Thus, we numerically solve the\nEq. (9) to calculate the precession and nutation resonance\nfrequencies for ferrimagnets. We consider the case where\nMA0= 10\u0016BandMB0= 2\u0016B, reminiscent of rare-earth–\ntransition-metal ferrimagnets as GdFeCo [2, 3] or TbCo\n[56–58]. However, we emphasize that the inertial relaxation\ntimes\u0011and\u00110are not known for these materials. The\ncalculated precession frequencies are shown in Fig. 5(a).\nThe effect of intra-sublattice inertial dynamics has already\nbeenstudiedinRef.[46]. Forferrimagnets, thenegativefre-\nquencymodeappearstohaveahigherfrequency(i.e.,larger\nnegative) than the positive one. However, both precession\nfrequenciesdecreasewithintra-sublatticerelaxationtime, \u0011\n[46]. We, therefore, have set the intra-sublattice relaxation\ntime\u0011to 100 fs and vary the inter-sublattice relaxation\ntime\u00110< \u0011. The upper precession resonance mode !p\u0000–\nthe exchange-like mode – increases with the inter-sublattice\nrelaxation time \u00110, while the ferromagnetic-like mode !p+\nshows a very small increase. Thus, for ferrimagnets, the\nchange in precession frequencies is more significant in the7\nexchange-like mode than in the ferromagnetic-like mode.\nAtthelimit \u00110!\u0011, theprecessionresonancefrequenciesal-\nmost coincide with the resonance frequencies calculated at\n\u0011=\u00110= 0, meaning that the inertial dynamics do not play\nany role for the precession resonance frequency. The lat-\nter can clearly be seen in Fig. 5(a). These observations are\nsimilar to the antiferromagnet as discussed earlier. The nu-\ntation resonance frequencies in Fig. 5(b) again decline with\nthe inter-sublattice relaxation time showing a divergence\nat the limit \u00110!\u0011. However, one can notice here two dis-\ntinguishable nutation resonance frequencies unlike almost\na single-valued nutation frequencies of antiferromagnets.\nIV. SUMMARY\nIn summary, we have formulated a linear-response theory\nof the ILLG equations for antiferromagnets with inter- and\nintra-sublattice inertial dynamics. The calculation of the\nsusceptibility tensor shows that the intra-sublattice terms\nappear in the diagonal elements, while the inter-sublattice\nterms appear in the off-diagonal elements. The dissipated\npower contains a precession resonance peak in the sub-THz\nregime for antiferromagnets, however, the introduction of\ninertial dynamics causes another peak, a nutation reso-\nnance peak at a higher, few THz frequency. Moreover, we\nobserve that the inter-sublattice inertial dynamics work op-\npositely to the intra-sublattice inertial one. By finding the\npoles of the susceptibility, we calculate the precession andnutation resonance frequencies. While the precession reso-\nnance frequencies decrease with intra-sublattice relaxation\ntime, the inter-sublattice inertial dynamics have the op-\nposite effect. In fact, we observe that the magnetic inertia\ndoesnothaveanyeffectontheantiferromagneticprecession\nresonance at the limit \u00110!\u0011. On the other hand, the THz\nnutation resonance frequency decreases slightly with the\nintroduction of inter-sublattice inertial dynamics, however,\nshowing a divergence at the limit \u00110!\u0011. Our derived an-\nalytical theory explains such inter-sublattice contributions.\nFinally, for ferrimagnets, we find a similar behavior for the\ninter-sublattice inertial dynamics. 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M.\nOppeneer, A.Kirilyuk, andV.Kapaklis,Phys.Rev.Mater.\n4, 104418 (2020)."    },    {        "title": "1312.6478v2.Magnetic_states_and_ferromagnetic_resonance_in_geometrically_frustrated_arrays_of_multilayer_ferromagnetic_nanoparticles_ordered_on_triangular_lattices.pdf",        "content": " 1 Magnetic states and ferromagnetic resonance in  geometrically  frustrated  arrays  of \nmultilayer  ferromagnetic nanoparticles  ordered on triangular lattices  \n \nV.L. Mironov1,2, E.V.  Skorohodov1, J.A.  Blackman3 \n1 Department of magnetic nanostructures, Institute f or physics of microstructures RAS,  \nNizhniy Novgorod, GSP -105, 603950,  Russia  \n2 Department of physics and nanoelectronics, Lobachevsky State University of Nizhniy Novgorod,  \nNizhniy Novgorod, Gagarin Avenue, 23, 603950, Russia  \n3 Department of Physics, Univ ersity of Leicester, Leicester, LE1 7RH, UK  \n(Submitted on 26 December 2013, Revised 4 February 2014 ) \nWe present a theoretical investigation of  magnetostatic interaction  effects  in geometrically  frustrated  arrays  of anisotropic  \nmultilayer  ferromagnetic nanoparticles  arranged in  different spatial ly configured  systems  with triangular  symmetry . We show \nthat the interlayer magnetostatic interaction  significantly expands the  opportunities to create  magnetically frustrated systems . \nThe e ffects of  the magnetostatic  interaction in magnetization reversal processes  and the  possibilit y to control  the ferromagnetic \nresonance  spectrum in such systems  are d iscussed . \n1. Introduction  \nThe effects  of magnetostatic interaction  in the regular arrays  of anisotropic  single -domain  ferromagnetic  nanoparticles \narranged in  two-dimensional lattice s with different spatial symmetry (in the literature , such systems are called  “artificial  \nspin ice” ) are the subject of the intensive investigations in the last decade [1-6]. The increasing interest in these  objects  is \nassociated primarily with the possibility to investigate the fundamental properties of geometrically frustrated magnetic \nsystems using  a relatively  simple model.   \nThe recent progress in the e -beam  nanolithography  techniques enables the fabrication  of super dense arrays of  \nnanoparticles , which  demonstrate  an unusual collective behavior connected with the competition between configuration \nentropy effects, dipolar interaction  and a rtificial anisotropy  [7-9]. It was shown that artificial  spin ice systems demonstrate  \nthe considerable changes in coercivity, switching fields and scenario of magnetization reversal. In particular, the dynamic \nswitching  in an external magnetic field is accompanied by the effects of the effective magnetic charges ordering [ 9] and the \nappearance of the exotic states called “magnetic monopoles” [ 10,11].  \nThe basic idea of our work is the investigation of the influence of magnetostatic interaction on the magnetic states and \nhigh frequency properties of the frustrated magnetic systems based on multilayer stacks of binary  nanoparticles . In this case \none can expect a considerable expansion of the spectrum of magnetic states and a significant increase in the averaged \nmagnetostatic energy  at a lattice site  due to the strong  dipolar coupling between particles  in the neighborin g layers.  \nFurthermore,  in multilayer systems the effects  of geometrical  frustration  can be significantly expanded due to the variation \nof the  magnetic  moments for the particles  located in different  layers by varying the materials and layer thicknesses .  \nIn the current letter we concentrate our attention on two aspects of the magnetostatic interaction in multilayer stack \narrays. First is the realization of exotic magnetic configurations in the magnetization reversal process, which remain stable \nafter removal  of the external magnetic field. The other interesting problem is the influence of intralayer  and interlayer  \nmagnetostatic  interaction s on the spectrum of ferromagnetic resonance and magnetostatic spin waves  in the ordered arrays \nof multilayer stacks  as th e prototype s of artificial 3-D magnonic crystals  [12-15]. In particular,  we show that  the spectrum \nof collective modes of ferromagnetic resonance in such systems strongly depends  on the spatial  configuration of the \nmagnetic  moments of particles and can be significantly  changed  by switching of magnetic states in an  external magnetic \nfield. From a practical point  of view such systems  are promising for  the development  of tunable microwave  devices  [16-18] \nfor civil and military applications .  \n2. The methods  of calculations  \nWe investigate d the effects of  magnetostatic  interaction and  ferromagnetic resonance in  arrays of the elliptical \nnanodisks ( a × b × h) arranged in a triangular grating (Fig. 1 ). To simplify the calculations we used the theoretical model of  \nanisotropic  dipoles common ly used  for the description of such systems [4, 19, 20]. We assumed that the magnetic field of \nthe nanoparticles corresponds to the field of  a uniformly magnetized sphere with built -in anisotropy corresponding to the \nshape anisotropy  of an elliptical disc. In this approximation the energy of  a system  can be presented as   \n 2 2 2\n3 5( ) 3( )( ) 1 1\n2 2i j i ij j ij\nxi xi yi yi zi zi i ex\ni j i i ij ijM M M R M RW N M N M N M M HR R            \n          , (1) \n                                                \nCorresponding author : mironov@ipmras.ru   2 where iM\n is magnetic moment of i-th particle; xiN, yiN and ziNare demagnetizing factors along the main axes of \nelliptical disc; ijRis the separation between disc’s centers; exH\n is an external magnetic field .  \nTo find the eigenfrequencies  of the ferromagnetic resonance  in the particle array s we solved  the system  of Landau -\nLifshitz equations without  external magnetic field. The oscillations  of the magnetic  moment of i-th particle  in the array  are \ndescribed by the following equation:  \ni iiH MtM \n ,      (2) \nwhere  is gyromagnetic ratio , iH\n is the effective  magnetic field, which  is generally  defined as  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1. The schematic drawing of the elliptical particles array on tri angular lattice . \ni j di\nj iH H H\n   .      (3) \nHere jH\n is the stray field from j-th particle at the location of i-th particle  and diH\n is demagnetizing field, which is defined \nas \ndi iH NM  \n, \nwhere N\n is the tensor of demagnetizing factors:  \n\n\n\n\n\nzyx\nNNN\nN\n0 00 00 0\n. \nThe FMR eigenfrequency of the single elliptical disc is defined by effective field s of shape anisotropy  \n0 ( )( )x y z y SN N N N M     ,      (4) \nwhere SM is the saturation magnetization.  \nTo find the spectrum  of eigenfrequencies  for the interacting particles  array  we solved  the linearized equation  (2) with \n0exH\n. The magnetic moment  of each particle and  acting  field were re presented as : \n,st\ni i iM M m t    \njst\nj j h H H\n , \nwhere  st\niM\n is an average  static  magnetic  moment of the particle ; im t is an  alternating  magnetic moment  \n st\ni im t M ; st\njH\n is a static field;  jh\n is a high frequency magnetic field  acting on the  i-th particle from the  j-th \nparticle , (jh<<st\njH). Then the  linearized system  of Landau -Lifshitz equations can be written as follows : r jM\n ijR \ni j \niM\n a  b  h z \nx y  3 st st st i\ni j i i j i\nj i j imM h Nm m H NMt \n                                    .   (5) \nFurther, we  always assume  that the effective field  associated with  the particle shape  anisotropy  more than the fields  of the \nother particles (( ) ; ( )st\nx y i z y i jN N M N N M H  \n), i.e. precession of the  magnetic mo ments occurs  mainly around  the \nanisotropy axes  of each particle. However t he spectrum of eigenfrequencies  in such systems  is determined by the  \nmagnetostatic  interaction and strongly depends on the spatial configuration  of the magnetic moments.  \nThe remagnet ization  effects in interacting arrays under an external magnetic field were investigated by Monte Carlo  \nsimulations based on the model of anisotropic spheres . In the calculations we  have neglected any deviation  of the vectors \niM\n out of the plane of  the elliptical  disc and used the simple model of uniaxial anisotropy. In this case the magnetic \nenergy of particles array can be represented as  \n2 *\n3 5( ) 3( )( ) 1sin ( )2i j i ij j ij\ni i i ex\ni j i i ij ijM M M R M RW KV M HR R \n          \n          ,  (6) \nwhere K is anisotropy constant; V is the volume of particle; i is the angle characterizing the direction of magnetic \nmoment of i-th particle; *\ni is the angle characterizing the direction of anisotropy axis of i-th particle . The normalized \nenergy can be r epresented in the following form:  \n2 * *sin ( )i i i ex\ni iW m H       .    (7) \nHere  is the normalized magnetostatic  energy  \n 0 0 0 0\n3 53 1\n2i ij j ij i j\ni j ij ijm r m r m m\nr r                \n ,     (8) \nwhere 0im is a unit vector of the magnetic mo ment of i-th particle, ijr is normalized separation between particle centers. \nThe p arameter  and normalized external magnetic field exH\n are expressed as follows : \n2\n3sM V\nR K ,      (9) \nanex\nexHHH*,       (10) \nwhere anH is effective field of anisotropy (an s H K M ); sM is a saturation magnetization of particle material. The \nparameter  is the ratio of the magnetostatic energy to the energy of anisotropy.  \nThe magnetization reversal processes and stationary distribution of the magnetic moments in the part icle’s arrays \ncorresponding to the minimum energy (7) were simulated by  the Monte Carlo method.  \n3. Magnetostatic interaction in one -layer s ystems  \nOne of the  simplest systems  of interacting  binary  particles is a  system  with basic  a building block of  three particles  \nlocated  in the plane  at an angle of  120°  (Fig. 2).  \n \n \n \n \n \n \n \n \n \n \n \nFig. 2. The p ossible configurations  of magnetic moments  in an array  of three particles  with triangular \nsymmetry. Parameters : v is the number of ways to  implement the  state; q is an effective  magnetic \ncharge  on the lattice site ; m is an effective  magnetic mom ent on the lattice site ;  is normalized \ndensity of magnetostatic energy . (A) (B) \nv         2                 6 \nq         3                 1 \nm        0                 2  \nε       1.01         - 0.34  4 As it is known this system has two  states  differ ing in  the spatial configuration of  the magnetic  moments [6]. These states \ndiffer in the  probability  of their realization at  random in itial conditions ( this quantity is proportional to the number of  ways \nto implement  this state  v), the effective  magnetic charge  on the lattice site  q, the effective  magnetic moment  on the lattice \nsite m, and the normalized density of magnetostatic energy ε. As is seen , the state ( A) corresponding to the highest  effective  \nmagnetic charge  and zero mean  magnetic moment  (“magnetic monopole ”) has the highest energy . The quasi -homogeneous \nstate ( B) with effective  charge q = 1 and  nonzero average  magnetic moment  has a lower energy.  \n3.1. Magnetization reversal  in the one -layer  three particles array  \nThe form of the magnetization reversal  of the three particle array  in an uniform external  magnetic field  depends on the \nratio, , of the anisot ropy energy to the magnetostatic  interaction . The hysteresis  curves  and schemes of transitions \nbetween  different states for  the array  of three interacting particles  are shown in Fig. 3.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3. The h ysteresis  curve s and the scheme s of transitions  between different states of three  particles array : (a) 0, \n(b) 0.1 , (c) 0.5 , (d) 1. The  configuration s of the magnetic moments  of the particles  corresponds \nto the points on  the hysteresis curve s. The d irection of the external  magnetic field is  shown in the diagram s by \narrows  located above the  array  of particles.  \n(1) (2) (3) (4) \n(1) (2) (3) (4) (b) 0.1  \n(1) (2) (1) (2) \n(c) 0.5  \n(1) \n(1) (3) \n(2) \n(2) (3) (a) 0 \n(1) (3) \n(2) \n(d) 1 \n(1) (2) (3)  5 If the anisotropy is dominant as in Fig. 3a (here, the magnet ostatic interaction is switched off completely, =0), the high \nenergy state is realized ( “magnetic monopole ” state (2) in Fig. 3a) at intermediate reversing field. In this case  we can \nprepare two stable sta tes (A) and (B), in the notation of Fig. 2, by an external magnetic field.  As the interaction energy is \nincreased in comparison with anisotropy energy, the high energy state (B) becomes unstable. For exam ple, the hysteresis \ncurve for 0.1  (in Fig. 3b ) demonstrates o nly a narrow step  corresponding to the state (B), while for =0.5 (Fig 3c) \nthere is a continuous transition between state (A) and its complete reversal. As the interaction ene rgy is further increased \n(negligible anisotropy),  1 , as in Fig. 3d, the hysteresis loop collapses. The ground state  of the array  at zero external  \nmagnetic field is the  vortex state ( state (2) in the transition scheme  of Fig. 3d). \n3.2. Ferromagnetic resonance  in single  layer systems  \nWe calculated  the dependence of the the FMR frequencies  of the three particle array on parameter r (the distan ce \nbetween center of particle and center of array). The increasing magnetostatic interaction  between particles leads  to a \nsplitting of  the FMR  spectrum. The depend ences of normalized FMR  frequencies  on the nondimensional parameter 3/V r  \nfor the region of parameter 0.1  are shown in Fig . 4.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4. The dependences of normalized FMR  frequencies  on the distance  between center of the particles  \nand array center for two possible magnetic moment configurations.  The FMR fre quencies are \nnormalized on the resonance  frequency of single particle.  \nAs is seen  from Fig.  4a the interaction between particles  in a high -energy  configuration  (A) leads to the splitting of  the \nFMR  spectrum  into two branches and shifting to lower frequencies. On the other hand, for the low-energy configuration  \n(B) the splitting into three branches  is observed  (Fig 4b). Two branches  are shifted to the higher frequencies  and one to \nlower frequencies . Since t he high -energy  configuration  (A) has a higher symmetry than configuration  (B) the lower branch \nof spectrum in Fig. 4a is doubly  degenerate .  \nThe three  elements  group  considered  above  can be combined into  the structures  with different symmetry , leading to \nadditional  changes  in the spectrum of  FMR. Here we  consider three  possible  spatial configurations  of association in  \nsymmetrical  two-dimensional arrays (Fig.  5) and discuss the  changes in the FMR spectra  from parameter r* (distance \nbetween centers  of particle groups ). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 5. Three spatial configurations with quasi -homogeneous state for association of three particle group . \nThe parameter r* is the distance between  canters  of particle groups. (I) \n(II) \n(III) r* 0,00 0,02 0,04 0,06 0,080,9951,0001,0051,0101,015 \n 0,00 0,02 0,04 0,06 0,080,9800,9850,9900,9951,000\n  \nV / r3 v / v0 \nV / r3 v / v0 \n(a) (b)  6  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 6. The dependences of normalized FMR  frequencies  on the distance between centers  \nof particle groups . \nIn the configuration  (I) arrays  interact mainly only by the two outermost particles.  This leads to an  additional  monotonic \nsplitting of  the high frequency  modes in the  FMR  spectrum  (see Fig.  6a and compare with Fig.  5b). A different situation  is \nobserved  for the configurations  (II) and (III).  The interaction  energy of  the arrays  in these configurations  is less  than in the \nconfiguration  (I), and as a result  lower frequency splitting  is observed  (see Fig. 6b and Fig. 6c). However  these \nconfigurations  implemented  a nonmonotonic changing FMR spectrum that is the effect of competition between interaction \nof three elements inside group and intergroup  interaction .  0,0 0,2 0,4 0,6 0,8 1,00,920,940,960,981,001,020,0 0,2 0,4 0,6 0,8 1,00,991,001,011,021,031,041,05 0,0 0,1 0,2 0,3 0,41,001,051,101,151,201,25\n  \n(a) v / v0 \nr / r* \n(b) \nr / r* v / v0 \n(c) v / v0 \nr / r*  7 4. The two-layer systems  \nThe magnetostatic interaction energy strongly depends on the distance between the particles and can be estimated as  \n2 2\nint 3M VWd . \nIn this connection the stack arrangement of e lliptical particles (one above the other , see Fig. 7a ) leads to an essential \nincrease in the interaction energy. In addition, it can be easy realize d technologically.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 7. The schematic image of two-layer stack  (a) and array of three two -layer particles  \nwith triangular symmetry  (b). \nWe investigated the  peculiarities of the magnetostatic  interaction  in an array  of two -layer  particles with  triangular  \nsymmetry ( Fig. 7b).  The dependences of magnetostatic interaction energies for different con figurations of magnetic \nmoments on the interlayer separation are presented in Fig. 8 (a). When the distance between the laye rs is large (parameter \nr / d is close to zero) the interaction between the layers is small and in this system three energy -degenerat e states (I), (II) \nand (III) are implemented. The degrees of degeneracy are (I) – 2; (II) – 2; (III) – 4 respectively (see Fig.  8b). The close \napproach of layers removes the  degeneracy and the energy spectrum is split into eight components due to the inter layer \ninteraction.  The energies of one group states are shifted to higher energies (u) and the energies of other state s are shifted to \nlower energies (d). In the region of parameters r / d 0.6 ÷ 1 we observe the crossing of energ y levels from completely \ndifferent configurations  (I, II and III)  as the interaction of particles in the layer becomes comparable with the interaction of \nthe particles located in different layers.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 8.  (a) is the  dependences of normalized magnetostatic energ ies for different configurations of \nmagnetic moments  on interlayer separation. (b), (c) and (d) a re possible configurations of \nmagnetic moments in the array of three two -layer particles.  The moment configurations \nfor the upper and lower layers are shown si de by side in different colors.  (b) d r d\n2 1 \n(a) \n(I) (II) \n(III) (u) (d) (u) (d) \n(u1) (u2) (d1) (d2) (c) (b) \n(d) (I) \n(III) (II) \nr / d  \n(a) (u) \n(d)  8 0,0 0,1 0,2 0,3 0,4 0,50,800,850,900,951,00 \n (a) \nV / d 3 v / v0 \n0,0 0,1 0,2 0,3 0,4 0,51,01,11,21,3\n  \nV / d 3 (b) v / v0 4.1. The ferromagnetic resonance in two -layer stack  \nWe calculated FMR  spectra  for different configurations of  a two -layer  stack. The  dependencies of  the FMR  frequency  \non the distance  between the layers  for the stack  in a st ate with  ferromagnetic and  antiferromagnetic order  are presented i n \nFig. 9. Qualitatively,  these curves  are similar to  the spectra of  two-dimensional array  of three particles (see Fig. 4), but the \ndegree of frequency splitting  for the stack  is much  more th an for two -dimensional  array.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 9.  The dependences of normalized FMR  frequencies  on the separation  between layers in the \ntwo-layer stack for different spatial configurations of moments. (a) is the case of \nferromagnetic ordering, (b) is for antiferromagnetic ordering.  \n4.2. Magnetization reversal  in two-layer  particle array  \nWe have calculated the hysteresis  curves  and the transition schemes between different states for two -layer  particle arrays  \nfor different values of the r / d ratio ( Fig. 10(a -c)). We show specifically results for the =0.5 case, for which a single layer \nthree island cluster displayed a continuous transition between type (A) moment configurations (see Fig. 3c). The effect of \nincreasing the ratio of the int erlayer to intralayer coupling can be seen in the sequence Fig. 3c (equivalent to r / d = 0), \nFig. 10a ( r / d = 0.45), Fig. 10b ( r / d = 0.58) Fig. 10c ( r / d = 0.83). At low interlayer interaction ( r / d = 0.45) we have the \nsituation similar to the one -layer system (Fig. 3c). In this case there is only one stable state with quasi -uniform orientation \nof magnetic moments in layers and ferromagnetic interlayer ordering. The in termediate case (Fig. 10b ) has  two stable states , \nwhich correspond to ferromagnetic and antiferromagnetic interlayer ordering. Increasing the interlayer coupling further \n(Fig. 10 c) leads to a closing up of the hysteresis loop, and the formation of an antiferromagnetic ordered st ate with zero \naveraged magnetic moment in zero field . Thus th e variation of distance between layers enables the effective controlling the \nmagnetic states in two -layer system .  9 \n(1) (2) (1) (2) \n(b) r / d = 0.58 \n     = 0.5 \n(1) (2) (1) (2) r / d = 0.83 \n     = 0.5 \n(c) \n(1) \n(a) (2) r / d = 0.45 \n     = 0.5 \n(1) (2)  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 10. The h ysteresis  curves  and the schemes of transitions  between different s tates of two -layer  particle \narray s for the case  α = 0.5 and two interlayer spacings:  (a) r / d =  0.45; (b) r / d =  0.58; ( c) r / d \n=  0.83. The moment configurations for the upper and lower layers are shown side by side in \ndifferent colors.   10 4.3. The f erromagnetic resonance in  two-layer systems  \nWe have  calculated the spectra for two stable states of two-layer  particles  array ordered on the lattice  with triangular \nsymmetry (Fig 11(a,b)). The calculations show ed that the two-layer arrays  have a more compl icated splitting  of the FMR  \nspectrum in comparison  with one -layer system . In particular the dependencies  of normalized FMR frequencies  of the array \nin the antiferromagnetic interlayer ordering have the  local extrem es at some distances  between the layers  (Fig.  11(b)). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 11. The depende nces of normalized FMR  frequencies  on the separation  between layers in the two -\nlayer three particle groups for different spatial configurations of moments.  (a) is for ferromagnetic \ninterlayer ordering.  (b) is for antiferromagnetic interlayer ordering . \n5. The three-layer systems  \nThe energy of interaction  in multilayer systems can be controlled  also by variation of the  magnetic  moment magnitude \nfor the particles  located  in different layers  [21]. Fig. 12 shows  a schematic drawing of a system of  three particles  arranged  \none above the other, and the  state diagram  of this system  depending on the  interlayer distance  and the ratio  of the magnetic \nmoments  of the particles. We assumed that  the moments  of the outer  particles  are the same. In such system  there are three  \nstable configurations with  different  magnetostatic  energy.  As can be seen  from Fig.  12b, at the critical point  with the \nparameters  12 13/d d = 0.5 and m2 / m1 = 0.2 there is  a frustrated  state,  in which the energ ies of all magnetic moment s \nconfigurations are equal.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 2. The three -layer stack (a) and the diagram of possible states in dependence of magnetic moments  \nrelation and separations between layers  (b). \nd12/d13 \nm2 / m1 (m1 = m3) (b) 12d\n23d13d\n3 2 1 (a) 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,40,900,951,001,05\n  \n(a) v / v0 \nr / d 0,0 0,2 0,4 0,6 0,8 1,01,0001,0251,050\n  v / v0 \nr / d (b)  11 0,00 0,02 0,04 0,06 0,081,001,021,041,06 \n (b) \nV / d 3 v / v0 \n0,00 0,02 0,04 0,06 0,080,960,981,00\n  \n(a) \nV / d 3 v / v0 \n0,0 0,1 0,2 0,3 0,4 0,50,80,91,01,11,21,3\n  \nV / d 3 (c) v / v0 \n0,0 0,4 0,8 1,2 1,60,500,751,001,251,501,752,00\n  \nV / d 3 v / v0 \n(d) 5.1. The ferromagnetic resonance in  three -layer stack  \nWe calculated the FMR  spectra  for different configurations of  a three -layer  stack.  Fig. 13 shows the  dependenc es of the \nFMR  frequency  on the distance  between the layers  for the stack  in a state with  ferromagnetic , antiferromagnetic  and mixed \nferromagnetic - antiferromagnetic order ing. Qualitatively,  these curves  are similar to  the spectra of  the two-dimensional \narray  of three particles ( Fig. 4), but the degree of frequency splitting  for the stack  is much  more than for two -dimensional  \narray.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 3. The dependences of normalized FMR  frequencies  on the separation  between layers in the three -layer \nstack for different spatial configurations of moments. (a) is the case of ferromagnetic ordering, (b) is \nfor antiferromagnetic  ordering , (c) is for mixed ferromagnetic - antiferromagnetic ordering and (d) is \nfor the stack with the reduced magnetic moment of the middle layer  (m1= m3, m2 / m1 = 0.5) . \nAlso we have  considered the stack with differe nt mag netic moments in the layers ( Fig. 13d). The r elation of magnetic \nmoment of central layer m2 to moments of outer laye rs (m1 = m3) is equal 0.5. In this case we have initial splitting due to \nthe difference of magnetic moments and t he dependences of normalized FMR  frequenc ies have a local extremum.  Thus \nallowing difference s between the magnetic moments of layers provides an  additional way of control ling the FMR spectrum.  \n5.2. Magnetization reversal  in three -layer particle array  \nThe results of hysteresis calculations for three -layer stacks of three i dentical particle arrays on a triangular lattice are \npresented in Fig. 14. We again consider the =0.5 case and display the hysteresis plots for different interlayer spacing: (a) \nr / d = 0.45;  (b) r / d = 0.58;  (c) r / d = 0.83. When the interlayer couplin g is not strong ( r / d = 0.45), the system implements \na simple magnetization reversal process through the state with antiferromagnetic ordering of the mag netic moments in \nneighboring layers (Fig. 14a). Increasing the interlayer interaction leads to additio nal configuration states encountered \nduring the hysteresis cycle, as shown in Fig. 14b for r / d = 0.58. In this case the state (1) becomes unstable. The transition \nfrom state (1) to state (2) involves just a flip of the moment of one particle in the centr al layer (indicated in red in scheme). \nThe other two particles in the central layer then experience a flip in the transition to state (3). The low average moment \nstate (3) is essentially the same as state (2) in Fig. 14a. The small average moments in the t wo plots are slightly different \nbecause of slight differences in deviations from easy axis alignment. The transition from state (3) to state (4) involves \nmoment reversals in all three layers, so we have the same state  as (3)  but with the opposite directed average magnetic \nmoment. Fig.  14c shows the behavior as the interlayer coupling is increased still further to r / d = 0.83. The transition from \nstate (1) to state (2) is similar to that in Fig. 14b, but in this case state (2) becomes unstable. Here we have  only one \nantiferromagnetic ordered stable state (3) similar to the state (3) in Fig.  14b. Again the transition from state (3) to state (4) \ninvolves moment reversals in all three layers, so we have practically the same state with the opposite directed aver age \nmagnetic moment.   12 As the number of layers is increased, the number of states traversed in a hysteresis cycle also incr eases, reflecting the \nmore complex sequence of moment reorientation processes that is taking place.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 4. The h ysteresis  curves  and the schemes of transitions  between different states of three -layer particles \narray  for  = 0.5 case:  (a) r / d = 0.45; (b) r / d = 0.58; (c) r / d = 0.83 . The c onfiguration  of the \nmagnetic moments  of the particles  corresponds to the points on  the hysteresis curve . The one -particle \nflip in transition schemes in the pictures (b) and (c) is indicated in red.  \n(1) \n(3) \n(2) \n(3) (2) (b) r / d = 0.58 \n     = 0.5 (1) \n(2) \n(1) (2) (a) r / d = 0.45 \n     = 0.5 \n(1) \n(3) \n(2) \n(2) (3) (c) r / d = 0.83 \n     = 0.5 (4) \n(4)  13  \n5.3. The ferromagnetic resonance in  three -layer system  \nThe Fig. 15 shows the  dependencies  of the FMR  frequenc ies on the distance  between the layers  for three -layer  particle \narray  in the state with  ferromagnetic (Fig. 1 5a), antiferromagnetic interlayer order  (Fig. 1 5b), in the state with partial \nferromagnetic and antiferromagnetic ordering (Fig. 1 5c) and for the system with different magnetic moments in \nantiferromagnetic configuration  (Fig. 1 5d). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 5. The dependences of normalized FMR  frequencies  on the separation  between layers in array of the \nthree -layer  stacks for different spatial configurations of moments. (a) is for the ferromagnetic ordering. \n(b) is for antiferromagnetic ordering.  (c) is for the state with partial ferromagnetic  and antiferromagnetic \nordering.  (d) is for the stack s with the reduced ma gnetic mo ment of the middle layer (m1= m3, m2 / m1 = \n0.5) in the state with  antiferromagnetic interlayer ordering . The middle layer with the reduced magnetic \nmoment is highlighted in red.  \nAnalogous to the two -layer system the dependencies of FMR spectra  on the distance between the layers for three -layer \nsystem are nonmonotonic as a consequence of competition between the interlayer  and interlayer interactions. Note that the \nuse of the stacks with d ifferent layer thickness or different layer materials substan tially extends the possibilities in \ncontrolling of ferromagnetic resonance in such systems.  \n6. Discussion   \nThus, the calculations showed that the magnetostatic interaction between the particles in the layer and the interlayer \ninteraction strongly affect t he magnetization reversal dynamics in an external magnetic field and lead to the complex \nsplitting of FMR spectra.  \nIn the calculations, we used a simple model of anisotropic point dipoles. However, the real particle s are distributed \nsystems with a complex  character of the magnetic interaction s and this leads to inhomogeneous magnetization reversal 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,40,9250,9500,9751,0001,025 \n r / d (a) v / v0 \n0,0 0,2 0,4 0,6 0,8 1,0 1,20,9751,0001,0251,0501,075 \n r / d (b) v / v0 \n0,0 0,2 0,4 0,6 0,8 1,0 1,20,9500,9751,0001,0251,050\n  \nr / d (c) v / v0 \n0,0 0,4 0,8 1,2 1,6 2,00,500,751,001,25\n  \nr / d v / v0 \n(d)  14 modes and complication of the FMR spectrum. Along with the uniform coherent precession modes in such  systems , there  \nexist nonhomogeneous modes [22 -24] associated  with the internal spin resonances and edge effects. However for the \nsystems consisting of small magnetic nanoparticles the FMR spectrum of ground modes of uniform prece ssion and the \nbasic mechanisms of the splitting of resonance frequencies are described adequately by the simple model  we have used . \nIt was shown, that the systems consisting of one -layer elements with only in -plane interaction s display  the complicated \nbut weak splitting of FMR spectra (Figs. 4,6). The arrangement of elliptical discs in multi layer stacks systems leads to the \nconsiderable increase of interparticle magnetostatic interaction and as a consequence to the appeara nce of unusual states \nduring magnetization reversal and significant changes in the FMR spectra. It allows us to construct the different systems \nwith fine tuning of ferromagnetic resonance. For example, we have demonstrated that in two -layer system (consisting of \nthree two -layer stacks, located at 120°) there are two stable states (Fig. 10). The first state has a quasi -uniform  moment’s \ndistribution (with ferromagnetic interlayer ordering)  and can be realized in a system with low interlayer interact ion \n(Fig.  10a). The second state has zero average magnetic moment (with antiferromagnetic interlayer ordering)  and can be \nrealized i n the system with high interlayer interaction (Fig. 10c). In the system with intermediate interlayer  interaction  there \nare both these states, which can be switched by magnetization reversal in an external magnetic field (Fig. 10b). The FMR \nspectra correspo nding to these two states are presented in Fig. 11 and have the fundamental differences. The quasi -uniform \nstate has the spectrum shifted to lower frequencies, while the state with zero average moment has th e spectrum with four \nbranches shifted to higher f requencies. Therefore such two -layer particles arrays with simply recognized magnetic states \ncan be used in the magnetic labeling systems.  \nWe have shown that adding a third layer in the stack leads to an extra frustration of magnetic state s and complicated  \nsplitting in FMR spectra. Depending on the interlayer coupling in such systems the states with ferro magnetic, \nantiferromagnetic and mixed ferromagnetic - antiferromagnetic interlayer ordering are realized (Fig.  14). This significantly \nexpands the possibil ities to control of the FMR spectrum in such systems (Fig. 15 (a-c)). Moreover the FMR spectrum \nsplitting in three -layer systems is considerably larger than in the case of two -layer systems.  \nThe great perspectives for the engineering of magnetic states and  FMR spectra are associated with the possibility of \nvarying the ratio of magnetic moments for the particles belonged to different layers and interlayer spacings. In particular, \nwe have considered the three -layer system with a reduced magnetic moment of the  particles in the middle layer. Such \nsystems can achieve an increase the splitting of the spectra due to the initial frequency splitting for the differed particles \nforming the array (Fig. 15d).  \n \n7. Conclusion  \n \nThus we have demonstrated that multilayer  ferromagnetic nanoparticles  significantly expand  the opportunities to create  \nthe magnetically frustrated systems . The different spatial arrangement, variation of magnetic moments and interlayer \nspacings enable the fine tuning of possible stable magnetic states  and effective control of FMR  spectrum in such systems  \nthat is very important from the standpoint of practical applications .  \nAcknowledgement s \nAuthors thanks K.D.  Bessmertni y for assistance and A.A.  Fraerman for the useful  discussion s. \nThis work was suppor ted by grants of Russian Foundation for Basic Research , the p rogram s of Russian Academy of \nSciences  and The Ministry of Education  and Science of Russian Federation . \nReferences   \n1. R.F. Wang, C. Nisoli, R.S. Freitas, J. Li, W. McConville, B.J. Cooley, M.S. Lun d, N. Samart h, C. Leighton, V.H. \nCrespi, P.  Schiffer - Artificial “spin ice” in a geometrically frustrated lattice of nanoscale ferromagnetic islands , Nature, \n439, 303 -306 (2006).  \n2. C. Nisoli, R. Wang, J. Li, W.F. McConville, P.E. Lammert, P. Schiffer, V.H. Crespi - Ground state lost but degeneracy \nfound: The effective thermodynamics of artificial spin ice , Phys. Rev. Lett., 98, 217203 (2007).  \n3. A. Remhof, A. Schumann, A. Westphalen, H. Zabel, N. Mikuszeit, E.Y. Vedmedenko, T. Last, U. Kunze - \nMagnetostatic int eractions on a square lattice // Phys. Rev. B, 77, 134409 1 -6 (2008).  \n4. A. Westphalen, A. Schumann, A. Remhof, H. 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"    },    {        "title": "2110.01318v3.Lumped_circuit_model_for_inductive_antenna_spin_wave_transducers.pdf",        "content": "Lumped circuit model for inductive antenna spin-wave\ntransducers\nFrederic Vanderveken,1, 2,\u0003Vasyl Tyberkevych,3Giacomo Talmelli,1, 2\nBart Sor\u0013 ee,1, 4, 5Florin Ciubotaru,1and Christoph Adelmann1,y\n1Imec, 3001 Leuven, Belgium\n2KU Leuven, Departement Materiaalkunde, SIEM, 3001 Leuven, Belgium\n3Department of Physics, Oakland University, Rochester, Michigan 48309, USA\n4KU Leuven, Departement Elektrotechniek, TELEMIC, 3001 Leuven, Belgium\n5Universiteit Antwerpen, Departement Fysica, 2000 Antwerpen, Belgium\n(Dated: December 1, 2021)\nAbstract\nWe derive a lumped circuit model for inductive antenna spin-wave transducers in the vicinity of a\nferromagnetic medium. The model considers the antenna's Ohmic resistance, its inductance, as well\nas the additional inductance due to the excitation of ferromagnetic resonance or spin waves in the\nferromagnetic medium. As an example, the additional inductance is discussed for a wire antenna\non top of a ferromagnetic waveguide, a structure that is characteristic for many magnonic devices\nand experiments. The model is used to assess the scaling properties and the energy e\u000eciency of\ninductive antennas. Issues related to scaling antenna transducers to the nanoscale and possible\nsolutions are also addressed.\nKeywords: spin waves, inductive antenna, equivalent circuit, equivalent impedance, radiation resistance\n1arXiv:2110.01318v3  [cond-mat.mes-hall]  30 Nov 2021I. INTRODUCTION\nFerromagnetic resonance (FMR) and spin waves are precessional collective excitations\nof the magnetization in ferro-, ferri-, or antiferromagnetic media.1{3Historically, FMR has\nfound applications in microwave resonators, oscillators, and \flters based on Yttrium iron\ngarnet (YIG).4,5In recent years, much research has been devoted to miniaturizing spin-wave\nand FMR-based devices and structures in the growing \feld of (nano-)magnonics, with po-\ntential applications in spintronic computation6{15and scaled microwave analog systems.16{18\nIn magnonic devices, FMR or spin waves are excited by microwave electrical signals,\nemploying transducers between electric (microwave) and magnetic (spin wave) domains.\nWith alternatives based on spin-orbit torques19or magnetoelectrics20{22only emerging, the\nvast majority of devices employs inductive antennas as transducers. Inductive antennas\nexcite FMR or spin waves using oscillating Oersted magnetic \felds generated by microwave\ncurrents and can also be utilized as a spin-wave detectors that inductively convert spin waves\nor FMR to microwave signals.\nFor practical applications, the power transmission from the peripheral microwave circuit\ninto the magnetic system is of great importance. Historically, microstrip waveguides in\ncombination with insulating ferrite \flms have been used typically in spin-wave devices. For\nsuch devices, distributed models of have been formulated that describe quantitatively the\nchange in microwave impedance of the microstrip due to the excitation of FMR or spin\nwaves.3Recently, micro- and nanomagnonic devices have however frequently used wire or\nU-shaped inductive antennas connected to coplanar waveguides since such structures can\nbe more easily miniaturized.14,18,23{32For such antennas, the microwave behavior is better\ndescribed by a lumped impedance of the inductive antenna and represented by an equivalent\ncircuit. In this framework, the coupling to FMR and the spin-wave system can be described\nby a radiation impedance, in an analogous way to conventional electromagnetic antennas.33,34\nThe equivalent circuit can be used to quantitatively assess the spin-wave excitation and as\na starting point for the design and optimization of inductive antennas. Thus, equivalent\ncircuits are crucial, when such inductive antennas are to be included in spin-wave-based\nmicrowave or spintronic (logic) systems. This approach can ultimately be used in SPICE-\nlike device models to simulate the behavior of magnonic or spintronic systems,35,65including\nperipheral microwave circuits.\n2Several works have studied the radiation impedance of inductive antennas for particular\ncases. Ganguly et al. \frst derived the spin-wave resistance and reactance for magneto-\nstatic surface spin waves excited by a microstrip with uniform current density by solving the\nMaxwell equations with appropriate boundary conditions in all layers.36,37Subsequent work\nhas addressed gradually more complex structures and presented alternative derivations. Em-\ntage reported a derivation based on surface permeabilities38whereas Kalinikos introduced\na Green's function method to determine the spin-wave power and impedance.39Also non-\nuniform current densities were considered,40{44as well as \fnite dimensions of the magnetic\nmedium.3,39,40,42,45Furthermore, the spin-wave radiation impedance has also been derived\nfor forward volume spin waves3,38,39,44{48as well as backward volume waves.3,43,44,46,48,49Nev-\nertheless, all these approaches require numerical methods to \fnd the impedance and do not\nallow for simple approximate expressions that can be implemented in electrical models and\nequivalent circuits.\nThis paper presents an analytical compact lumped element model of an inductive antenna\nspin-wave transducer emitting dipolar-exchange spin waves. The model includes magnetic\nrelaxation and can describe arbitrary static magnetization orientations. It allows for a com-\nplete representation of the transducer as an electric element, which can e.g. be used in\nmicrowave circuit simulations to improve the impedance matching with a connected waveg-\nuide. The model is then used to study the consequences of scaling the dimensions of an\ninductive wire antenna and the in\ruence of the magnetic material parameters. Finally, the\nmaximum power transfer from the microwave to the magnetic domain is quanti\fed for an\ninductive wire antenna, and implications of the wire antenna and waveguide scaling on the\nmaximum power transfer are discussed.\nII. LUMPED EQUIVALENT CIRCUIT MODEL OF INDUCTIVE ANTENNA\nSPIN-WAVE TRANSDUCERS\nThe paper is organized as follows: this section presents the derivation of the equivalent\nimpedance of an arbitrarily shaped antenna that is placed in the vicinity of a magnetic\nmaterial. Thereafter, the impedance model is applied in Sec. III to the special cases of\na magnetic thin \flm or waveguide and a straight wire antenna, con\fgurations that are\ncommonly used in magnonic experiments. Section IV examines the impedance spectra and\n3the scaling behavior of wire antennas on top of a CoFeB waveguide as a concrete example.\nFinally, Sec. V discusses the optimum power transfer e\u000eciency from the microwave to the\nmagnetic domain, including for the above special case, and the dependence on antenna and\nwaveguide dimensions.\nA. General geometry and model assumptions\nWe consider a ferro- or ferrimagnetic medium, in which FMR or spin waves can be excited.\nThe static magnetization of the medium and the static magnetic bias \feld are assumed to be\nparallel. Their orientation with respect to the geometry of the structure and the direction\nof spin-wave propagation can be variable and strongly determines the spin-wave properties.\nThe model presented here is valid for all con\fgurations. Several common speci\fc cases are\ndiscussed below.\nFMR or spin waves are excited by an inductive antenna with arbitrary shape and current\ndistribution. The microwave current in the antenna generates an oscillating Oersted mag-\nnetic \feld, which interacts with the magnetic layer and excites the magnetization dynamics,\ni.e.FMR or spin waves. Hence, there is a net energy \row from the microwave current inside\nthe antenna to the magnetic medium. Our model treats spin waves as linear excitations of\nthe magnetization and therefore neglects nonlinear e\u000bects. Spin waves in the linear regime\nare weak perturbations of the magnetization and thus carry small amounts of energy. In\nmagnonic device applications, antenna transducers convert electrical signals to the mag-\nnetic domain (and vice versa ). For ultralow power applications, large energy \rows into the\nspin-wave system are neither required nor desirable. However, even for signal conversion,\nhigh transducers e\u000eciencies are essential to avoid large losses during transduction and the\nassociated large external power requirements of the devices.\nFor simplicity, the antenna length `is assumed to be much smaller than the wavelength of\nan electromagnetic wave at the excitation frequency, i.e.`\u001c\u0015EM. In this limit, the phase of\nthe electric current density along the antenna is constant and the transducer can be described\nby a lumped equivalent circuit rather than by a distributed-element model. Hence, the active\nand reactive parts of the energy \row can be captured in a lumped equivalent impedance.\n4B. Impedance of an inductive antenna\nA microwave (radio-frequency, RF) current Iinside an inductive antenna generates an os-\ncillating magnetic Oersted \feld Ha(r) =ha(r)Ithat can excite both resonant and nonreso-\nnant (evanescent) spin waves in an adjacent magnetic medium. For typical (nano-)magnonic\ndevices, the Oersted \feld can be calculated in the magnetostatic approximation, which is\nvalid when the structure's dimensions and the spin-wave wavelength are much smaller than\nthe electromagnetic wavelength in vacuum ( \u001810 cm at GHz frequencies). Then, the in\ru-\nence of the time-varying electric \feld in the Maxwell equations can be neglected and the\nmagnetic \feld can be calculated by the Biot{Savart law. For an arbitrary current distribu-\ntionJ(r) =j(r)I, the magnetic \feld distribution hacan then be written as\nha(r) =1\n4\u0019Z\nj(r0)\u0002r\u0000r0\njr\u0000r0j3d3r0(1)\n=Z\nrrG(r;r0)\u0002j(r0)d3r0; (2)\nwith\nG(r;r0) =1\n4\u00191\njr\u0000r0j(3)\nthe Green's function of the Laplace equation r2\nrG(r;r0) =\u0000\u000e(r\u0000r0) and\u000e(r) the Dirac\ndelta function. For simple geometries, the magnetic \feld distribution generated by the\nantenna hacan also be obtained directly from Amp\u0012 ere's law, rr\u0002ha(r) =j(r) together\nwith symmetry considerations, without the use of Eq. (2). Below, in the description of the\nmagnetization dynamics, we assume that ha(r) is known. The feedback of the magnetization\ndynamics (FMR or spin waves) on the antenna current distribution is thus neglected in this\nmodel.\nIn addition to the Oersted magnetic \feld generated by the inductive antenna, magneti-\nzation dynamics ( e.g. FMR or spin waves) also generate a time-varying dipolar magnetic\n\feld. Both \felds in\ruence the voltage across the antenna via electromagnetic induction, i.e.\nvia an electromotive force (EMF). The impedance of the antenna can then be found by the\nratio of the voltage across the antenna and the current, i.e.Zeq=V=I. The total voltage\ndrop is the sum of EMF contributions and the Ohmic resistance of the antenna. Hence, its\nequivalent impedance can be written as\nZeq=R\n+Vind\nI; (4)\n5withR\nthe Ohmic antenna resistance and Vindthe EMF-induced voltage. Here, we neglect\nelectromagnetic radiation losses in the nonmagnetic surroundings, which is valid in the\nmagnetostatic limit. However, these can be added to the model as an additional impedance\ncontribution.33,34\nThe EMF-induced voltage originates from the time-varying magnetic \ruxes generated by\nthe Oersted and dipolar magnetic \felds due to FMR or spin waves. By applying Stokes'\ntheorem to the Maxwell{Faraday equation, the induced voltage at angular frequency !=\n2\u0019fcan be written as a function of the total \rux, leading to\nVind=I\nCaE(r)dl=\u0000i!Z\nSaB(r)d2a=\u0000i!\b; (5)\nwithSathe surface enclosed by the current loop, \b the magnetic \rux through this loop,\nCa=@Sa, andBthe magnetic induction. Considering the magnetic vector potential A\nde\fned by B=r\u0002A, the magnetic \rux can be written as\n\b =I\nCaA(r)dl: (6)\nFor a current distribution J(r) =j(r)I, the line integral can be converted to a volume\nintegral by averaging the magnetic vector potential weighted by the current density. This\nresults in\n\b =Z\nA(r)\u0001j(r)d3r\n=ZZ\nA(r)\u0001j(r0)\u000e(r\u0000r0)d3rd3r0\n=\u0000ZZ\nA(r)\u0001j(r0)r2\nrG(r;r0)d3rd3r0\n=\u0000ZZ\nr2\nrA(r)\u0001j(r0)G(r;r0)d3rd3r0: (7)\nIn the Lorenz gauge, the electrical current and the curl of the magnetization Mcan be\ninterpreted as sources of the magnetic vector potential that satis\fes the Poisson equation\nr2\nrA(r) =\u0000\u00160[j(r)I+rr\u0002M(r)]: (8)\nAfter substituting this into Eq. (7), the total \rux \b can be written as the sum of the \rux\noriginating from the current \b 0and the \rux originating from the curl of the magnetization\n\bm\n\b = \b 0+ \bm=L0I+LmI; (9)\n6withL0the conventional self-inductance of the antenna and Lmthe additional inductance\ngenerated by FMR or spin waves. The self-inductance is given by\nL0=\u00160ZZ\nj(r)\u0001j(r0)G(r;r0)d3rd3r0(10)\nand can be readily calculated when the current distribution is known. To obtain an expres-\nsion forLm, the magnetic \rux needs to be considered. Integration by parts and substitution\nof Eq. (2) leads to\n\bm=\u00160ZZ\n[rr\u0002M(r)]\u0001j(r0)G(r;r0)d3rd3r0\n=\u00160ZZ\nM(r)\u0001[rr\u0002j(r0)G(r;r0)]d3rd3r0\n=\u00160Z\nM(r)\u0001\u0014Z\nrrG(r;r0)\u0002j(r0)d3r0\u0015\nd3r\n=\u00160Z\nM(r)\u0001ha(r)d3r: (11)\nFor linear magnetization dynamics described by the linearized Landau|Lifshitz|Gilbert\nequation, the dynamic magnetization can be written as50\nM(r) =Z\n^\u001f!(r;r0)Ha(r0)d3r0\n=IZ\n^\u001f!(r;r0)ha(r0)d3r0: (12)\nHere, ^\u001f!(r;r0) is the dynamic magnetic susceptibility tensor and hathe dynamic Oersted\n\feld. Substitution of Eq. (12) into Eq. (11) results in an excess inductance generated by\nFMR or spin waves that is given by\nLm=\bm\nI=\u00160ZZ\nhT\na(r0)^\u001f!(r;r0)ha(r)d3rd3r0: (13)\nThis is the general expression to determine the spin-wave generated partial inductance in-\ntroduced in an arbitrary shaped conductive wire. Hence, to \fnd Lm, one needs to know the\nantenna's \feld distribution and the spin-wave susceptibility. Note that the self-inductance\ncan also be expressed in a similar form\nL0=\u00160ZZ\nhT\na(r0)^\u001f0(r;r0)ha(r)d3rd3r0(14)\nwith ^\u001f0(r\u0000r0) =\u000e(r\u0000r0)Ithe vacuum susceptibility and Ithe identity matrix.\n7public\n2\nR0\nL0\nRs\nRm\nXmL0 RΩ RmXmFIG. 1: Lumped equivalent circuit representation of an inductive antenna spin-wave transducer.\nThe equivalent circuit consists of an Ohmic series resistance ( R\n), a self-inductance ( L0), as well\nas a radiation resistance ( Rm) and reactance ( Xm) that capture the power coupling to FMR or\nspin waves.\nBased on this model, an inductive antenna can therefore be represented by a circuit\ncomprising its Ohmic resistance R\n, its self-inductance L0, as well as an additional partial\ninductance Lmthat stems from the coupling to the magnetic system. As shown below,\nLm=L0\nm+iL00\nmis in general complex because the dynamic magnetic susceptibility tensor\n^\u001f!(r;r0) is complex. The impedance Zm=i!Lmcan therefore be divided into a resistance\nRm=\u0000!L00\nmand a reactance Xm=!L0\nm. A schematic of the equivalent circuit is shown in\nFig. 1. In this equivalent circuit, the radiation resistance Rmdescribes the dissipation due\nto spin-wave or FMR excitation and can therefore be used to describe the energy transfer\nbetween electric and magnetic domains. This will be used below to discuss the energy\ne\u000eciency and the scaling behavior of inductive antennas.\nIII. INDUCTIVE ANTENNA ON TOP OF A FERROMAGNETIC WAVEGUIDE\nWITH UNIFORM MAGNETIZATION DYNAMICS\nA. Additional inductance due to spin-wave or FMR excitation\nAn inductive antenna on top of a ferro- or ferrimagnetic thin-\flm waveguide is a common\nbasic structure in many (nano-)magnonic experiments. We therefore apply the above general\nmodel to the case of a magnetic waveguide with thickness t. The spin waves are assumed\nto propagate along the x-axis, whereas the z-axis is chosen perpendicular to the waveguide\nsurface. We assume that the waveguide thickness tis much smaller than the spin-wave\nwavelength \u0015,i.e.kt\u001c1, withk= 2\u0019=\u0015, as this results in uniform magnetization dynamics\n8over the thickness. Furthermore, we also assume that there is no lateral mode formation, and\nthe magnetization dynamics are uniform over the width of the waveguide. This is realized in\nnarrow waveguides when the spin-wave wavelength is much larger than the waveguide width.\nFor wider waveguides, the boundary conditions at its edges leads to the formation of spin-\nwave modes with nonuniform magnetization dynamics over the waveguide width, which are\ndi\u000ecult to treat analytically. However, for \frst-order spin-wave modes in su\u000eciently wide\nwaveguides (with extended thin \flms as a limit), edge e\u000bects become again negligible, and the\nmagnetization dynamics become approximately uniform. Hence, the analytical calculations\nare appropriate both for narrow and wide waveguides with widths Wso thatkW\u001c1 or\nkW\u001d1. For waveguides with kW\u00181, mode formation cannot be neglected, and numerical\ncalculations are required to assess the inductance of an inductive antenna.\nFor uniform magnetization dynamics, the integration over the width and thickness direc-\ntions can be replaced by a multiplication with the structure's cross section\nLm=\u00160tWZZ\nhT\na(x0)^\u001f!(x;x0)ha(x)dxdx0; (15)\nwithtthe thickness and Wthe width of the waveguide.\nThe result can be simpli\fed by writing the expression in reciprocal space. The dynamic\nsusceptibility is translationally invariant, i.e.it only depends on the distance between two\npoints randr0, and can therefore be written as ^ \u001f!(r;r0) = ^\u001f!(r\u0000r0;0). Taking the Fourier\ntransform, applying Plancherel's theorem, and considering translational invariance results\nin\nLm=\u00160tWZ\nhT\na(k)^\u001f!(k)h\u0003\na(k)dk\n2\u0019: (16)\nFor linear magnetization dynamics, the susceptibility in reciprocal space can be written as\n^\u001f!(k) =^\n(k)2\n!2\nr(k)\u0000!2+i!\u0000(k); (17)\nwith!r(k) = 2\u0019fr(k) the spin-wave dispersion relation, \u0000( k) the spin-wave damping rate,\nand ^\n(k) a tensor dependent on the magnetic bias \feld and wavenumber k. Analytical\nexpressions of these quantities can be found in Appendix A.\nFor low damping, i.e. for Gilbert damping constants \u000b\u001c1, Eq. (17) can be further\napproximated by\n^\u001f!(k)\u0019^\n(k)\n!r(k)\u0000!+i\u0000(k): (18)\n9Using this approximation, an analytical solution to the integral in Eq. (16) can be found\nLm=i\u00160tW\njvg(k!)jh\nhT\na(k!)^\n!(k!)h\u0003\na(k!)i\n; (19)\nwithk!the wavevector at the spin-wave resonance for angular frequency != 2\u0019f,i.e.\n!r(k!) =!, andvg(k) =@!r(k)=@k the spin-wave group velocity. Note that reciprocal\nspin-wave propagation was assumed in the integration, which means that both propagation\ndirections of excited spin waves \u0006k!provide equal contributions to Lm. This is a valid\nassumption if the waveguide thickness is much smaller than the spin-wave wavelength, i.e.\nk!t\u001c1.\nEquation (19) also takes into account nonresonantly excited (evanescent) spin waves.\nFurthermore, unlike the self-inductance L0, the spin-wave inductance Lmis generally a\ncomplex quantity with both real and imaginary parts, i.e.Lm=L0\nm+iL00\nm. The real\npart leads to a reactance in the equivalent circuit, Xm=!L0\nm, whereas the imaginary part\ncorresponds to a resistance, Rm=\u0000!L00\nm, as discussed above. Combining the previous\nresults in Eq. (4) in the framework of the equivalent circuit in Fig. 1 leads to an impedance\nof an antenna above a magnetic thin \flm or waveguide of\nZeq=R\n\u0000i!(L0+Lm) =R\n\u0000Rm\u0000i!(L0+L0\nm); (20)\nwithLmgiven by Eq. (19).\nB. Two-antenna con\fgurations: Mutual inductance due to propagating spin-\nwaves\nThe above derived model can be extended to describe the mutual inductance between\ntwo inductive antennas connected via a magnetic waveguide. As explained above, the mag-\nnetization dynamics generates a dipolar magnetic \feld which induces an additional voltage\nin the antenna. The magnetization dynamics is not localized near the exciting inductive\nantenna but spreads over the magnetic medium. This can be interpreted as spin waves that\npropagate in the magnetic medium away from the source. A second \\receiving\" inductive\nantenna that is placed at some distance Dfrom the excitation antenna then also feels the\ndynamic dipolar \feld associated with the magnetization dynamics. Consequently, this \feld\nalso induces a voltage in the receiving antenna. The interaction between the two anten-\nnas can be represented by a mutual inductance M. The derivation of Mis analogous to\n10public3Wtw\nd\nCoFeBxzy\nsHaFIG. 2: Schematic of the studied geometry: wire antenna (yellow) on top of a ferromagnetic narrow\nwaveguide or thin \flm (blue). The static magnetization inside the waveguide or thin \flm can take\nany arbitrary direction in the general model.\nthe above derivation of the additional magnetic inductance Lm. However, here, a space\nseparationDbetween the two antennas should be considered in equation Eq. (15)\nM=\u00160tWZZ\nhT\na(x0)^\u001f!(x;x0)ha(x\u0000D)dxdx0; (21)\nIn reciprocal space, this leads to a phase accumulation of kDduring propagation and gives\nM=\u00160tWZ\neikDhT\na(k)^\u001f!(k)h\u0003\na(k)dk\n2\u0019\n=\u00160tWe\u0000kiDZ\neikrDhT\na(k)^\u001f!(k)h\u0003\na(k)dk\n2\u0019: (22)\nHere,k=kr+ikiis the wavevector of the propagating spin wave. E\u000bects of magnetic\ndamping are captured by the imaginary component of the complex wavevector, resulting in\nan exponential decay of Mwith increasing inter-antenna distance D. We note that near\nFMR (k\u00190), the mutual inductance becomes equal to the additional spin-wave induced\ninductance of the excitation antenna, i.e. M\u0019Lm.\nC. Radiation impedance of a wire antenna on top of a ferromagnetic narrow\nwaveguide with uniform magnetization dynamics\nAs a more concrete structure, we now consider a straight wire antenna with rectangular\ncross section at a distance sabove a magnetic thin \flm, as used in various spin-wave prop-\nagation experiments.14,18,23{32The structure and the geometry of the studied structure are\ndepicted in Fig. 2. To simplify the system, we assume a uniform current density inside the\nantenna, i.e.we neglect the impact of the skin e\u000bect. This is justi\fed as long as the antenna\n11width or thickness are much smaller than the skin depth \u000e=q\n\u001a\n\u0019f\u0016at frequency f. Here,\n\u001ais the resistivity of the antenna metal and \u0016=\u00160\u0016rits permeability. For Cu, the skin\ndepth is of the order of 1 \u0016m at GHz frequencies (larger for poorer conductors) and thus\nthe approximation is acceptable for the nanoscale antennas considered below. The Ohmic\nresistance of the antenna is then described by Pouillet's law\nR\n=\u001a`\ndw; (23)\nwith`,d, andwthe antenna length, thickness, and width, respectively. If the skin e\u000bect\ncannot be neglected, an e\u000bective area Ae\u000bbased on the current distribution inside the\nantenna needs to replace dwin Eq. (23). The self-inductance for this geometry can be\napproximated by51\nL0=\u00160`\n2\u0019\u0014\nln\u00122`\nw+d\u0013\n+1\n2\u0015\n: (24)\nTo evaluate the additional inductance Lmin Eq. (16) that originates from the excitation\nof FMR or spin waves, the Oersted \feld of the antenna, Ha, needs to be determined. A\nsimple analytical expression for Hacan be found when two approximations hold: (i) the\n\flm thickness tis smaller than the sum of the antenna thickness and the separation d+s\nwhich, (ii) is in turn assumed to be smaller than the antenna width w,i.e.t\u001cd+s\u001cw.\nFor this geometry, the Oersted \feld inside the magnetic waveguide can be approximated by\nHa(r) =I\n2(w+\u0019s)\u0005\u0010x\nw\u0011\n^ex; (25)\nwith \u0005 (x) the rectangular function (de\fned as 1 for jxj\u00141\n2and 0 forjxj>1\n2) and ^exthe\nunit vector in the x-direction. The spatial Fourier transform of this \feld is given by\nHa(k) =I\nk(w+\u0019s)sin\u0012kw\n2\u0013\n^ek: (26)\nWhenw\u001cs, the Oersted \feld distribution becomes a sinc-function\nha(k) =1\n2sinc\u0012kw\n2\u0013\n^ek: (27)\nUsing Eq. (16) and further assuming s\u001cw, the partial spin-wave inductance can then be\nwritten as\nLm=\u00160tWZ\n\u001f!;xx(k)\u00141\n2sinc\u0012kw\n2\u0013\u00152dk\n2\u0019: (28)\n12Multiplying the inductance with the angular frequency results in the spin-wave impedance\nZm=Rm+iXmwith the spin-wave resistance\nRm=!\u00160tW\n8\u0019Z\n\u001f00\n!;xx(k) sinc2\u0012kw\n2\u0013\ndk (29)\nand the spin-wave reactance\nXm=!\u00160tW\n8\u0019Z\n\u001f0\n!;xx(k) sinc2\u0012kw\n2\u0013\ndk; (30)\nwith the complex susceptibility ^ \u001f!;xx(k) = ^\u001f0\n!;xx(k)\u0000i^\u001f00\n!;xx(k) derived in Appendix A.\nIV. SCALING BEHAVIOR OF THE SPIN-WAVE IMPEDANCE OF INDUCTIVE\nWIRE ANTENNA TRANSDUCERS\nA. Impedance spectra of inductive wire antenna transducers\nThe spin-wave resistance Rmand reactance Xmin Eqs. (29) and (30) are critical param-\neters that determine the measured signals in magnonic experiments using inductive wire an-\ntennas and ferri- or ferromagnetic waveguides. Their scaling behavior is key when magnonic\ndevices are miniaturized to the micro- and nanoscale. We therefore discuss here the general\nbehavior of RmandXmfor a concrete example based on a thin CoFeB waveguide, assuming\nlaterally uniform magnetization dynamics. Since both the spin-wave dispersion relations and\nthe susceptibility ^ \u001f!;xx(k) depend on the relative orientations between the static magnetiza-\ntion, the normal to the waveguide plane, and the spin-wave wavevector kthat points along\nthe waveguide, three di\u000berent con\fgurations need to be distinguished ( cf.Fig. 3): (i) the\nbackward-volume con\fguration with the magnetization and the wavevector both in-plane\nand parallel; (ii) the forward-volume con\fguration with the magnetization out-of-plane and\nthe wavevector in-plane; and (iii) the Damon{Eshbach con\fguration with the magnetization\nand the wavevector both in-plane and perpendicular.\nIn the following, we discuss, as an example, the spectral characteristics of the spin-\nwave resistance Rmand reactance Xmof a wire antenna above a ferromagnetic CoFeB\nwaveguide24,31,52{54with laterally uniform magnetization dynamics. The used material and\ngeometric parameters are: saturation magnetization Ms= 1.3 MA/m; exchange constant\nAex= 18 pJ/m; Gilbert damping \u000b= 0.004; static bias \feld \u00160H0= 50 mT; waveguide\n13M Mk k(a) (c)\n(b) (d)Ww\ntWw\ntFIG. 3: Spin-wave impedance of a 300 nm wide antenna on top of a 1 µm wide and 20 nm thick\nCoFeB waveguide (see the text for material parameters) in the (a)Damon{Eshbach and (c)the\nforward-volume con\fgurations. (b)and(d)show the real ( Rm) and imaginary ( Xm) components\nof the complex impedances as a function of frequency fin the Damon{Eshbach and forward-volume\ncon\fgurations, respectively. In addition, the spin-wave dispersion relations are shown as yellow\nlines. The FMR frequency is located at zero wavenumber; for the parameters considered, it is\n1.4 GHz and 8.1 GHz for the forward-volume and Damon{Eshbach con\fgurations, respectively.\nwidthW= 1µm; CoFeB thickness t= 20 nm; and antenna width w= 300 nm. We focus\non the Damon{Eshbach and forward-volume con\fgurations as spin waves are only e\u000eciently\nexcited in such geometries by an inductive antenna transducer. By contrast, the excitation\ne\u000eciency of backward-volume spin waves55{57is much lower since the in-plane component\nof the antenna excitation \feld is parallel with the static magnetization and thus does not\ngenerate any torque. Only the much smaller out-of-plane component of the Oersted \feld\ncan excite spin waves in the backward-volume con\fguration.57While the above approach is\nalso valid for the case where the z-component of the Oersted \feld is not neglected, we omit\nfor simplicity the treatment of the backward-volume con\fguration in the following since the\nexcitation e\u000eciency for this geometry is zero in our approximation.\nThe resulting spin-wave impedances in the two considered geometric con\fgurations are\nplotted in Fig. 3. Several aspects can be identi\fed from these plots, which are common to\n14both con\fgurations. Below the FMR frequency, the spin-wave resistance is zero, whereas\nthe reactance remains nonzero for considerably smaller frequencies. This means that, below\nFMR, no net power transfer takes place from the electric to the magnetic domain. Instead,\nresonant power oscillations between the two domains occur that originate from evanescent\nspin waves. Near FMR (which is broadened by damping), the magnetic resonance conditions\nare met and net power transfer from the electric into the magnetic system occurs. At the\nsame time, the reactance reaches a maximum at the FMR frequency.\nAbove the FMR frequency, the resistance Rmincreases further until it reaches a max-\nimum. The maximum is due to the interplay of three factors in Eq. (29): the angular\nfrequency!, the complex part of the susceptibility, and the spectrum of the Oersted \feld\ngenerated by the antenna. At frequencies just above FMR, the susceptibility and Oersted\n\feld spectrum are quasi-constant and the resistance increases linearly with frequency, as it\nis proportional to !(cf.Eq. (16)). At higher frequencies, and thus also at higher wavenum-\nbers, the Oersted \feld spectrum dominates. This results in strongly damped oscillations\nand an overall decrease of the resistance at higher frequencies.\nBy contrast, the reactance Xmis strongly reduced above the FMR frequency until it\nreaches a negative minimum. This decrease with frequency can be attributed to the real\npart of the susceptibility. At a speci\fc frequency, i.e. the frequency for which \u001f0(k) =\n0, the reactance becomes zero and no evanescent spin waves are generated. Below this\nfrequency, the reactance is positive and has inductive behavior, whereas above this frequency\nthe reactance is negative and has capacitive behavior. At even higher frequencies, the\nOersted \feld spectrum also becomes dominant for the reactance, which results in damped\noscillations and an overall decay.\nA comparison of Figs. 3(a) and 3(b) reveals that the impedances for both forward-volume\nand Damon{Eshbach spin waves share the general trends. The main di\u000berence lies in the res-\nonance frequency (FMR), which has a value of 1.4 GHz in the forward-volume and 8.1 GHz in\nthe Damon{Eshbach con\fguration, as well as in the magnitude of the spin-wave impedance.\nFor identical CoFeB parameters, the spin-wave resistance and reactance are about an or-\nder of magnitude larger for the Damon{Eshbach con\fguration than for the forward-volume\ncon\fguration which is due to di\u000berences in the respective susceptibilities and the higher\nworking frequencies in the Damon{Eshbach con\fguration. We remark that impedance val-\nues in the backward-volume con\fguration are again more than one order of magnitude lower\n15than in forward-volume con\fguration due to the weak z-component of the Oersted \feld that\nis required to excite spin waves.\nB. Scaling behavior\nIn recent years, numerous experiments have been conducted to study spin waves in\nnanoscale magnetic structures.15,24,58{60When such spin waves are excited by inductive an-\ntennas, the scaling behavior of the spin-wave impedance of the antenna{waveguide system\nis key to understand the experimental signals and their dependence on the device geometry\nand dimensions. The scaling behavior of the system can be divided in two parts: (i) the\ndependence of the spin-wave impedance on the antenna and waveguide dimensions and (ii)\nthe power transfer between the electrical source and the spin-wave system, which depends\non the entire equivalent circuit, as represented in Fig. 1. Here, we \frst discuss (i) whereas\n(ii) will be addressed in the next section. Since the spin-wave impedance is much larger in\nthe Damon{Eshbach con\fguration than in the forward-volume con\fguration, we focus in\nthe following on the former. However, the above results indicate that the two con\fgura-\ntions share general trends, the conclusions are also qualitatively valid for spin waves in the\nforward-volume geometry. All results are again based on the CoFeB material parameters\nlisted above.\nThe three main geometrical parameters of the antenna{waveguide system that in\ruence\nthe spin-wave impedance are the magnetic waveguide thickness t, the waveguide width W,\nand the antenna width w. Equation (16) indicates that, for plane waves, the impedance is\nsimply proportional to the waveguide width Was is the case for very wide waveguides and\nnarrow waveguides with a waveguide width in the order of the exchange length. For inter-\nmediate cases with Wcomparable to the spin-wave wavelength \u0015, the lateral con\fnement of\nthe spin waves leads to mode formation, which complicates the expressions for the suscep-\ntibility and spin-wave excitation e\u000eciency. In this case, the spin-wave impedance and the\nin\ruence of the waveguide width typically needs to be determined numerically. Nevertheless,\nsome qualitative predictions can be made by considering the overlap integral approach to\ndetermine the excitation e\u000eciency of a particular mode. The higher the overlap between the\nOersted excitation \feld and the spin-wave mode, the stronger the excitation and thus the\nlarger the spin-wave resistance. Hence, increasing the waveguide width results in a higher\n16M\n4 6 810 12 14 16 18 20 22 2401020304050607080\n 10 nm\n 30 nm\n 50 nm\nFrequency (GHz)Resistance (mOhm)\n0102030405060\nWavenumber (Rad/ mm)\n4 6 810 12 14 16 18 20 22 24-80-60-40-200204060\n 10 nm\n 30 nm\n 50 nm\nFrequency (GHz)Reactance (mOhm)\n0102030405060\nWavenumber (rad/ mm)\n4 6 810 12 14 16 18 20 22 24-80-60-40-200204060\n 100 nm\n 300 nm\n 500 nm\n 1 mm\n 5 mm\nFrequency (GHz)Reactance (mOhm)\n01020304050\nWavenumber (rad/ mm)(a)\n(b)(d)\n(c) (e)Ww\n4681012141618202224020406080  100 nm\n 300 nm\n 500 nm\n 1 mm\n 5 mm\nFrequency (GHz)Resistance (mOhm)\n01020304050\nWavenumber (rad/mm)\ntFIG. 4: Dependence of the spin-wave impedance on the frequency fin the (a)Damon{Eshbach\ncon\fguration for di\u000berent geometrical antenna and waveguide parameters. (b)Dependence of the\nspin-wave resistance and (c)spin-wave reactance on the magnetic waveguide thickness t(W= 1\nµm,w= 300 nm). (d)Dependence of the spin-wave resistance and (e)spin-wave reactance on the\nantenna width w(W= 1µm,t= 30 nm). The FMR frequency is located at zero wavenumber,\ni.e.at a frequency of 8.1 GHz for the parameters considered.\nvolumetric overlap integral and thus higher spin-wave resistances. The in\ruence of the mode\npro\fle can be qualitatively captured by a form factor that takes into account the e\u000bective\nmode amplitude, for example the pro\fle rms-value instead of its peak value. Therefore, the\nmode formation typically results in a slightly smaller net overlap integral as compared to\nthe overlap integral with a plane wave.\n17By contrast, the in\ruence of the magnetic waveguide thickness ton the spin-wave\nimpedance is more complex. On one hand, Eq. (16) contains a prefactor tW, which indi-\ncates that the spin-wave impedance is (linearly) increasing with increasing magnetic volume\n(and therefore also with magnetic waveguide thickness t). However, the thickness talso\nstrongly in\ruences the spin-wave dispersion relation and therefore the spin-wave suscepti-\nbility. Figure 4(b) depicts the spin-wave resistance in the Damon{Eshbach geometry ( cf.\nFig. 4(a)) for three di\u000berent magnetic waveguide thicknesses ( W= 1µm,w= 300 nm).\nThe data clearly indicate that a larger \flm thickness results in a larger spin-wave resistance.\nAs mentioned above, this can be partly ascribed to a larger magnetic volume, which can\nstore more magnetic power when the \flm thickness is increased. In addition, talso a\u000bects\nthe spin-wave dispersion, leading to a larger spin-wave group velocity (a steeper slope of the\ndispersion relation) for thicker \flms. This means that for a given frequency, the correspond-\ning spin-wave wavenumbers are lower for thicker waveguides. Since the driving Oersted \feld\nspectrum has larger values at lower wavenumbers, this results in an additional increase of\nthe spin-wave resistance for thicker waveguides. Similar conclusions can be drawn for the\nspin-wave reactance in Fig. 4(c). Here, higher waveguide thicknesses also result in increased\nspin-wave reactance values originating from the increased magnetic volume and the steeper\ndispersion relation for thicker waveguides.\nA third important parameter is the antenna width w. Whereas wdoes not a\u000bect the\nspin-wave dispersion relation and susceptibility, it modi\fes the driving Oersted \feld spec-\ntrum that determines the overlap integral with the magnetization dynamics in Eq. (16).\nFigure 4(d) shows the spin-wave resistance for \fve di\u000berent antenna widths ( W= 1µm,t=\n30 nm). The data indicate that a smaller antenna width results in a larger peak spin-wave\nresistance. This can be understood by considering that the sinc-function that describes the\nOersted \feld spectrum becomes broader for smaller antenna widths. For a wide antenna, the\nsinc-function strongly decays already at small wavenumbers, resulting in a reduced width of\nthe spin-wave resistance peak and, in general, in smaller resistances for higher frequencies. A\nsimilar reasoning can be made for the spin-wave reactance in Fig. 4(e). Here, larger antenna\nwidth also results in smaller peaks due to the narrower antenna spectrum.\nBesides geometrical parameters, the spin-wave impedance also depends on the properties\nof the waveguide material. The saturation magnetization Msis important as it strongly\nin\ruences the FMR frequency and the spin-wave dispersion relation. A higher saturation\n18magnetization leads to a higher the FMR frequency, thereby shifting the impedance curve\nto higher frequencies and higher values due to the proportionality to !. In addition, an\nincreasedMsalso results in a steeper dispersion relation, analogously to an increased waveg-\nuide thickness, as discussed above. Thus, the e\u000bect of an increased Msis also similar to\nthe e\u000bect of increasing the waveguide thickness, resulting in higher and broader impedance\npeaks for higher saturation magnetization values. Another important material parameter is\nthe magnetic damping \u000b, which in\ruences the magnetic susceptibility. A larger \u000bresults\nin a broader susceptibility spectrum and a smaller susceptibility peak value. Despite the\nreduction of the susceptibility peak value, this does not automatically result in a reduced\nimpedance since the magnetic inductance depends on the spectral overlap integral between\nthe susceptibility and the Oersted \feld ( cf.Eq. (16)). Hence, the broader the susceptibil-\nity and the Oersted \feld spectrum, the higher the overlap integral and thus the spin-wave\nimpedance.\nV. POWER TRANSMISSION EFFICIENCY OF AN INDUCTIVE WIRE AN-\nTENNA TRANSDUCER ON A FERROMAGNETIC NARROW WAVEGUIDE\nAbove, we have discussed the frequency dependence of the spin-wave impedance on waveg-\nuide and antenna geometry, as well as on the magnetic material properties. From a spintronic\napplication point of view, a key parameter is the power coupling e\u000eciency that describes\nthe ratio of the power that is transferred to the magnetic system and the total microwave\npower that is incident on the antenna transducer.\nIn a microwave circuit, the power transmission from a source into a load (here the induc-\ntive antenna) is determined by the matching conditions between source and load impedances.\nDi\u000berent matching approaches exist. Maximum power dissipation in the load is obtained\nwhen the antenna impedance is equal to the complex conjugate of the source impedance\nZeq=Z\u0003\nS.61Note that, in this case, the maximum transferred power is half of the source\npower. Zero re\rection at the load occurs for Zeq=ZS. By contrast, the maximum power\ntransmission e\u000eciency is obtained for e\u000bective open circuit conditions, i.e.forRS\u001cReq,\nreaching an e\u000eciency of 1 when RS=Req!0.\nIn practice, experiments often utilize a source with a real 50 \n or 75 \n source resistance,\nalthough this no necessary condition. A detailed discussion on impedance matching to\n19common 50 \n or 75 \n microwave instrumentation is beyond the scope of the article (see\ne.g.Refs. 62{64) and has been recently addressed for di\u000berent inductive antenna designs.65\nHowever, some general power coupling considerations can be made based on the equivalent\ncircuit in Fig. 1, which allow for the discussion of the scaling properties of inductive antennas\nand the associated power coupling scaling. The equivalent circuit indicates that power can be\ndissipated both by emission of spin-waves or excitation of FMR ( Pm) as well as by Ohmic\nlosses in the antenna ( P\n). Both losses are proportional to the square of the microwave\ncurrent in the antenna and thus their ratio does not depend on the total power dissipated\nin the antenna and thus also not on the detailed impedance matching approach. Therefore,\nthe power transfer e\u000eciency into the spin-wave system can be expressed as\n\u0011=\u0010Pm\nP\n+Pm=\u0010RmI2\nrf\nR\nI2\nrf+RmI2\nrf=\u0010Rm\nR\n+Rm: (31)\nHere, 0\u0014\u0010\u00141 represents the power coupling e\u000eciency into the antenna with respect\nto the source power, which depends on the matching conditions. Under complex conjugate\nmatching conditions (maximum power transfer) or for re\rectionless matching to a real source\nimpedance, \u0010=1\n2, whereas\u0010= 1 near e\u000bective open circuit conditions. However, in any\ncase, the antenna transduction e\u000eciency is limited by the ratio of the spin-wave and Ohmic\nresistances, given by Eqs. (29) and (23), respectively. The scaling behavior of the power\ntransmission e\u000eciency of an inductive spin-wave antenna depends thus on the scaling of its\nOhmic resistance as well as of its spin-wave impedance, as discussed in the previous section.\nWe now quantitatively assess \u0011in the Damon{Eshbach con\fguration for an antenna\nthickness of d= 40 nm, using the Cu resistivity of \u001a= 17 n\nm, as well as the above\nCoFeB material parameters. We further assume that the wire antenna length `is equal\nto the waveguide width W, which is a lower limit for experimental realizations, as well as\n\u0010=1\n2(maximum power transfer or re\rectionless matching to real source power). Figure 5(a)\nshows\u0011for di\u000berent antenna widths w. The data indicate that the maximum power coupling\ne\u000eciency follows the spectral behavior of the spin-wave radiation resistance Rm. Note that\nthe Ohmic antenna resistance R\nis assumed to be frequency independent as the skin e\u000bect\nis neglected. Although Fig. 4(b) shows that the spin-wave radiation resistance Rmincreases\nwith decreasing w, especially at high frequencies, the Ohmic antenna resistance R\nalso\nincreases with decreasing w. As a consequence, the power transfer e\u000eciency \u0011decreases\nwith decreasing wdue to a faster increase of R\n, despite the larger bandwidth. By contrast,\n20FMR\n5 10 15 20 25-40-30-20-100Efficiency h (db)\nFrequency (GHz) 100 nm\n 300 nm\n 500 nm\n 1 um\n 5 um\n0.010.1110100\nEfficiency h (%)\n5 10 15 20 25-40-30-20-100Efficiency h (dB)\nFrequency (GHz) 10 nm\n 30 nm\n 50 nm\n0.010.1110100\nEfficiency h (%)(a) (b)\nFMRFIG. 5: Power transmission e\u000eciency \u0011into the magnetic system ( \u0010=1\n2, maximum power transfer)\nin the Damon{Eshbach con\fguration (a) as a function of the inductive antenna width wbetween\n100 nm and 5 µm (W= 1µm,t= 20 nm) and (b) as a function of waveguide thickness tbetween\n10 nm and 50 nm ( W= 1µm,w= 300 nm). The FMR frequency at 8.1 GHz is indicated in both\ngraphs as a reference.\nthe magnetic waveguide thickness tonly a\u000bects Rmand therefore a larger tnot only increases\nRmbut also improves \u0011. Similar e\u000bects can be found for the dependence on the magnetic\nmaterial properties, which can be optimized to enhance Rmand therefore \u0011. By contrast,\nchanging the waveguide width or the antenna length does not alter the e\u000eciency \u0011because\nthe spin-wave and Ohmic resistance are equally proportional to both dimensions. For the\nchosen parameters and nanoscale dimensions, the maximum power transfer e\u000eciency into\nthe spin-wave system is of the order of 1 to 3% ( \u000020 to\u000015 dB).\nThese results quantitatively illustrate the scaling behavior of inductive antennas as spin-\nwave transducers. As observed in many experiments, the power transfer e\u000eciency typically\ndecreases when the device dimensions are reduced, in particular the antenna width as well\nas the waveguide thickness. In practice, poor matching between the source impedance and\nthe equivalent antenna impedance Zeqin Eq. (20) further reduce the power emitted into\nthe spin-wave system. The results in the previous section show that absolute spin-wave\nradiation resistances Rmare of the order of m\n. Nonetheless, for the given CoFeB materials\nparameters, the results indicate that high power transfer e\u000eciencies \u0011can be obtained for µm\ndimensions (antenna widths) when source and antenna impedances can be matched, i.e.for\nZeq=Z\u0003\nS, in agreement with a recent report.65For example, \u0011can be as high as 11% ( \u00009:5\n21dB) forW=`= 5µm,w= 300 µm, andt= 20 nm. However, when the antenna width is\nscaled to nm dimensions, \u0011is reduced to 1{3% ( \u000020 to\u000015 dB) and below, indicating that\nwire antennas become increasingly ine\u000ecient when the bandwidth is increased by reducing\ntheir width.\nVI. CONCLUSIONS\nIn conclusion, we have derived an equivalent circuit for inductive antennas as transduc-\ners between microwave currents and FMR or spin waves. Such antennas have been used\ncommonly in magnonic experiments. Furthermore, we have derived analytical equations,\nassuming an arbitrary antenna shape as well as an arbitrary current density, for the dif-\nferent components of the equivalent circuit, which comprises an Ohmic resistance R\n, a\nself-inductance L0, as well as an additional complex partial inductance Lmthat stems from\nthe coupling to the magnetic system. In addition, both exchange and dipolar interaction\nhave been considered to describe the magnetization dynamics. The model has then been\nused to present a case study for a thin CoFeB waveguide with a straight wire antenna on top.\nBoth the spin-wave resistance and reactance have been determined and general spectral and\ngeometrical trends have been identi\fed. The results have been used to assess the maximum\ntransduction power e\u000eciency for such a system, with a focus on its scaling behavior.\nThese results have multiple implications when magnonic devices including inductive an-\ntennas are scaled to nm dimensions. In real-world applications, the power transfer e\u000eciency\nis a key parameter to determine the performance of any magnonic device. As shown above,\nspin-wave radiation resistances for sub- µm straight wire antenna dimensions are of the order\nof a few 10 m\n. When the antenna width (and thickness) is scaled, the Ohmic resistance can\nbecome rapidly an order of magnitude or more larger, leading to a reduced power transfer\ne\u000eciency and to more relative Ohmic power dissipation. In addition, the matching of small\nantenna impedances to conventional 50 \n or 75 \n sources can be challenging and reduces\nthe available bandwidth considerably. The matching can be improved by increasing, e.g.\nthe waveguide width and the antenna length.65While this does not a\u000bect the power transfer\ne\u000eciency, it increases the overall resistance (both Ohmic and spin-wave) and can be used\nto match the load impedance better to e.g.50 \n source impedance; however, this also in-\ncreases the structure (device) size. We note that analogous arguments apply to the case of\n22the two-antenna mutual inductance introduced in Sec. III B and the associated equivalent\ntwo-port networks.\nAs mentioned before, the spin-wave impedance is strongly determined by the spectral\noverlap integral between the magnetic susceptibility and the magnetic Oersted \feld. In\nthe above discussion, a rectangular wire has been considered which results in a sinc-like\nOersted \feld in reciprocal space. It is also possible to design di\u000berent antenna shapes such\nas U-shaped antennas or IDTs, i.e.multiple wire antennas, or envisage even more exotic\ndesigns, which improve the spin-wave inductance Lmwith respect to the straight wire case\nconsidered here.65In all such cases, the above derived model remains valid and can be\napplied to study the maximum e\u000eciency of a particular antenna design and its scalability.\nThe only parameter that needs to be modi\fed is the spectrum of the Oersted in the overlap\nintegral. The equivalent circuit and the model can thus be used to determine the coupling\ne\u000eciency between the microwave source and the antenna (by impedance matching) as well\nas between the antenna and the magnetization dynamics (FMR, spin waves). While it is\nbeyond the scope of the paper to discuss the particular spin-wave impedance value for the\ndi\u000berent antenna designs, the general trends and thinking strategies outlined above remain\nvalid for di\u000berent types of antenna transducers. Therefore, this model can be used as the\nstarting point for the design and optimization of antenna transducers for (nano-)magnonic\ndevices.\nAppendix A: Dynamic susceptibility\nThe magnetization dynamics can be described by the LLG equation66,67\ndM\ndt=\u0000\r0(M\u0002He\u000b) +\u000b\nMa\u0012\nM\u0002dM\ndt\u0013\n; (A1)\nwithM=M0+m,\r0=j\rj\u00160,j\rjthe absolute value of the gyromagnetic ratio, He\u000bthe\ne\u000bective magnetic \feld and \u000bthe magnetic damping constant. M0andmare the static\nand dynamic components of the magnetization M, respectively. In this work, the e\u000bective\n\feld consist of a static bias \feld H0, a dynamic antenna \feld ha, spin-wave dipolar \feld hd\nand spin-wave exchange \feld hex. For weak magnetization dynamics, i.e.jmj\u001cjM0j, the\n23LLG equation can be linearized and becomes\ni!m(k;!) =\u0000\r0(M0\u0002[hd(k;!) +hex(k;!)] +M0\u0002ha(k;!) +m(k;!)\u0002H0)\n+i!\u000b\nM0(M0\u0002m(k;!)); (A2)\nwhere complex notation was used and kand!denote the spin-wave wavevector and angular\nfrequency, respectively. The dynamic dipolar magnetic \feld is given by68\nhd(r) =Z\nV^\u0000(r;r0)m(r;r0)dr0; (A3)\nwithVthe volume of the magnetic material and ^\u0000(r;r0) the magnetostatic Green's function\ngiven by\n^\u0000(r;r0) =\u0000r rrr01\njr\u0000r0j: (A4)\nFor a plane wave in a thin \flm, the averaged dipolar magnetic \feld inside the \flm\nbecomes68{70\nhd(k;!) =\u00002\n6664Psin2(\u0012)Pcos(\u0012) sin(\u0012) 0\nPcos(\u0012) sin(\u0012)Pcos2(\u0012) 0\n0 0 1 \u0000P3\n7775m(k;!); (A5)\nwith\u0012the (in-plane) angle between the static magnetization and the wavevector and\nP= 1\u00001\u0000e\u0000kt\nkt: (A6)\nThe dynamic exchange \feld is given by\nhex(k;!) =\u0015exr2m(r): (A7)\nFor a plane wave, this becomes\nhex(k;!) =\u0000\u0015exk2m(k;!); (A8)\nwith\n\u0015ex=s\n2Aex\n\u00160M2\na(A9)\nandAexthe exchange sti\u000bness constant. To simplify the expressions, the following tensor is\nintroduced\n^F=2\n6664Psin2(\u0012) +\u0015exk2Pcos(\u0012) sin(\u0012) 0\nPcos(\u0012) sin(\u0012)Pcos2(\u0012) +\u0015exk20\n0 0 1 \u0000P+\u0015exk23\n7775: (A10)\n24The general form of the linearized LLG equation then becomes\nm= ^\u001f!ha (A11)\nwith\n^\u001f!= (A12)\n\u0000!M2\n6664i!+!MFxy\u0010z [!0+!MFyy+i!\u000b]\u0010z\u0000[!0+!MFzz+i!\u000b]\u0010y\n[!0+!MFxx+i!\u000b]\u0010z i!+!MFxy\u0010z [!0+!MFzz+i!\u000b]\u0010x\n[!0+!MFxx+i!\u000b]\u0010y\u0000!MFxy\u0010x\u0000[!0+!MFyy+i!\u000b]\u0010x\u0000!MFxy\u0010y i!3\n7775\u00001\n;\n!M=\r0M0,!0=\r0H0, and\n\u0010=2\n6664cos( ) cos(\u0012)\ncos( ) cos(\u0012)\nsin( )3\n7775: (A13)\nHere, is the angle between the static magnetization M0and the \flm. When the static\nmagnetization direction coincides with one of the coordinate axes, the expression can be\nsimpli\fed and written as\n2\n4!1+i!\u000b\u0000i!\ni! ! 2+i!\u000b3\n52\n4mk\nml3\n5=!M2\n4hk\nhl3\n5; (A14)\nwhich becomes\n2\n4mk\nml3\n5= ^\u001f!2\n4hk\nhl3\n5 (A15)\n=!M\n(!1!2\u0000!2) +i\u000b!(!1+!2)2\n4!2+i\u000b! i!\n\u0000i! ! 1+i\u000b!3\n52\n4hk\nhl3\n5 (A16)\n=^\n2\n!2\nr\u0000!2+i!\u00002\n4hk\nhl3\n5; (A17)\nwith!r=p!1!2the spin-wave resonance frequency or dispersion relation and \u0000 =\n\u000b(!1+!2) the spin-wave damping rate. The factors !1and!2are con\fguration dependent\nand are given for the three speci\fc cases below. The susceptibility can be separated into a\nreal and imaginary part as given below\n^\u001fkk=!M(!2(!2\nr\u0000!2) +\u000b2!2(!1+!2))\n(!2\nr\u0000!2)2+!2\u000b2(!1+!2)2\u0000i\u000b!!M(!2+!2\n2)\n(!2\nr\u0000!2)2+!2\u000b2(!1+!2)2(A18)\n25^\u001fll=!M(!1(!2\nr\u0000!2) +\u000b2!2(!1+!2))\n(!2\nr\u0000!2)2+!2\u000b2(!1+!2)2\u0000i\u000b!!M(!2+!2\n1)\n(!2\nr\u0000!2)2+!2\u000b2(!1+!2)2: (A19)\nThe group velocity is de\fned as\nvg=@!r\n@k=@p!1!2\n@k=1\n2!1!2\u0014\n!2@!1\n@k+!1@!2\n@k\u0015\n: (A20)\n1. Spec\fc cases\nFor\u0012= 0 and = 0 (backward-volume con\fguration), k=y,l=zand\n8\n><\n>:!1=!0+!M(\u0015exk2)\n!2=!0+!M(\u0015exk2+ 1\u0000P):\nFor\u0012=\u0019=2 and = 0 (Damon{Eshbach con\fguration), k=z,l=xand\n8\n><\n>:!1=!0+!M(\u0015exk2+ 1\u0000P)\n!2=!0+!M(\u0015exk2+P):\nFor\u001e=\u0019=2 (forward-volume con\fguration), k=x,l=yand\n8\n><\n>:!1=!0+!M(\u0015exk2+P)\n!2=!0+!M(\u0015exk2):\nAcknowledgments\nThis work has received funding from the European Union's Horizon 2020 research and\ninnovation program within the European Innovation Council Path\fnder FET-OPEN project\nCHIRON under grant agreement No. 801055. It has also been supported by imec's indus-\ntrial a\u000eliate program on beyond-CMOS logic. F.V. acknowledges support by the Research\nFoundation { Flanders (FWO) through a PhD fellowship.\nII. AUTHOR CONTRIBUTIONS\nF.V. and V.T. developed the theoretical model. F.V., F.C. and C.A. described the scaling\nbehavior. 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Guinea2\n1Department of Physics, University of G¨ ottingen, 37077 G¨ o ttingen, Germany.\n2Instituto de Ciencia de Materiales de Madrid. CSIC. Sor Juan a In´ es de la Cruz 3. 28049 Madrid. Spain.\nThe effect of electronic interactions in graphene with vacan cies or resonant scatterers is inves-\ntigated. We apply dynamical mean-field theory in combinatio n with quantum Monte Carlo sim-\nulations, which allow us to treat non-perturbatively quant um fluctuations beyond Hartree-Fock\napproximations. The interactions narrow the width of the re sonance and induce a Curie magnetic\nsusceptibility, signaling the formation of local moments. The absence of saturation of the suscepti-\nbility at low temperatures suggests that the coupling betwe en the local moment and the conduction\nelectrons is ferromagnetic.\nIntroduction. Since its isolation1,2, single layer\ngraphene has attracted a great deal of attention, due to\nnovelfeatures. The masslessnatureof the chargecarriers\nimplies that the density of states vanishes at the Fermi\nenergy in a neutral layer3. The low density of states in\nits vicinity allows for the formation of sharp resonances\ndue to vacancies4or impurities like hydrogen which form\na strong covalent bond with the carbon atoms, the so\ncalled resonant impurities5. These resonances have been\nobserved in graphite surfaces6.\nThe enhancement in the density of states by those res-\nonances and the electron electron interaction favor the\nformation of local moments. The states associated to the\nresonancesdifferfromthoseinducedbycoupledmagnetic\ndopants in a number of ways: i) They are built up from\nthe same πorbitals as the conduction band of graphene,\nii) The resonance state is orthogonal to the conduction\nstates, and the hopping between the resonance and the\nextended states vanishes, and iii) They extend over a\nlarge region near the defect, as there is no gap in the\nspectrum to confine them.\nMean field arguments based on the enhancement of\nthe local density of states favor the formation of a\nstatic magnetic moment, as shown in a number of\ncalculations7–12. It is known that quantum fluctuations,\nnot included in Hartree-Fock approximations, determine\nthe low temperature properties of magnetic impurities\nin solids, described by the ferro- or antiferromagnetic\nKondo model13. The differences between the resonant\nlevels in graphene and magnetic impurities in metals and\ngraphene14–16imply that mean field resultscannot be ex-\ntrapolated in a straightforwardway to low temperatures.\nIn the following, weanalyzethe electronicpropertiesof\ngraphene by non perturbative methods beyond a static\nmean field approximation. Our results show that, for\nreasonable interaction strengths, the main features are\nwell described by assuming the existence of a fluctuat-\ning magnetic moment. This moment is not quenched\nat the lowest accessible temperatures, suggesting a fer-\nromagnetic coupling with the conduction band. For a\nfinite concentration of resonant impurities, the absence\nof competition between the antiferromagnetic Kondo ef-\nfect and the RKKY interaction might lead to ferromag-\nFIG. 1. (Color online). Sketch of the model studied in the\ntext. The presence of a vacancy, or a resonant scatterer, pas -\nsivates one of the lattice sites. The resulting localized st ate\nnear the Dirac energy becomes spin polarized.\nnetism, provided that the concentration of impurities is\nlarge enough17–20.\nThe model. We describe the πband of graphene by\na nearest neighbor tight binding model, with a hopping\nparameter t. The effect of a resonant scatterer strongly\nbound to a given site is taken into account by shifting\nthe on-site energy by an amount ǫ0. In the limit |ǫ0| ≫t\nthe model describes an unrelaxed vacancy. For |ǫ0|/greaterorsimilart,\na resonance near the Dirac energy builds up. This reso-\nnance moves to the Dirac energy and its width vanishes\nin the limit |ǫ0|/t→ ∞. The main features of the model,\nincluding the possibility of a local moment at the reso-\nnance, are shown in Fig. 1.\nWe assume that the long range part of the electron-\nelectron interaction is screened, and we describe the elec-\ntron electron interaction by an on-site Hubbard repulsive\nterm,U. The full hamiltonian is\nH=−t/summationdisplay\n/angbracketlefti,j/angbracketrightσc†\niσcjσ−ǫ0/summationdisplay\nσc†\n0σc0σ\n+U/summationdisplay\ni/parenleftbigg\nni↑−1\n2/parenrightbigg/parenleftbigg\nni↓−1\n2/parenrightbigg\n,(1)\nwherec(†)\niσannihilates (creates) an electron with spin\nσ=↑,↓on lattice site iand where niσ=c†\niσciσdenotes\nthe corresponding number density. The hopping tis only2\nA\nB\nImpurity\nΣ\nFIG. 2. (Color online). Sketch of the lattice system with two\nsub-lattices A and B (left panel). The impurity is denoted\nby an open circle. We map the lattice on a two-site cluster\nconsisting of only one unit cell (right panel). The cluster i s\nembedded in an interacting medium which determined by a\nlocal self-energy Σ. The self-energy is calculated either b y\nDMFT or by second-order perturbation theory.\nfinite forneighboringlatticesites(denoted by ∝angbracketlefti,j∝angbracketright). The\nlattice site i= 0 corresponds to the impurity where the\nlocal on-site energy ǫ0is non-zero. We study the model\nat half filling, i.e., at chemical potential µ= 0.\nWe approximate the interacting lattice problem by as-\nsuming that the impurity site where ǫ0∝negationslash= 0 and a neigh-\nboring site are attached to an effective medium described\nby a local self-energy Σ as sketched in Fig. 2. The effect\nof the impurity on the bath vanishes in the thermody-\nnamic limit and can therefore be neglected21. The re-\nsulting two-site cluster problem is solved numerically by\nquantum Monte Carlo (QMC) simulations. We employ\nQMC methods in continuous imaginary time which per-\nform a systematic expansion in the interaction term of\nthe Hamiltonian22,23. The QMC solution of the cluster\nproblemisnumericallyexactandfully incorporatesinter-\nactions and quantum fluctuations. The analysis can be\nregarded as the solution of a quantum impurity problem\nwhere the interaction effects are included at the impurity\nsite and at a close neighbor, and approximated by means\nof a local self-energy in the surrounding medium.\nIn order to calculate the self-energy necessary to de-\ntermine the effective medium, we start by simulating the\nhomogeneous lattice system with ǫ0= 0. We calculate\nthe self-energy necessary to obtain the effective medium\nusing either Dynamical Mean Field Theory(DMFT)24or\nsecond-orderperturbationtheoryin U/t. DMFT fullyin-\ncorporates quantum fluctuations local to the cluster but\nignores spatial fluctuations. The medium depends on the\nself-energy of the cluster system and has to be calculated\nself-consistently by an iterative procedure using QMC\nsimulations. Subsequently, the impurity is added to the\nsystem and one additional QMC simulation is performed\nusing the medium of the converged DMFT calculation.\nIn order to check the quality of the DMFT approxi-\nmation, we additionally calculate the self-energy of the\nhomogeneous lattice system using second-order pertur-\nbation theory in U/t. This self-energy – instead of the\nself-consistently determined DMFT solution – is used to00.20.40.60.811.2\n−10−8−6−4−2 0 2 4 6 8D(ω)t\nω/t00.20.40.60.811.2\n0 0.5U=t\nU= 2t\nU= 3t\nU= 4t\n00.20.40.60.81\n−10−8−6−4−2 0 2 4 6 8D(ω)t\nω/t00.20.40.60.81\n0 0.5ǫ0=t\nǫ0= 2t\nǫ0= 3t\nǫ0= 4t\nFIG. 3. (Color online). Density of states D(ω) at the site next\nto the impurity at βt= 10 calculated by DMFT in combina-\ntion with QMC. The analytic continuation was performed by\nmaximum entropy. Top: density of states for ǫ0/t= 3 and\ndifferent values of U/t. Bottom: density of states for U/t= 4\nand different values of ǫ0/t. The insets show a blowup of the\nregion near the Dirac energy.\ncalculate the bath of the impurity problem which is then\nagain solved by QMC. The perturbative self-energy does\nnot include quantum fluctuations to all orders. How-\never, it incorporates non-local effects of the actual lattice\nstructure which are neglected by DMFT. Thus we are\nable to test the influence of non-local correlations and\nthe accuracy of the DMFT approximation.\nQMC methods map quantum-mechanical systems on\na classical one at the expense of an additional dimension\nwhich – in most cases – is an imaginary time dimension.\nThus, QMC can only provide dynamical data for imag-\ninary times or frequencies. The necessary analytic con-\ntinuation to physically relevant real times or frequencies\nis usually performed by maximum entropy techniques25.\nWe use a standard maximum entropy implementation26\nto calculate the interacting density of states for real fre-\nquencies.\nResults. The density of states at the site next to the3\n00.20.40.60.811.2\n−8−6−4−2 0 2 4 6D(ω)t\nω/t00.20.40.60.811.2\n0 0.5ǫ0=t\nǫ0= 2t\nǫ0= 3t\nǫ0= 4 /D8\nFIG. 4. (Color online). Density of states at the site next to t he\nimpurity at βt= 10 calculated by second-order perturbation\ntheory. Parameters are βt= 10,U/t= 1, and different values\nofǫ0/t. The inset is a blowup of the region near the Dirac\nenergy.\nimpurity is shown in Fig. 3. We find a resonance whose\nwidth decreases either by increasing ǫ0/tor by increasing\nU/t. The reduction in width by the interactions is a\ncharacteristic feature of magnetic impurity problems13.\nIt indicates the decoupling of the impurity degrees of\nfreedom from the conduction band. For ǫ0≥2tand\nU≥2twe find a second resonanceto the left of the Dirac\npoint which is shifted further to the left for increasing\nvalues of ǫ0/torU/t. There appear also satellite peaks\nat high energies. These peaks are related to transitions\ninvolving configurations where the resonance hosts zero\nor two electrons. The positions of these peaks are shifted\nby an amount proportional to ǫ0, confirming that they\nare related to the impurity.\nWe have repeated the previous calculations using an\ninput self-energy obtained from second-order perturba-\ntion theory. The results are shown in Fig. 4 and are\nconsistent with those shown in Fig. 3 for U/t/lessorsimilar3. One\nclearly sees the formation of a resonance that becomes\nsharper and shifts towards ω/t= 0 for increasing ǫ0/t.\nWe also have a peak to the left of ω/t= 0 that is shifted\nwith increasing ǫ0.\nWe calculate the magnetic susceptibility of the impu-\nrity in the imaginary-time framework of the QMC via\nχS=/integraldisplayβ\n0dτ∝angbracketleft(n↑(τ)−n↓(τ))(n↑(0)−n↓(0))∝angbracketright,(2)\nwhereτdenotes imaginary time, nσ(τ) =e−τHnσeτH,\nandβ= 1/kBT. As usual, Tdenotes temperature and\nkBBoltzmann’s constant. Fig. 5 shows χSas function\nof the inverse temperature. A fully localized spin pos-\nsesses the susceptibility χloc=βt. The numerical results\nare consistent with a Curie dependence on temperature,\nχ∝1/T, suggesting the formation of a local moment.\nThis moment is not fully localized at the positions clos-246810121416\n4 6 8 10 12 14 16 18 20χS·t\nβ·tU=t\nU= 2t\nU= 3t\nU= 4t\n2468101214161820\n4 6 8 10 12 14 16 18 20χS·t\nβ·tǫ0=t\nǫ0= 2t\nǫ0= 3t\nǫ0= 4t\nFIG. 5. (Color online). Local magnetic susceptibility χSas\nfunction of the inverse temperature calculated by DMFT in\ncombination with QMC. Top: susceptibilities for ǫ0/t= 3 and\ndifferent values of U/t. Bottom: susceptibilities for U/t= 4\nand different values of ǫ0/t.\nest to the impurity, and the measured χSis smaller than\n≈χloc/2. As the interaction increases the magnetic mo-\nment becomes better defined and more localized near the\nimpurity. We find no sign of saturation of the suscepti-\nbility down to the lowest temperatures. This result is\nconsistent with a ferromagnetic Kondo coupling.\nDiscussion. We have studied the effects of interactions\nin the presence of a resonance in the graphene electronic\nspectrum. We use a local approach, and fully include\nquantum fluctuations. The interaction is described by a\nlocalHubbardterm, andwedonotconsidertheimperfect\nscreening in graphene near the neutrality point which is\nexpected in suspended layers. Given this interaction, our\ncalculation can be regarded as the solution of a quantum\ncluster problem. The effect ofinteractionsin the medium\nsurroundingtotheclusterisdescribedbymeansofalocal\nself-energy. The results obtained when this self-energy\nis calculated by DMFT and by perturbation theory are\nmutually consistent, in the regime where perturbation\ntheory is valid. The off diagonal corrections to the self-\nenergynot includedherearemostlydue tothe longrange\npart of the interaction27. They lead to an increase in the4\nFermi velocity28. The density of states at low energies is\nreduced, favoringfurtherthe formationoflocalmoments.\nWe find that interactions reduce the width of the res-\nonance induced by the impurity, and lead to a mag-\nnetic susceptibility which grows at low temperatures as\nχS∝1/T. The local moment is not quenched at the low-\nest temperatures studied. This is consistent with the ex-\nistence of a ferromagnetic coupling between the moment\nand the valence electrons. Note that the exchange mech-\nanism which leads to an antiferromagneticinteraction for\nmagnetic impurities coupled to a conduction band does\nnot exist in the case of a resonance built up from the or-\nbitals which also give rise to the conduction band. The\nexistenceofthevacancydoesnot leadtovirtualhoppings\nbetween extended and localized states. An electron oc-\ncupying the resonant state interacts with a conduction\nelectron through the onsite Hubbard term. This cou-\npling favors a ferromagnetic alignment of the spin of the\nelectron in the resonance and the spin of the conduction\nelectrons.\nThe scaleat which the localmoment studied hereleads\ntosignificanteffectsdepends onthe valueof U/t, which isnot very precisely determined in graphene. Calculations\nbased on the Local Density Functional Approximation\nsuggest29U/t∼1, while quantum many body calcula-\ntionsforaromaticmoleculesgive30–32U/t∼3. Thevalue\nofU/tis bounded by U/t≈4.5, above which graphene\nshould become antiferromagnetic33.\nAcknowledgements. We appreciate helpful discussions\nwith M. I. Katsnelson and A. H. Castro Neto. F.\nG. and H. O. are supported by by MICINN (Spain),\ngrants FIS2008-00124 and CONSOLIDER CSD2007-\n00010. 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Troyer, accepted for publication in Computer Physics\nCommunications (2010)."    },    {        "title": "1310.6108v1.Exchange_dominated_Standing_Spin_Wave_Excitations_under_microwave_irradiation_in_Ni80Fe20_Thin_Films.pdf",        "content": "Exchange-dominated Standing Spin Wave Excitations under microwave irradiation in Ni80Fe20 Thin Films  Ziqian Wang, Xiaofeng Zhu, Xiaoshuang Chen, and Wei Lu    We investigated the microwave-assisted DC voltages of ferromagnetic resonances and exchange-dominated standing spin wave excitations in two different in-plane magnetized permalloy thin films via homodyne detection. The line shapes of ferromagnetic resonance spectra and the dispersion curves of ferromagnetic resonance and standing spin wave are in agreement of previous studies, while further investigations of DC voltage spectra for these two excitations reveal that 1. unlike ferromagnetic resonance signals, the anti-symmetrical line shapes of standing spin wave excitations are not depend on the electromagnetic relative phase of assisted microwave, and 2. linewidths of their DC voltage spectra are distinct. The complicated spin dynamics of standing spin wave is consequently discussed by applying Landau-Lifshitz-Gilbert equation in term of exchange interaction.      _________________________________________________________ 1National Laboratory of Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences  500 Yutian Rd, Shanghai, 200083, China  DC effect in ferromagnetic conductors under microwave radiation has been firstly studied by H. J. Juretschke since 1960.1-6 This solid-state phenomenon, generated by the galvanomagnetic effects and magnetization dynamics, 2,7 is usually electrically detected as non-zero time-averaged DC voltage by homodyne detection. 8-11 Homodyne detection of DC effect bears an important role for both scientific points of view and technological perspectives. Firstly, the signal spectrum of this effect is found to be very informative to further understand the spin dynamics: measured signal spectra have been applied to probe the microwave assisted magnetization switching, 12 which is considered as a pivotal factor for the next generation magnetic recording technology with faster speed. 13,14 In addition, researchers can obtain some important physical process, such as spin pumping, in ferromagnetic materials via this technology. 15-17 Furthermore, DC response of microwave reveals the electromagnetic features of assisted microwaves since both of the line shapes and amplitudes of signals are sensitive to the assisted microwave’s frequency, polarization and electromagnetic relative phase (the phase difference between e-component and h-component of electromagnetic wave): 8,10 a novel microwave imaging technology is developed through analyzing the microwave induced DC signals. 18  Most of recent studies and applications of electrically detected signals are based on the DC response of ferromagnetic resonance (FMR). 10,18,19 FMR is a typical excitation for uniform precession mode: every electron spin precesses at the same phase, frequency and amplitude. Spin waves, which exhibits as a non-uniform mode, is also possibly observed via microwave stimulation. However, fewer results have been reported for DC effect of spin wave due to the complicated spin-spin exchange and dipole interactions. Furthermore, the spin wave excitation is easily mixed with FMR. It is widely accepted that the interaction between spin and electromagnetic field is described by Landau-Lifshitz-Gilbert (LLG) equation, 20 the nonlinearlity of LLG equation indicates that simply extracting the spin wave signals from a whole DC signal spectrum by simply subtracting FMR component is not a property endeavor.  In ferromagnetic thin films, spin waves usually perform as standing spin waves (SSW) by taking boundary condition into account, 21-24 and, from a macroscopic view, result in magnetization precession. Although usually weaker than FMR signals, the excitation of spin waves are possible to be electrical detected since the time-dependent AMR is generated by precessional magnetization, the time-varying current is induced by microwave as well. To provide a general picture to show how the SSW mode evolves while it is excited, we are now focusing on the DC response of exchange-dominated SSW in ferromagnetic thin films. For overcoming the above predicament, two ferromagnetic films are fabricated as very thin microstrips for larger wave vector in order to avoid the coupling of SSW excitation and FMR, and the strip structures are designed for convenience of homodyne detection. The confusing line shapes of excited SSW spectra show that SSW mode is significantly different from FMR, not only in its non-uniformity of precession magnitudes. Similar phenomena occur in both of our two samples.  Results Measurement setup, sample structures and magneto-resistance. We show the preparation of our work in Figure 1. Fig. 1a is the schematic diagram of measurement, the coordinate system we select in our work is also shown in Fig. 1a and b. This work is performed on two Ni80Fe20 (permalloy, Py) thin films (labeled strip 1 and strip 2) that are illustrated in Fig. 1c, d.  The dimensions of strip 1 are: length=2400µm, width=200µm and thickness=50nm (prepared as described elsewhere) 8 while strip 2 is with a dimension of 300×7×0.1 µm3 (prepared as described elsewhere). 9 These two strips are fabricated by lithography and liftoff technology on GaAs substrates.  Unlike strip 1, strip 2 is buried by a coplanar waveguide (CPW) fabricated by a Cu/Cr (100nm) bilayer. A 200nm SiO2 layer is also grown between CPW and strip 2 for electrical isolation. Two different methods are used to impose microwave for each strip: A rectangular waveguide is applied to transmit modulated microwave and ensure them normally propagate into strip 1, this rectangular waveguide is also applied as a holder for these two samples; for strip 2, localized microwave is generated by the CPW. The DC voltage signals of SSW and FMR are extracted by a lock-in amplifier connecting with two electrodes at both sides of each strip’s length via gold bonding wires and coaxial cables. An electromagnet is employed to provide an external static magnetic field µ0Hex in xz-plane with maximum amplitude of 1.5T, see in Fig. 1a. Signals are obtained on field-swept mode while the frequency of microwave is fixed when sweeping Hex. Hex is in-plane with an angle θ to each strip’s long axis. All data were obtained at room temperature.  The amplitude of FMR DC response is proportional to AMR ratio. 25-28 The AMR ratio for strip 1 (see in Fig. 1e) and strip 2 (see in Fig. 1f) are obtained by measuring the magnetoresistance R. AMR follows a simple relation of θ according to R(θ) = R0 − ΔRsin2θ when the amplitude of Hex is higher than a characteristic value to overcome the demagnetization field in x-direction and pull the magnetization M to θ direction. Fig. 1e, f reveal that the characteristic values are 20 Gauss for strip 1 and 140 Gauss for strip 2. The AMR ratio ΔR/ (R0−ΔR) are 0.0167 for strip 1 and 0.0054 for strip 2.  Distinct line shapes of FMR and SSW DC response. Figure 2 shows the typical (a) FMR and (b) SSW DC responses (VFMR and VSSW) in strip 1. The line shape of Fig. 2a is described as a sum of symmetrical component Lsym=ΔHFMR/[(Hex−HFMR)2+ΔH2] and anti-symmetrical component La-sym=(Hex−HFMR)/ [(Hex−HFMR)2 +ΔHFMR2], and is well fitted by VFMR~La-sym×sinΨ+Lsym×cosΨ.() Here Hex=|Hex|, HFMR is the resonant field of FMR, ΔH is the VFMR linewidth that related to Hex and damping, andΨ represents the relative phase of introduced microwave. Line shape of VSSW exhibits as anti-symmetry, as shown in Fig. 2b. The fitting curve in Fig.2b is obtained by applying VSSW ~ (Hex−HSSW)/[(Hex−HSSW)2+ΔHSSW 2], where HSSW is the value of Hex while SSW is excited under certain microwave frequency and the phenomenological coefficient ΔHSSW refers to the linewidth of VSSW.  FMR and SSW DC spectra. Figure 3 shows the spectra of DC voltage VDC versus Hex imposed by microwave with different frequencies for (a) strip 1 and (b) strip 2 in field-swept mode. VDC is well fitted by applying the expression of VDC = VFMR +VSSW. One SSW excitation and one FMR can be observed in each spectrum. All measured results in Figure 3 indicate that SSW line shapes are anti-symmetrical while FMR line shapes are combined by Lsym and La-sym. The observed SSW signals in this article are 1st SSW. We does not observe higher ordered SSW excitations with larger wave vectors due to the confinement of equipment. Spectra of SSW excitation and FMR are split away in Fig. 3a, b. As noted in Figure 3, Hexchange=HFMR − HSSW is represented as the magnitude of a static magnetic field Hexchange, which refers to the exchange interaction between spins.  Hexchange is proportional to the wave vector of SSW. SSW in thicker Py sheet have larger wave vector, which results in a higher Hexchange in strip 1 (1050Gauss) than that of strip 2 (300Gauss). Because of the smaller Hexchange, SSW excitations are slightly coupled with FMR in (b), while in (a), SSW and FMR curves are independent. The power of microwave induced to strip 1 (25dBm) on the output of an rf signal generator is higher than that to strip 2 (10dBm), also, AMR ratio in strip 1 is higher than that of strip 2. However spectra in strip 2 possess DC voltage amplitudes and better signal noise ratio (SNR), this is because CPW can introduce microwave to strip more efficiently than rectangular waveguide.  Dispersion and line widths. Figure 4 shows (a) the measured resonant Hex (including HSSW and HFMR for both strip 1 and 2), the comparison of ΔHSSW and ΔHFMR of (b) strip 1 and (c) strip 2 as functions of fmw. Dispersion curve of FMR in (a) is well fitted by 2π f=µ0γ ×[Hex×(Hex+M/4π)]1/2 with magnetization moment M=|M|, while SSW dispersions of strip 1 and 2 are fitted by 2π f=µ0γ ×[(Hex+Hexchange)×(Hex+Hexchange +M/4π)]1/2. In Fig. 2b, c we may observe that ΔHSSW ≠ ΔHFMR.  Discussion  Two factors that contribute to the DC response are represented in Fig. 5a: the microwave-induced time-varying current imw=εmwexp(iωt−iΨ)/(R0−ΔR) where εmw is the electrical field of microwave that parallel to the strip’s length and ω=2πf, and in-plane magnetized oscillating AMR R(t)=R0−ΔR+ΔR×sin2(θ+mxexp(iωt−iϕ)/M) where mx is the time-dependent x-component of precessional magnetization moment and ϕ is the phase different between precession and microwave magnetic field hmw. By solving LLG equation: 20 ()0dddt M dtαγµ=+ + × −×nnex d mw n nMMHh h M M,                                                                                    (1) mx and ϕ are obtained. Here hd is the time-varying demagnetization field along y-axis due to the oscillating y-component of M, γ is the gyromagnetic ratio and α is the Gilbert damping coefficient. In non-resonant uniformly precession mode, DC response is weak for the smaller precessional amplitude, as shown in Fig. 5b. The spin precession is strongly excited in FMR and results obviously voltage signal, see in Fig. 5c. FMR DC signal is resulted by VFMR= <imw×R(t)>, where < > denotes the time averaging. In Hex-swept mode, the derived line shape of FMR is contributed by Lsym and La-sym by solving LLG equation, the measured FMR spectra in Figure 2, 3 and the dispersion in Fig. 4a verified this model. Similarly, VSSW is generated by imw and R(t) due to the coherent spin precession, as shown in Fig. 5d. Otherwise, if spin in SSW precesses incoherently, the DC response is weaker, see in Fig. 5e.  We begin the discussion on SSW with Hexchange. In non-uniformly precession mode for a certain Mn in SSW, exchange interaction is induced by the nearest neighbor spins Mnb =Mn-1 +Mn+1. Consequently, LLG equation is transformed as:   ()00dddt M dtαγµ γµ λ=+ + × −× + ×nnex d mw n n nb nMMHh h M M MM                                           (2) here λ is the Weiss molecular field constant. The final term in the right side of Eq. 2 expresses the exchange interaction on Mn=(mx_nexp(iωt−iϕ), my_nexp(iωt−iϕ+iφ), M) from Mnb=(mx_nbexp(iωt−iϕ), my_nbexp(iωt−iϕ+iφ), M), here φ represents the phase between x- and y-components of precessional M.  We are not going to discuss φ in this article since in the in-plane magnetization configuration, DC response not determined by my. Focusing on the exchange term of Eq. 2, the influence of Mnb acts on Mn is equivalent to a static magnetic field with its direction to z-axis. Especially, if each precessional motion in SSW follows sinusoidal distribution, the static field are the same for each spin and it is equal to Hexchange=( 0, 0, 2λM(1-cos(π/N) ). Noting that N is the number of spin within half wavelength of SSW, as shown in Fig. 1a. Therefore, the impact of exchange interaction on SSW mode can be treat as adding Hexchange into Hex for spins, if the assumption “each spin precesses at the same ϕ and different mx in SSW” is true. Seemingly, the dispersion curves in Fig. 4a confirms this assumption. However, it is difficult by using the “symmetrical and anti-symmetrical” model to analyze the line shapes of VSSW since the expected symmetrical component is not observable. The resultant relative electromagnetic phase by applying the above model on SSW signal is Ψ = ±90○. This conclusion is misleading: Ψ is fixed in field swept mode while the calculated Ψ are not at ±90○\t\r form VFMR analysis. As a result, shapes of VSSW spectra are not determined by Ψ .  Noting that what are also indicated by solving LLG equation: for two spins share the same α, time-varying fields h (e.g. hmw) and time-independent fields Hdc (such as Hex and Hexchange), their ϕ and mx will be equal; in contrast, once Hdc are different while h are the same, their ϕ will be different. These two derivations from LLG equation are described in the diagram of mx and ϕ as a function of Hdc in Fig. 6a. Thereby, “precession at same ϕ” indicates the same mx for each spin under same Hdc and h. Thus, at least in resonant condition, this standing wave mode cannot exist while precession is enhanced by sweeping Hex into the vicinity of HSSW. The increased non-uniformity of mx in SSW indicates the non-uniformity of ϕ, as illustrated in the resonant region of Fig. 6a. Theoretically, if each spin coherently precesses, the exchange interaction can be mathematically equivalent to Hexchange without time-dependent item. In the ideal condition that mx distribution is sinusoidal, it is impossible for each spin precesses with different mx since their Hdc and h are the same. Even though, if mx distribution in SSW is not sinusoidal, Hdc=| Hdc | for each spin is distinct, while still, contains only time-independent item. As what has been derived from LLG equation, precessional phases for spins are still impossible to be uniform. Experimentally, if SSW always performs as standing wave mode, whatever in resonant or non-resonant condition, the line shapes of SSW DC response is expected to be the same as FMR: a combination of symmetric and anti-symmetric components. Nonetheless, the resultant data goes against this expectation since the shapes of SSW spectra are anti-symmetric. Two possible spintronic states of resonant SSW are: 1. SSW exhibits as a state with non-uniform mx and non-uniform ϕ for each spin, as shown in Fig. 5e and 2. SSW degenerates to a non-resonant uniform precession state with weak mx, as shown in Fig. 5c. The anti-symmetric SSW spectra are in good agreement of this assumption: VSSW=0 at Hex=HSSW while for these two possible states, there DC response are weak.  In contrast, non-resonant SSW performs as a standing wave because even weak mx for every spin might be distinct with each other, precessional motion in SSW is coherent since in the non-resonant condition, the value of ϕ is approximately 0 for Hdc>HFMR and -180° for Hdc<HFMR where Hdc=|Hdc|. In the non-resonant region, VSSW should be contributed by both symmetric and anti-symmetric components while the shape line is dominated by anti-symmetric component in non-resonant condition, as shown in Fig. 6b.  That is why the shapes of SSW DC spectra are anti-symmetrical. In the resonant region, SSW cannot maintain its standing wave state, which results in the decrement of VSSW when Hex is approaching to HSSW; while in the non-resonant state, microwave-assisted SSW is a standing wave with similar line shape of DC response that be dominated by anti-symmetric component. It is also implied that, although SSW DC spectra are anti-symmetric and they can be perfectly fitted by VSSW ~ (Hex−HSSW)/[(Hex−HSSW)2+ΔHSSW 2], this formula is still empirical without the same physical meaning of La-sym. Furthermore, because of the influence of exchange interaction, the linewidth ΔHSSW different from ΔHFMR at the same fmw, as shown in Fig. 4b, c.  In summary, spin wave mode and uniform precession mode are different, especially when they are excited by microwave. Non-resonant standing spin wave is indeed in the spin wave state, while SSW cannot exhibits as a spin wave in resonant condition because of the requirement of LLG equation. In contrast, each spin in uniform precession mode shares the same precessional amplitude and phase, whatever in the resonant (FMR) or non-resonant state.  These are why the observed SSW spectra are anti-symmetrical and distinct from FMR spectra. 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Spin pumping and anisotropic magnetoresistance voltages in magnetic bilayers: Theory and experiment. Physical Review B 83, doi:10.1103/PhysRevB.83.144402 (2011). 26 Liu, Y. F., Cai, J. W. & Sun, L. Large enhancement of anisotropic magnetoresistance and thermal stability in Ta/NiFe/Ta trilayers with interfacial Pt addition. Applied Physics Letters 96, doi:10.1063/1.3334720 (2010). 27 Tripathy, D., Vavassori, P., Porro, J. M., Adeyeye, A. O. & Singh, N. Magnetization reversal and anisotropic magnetoresistance behavior in bicomponent antidot nanostructures. Applied Physics Letters 97, doi:10.1063/1.3474802 (2010). 28 Wang, S. et al. Anisotropic magnetoresistance of Ni81Fe19 films on NiFeNb buffer layer. Journal of Alloys and Compounds 575, 419-422, doi:10.1016/j.jallcom.2013.05.200 (2013).         Figure Captions Figure 1 | measurement setup, sample structures and magnetoresistance (a) Schematic diagram of the experimental setup, a coordinate system selected for further calculation is also illustrated: x-axis lies parallel to Hex while y-axis is set as normal to strip’s plane. The microwave-induced time-varying current imw flows along strip’s length. (b) Description of precessional N electron spins in a SSW. They are marked from M1, to MN. (c) Sample structure of a single Py thin strip (marked as strip 1). Modulated radio-frequency (rf) signal is introduced through a rectangular waveguide. (d) Structure of the sample with a Py thin strip (strip 2) that underneath a co-planar waveguide. Magnetoresistance measurements at orientations θ=0 and 90○ for (e) strip 1 (R0=89.22 Ω, ΔR=1.49 Ω) and (f) strip 2 (R0=121.53Ω, ΔR=0.66 Ω).  Figure 2 | distinct line shapes of FMR and SSW DC responses. The experimental (dots) and fitted (solid lines) signal of (a) a typical FMR DC response versus external field and (b) a characteristic anti-symmetric DC voltage signal in the vicinity of SSW excitation obtained at magnetic field angle θ=45○ and rf signal frequency f=11.7GHz with a modulation frequency fm=5.27kHz in strip 1. The colored regions in panel a show the fitting of symmetrical (blue) and anti-symmetrical (red) components that contribute to the line shape of FMR spectrum.  Figure 3 | FMR and SSW DC voltage spectra. The experimental (grey dots) and fitted (colored lines) of (a) SSW and FMR DC voltage spectra of strip 1 obtained at microwave power P=25dBm and frequencies ranging from 12.3GHz to 17.2GHz in field-swept mode and (b) SSW and FMR spectra of strip 2 obtained at microwave power P=10dBm and frequencies ranging from 6.5GHz to 9.5GHz in field-swept mode. θ is always fixed at 45○. The exchange interaction built-in field Hexchange is also shown in panel a and b. \t\r Figure 4 | dispersion and linewidths. (a) Measured dispersion (solid squares for strip 1 and hollow squares for strip 2) of FMR (orange squares) and SSW (green squares) of these two strips with their fitting results (solid lines).  The linewidth ΔH for FMR (orange) and SSW (green) of (b) strip 1 and (c) strip 2 at different microwave frequencies.  \t\r Figure 5 | diagrams of discussions. (a) The illustration shows how VDC generates and 4 states that might caught VDC: (b) resonant uniformly precession mode (FMR), (c) non-resonant uniformly precession mode, (d) standing wave mode and (e) non-uniform precession mode with the existence of phase difference within spins.   Figure 6 | diagrams of discussions. (a) Theoretical ϕ and mx diagram for a precessional spin excited by static and time-varying magnetic field. (b) Theoretical line shape of FMR DC response. Noting that in the non-resonant region of b, the line shape is dominated by the anti-symmetric component while in the resonant region, the voltage is contributed by both Lsym and La-sym.    Figure 1. \n    Figure 2.  \n              Figure 3. (double column) \n  Figure 4. \n    Figure 5. \n  Figure 6. \n  "    },    {        "title": "1204.0038v2.Network_analyzer_measurements_of_spin_transfer_torques_in_magnetic_tunnel_junctions.pdf",        "content": "1 Network Analyzer Measurements of Spin Transfer Torques in Magnetic Tunnel \nJunctions  \n \nLin Xue1, Chen Wang1, Yong -Tao Cui1, J. A. Katine2, R. A. Buhrman1 and D. C. Ralph1,3 \n1Cornell University, Ithaca, N ew York 14853, USA \n2Hitachi Global Storage Technologie s, San Jose, California 95135, USA  \n3Kavli Institute at Cornell, Ithaca, N ew York 14853, USA \n \nAbstract   \nWe demonstrate a simple network -analyzer technique to make quantitative measurements \nof the bias dependence of spin torque in a magnetic tunnel junction.   We apply a \nmicrowave current to exert an oscillating spin torque near the ferromagnetic resonance \nfrequency of the tunnel junction’s free layer. This produces an oscillating resistance that, together with an applied direct current, generates a microwave signal that we measure with the network analyzer. An analysis of the resonant response  yields the  strength and \ndirection of t he spin torque at non- zero bias.  We compare to measurements of the spin \ntorque vector by time -domain spin- torque ferromagnetic reso nance.  \n  2 Spin transfer torque provides the possibility of efficiently manipulating the \nmagnetic moment in a nanoscale magnetic device using applied current.1-3 \nUnderstanding  the strength of the spin torque, and particularly its bias dependence, is  \nimpor tant for applications that include spin torque magnetic random access memory and \nfrequency -tunable microwave oscillators.4 Several different techniques have been \ndeveloped to measure the bias dependence of the spin torque vector in magnetic tunnel \njunctions (MTJs), with results that in some cases are inconsistent with each other.  These include measurements of the bias dependence of the magnetic precession frequency and linewidth,\n5-10 DC-voltage -detected spin torque ferromagnetic resonance (ST -FMR),11-13 \nfits to the statistics of magnetic switching as a function of current and magnetic field,14,15 \nanalyses of the current dependence of magnetic astroids and switching phase diagrams,\n16,17 and time -domain detection of ST -FMR.18  Of these, in the high bias \nregime that is relevant for applications we believe that  the time -domain ST -FMR \ntechnique is the most accurate and trustworthy, since it measures directly the amplitude and phase of small -angle magnetic precession in response to an oscillating spin torque \nand ther efore is least susceptible to artifacts associated with heating, spatially nonuniform \nmagnetic dynamics, and changes in the DC resistance in response to spin torque.\n13,18 \nHowever, time -domain ST -FMR requires expensive, specialized equipment ( i.e., a \nhigh-bandwidth oscilloscope and multiple pulse generators). Here we show that it is \npossible to use a simple network analyzer measurement to determine the bias dependence \nof the spin torque vector, by s tudying the resonant response of a magnetic tunnel junction 3 subject to both DC and microwave currents. We find excellent agreement with \ntime-domain ST -FMR measurements18 made on the same devices.   \nThe MgO -based magnetic tunnel  junctions (MTJs) that  we study came from the \nsame batches measured in references [ 18] and [ 19], with resistance -area products for the \ntunnel barriers equal to RA = 1.5 Ω ·μm2 and 1.0 Ω ·μm2. We will present data for one \nsample w ith RA = 1.5 Ω ·μm2, a resistance of 272 Ω in the parallel state, and a tunneling \nmagnetoresistance (TMR) of 91%, but we found similar behavior in three other samples. \nThe device on which we will focus has the  layer structure (in nm): bottom electrode, \nIrMn  pinned synthetic antiferromagnet (S AF) [IrMn(6.1)/CoFe(1.8)/Ru /CoFeB( 2.0)], \ntunnel barrier [MgO x], magnetic free layer [CoFe(0.5)/CoFeB(3.4)], capping layer \n[Ru(6.0)/Ta(3.0)/Ru(4.0)]. Both  the pinned layer and the free layer were patterned into a \ncircular  cross section with a nominal  90 nm diameter. All the measurements were done at \nroom temperature . We confirmed that the device properties did not degrade during the \nprocess of measurement20 by checking that the device resistance and TMR remained \nunchanged. We will use a sign convention that positive values of current correspond to \nelectron flow from the free layer to the reference layer (giving spin torque favoring antiparallel alignment).  \nWe performed measurements with a commercial network analyzer (Agilent  \n8722ES, 50 MHz – 40 GHz) using the circuit in Fig. 1. We measured the microwave \nresponse in a reflection geometry, using a bias tee to allow simultaneous application of a DC bias to the MTJ. Before routing the reflected microwave signal to the network 4 analyzer, we amplified it using  a 15-dB amplifier in combination with a directional \ncoupler. The microwave gain of the amplifier and transmission losses in other circuit \ncomponents were calibrated by standard methods. Figure 2 shows an example of the real and imaginary parts of the reflected signal as a function of frequency, in the frequency \nrange exhibiting spin -torque -driven magnetic resonance.  These data correspond to a DC \ncurrent of - 0.4 mA and an applied magnetic field H  = 20 0 Oe oriented 70 degrees fro m \nthe exchange bias of the SAF reference layer, so that the initial offset angle of the two \nmagnetic layers is approximately \nθ = 61°. The microwave excitation signal V in that we \napplied to the sample had an amplitude always less than 22 mV .  Within the model \ndiscussed below this results in magnetic precession angles < 3 °, and we verified that the \noutput signals scaled linearly with V in as expected in the linear -response regime.  \nTo interpret these data, and to use them to measure t he strength of the spin \ntransfer torque, we analyze the reflected microwave signal refV within a macrospin \nmodel of the magnetic dynamics, combining the Landau- Lifshitz -Gilbert- Slonczewski \nequation of motion for a magnetic tunnel junction subject to an oscillating spin torque together with appropriate microwave circuit equations. (See ref.  [19] for details.)  The \nresulting (complex -valued) reflection coefficient corresponding to the resonant magnetic \nresponse can be written  \n()\n()()\n()()0\n11\n0050 50 \n50 50 ref\nDC\ninV RSIVR Rχω−Ω Ω≡= ++Ω +Ω,     (1) \nwhere  5 ()|| 0 eff\n0() () /\n1        ,50 in\nx\nI Sm mRV\nR NM RiR M Vol i V Vθ θχω ω\nτγ τ γ\nθ ωω σ ω⊥≡∆\n∂ ∂ ∂=−+ ∂ + Ω −− ∂ ∂   (2) \nand the resonance frequency and current -dependent resonant linewidth are  \n()\n()eff\neff 050 1\n50 m x y DC\nI V SRM NN IM M Vol R Vθ∂τ ∂τωγ∂θ θ ∂⊥⊥  Ω∂≈− +   +Ω ∂   ,  (3) \n()\n()eff || ||\n0() 50 1.2 2 50 xy\nDC\nI s VMNN RIM Vol R Vθαγ τ ∂τ γσθ θ∂ + ∂ Ω∂≈ −+ ∂ +Ω ∂   (4) \nHere R0 is the dif ferential resistance of the MTJ, () Rω∆  is the oscillating part of the \nDC resistance,  θ is the angle between the magnetizations of the two electrodes of the \nMTJ,  α is the Gilbert damping parameter, SM Vol  is the total magnetic moment of the \nfree layer, ||(,)Vτθ  and (,)Vτθ⊥  are the “in -plane” and “perpendicular” components of \nthe spin torque, V is the voltage across the MTJ including both DC and high- frequency \nterms, 2/Bγµ= is the absolute value of the gyromagnetic ratio, eff 4/xN HMπ= + , \neff/yN HM≈ , eff 4Mπ  is the strength of the easy -plane anisotropy field, and H  is the \ncomponent of applied magnetic field along the precession axis. When both in- plane and \nperpendicular components of torque are present, both the rea l and imaginary parts of the \nresonant signal consist of a sum of frequency -symmetric and antisymmetric Lorentzian \ncurves. Both torque components can therefore be extracted by fitting the symmetric and \nantisymmetric parts of either the real or imaginary res ponse .  \nThe solid lines in Fig. 2(a) and (b) show an example of the good agreement we \nfind when fitting Eq. (1) to our resonance measurements. We observe two resonances in each panel in Fig. 2, one with large amplitude near 5.9 GHz and a second with smalle r 6 amplitude near 7.5 GHz.  We perform separate fits to the real and imaginary curves, \nemploying four free parameters for each resonance in a fit: the center frequency of the \nresonance, the amplitude of the frequency -symmetric and antisymmetric Lorentzians,  and \nthe linewidth (taken to be the same for both the symmetric and antisymmetric \ncomponents). We allow for a small nonzero constant slope in the non- resonant \nbackground signals (dashed lines in Fig. 2) that may be associated with an imperfect capacitance calibration.   \nThe dependences  on H and I\nDC for the real part of the resonances are shown in Fig. \n3(a) and 3(b).  As in Fig. 2(a), the spectra contain one primary dip in Re( S11) together \nwith a smaller side resonance at a higher frequency.  The primary res onance shifts with \nH as expected from the Kittel formula while the secondary signal shifts more slowly and \ndecreases in amplitude with increasing field strength.  We suspect that the secondary peak may involve coupled motion of the magnetic layers in the  synthetic antiferromagnet  \npolarizing layer .  To avoid having this mode interfere with our measurements of spin \ntorque, we select values of magnetic field and magnetic field angle such that the secondary mode has small amplitude and maximum separation in fre quency from the \nprimary mode.  These are the same selection criteria used in ref. [ 18]. \nBased on Equations (1) and (2), for any value of bias we can determine the spin \ntransfer “torkances”\n21 ||/V\nθ∂τ ∂  and /Vθ∂τ ∂⊥  from fits to the frequency -symmetric \nand antisymmetric parts of the primary resonance in either Re( S11) or Im(S 11).  In \ncalculating the torkances from the resonant amplitudes we use the following parameters: 7 SM Vol = 1.8 × 10-14 emu ( ±15% ),18 eff 4Mπ = 13 ± 1 kOe determined from high -field \nmeasurements of the resonance frequency, and α = 0.016 ± 0.001 determined by \nmeasuring the resonance l inewidth at positive and negative biases and interpolating to \nzero bias. In Fig. 3(c) we plot the bias dependence of the resulting torkances as found by \nthe network analyzer technique.  We normalize the results by sinθ since the spin  \ntorque of a MTJ is predicted to have this angular dependence.21  We note that the \ntorkance values determined by independent fits to the real and imaginary parts of the \nresonance agree, as is required in order that our analysi s procedure be self -consistent. \nFigure 3(c) also shows a comparison to measurements on the same sample using the time-domain ST -FMR technique introduced in ref.  [18], whereby the magnetic precession \ndriven by a resonant spin t orque is detected by a fast oscill oscope. We find excellent \nagreement between the two types of measurements.  The in -plane component of the \ntorkance, \n||/V\nθ∂τ ∂ , measured by the two techniques agrees in magnitude near zero bias  \nwith the same moderate dependence on bias, with no adjustment of parameters for either \ntechnique.  The perpendicular component /Vθ∂τ ∂⊥  displays the same approximately \nlinear bias dependence at low bias. In Fig. 3(d), we plot the full bias dependent  torques \n()Vτ and ()Vτ⊥, obtained by numerical integration of the torkances.  \nNeither the network -analyzer ST -FMR technique nor the time -domain ST -FMR \ntechnique can be used at V  = 0, because a non -zero DC bias i s required to generate the \noscillatory voltage signal that is measured (see Fig. 3(c)).  (For measurements near zero \nbias, DC -voltage -detected ST -FMR can provide accurate torque measurements without 8 artifacts in the mixing signal.11-13) The time -domain ST -FMR technique allows \nmeasurements to higher biases, because it is naturally implemented using short bias \npulses that are less likely to produce dielectric breakdown in the tunnel barrier, compared  \nto the constant DC biases used in our network analyzer technique. However, in the bias range shown in Fig. 3(c,d) the network analyzer method provides a more convenient approach in that it does not require specialized, expensive equipment, while it yields  a \nsensitivity comparable to time -domain ST -FMR.  \nIn summary, we  demonstrate that it is possible to use a simple network -analyzer \ntechnique to measure the strength and direction of the spin transfer torque vector as a function of bias in magnetic tunnel junctions . This technique provides roughly similar \nsensitivity  as the time -domain ST -FMR method,\n18 making it useful as a simple and rapid \nmeans for characterizing spin -torque devices.  \n Cornell acknowledges support from ARO, NSF (DMR -1010768), ONR and the \nNSF/NSEC program through the Cornell Center for Nanoscale Systems. We also acknowledge NSF support through use of the Cornell Nanofabrication Facility/NNIN and the Cornell Center for Materials Research facilities  (DMR -1120296).  \n 9 \n \nFIG. 1. (color online) The network analyzer circuit used in the measurement.  \n \n \n \nFIG. 2. (color online) The measured (a) real part and (b) imaginary part of the reflection  \nsignal (S11) for IDC = -0.4 mA and a magnetic field H  = 200 Oe applied 70° from the \nexchange bias direction, giving θ= 61°. The solid lines are a fit to Eq. (1). The dashed \nlines are the nonresonant backgrounds used in the fits.  \n 10 \n \nFIG. 3. (color online) (a) Measured frequency dependence of  the real part of S 11 for \nseveral values of magnetic field applied 70°  from the exchange bias direction, with I DC = \n-0.4 mA. The curves are offset by 0.01 vertically. (b) Measured frequency dependence of \nthe real part of S 11 for several values of DC curren t, with  H = 200 Oe applied 70° from \nthe exchange bias direction. The curves are offset vertically by 0.01 . (c) Bias dependence \nof the in- plane and perpendicular components of the torkance /Vθτ∂∂  determined by \nfitting to the frequency d ependence of Re( S11) (red circles) and Im( S11) (blue diamonds) \nat different values of the DC bias.  These data correspond to H  = 200 Oe applied  70° \nfrom the exchange bias direction, giving θ= 61°.  For comparison we also show in g ray 11 the results on the same device from time -domain ST -FMR measurements (triangles : for H \n= 250 Oe applied 95° from the exchange bias direction giving θ= 85°; squares : H = 200 \nOe applied at 68° giving θ= 64°). (d) Integrated in -plane and perpendicular components \nof the spin torque  vector  determined by integrating the  network -analyzer data in (c), with \nrepresentative error bars .  \n 12 References :  \n \n1. J. C. Slonczewski, J. Magn. Magn. Mater . 159, L1 (1996).  \n2. L. Berger, Phys. Rev. B  54, 9353 (1996).  \n3. D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater . 320, 1190 (2008).  \n4. J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater . 320, 1217 (2008).   \n5. S. Petit, C. Bar aduc, C. Thirion, U. Ebels, Y . Liu, M. Li, P. Wang, and B. Dieny . Phys. \nRev. Lett.  98, 077203 (2007). \n6. S. Petit, N. de Mestier, C. Baraduc, C. Thirion, Y. Liu, M. Li, P. Wang, and B. Dieny , \nPhys. Rev. B  78, 184420 (2008).  \n7. A. M. Deac, A. Fukushima, H. Kubota, H. Maehara, Y . Suzuki, S. Yuasa, Y . Nagamine, \nK. Tsunekawa, D. D. Djayaprawira, and N. Watanabe , Nature Phys . 4, 803 (2008).  \n8. M. H. Jung, S. Park, C.- Y . You, and S. Yuasa , Phys. Rev. B  81, 134419 (2010).  \n9. O. G. Heinonen, S. W. Stokes, and J. Y . Yi, Phys. Rev. Lett . 105, 066602 (2010).  \n10. P. K. Muduli, O. G. Heinonen, and J. Akerman, Phys. Rev. B  83, 184410 (2011).  \n11. J. C. Sankey, Y .- T. Cui, J. Z. Sun, John C. Slonczewski, R. A. Buhrman, and D. C. \nRalph, Nature Phys . 4, 67 (2008).  \n12. H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama, S. Yuasa, K. Ando, H. \nMaehara, Y . Nagamine, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and Y . Suzuki , \nNature Phys . 4, 37 (2008).  \n13. C. Wang, Y . -T. Cui, J. Z. Sun, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Phys. \nRev. B  79, 224416 (2009).  \n14. Z. Li, S. Zhang, Z. Diao, Y . Ding, X. Tang, D. M. Apalkov, Z. Yang, K. Kawabata,  13  \nand Y . Huai, Phys. Rev. Lett . 100, 246602 (2008).  \n15. S. -C. Oh, S.- Y . Park, A. Manchon, M. Chshiev, J.- H. Han, H. -W. Lee, J.- E. Lee, K.- T. \nNam, Y . Jo, Y .- C. Kong, B. Dieny, and K. -J. Lee, Nature Phys . 5, 898 (2009). \n16. T. Devolder, Joo- V on Kim, C. Chappert, J. Hayakawa, K. Ito, H. Takahashi, S. Ikeda, \nand H. Ohno, J. Appl. Phys . 105, 113924 (2009).  \n17. S.- Y . Park, Y . Jo, and K. -J. Lee, Phys. Rev. B  84, 214417 (2011).  \n18. C. Wang, Y.- T. Cui, J.A. Katine, R.A. Buhrman, and D.C. Ralph, Nature Phys . 7, 496 \n(2011).  \n19. L. Xue, C. Wang, Y. -T. Cui, J.A. Katine, R.A. Buhrman, and D.C. Ralph, Appl. Phys. \nLett. 99, 022505 (2011).  \n20. D. Houssameddine, S. H. Florez, J. A. Katine, J.- P. Michel, U. Ebels, D. Mauri, O. \nOzatay, B. Delaet, B. Viala, L. Folks, B. D. Terris, and M.- C. Cyrille , Appl. Phys. Lett. 93, \n022505 (2008).  \n21. J. C. Slonczewski and J. Z. Sun, J. Magn. Magn. Mater . 310, 169 (2007).  "    },    {        "title": "2201.10995v1.Instability_of_the_ferromagnetic_quantum_critical_point_in_strongly_interacting_2D_and_3D_electron_gases_with_arbitrary_spin_orbit_splitting.pdf",        "content": "arXiv:2201.10995v1  [cond-mat.str-el]  26 Jan 2022Instability of the ferromagnetic quantum critical point in strongly interacting 2D and\n3D electron gases with arbitrary spin-orbit splitting\nDmitry Miserev,1∗Daniel Loss,1and Jelena Klinovaja1\n1Department of Physics, University of Basel,\nKlingelbergstrasse 82, CH-4056 Basel, Switzerland\n(Dated: January 27, 2022)\nIn this work we revisit itinerant ferromagnetism in 2D and 3D electron gases with arbitrary\nspin-orbit splitting and strong electron-electron intera ction. We identify the resonant scattering\nprocesses close to the Fermi surface that are responsible fo r the instability of the ferromagnetic\nquantum critical point at low temperatures. In contrast to p revious theoretical studies, we show\nthat such processes cannot be fully suppressed even in prese nce of arbitrary spin-orbit splitting.\nA fully self-consistent non-perturbative treatment of the electron-electron interaction close to the\nphase transition shows that these resonant processes alway s destabilize the ferromagnetic quantum\ncritical point and lead to a first-order phase transition. Ch aracteristic signatures of these processes\ncan be measured via the non-analytic dependence of the spin s usceptibility on magnetic field both\nfar away or close to the phase transition.\nI. Introduction\nItinerant ferromagnetism in two-dimensional (2D) and\nthree-dimensional (3D) electron gas has been observed\nin various materials, such as manganese perovskites [ 1–\n4], transition-metal-doped semiconductors [ 5–7], mono-\nlayers of transition metal dichalcogenides [ 8–12], and\nmany others [ 13–20]. The physical mechanisms lead-\ning to the ferromagnetic ground state depend strongly\non the materials. Doping by transition metals results\nin strong interaction between the itinerant spins and\nthe magnetic moments of the transition metal ions, the\nmechanism known as double exchange or Zener mech-\nanism [21–23]. In this case the ferromagnetism is not\nintrinsic but rather induced by the magnetic moments\nof the dopants. In contrast, in this work, we are in-\nterested in the itinerant ferromagnetism emerging from\nstrongelectron-electroninteractionsbetween the delocal-\nized charge carriers. This mechanism is often referred to\nas Stoner mechanism [ 24,25].\nIn the original work of Stoner [ 24] the phase transition\nto the ferromagnetic phase is of second order, i.e. contin-\nuous. The ferromagnetic quantum critical point (FQCP)\nof the spin-degenerate electron gas is analyzed in the lit-\nerature via the effective Ginzburg-Landau-Wilson theory\n[26–30] describing the fluctuating magnetic order param-\neter. However, this theory relies on the analyticity of the\neffective Lagrangian [ 29,30] which does not hold for in-\nteracting 2D and 3D electron gases [ 31]. The negative\nnon-analytic corrections, originating from the resonant\nbackscattering of itinerant electrons close to the spin-\ndegenerate Fermi surface, emerge already in second or-\nder perturbation in the electron-electron interaction [ 31–\n39]. If these non-analyticities survive in a small vicinity\nof the ferromagnetic quantum phase transition (FQPT),\nthey destabilize the FQCP at zero temperature and lead\nto a first-order FQPT in 2D and 3D electron gases [ 32–\n39]. This phenomenon is an example of the fluctuation-\ninduced first-order transition first predicted in high en-ergy physics [ 40]. In condensed matter, this effect is re-\nsponsible for weak first-order metal-superconductor and\nsmectic-nematic phase transitions [ 41].\nThe problem with FQPTs in clean metals is that\nit happens at very strong electron-electron interaction\nwhere the perturbative approach [ 31–39] is no longer\nvalid. Ithasbeenpointedoutintheliteraturethathigher\norder scattering processes in the spin-degenerate elec-\ntron gas [ 42,43] may change the sign of the non-analytic\nterms making them irrelevant in the infrared limit and\nthus, stabilizing the FQCP. So far, the greatest advances\nin understanding the strongly interacting regime are at-\ntributed to the effective spin-fermion model [ 44] where\nthecollectivespinexcitationsinstronglyinteractingelec-\ntron gases are coupled to the electron spin. The nega-\ntivenon-analyticitiescalculatedwithinthismodelremain\nrelevant, although strongly suppressed near the FQPT\n[42,43].\nNumerical simulations of the low-density spin-\ndegenerate 2D electron gas (2DEG) confirm a first-order\nFQPT in the liquid phase with further transition to the\nWigner solid at even lower densities [ 45–48]. The situa-\ntion is less definite in a 3D electron gas (3DEG) where\nvarious advanced numerical techniques predict either a\nfirst- or second-order FQPT depending on the numerical\nscheme [49–54]. This disagreement between different nu-\nmerical results for 3DEGs might follow from the much\nweaker character of the non-analytic terms destabilizing\nthe FQCP compared to the 2D case [ 31].\nThe problem becomes even more complicated if a spin-\norbit (SO) splitting of the Fermi surface is present. The\nmain effect of the SO splitting is the spin symmetry\nbreaking restricting possible directions of the net mag-\nnetization in the magnetically ordered phase [ 55,56]. In\nparticular, the Rashba SO splitting in 2DEGs restricts\npossible net magnetization directions in the 2DEG plane\n[55,56]. So far, SO splitting is considered in the litera-\nture as a promising intrinsic mechanism cutting the non-\nanalyticity and stabilizing the FQCP in the interacting2\nelectron gas [ 57].\nIn this work, we consider the general case of a D-\ndimensional electron gas, D >1, with arbitrarySO split-\nting and identify the resonant scattering processes close\nto the Fermi surface which result in the non-analytic cor-\nrections with respect to the magnetization. Our results\nare in perfect agreement with the previously considered\ncaseoftheRashba2DEG[ 55,56]. However,weshowthat\neven arbitrary SO splitting is not able to cut negative\nnon-analytic corrections in 2DEGs and 3DEGs. Thus, in\ncontrast to Ref. [ 57], we find that SO splitting cannot be\nconsidered as a possible intrinsic mechanism stabilizing\nFQCP in a uniform electron gas.\nIn this work we apply the dimensional reduction of\nthe electron Green function which we developed earlier\nin Ref. [58]. This procedure allows us to reduce Dspa-\ntial dimensions to a single effective spatial dimension and\nsignificantly simplifies the derivation of the non-analytic\ncorrections in the perturbative regime for arbitrary SO\nsplitting. We confirm the validity of our approach by\ncomparison with known results [ 43,55,56]. In order to\naccess the strongly interacting regime, we treat the res-\nonant scattering processes near the Fermi surface within\nthe self-consistent Born approximation and solve it in\nthe limit of strong interaction. Within this approach, we\nfind the non-Fermi liquid electron Green function which\ndiffers significantly from the Green function calculated\nwithin the effective spin-fermion model [ 42,43]. Within\nour model, the non-analyticities are strongly enhanced\nclose to the FQPT and remain negative at arbitrary SO\nsplitting. Thus, we conclude that the FQCP in strongly\ninteracting 2DEGs and 3DEGs is intrinsically unstable.\nIn order to test our theoretical model experimentally,\nwe suggest to measure the spin susceptibility in the para-\nmagnetic phaseclose to the FQPT. Accordingto ourpre-\ndictions, thespinsusceptibility χij(B)closetotheFQPT\ntakes the form χij(B)−χij(0)∝ |B|D−1\n2modulo powers\nof ln(EF/|B|), while the spin-fermion model predicts a\nmuch weaker scaling: χij(B)−χij(0)∝ |B|3\n2for 2DEGs\nandχij(B)−χij(0)∝ |B|2lnln(EF/|B|) for 3DEGs [ 43],\nwhereEFis the Fermi energy, |B|is measured in units\nof energy. In the presence of SO splitting we also predict\na non-trivial tensor structure of χij(B)−χij(0), which\ncan also be used to identify the structure of the SO cou-\npling. The candidate materials for experiments are the\npressure-tuned3DmetalsZrZn 2[59], UGe 2[60], 2DAlAs\nquantum wells [ 61] and many more [ 62].\nThe paper is organized as follows. In Sec. II we\nintroduce the non-interacting electron gas in D >1\nspatial dimensions with arbitrary spin splitting. In Sec.\nIII we derive the asymptotics of the free electron Green\nfunction at large imaginary time τ≫1/EFand large\ndistance r≫λF, whereEFis the Fermi energy, λFis\nthe Fermi wavelength. In Sec. IV we use second order\nperturbation theory to derive the non-analytic correction\nto the thermodynamic potential Ω with respect to the\narbitrary spin splitting. In Sec. V we calculate the\nelectron Green function in the limit of strong electron-electron interaction and find a sector in the phase space\nwhere the Green function is non-Fermi-liquid-like. In\nSec. VI we calculate the thermodynamic potential Ω\nin the regime of strong interaction and find that the\nnon-analytic corrections are negative and parametrically\nlarger than the ones predicted in the weakly interacting\nlimit. In Sec. VII we derive the non-analytic corrections\nto the spin susceptibility both far away and close to\nthe FQPT. Conclusions are given in Sec. VIII. Some\ntechnical details are deferred to the Appendix.\nII. Non-interacting electron gas with arbitrary spin\nsplitting\nIn this section we consider a non-interacting single-\nvalley electron gas in D >1 spatial dimensions with ar-\nbitrary spin splitting. The case of D= 1 is not included\nin this paper due to the Luttinger liquid instability of\none-dimensional Fermi liquids with respect to arbitrarily\nsmall interactions [ 63]. The electron gas is described by\nthe following single-particle Hamiltonian:\nH0=p2\n2m−EF−σ·β(p), (1)\nwherepis aD-dimensional momentum, mthe effec-\ntive mass, EFthe Fermi energy, β(p) the spin splitting,\nσ= (σx,σy,σz) the Pauli matrices. The spin splitting is\nconsidered small compared to the Fermi energy:\nβ(p)≡ |β(p)| ≪EF, (2)\nbut otherwise arbitrary. Therefore, the spin splitting\nclose to the Fermi surface can be parametrized by the\nunit vector np=p/palong the momentum p:\nβ(p)≈β(np),np=p\np, p=kF=/radicalbig\n2mEF.(3)\nHere we introduced kFas the Fermi momentum at zero\nspin splitting β(p) = 0.\nThe eigenvectors |σ,np∝an}bracketri}htof the Hamiltonian H0corre-\nspond to the eigenvectors of the operator σ·β(np):\nσ·β(np)|σ,np∝an}bracketri}ht=σβ(np)|σ,np∝an}bracketri}ht, (4)\nwhereσ=±1 andβ(np) =|β(np)|. The explicit form\nof the spinors is given by\n|σ,np∝an}bracketri}ht=(β−(np), σβ(np)−βz(np))T\n/radicalbig\n2β(np)[β(np)−σβz(np)],(5)\nwhere the superscriptTmeans transposition, β±(np) =\nβx(np)±iβy(np). Two spinors with the same npand\nopposite σare orthogonal:\n∝an}bracketle{t+,np|−,np∝an}bracketri}ht= 0. (6)\nThis forbids the forward scattering between the bands\nwith opposite band index σ.3\nIn this paper we need the backscattering matrix ele- ments:\nMσσ′(np) =∝an}bracketle{tσ,np|σ′,−np∝an}bracketri}ht. (7)\nUsing Eq. ( 5), we find the matrix elements explicitly:\nMσσ′(np) =β+(np)β−(−np)+σσ′[β(np)−σβz(np)][β(−np)−σ′βz(−np)]/radicalbig\n4β(np)β(−np)[β(np)−σβz(np)][β(−np)−σ′βz(−np)]. (8)\nFSσ\nkkσkkσn( )δpp\n(a) (a).FSσ\nrnr\n-nr(nr)kkσ\n(-nr)kkσ\n(b) (b).\nFIG. 1. (a) Expansion of the momentum pclose to the\nFermi surface FSσ:kσis the normal projection of ponFSσ,\nn(kσ) is the outward normal at kσ∈ FS σ,δp≪kF. (b)\nTwo points kσ(nr) andkσ(−nr) on a nearly spherical Fermi\nsurfaceFSσwhere the outward normals are equal to nrand\n−nr, respectively. The two red patches on FSσcorrespond\ntothe vicinities U(±nr) of thepoints kσ(±nr) which give the\nleading contribution to the τ≫1/EFandr≫λFasymp-\ntotics of the Green function, see Eq. ( 18).\nThe spin splitting β(np) results in two Fermi surfaces\nlabeled by σ=±1 with the Fermi momenta being de-\npendent on np:\nkσ(np) =/radicalBig\n2m(EF+σβ(np))≈kF+σβ(np)\nvF,(9)\nwherevF=kF/mis the Fermi velocity at β(np) = 0.\nHere we used Eq. ( 2) in order to expand the square root.\nIII. Asymptotics of the free electron Green\nfunction\nIn the following it is most convenient to work with the\nelectron Green function in the space-time representation.\nIn this paper we operate with the statistical (Matsubara)\nGreen function Gσ(τ,r), where τis the imaginary time,\nrtheD-dimensional coordinate vector, and σ=±1 the\nband index. In this section we derive the asymptotics of\nthe free electron Green function G(0)\nσ(τ,r) atτ≫1/EF\nandr≫λF, whereλF= 2π/kFis the Fermi wavelength.\nSimilar derivations can be found in Ref. [ 64] in applica-\ntion to the Fermi surface imaging.\nThe asymptotics of G(0)\nσ(τ,r) atτ≫1/EFandr≫λFcomes from the sector ( ω,p) close to the Fermi sur-\nface:\nω≪EF,p=kσ+n(kσ)δp, δp≪kF,(10)\nwherekσ∈ FSσis a point on the spin-split Fermi sur-\nfaceFSσwith index σ,n(kσ) is the outward normal at\nthis point, δp >0 (δp <0) corresponds to empty (oc-\ncupied) states at zero temperature, see Fig. 1(a). The\nfree electron Green function G(0)\nσ(iω,p) is given by the\nquasiparticle pole:\nG(0)\nσ(iω,p)≡G(0)\nσ(iω,δp,n) =|σ,n∝an}bracketri}ht∝an}bracketle{tσ,n|\niω−vσ(n)δp,(11)\nwhere we shortened n(kσ) tonhere,|σ,n∝an}bracketri}htis the spinor\ngiven by Eq. ( 5),vσ(n) is the Fermi velocity at kσ. Here\nwe also linearized the dispersion with respect to δpbe-\ncauseδp≪kF. At the same time, the finite curvature of\nthe Fermi surface is important for the asymptotic form\nofG(0)\nσ(τ,r).\nThe space-time representation of the free electron\nGreen function is given by the Fourier transform:\nG(0)\nσ(τ,r) =∞/integraldisplay\n−∞dω\n2π/integraldisplaydp\n(2π)Dei(p·r−ωτ)G(0)\nσ(iω,p),(12)\nwhereG(0)\nσ(iω,p) is given by Eq. ( 11) at the infrared sec-\ntor defined in Eq. ( 10). We integrate over the Matsub-\nara frequencies because here we consider the case of zero\ntemperature T= 0. The integral over ωis elementary:\nG(0)\nσ(τ,δp,n) =∞/integraldisplay\n−∞dω\n2πe−iωτ|σ,n∝an}bracketri}ht∝an}bracketle{tσ,n|\niω−vσ(n)δp\n=−sgn(τ)θ(δpτ)e−vσ(n)δpτ|σ,n∝an}bracketri}ht∝an}bracketle{tσ,n|,(13)\nwhere sgn( τ) returns the sign of τ,θ(z) is the Heaviside\nstep function, i.e. θ(z) = 0 (θ(z) = 1) if z <0 (z >0).\nTheintegrationover pisconvenienttoperformviathin\nlayers located at the distance δpfrom the Fermi surface\nFSσ. Asδp≪kF, the measure can be approximated as\nfollows:\ndp≈dkσdδp,kσ∈ FSσ. (14)\nThe momentum is expanded via Eq. ( 10), see also\nFig.1(a), i.e. p=kσ+n(kσ)δp, where the normal\nn(kσ) is taken at the point kσ∈ FSσ.4\nNext, we apply the stationary phase method in order\nto evaluate the integral over kσ∈ FS σ. For this, we\nfirst find the stationary points where kσ·rreaches its\nextrema. This happens when dkσ·r= 0, where dkσis\nan arbitraryelement of the tangent space attached to the\nFermi surface FSσat the point kσ∈ FSσ. This condi-\ntion is satisfied at such points kσ∈ FSσat which thenormalsn(kσ) are collinearwith the coordinatevector r.\nAs the Fermi surfaces are nearly spherical, see Eq. ( 9),\nthere are exactly two points kσ(±nr)∈ FSσwhere the\noutward normals are equal to ±nr,nr=r/ris the unit\nvector along r, see Fig. 1(b). Thus, we find that the in-\ntegral over pyields the sum of two integrals over small\nvicinities U(±nr)⊂ FSσof the points kσ(±nr)∈ FSσ,\nsee Fig.1(b):\nG(0)\nσ(τ,r)≈/integraldisplay\nkσ∈U(nr)dkσ\n(2π)D−1ei(kσ−kσ(nr))·r∞/integraldisplay\n−∞dδp\n2πeiδprG(0)\nσ(τ,δp,nr)eikσ(nr)·r\n+/integraldisplay\nkσ∈U(−nr)dkσ\n(2π)D−1ei(kσ−kσ(−nr))·r∞/integraldisplay\n−∞dδp\n2πe−iδprG(0)\nσ(τ,δp,−nr)eikσ(−nr)·r,nr=r\nr. (15)\nHerekσ(±nr) are the points on FSσwhere the outward\nnormals are equal to ±nr, see Fig. 1(b). The integration\noverδpis extended to the interval δp∈(−∞,∞) due\nto quick convergence on the scale δp∼1/r≪kF. The\nintegrals over kσin the vicinities U(±nr)⊂ FSσof the\npointskσ(±nr) are Gaussian and they are convergent\ndue to the finite Gaussian curvature of nearly spherical\nFermi surface FSσat the points kσ(±nr), see Appendix\nAfor more details . The integrals over δpyield the one-\ndimensional Fourier transforms:\nG(0)\nσ(τ,x,n) =∞/integraldisplay\n−∞dδp\n2πeiδpxG(0)\nσ(τ,δp,n)\n=1\n2π|σ,n∝an}bracketri}ht∝an}bracketle{tσ,n|\nix−vσ(n)τ, (16)\nwherenhere is an arbitrary unit vector and x∈\n(−∞,∞) an effective one-dimensional coordinate. Keep-\ning only the linear order in SO splitting, we show in the\nAppendix Athat\nkσ(±nr)·r≈ ±kσ(±nr)r, (17)\nwherekσ(n) is given by Eq. ( 9) for arbitrary unit vector\nn. The Gaussian integrals over U(±nr) are proportional\nto 1/r(D−1)/2, see Appendix Afor the details. Substi-\ntuting Eqs. ( 16) and (17) into Eq. ( 15) and evaluating\nthe Gaussian integrals over U(±nr), we find the infrared\nlong-range asymptotics of the free-electron Matsubara\nGreen function:\nG(0)\nσ(τ,r)≈/parenleftbigg1\nλFr/parenrightbiggD−1\n2/bracketleftbiggei(kσ(nr)r−ϑ)\n2π|σ,nr∝an}bracketri}ht∝an}bracketle{tσ,nr|\nir−vFτ\n−e−i(kσ(−nr)r−ϑ)\n2π|σ,−nr∝an}bracketri}ht∝an}bracketle{tσ,−nr|\nir+vFτ/bracketrightbigg\n,nr=r\nr,(18)\nϑ=π\n4(D−1), (19)wherekσ(n) is given by Eq. ( 9) for arbitrary unit vec-\ntorn,λFis the Fermi wavelength, vF=kF/mis the\nFermi velocity at zero spin splitting, and nr=r/ris\nthe unit vector along r. Here we stress that Eq. ( 18) is\ntrue only if the SO splitting is small compared to EF,\nsee Eq. ( 2). We also neglected the weak dependence of\nthe Fermi velocity on the spin splitting in the denomi-\nnators in Eq. ( 18) because it does not provide any non-\nanalyticities. We see from Eq. ( 18) that the Green func-\ntion contains the oscillatory factors that are sensitive to\nthe spin splitting through kσ(±nr), see Eq. ( 9). As we\nwill see later, these oscillatory factors are responsible for\nthe non-analytic terms in the thermodynamic potential\nΩ.\nIn the Appendix Awegeneralizethis calculationto the\ncase of a strongly interacting electron gas with a singu-\nlarity (not necessarily a pole) at the Fermi surface of ar-\nbitrary geometry. The Appendix Aalso contains details\nfor nearly spherical Fermi surfaces. The case of spherical\nFermi surfaces is covered in Ref. [ 58].\nIV. Non-analyticities in Ω: limit of weak\ninteraction\nIn this section we calculate the non-analytic correc-\ntions to the thermodynamic potential Ω in the limit of\nweak electron-electron interaction. The calculation is\nperformed within second order perturbation theory and\nitisvalidintheparamagneticFermiliquidphasefaraway\nfrom the FQPT. However, this calculation is important\nbecause it allows us to identify the resonant scattering\nprocesses close to the Fermi surface which are respon-\nsible for the non-analytic terms in Ω. In the following\nsections we treat these processes non-perturbatively and\nfind that the non-analytic terms are strongly enhanced\nclose to the FQPT. The results of this section extend\nexisting theories [ 32–34,43,55] to the case of arbitrary5\n(a) (b)\nFIG. 2. (a) First-order interaction correction to Ω, see\nEq. (20). (b) Second-order interaction correction to Ω con-\ntributing to the non-analyticity. Solid lines correspond t o\nthe electron propagators G(0)\nσ(τ,r), see Eq. ( 18); wiggly lines\nstand for the Coulomb interaction, see Eq. ( 22).\nspin splitting. In particular, we show that arbitrary SO\nsplitting is not able to gap out all soft fluctuation modes\nandthe non-analyticityinΩwith respecttothe magnetic\nfieldBsurvives, in contrast to predictions of Ref. [ 57].\nIn this section we are only after the non-analytic terms\nin Ω, all analytic corrections will be dropped.\nA. First-order interaction correction to Ω\nLet us start from the first-order interaction correction\nto Ω, see Fig. 2(a):\nΩ(1)=1\n2/summationdisplay\nσ,σ′/integraldisplay\ndzV0(z)Pσσ′(z), (20)\nPσσ′(z) =−Tr/braceleftBig\nG(0)\nσ(z)G(0)\nσ′(−z)/bracerightBig\n,(21)\nwherez= (τ,r), Trstandsforthespintrace,and Pσσ′(z)\nis the particle-hole bubble. Here, V0(z) is the Coulomb\ninteraction,\nV0(τ,r) =e2\nǫrδ(τ), (22)\nwhereeis the elementary charge, ǫthe dielectric con-\nstant, and δ(τ) is due to the instantaneous nature of\nthe Coulomb interaction (the speed of light is much\nlarger than the Fermi velocity). Using the asymptotics\nof the Green function G(0)\nσ(τ,r), see Eq. ( 18), we find the\nasymptotics of the particle-hole bubble:\nPσσ′(τ,r) =PL\nσσ′(τ,r)+PK\nσσ′(τ,r), (23)\nPL\nσσ′(τ,r)≈δσσ′\n2π2/parenleftbigg1\nλFr/parenrightbiggD−1v2\nFτ2−r2\n(r2+v2\nFτ2)2,(24)\nPK\nσσ′(τ,r)≈1\n4π21\nr2+v2\nFτ2/parenleftbigg1\nλFr/parenrightbiggD−1×/bracketleftBig\ne−2iϑeir(kσ(nr)+kσ′(−nr))|Mσσ′(nr)|2\n+e2iϑe−ir(kσ(−nr)+kσ′(nr))|Mσσ′(−nr)|2/bracketrightBig\n,(25)\nwhere the matrix elements Mσσ′(±nr) are given by\nEq.(8). HerePL\nσσ′(τ,r) is the Landaudamping contribu-\ntion to the particle-hole bubble coming from the forward\nscattering. It is clear that this contribution is insensitive\ntothespinsplitting. Thesecondcontribution, PK\nσσ′(τ,r),\nis the Kohn anomaly coming from the backscattering\nwith the momentum transfer close to 2 kF. The Kohn\nanomaly is sensitive to the spin splitting through the os-\ncillatoryfactorscontainingthe Fermi momenta kσ(±nr),\nsee Eq. ( 9).\nAsonly theKohnanomalyissensitiveto thespin split-\ntingβ(np), we can simplify Eq. ( 20):\nΩ(1)=1\n2/summationdisplay\nσ,σ′/integraldisplay\nSD−1dnr∞/integraldisplay\n0drrD−1e2\nǫrPK\nσσ′(0,r),(26)\nwhereSD−1is the (D−1)-dimensional unit sphere, dr=\nrD−1drdnr. The integral over ris divergent at small r\n(the ultraviolet divergence)because it takesthe following\nform:\n∞/integraldisplay\n0dr\nr3eir∆→ ∞, (27)\nwhere ∆ is either equal to ∆ = kσ(nr)+kσ′(−nr) or to\n∆ =−kσ(−nr)−kσ′(nr). This divergence comes from\nthe asymptotics of the particle-hole bubble, see Eq. ( 25),\nthat is only valid at r≫λF. Therefore, the lower limit\nforrin Eq. (27) is bounded by r∼λF. This divergence\ncan also be cured via the analytical continuation to the\nEuler gamma function Γ( x) using the following identity:\nIα(∆) =∞/integraldisplay\n0dr\nrαeir∆=π|∆|α−1\nsin(πα)Γ(α)e−iπ\n2(α−1)sgn(∆).(28)\nIn our case α= 3 and the integral is indeed divergent\ndue to sin(3 π) = 0 in the denominator. Therefore, we\nconsider α= 3+δand take the limit δ→0:\n∞/integraldisplay\n0dr\nr3eir∆=∆2\n2/parenleftbigg1\nδ+ln|∆|−iπ\n2sgn(∆)/parenrightbigg\n.(29)\nNow it is clear that the physical dimension of ∆ under\nthe logarithm has to be compensated by the ultraviolet\nscalep0∼2kFwhich is equivalent to cutting the lower\nlimit in Eq. ( 27) atr∼λF:\n∞/integraldisplay\n∼λFdr\nr3eir∆=∆2\n2/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆\np0/vextendsingle/vextendsingle/vextendsingle/vextendsingle−iπ\n2sgn(∆)/parenrightbigg\n.(30)\nNow, we come back to Ω(1)where ∆ is either equal to\n∆ =kσ(nr)+kσ′(−nr) or to ∆ = −kσ(−nr)−kσ′(nr),6\n(a) (b)\nFIG. 3. Other second-order diagrams that do not contribute\nto the non-analyticities in Ω, see Eqs. ( 45), (46).\nso using Eq. ( 9) we find:\n|∆| ≈2kF+σβ(±nr)+σ′β(∓nr)\nvF.(31)\nAs the spin splitting is much smaller than the Fermi\nenergy, we can expand the logarithm ln |∆/p0|in the\nanalytic Taylor series with respect to the spin split-\nting. Hence, we see that Ω(1)does not contain any non-\nanalyticities for arbitrary spin splitting.\nHere we have performed the calculations for the long-\nrange Coulomb interaction Eq. ( 22). Finite electron den-\nsity results in the Thomas-Fermi screening of the long\nrange Coulomb tail on the scale of the screening length\nr0. The weak coupling limit that we consider in this\nsection corresponds to r0≫λF. However, the inte-\ngral over rin Ω(1)converges at r∼λF≪r0, because\n∆≈2kFhere. Therefore, we can indeed neglect the\nThomas-Fermi screening in this section.\nB. Second-order interaction corrections to Ω\nWe see from the calculation of Ω(1)that the non-\nanalytic terms may come from the oscillatory integrals\nliketheoneinEq.( 28). However,wehavetosubtractthe\n2kFfactor first, such that ∆ in Eq. ( 28) becomes propor-\ntional to the spin splitting. One way to achieve this is toconsider Ω(1), see Eq. ( 20), with the interaction V(τ,r)\nwhich has oscillatory components e±i2kFr. In fact, the\nelectron-electron interaction acquires such components\nupon the dynamic screening by the particle-hole bubble.\nOne consequence of this is the Thomas-Fermi screening\nwhich we already discussed and concluded that it is not\nimportant if the interaction is weak. However, there is\nanothermuch moreimportant consequence ofsuch dress-\ning that results in 2 kFharmonics in the interaction due\nto backscattering of electrons near the Fermi surface, the\neffect known as Friedel oscillations. As we consider the\ncorrelations at large r∼vF/β≫λF, whereβis a char-\nacteristic value of the spin splitting at the Fermi surface,\nthe interaction matrix elements at the momentum trans-\nfer 2kFare effectively local, so we can use the effective\ncontact interaction:\nV2kF(z) =uδ(r)δ(τ) =uδ(z), (32)\nu≈V0(2kF) =2πD\n2Γ(D−1)\nΓ/parenleftbigD\n2/parenrightbige2\nǫ(2kF)D−1,(33)\nwhereδ(r) is the D-dimensional delta function, V0(q)\nis the Fourier transform of the Coulomb interaction\nEq. (22).\nIf we dress the interaction line in Fig. 2(a) by a single\nparticle-holebubble, we get the second-orderdiagram for\nΩ shown in Fig. 2(b):\nΩ(2)=−1\n4/summationdisplay\nσi/integraldisplay\ndz1dz2dz3V2kF(z2)V2kF(z3−z1)\n×Pσ1σ2(z1)Pσ3σ4(z3−z2).(34)\nUsing the contact approximation Eq. ( 32), we simplify\nΩ(2)to the following expression:\nΩ(2)=−u2\n4/summationdisplay\nσi/integraldisplay\ndzPσ1σ2(z)Pσ3σ4(z),(35)\nwherez= (τ,r). From Eqs. ( 23)–(25) we find that only\ntheproductofKohnanomaliescontainsslowlyoscillating\nterms on the scale vF/β, whereβstands for the charac-\nteristic spin splitting at the Fermi surface:\nPσ1σ2(z)Pσ3σ4(z) =eir∆σ1σ2σ3σ4(nr)|Mσ1σ2(nr)Mσ3σ4(−nr)|2+e−ir∆σ1σ2σ3σ4(−nr)|Mσ1σ2(−nr)Mσ3σ4(nr)|2\n(2π)4(λFr)2(D−1)(r2+v2\nFτ2)2+...,(36)\n∆σ1σ2\nσ3σ4(nr) =kσ1(nr)+kσ2(−nr)−kσ3(−nr)−kσ4(nr)≈(σ1−σ4)β(nr)+(σ2−σ3)β(−nr)\nvF, (37)\nwhere dots in Eq. ( 36) stand for the rapidly oscillating\nterms on the scale of 2 kFand 4kFand also the for-\nward scattering contribution which does not contain any\nnon-analytic dependence on the spin splitting. We used\nEq. (9) to express ∆σ1σ2σ3σ4(nr) in terms of the spin split-ting.\nThen we substitute Eq. ( 36) into Eq. ( 35) and evalu-\nate the integral over z= (τ,r). The integral over τis7\nelementary:\n∞/integraldisplay\n−∞dτ\n(r2+v2\nFτ2)2=π\n2vFr3. (38)\nThe integral over rcan be represented using the integral\nIα(∆) defined in Eq. ( 28):\nΩ(2)=−u2\n26π3vFλ2(D−1)\nF/summationdisplay\nσi/integraldisplay\nSD−1dnr\n|Mσ1σ2(nr)Mσ3σ4(−nr)|2Re/parenleftbig\nID+2/parenleftbig\n∆σ1σ2\nσ3σ4(nr)/parenrightbig/parenrightbig\n,(39)\nwhere Re stands for the real part. Here we used that\nIα(−x) =I∗\nα(x), where the star corresponds to the com-\nplex conjugation.\nAt this point it is convenient to introduce the dimen-\nsionless interaction parameter g:\ng=uNF=umkD−2\nF\n2D−1πD\n2Γ/parenleftbigD\n2/parenrightbig, (40)\nwhereNFis the density of states per band at the Fermi\nlevel. Substituting Eq. ( 33) into Eq. ( 40), we find anestimate for the dimensionless coupling constant g:\ng≈Γ(D−1)\n22D−3Γ2/parenleftbigD\n2/parenrightbig1\nkFaB, aB=ǫ\nme2,(41)\nwhereaBis the effective Bohr radius. The weak coupling\nregime correspondsto high densities such that kFaB≫1\norg≪1.\nThenEq.( 39)canberepresentedinthefollowingform:\nΩ(2)=LDvD+1\nF\n2D+2/summationdisplay\nσi/integraldisplay\nSD−1dnr\n×|Mσ1σ2(nr)Mσ3σ4(−nr)|2/vextendsingle/vextendsingle∆σ1σ2\nσ3σ4(nr)/vextendsingle/vextendsingleD+1,(42)\nLD=g2\n32/parenleftbigg2\nπvF/parenrightbiggDΓ2/parenleftbigD\n2/parenrightbig\nΓ(D+2)1\ncos/parenleftbig\nπD\n2/parenrightbig. (43)\nWe perform the summation over the band indexes σiex-\nplicitly:\nΩ(2)=LD/integraldisplay\nSD−1dn/bracketleftBig\n|M+−(n)M+−(−n)|2|β(n)−β(−n)|D+1+|M++(n)M−−(n)|2|β(n)+β(−n)|D+1\n+2/parenleftBig\n|M++(n)M−+(n)|2+|M−−(n)M+−(n)|2/parenrightBig\n|β(n)|D+1/bracketrightBig\n. (44)\nHere we dropped the index rinnr, such that ncan\nbe also interpreted as the unit vector np=p/p,p≈\nkF, in the momentum space. This interpretation makes\nsense because the asymptotics of the Green function, see\nEq. (18), comes from small vicinities of two points on the\nFermi surface whose outward normals are collinear with\nr. So,randpare in a way pinned to each other.\nFinally, we have to check that the second-order di-\nagrams in Fig. 3(a),(b) do not contribute to the non-\nanalytic terms in Ω:\nΩa=u2\n2/summationdisplay\nσi/integraldisplay\ndz\nTr/braceleftBig\nG(0)\nσ1(0)G(0)\nσ2(z)G(0)\nσ3(0)G(0)\nσ4(−z)/bracerightBig\n,(45)\nΩb=u2\n4/summationdisplay\nσi/integraldisplay\ndz\nTr/braceleftBig\nG(0)\nσ1(z)G(0)\nσ2(−z)G(0)\nσ3(z)G(0)\nσ4(−z)/bracerightBig\n.(46)\nHereG(0)\nσ(0) =G(0)\nσ(τ=−0,r= 0) due to the ordering\nof the field operatorswithin the interaction Hamiltonian:\nG(0)\nσ(0) =/integraldisplaydp\n(2π)Dθ(kσ(np)−p)|σ,np∝an}bracketri}ht∝an}bracketle{tσ,np|,(47)where|σ,np∝an}bracketri}htare the eigenvectors of the single-particle\nHamiltonian, see Eq. ( 5), andkσ(np) is given by Eq. ( 9).\nThe diagram Ω ahas a single particle-hole bubble in\nit due to the Green functions G(0)\nσ2(z) andG(0)\nσ4(−z), see\nEq. (45). The product of these Green functions contains\nweakly oscillating terms and ≈2kFharmonics. As in the\ncase of Ω(1), the 2kFharmonics do not produce any non-\nanalyticities. Theweaklyoscillatingtermsoriginatefrom\nthe Landau damping part of the particle-hole bubble but\nthese terms vanish due to the integral over τ:\n∞/integraldisplay\n−∞dτ\n(vFτ±ir)2= 0. (48)\nThe diagram Ω bis more complicated. Let us\nconsider two matrix products G(0)\nσ1(z)G(0)\nσ2(−z) and\nG(0)\nσ3(z)G(0)\nσ4(−z), spin traces are not taken here. As\nusual,weareaftertheslowlyoscillatingtermsinEq.( 46).\nOnepossibility forthis isthe productofthe forwardscat-\ntering contributions coming from G(0)\nσ1(z)G(0)\nσ2(−z) and\nG(0)\nσ3(z)G(0)\nσ4(−z). From Eq. ( 18) it is clear that the for-\nward scattering contributions are non-zero only if σ1=\nσ2andσ3=σ4, matrixproductsofcorrespondingprojec-8\ntors vanish otherwise. However, in this case the oscillat-\ning factors are canceled exactly and thus, this contribu-\ntion is analytic. Another way to obtain slowly oscillating\nterms in Eq. ( 46) is the product of Kohn anomalies con-\ntained in G(0)\nσ1(z)G(0)\nσ2(−z) andG(0)\nσ3(z)G(0)\nσ4(−z). In this\ncase, we have to look at the spin trace in Eq. ( 46) which\nis non-zero only if σ1=σ4andσ2=σ3. This condition\nbecomes obvious if we notice that the product of Kohn\nanomalies of G(0)\nσ1(z)G(0)\nσ2(−z) andG(0)\nσ3(z)G(0)\nσ4(−z) is ac-\ntually equivalent to the product of the forwardscattering\ncontributions of G(0)\nσ2(−z)G(0)\nσ3(z) andG(0)\nσ4(−z)G(0)\nσ1(z)\nwhich is analytic for the reasons we discussed above.\nHence, only the diagram in Fig. 2(b) contains non-analytic terms and, therefore, Eq. ( 44) describes the non-\nanalytic corrections to Ω due to arbitrary spin splitting\nβ(p) within second-order perturbation theory.\nEven though Eq. ( 44) is true in arbitrary number Dof\nspatialdimensions, we giveexplicit expressionsfor D= 2\nandD= 3. For 2DEG the coefficient L2is negative, see\nEq. (43) forD= 2:\nL2=−g2\n48π2v2\nF. (49)\nThe integral over dncan be parametrized by a single\nangleφ∈(0,2π], so the non-analytic correction Eq. ( 44)\nfor 2DEG then reads:\nΩ(2)=−g2\n24πv2\nF2π/integraldisplay\n0dφ\n2π/bracketleftBig\n|M+−(φ)M−+(φ)|2|β(φ)−β(φ+π)|3+|M++(φ)M−−(φ)|2|β(φ)+β(φ+π)|3\n+2/parenleftBig\n|M++(φ)M−+(φ)|2+|M−−(φ)M+−(φ)|2/parenrightBig\n|β(φ)|3/bracketrightBig\n, D= 2. (50)\nOur result Eq. ( 50) agrees with previous studies [ 43,55,\n56] and extends them to the case of arbitrary spin split-\nting. Equation ( 50) together with Eq. ( 8) for the matrix\nelements allows one to find the non-analytic terms in Ω\ndirectly from the spin splitting β(n).\nThecaseof D= 3ismarginalbecausethe non-analytic\nterms in Eq. ( 44) are proportional to the fourth power\nof the spin splitting. The non-analyticity itself comes\nfrom the divergence of the LDprefactor at D= 3, see\nEq. (43), which results in an additional logarithm. This\nis best seen from the dimensional regularization:\nD= 3−δ, δ→+0. (51)The dimension Denters Eq. ( 44) in the following form:\nLD∆D+1=−g2∆4\n192π3v3\nF/parenleftbigg1\nδ−ln∆/parenrightbigg\n+O(δ),(52)\nwhere ∆ takes one of the following values: ∆ = |β(n)±\nβ(−n)|or ∆ =|β(n)|. Here we expanded the expression\natδ→+0. The divergent 1 /δcontribution is actually\nanalytic and can be represented by lnΛ factor, Λ ∼EF,\nwhich compensates the physical dimension of ∆:\nLD∆D+1→g2\n48π2v3\nF∆4\n4πln/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆\nΛ/vextendsingle/vextendsingle/vextendsingle/vextendsingle,Λ∼EF.(53)\nUsing the regularization Eq. ( 53), we find the non-\nanalytic correction to the spin-split 3DEG:\nΩ(2)=g2\n48π2v3\nF/integraldisplay\nS2dn\n4π/bracketleftbigg\n|M+−(n)M−+(n)|2|β(n)−β(−n)|4ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ(n)−β(−n)\nΛ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+|M++(n)M−−(n)|2|β(n)+β(−n)|4ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ(n)+β(−n)\nΛ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+2/parenleftBig\n|M++(n)M−+(n)|2+|M−−(n)M+−(n)|2/parenrightBig\n|β(n)|4ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ(n)\nΛ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg\n,Λ∼EF, D= 3. (54)\nHere, integration over the unit sphere S2meansdn=\nsinφ1dφ1dφ2,φ1∈[0,π],φ2∈(0,2π]. The non-analytic\ncorrection is negatively defined for arbitrary spin split-\nting due to the logarithms. In particular, if β(n) =B,we get the well-known result, see Ref. [ 43]:\nΩ(2)=g2B4\n3π2v3\nFln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2B\nΛ/vextendsingle/vextendsingle/vextendsingle/vextendsingle,Λ∼EF.(55)9\nC. Large SO splitting and small magnetic field\nHere, we consider the important special case of arbi-\ntrary SO splitting and small magnetic field:\nβ(np) =βSO(np)+B, B≪βSO,(56)\nwherenp=p/p,p≈kF,βSOis a characteristic value\nof the SO splitting at the Fermi surface. As any SO\nsplitting respects time reversal symmetry, it has to be an\nodd vector function of np:\nβSO(−np) =−βSO(np). (57)\nAs we consider B≪βSO, then we can expand β(n) with\nrespect to B:\nβ(n)≈βSO(n)+βSO(n)·B\nβSO(n), (58)\nwhereβSO(n) =|βSO(n)|. Together with the symmetry\ncondition Eq. ( 57), we conclude that only the very first\nterm in Eq. ( 44) contributes to the non-analyticity with\nrespect to Bdue to the following identity:\nβ(n)−β(−n)≈2βSO(n)·B\nβSO(n). (59)\nAs we only consider the leading non-analyticity, we cal-\nculate the matrix elements at B= 0:\nMσσ(n) = 0, Mσ−σ(n) =−1. (60)\nSubstituting Eqs. ( 59), (60) in Eq. ( 44), we find the non-\nanalytic in Bcorrection to Ω in case of arbitrary SO\nsplitting:\nδΩ(B) =LD/integraldisplay\nSD−1dn/vextendsingle/vextendsingle/vextendsingle/vextendsingle2βSO(n)·B\nβSO(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingleD+1\n,(61)\nwhereδΩ(B) indicates that only the non-analytic terms\nwith respect to Bare included. Thus, we see that the\nnon-analyticity in magnetic field Bcannot be eliminated\neven by arbitrary SO splitting, in contrast to the predic-\ntions of Ref. [ 57].\nThe elementary processes that are responsible for the\nnon-analyticity given by Eq. ( 61) are shown in Fig. 4.\nThese processes describe the resonantscattering of a pair\nofelectronswith the bandindex σandoppositemomenta\n±kσinto a pair of electrons in the other band with index\n−σand momenta ±k−σthat are collinear with momenta\nof initial electrons ±kσ. The momentum transfer in such\na scattering processes is close to 2 kF, see Fig. 4(b). The\nscattering with small momentum transfer is forbidden\ndue to the orthogonality condition Eq. ( 6). The consid-\nered processes are resonant due to the time reversal sym-\nmetry, see Eq. ( 57). The collinearity condition comes\nfrom the local nesting when the momentum transfer be-\ntween the resonantly scattering states also matches small\nvicinities around these states. This matching is satisfied\nwhen the outward normals in the scattering states are(a)≈2kFk+ -k-\n-k+ k-\n(b)\nFIG. 4. (a) Fermi surfaces at arbitrary SO splitting, red\n(blue) color corresponds to σ= +1 (σ=−1). The arrows\nshow the resonant scattering processes. (b) The interactio n\nmatrix element corresponding to the resonant scattering pr o-\ncesses at finite SO splitting. Here a pair of electrons with\nthe band index σ= +1 and opposite momenta ±k+scatter\ninto a pair with momenta ±k−that are collinear with ±k+.\nThese processes are resonant due to the time reversal sym-\nmetry, see Eq. ( 57). The collinearity of k+andk−is due\nto thelocal nesting discussed in the main text after Eq. ( 61).\nThese processes are responsible for the non-analyticity in Ω\nwith respect to small magnetic field B, see Eq. ( 61).\ncollinear such that the mismatch comes only from dif-\nferent curvatures of the Fermi surface in the considered\npoints. The local nesting strongly enhances correspond-\ning scattering processes because not only the considered\nstates are in resonance but also small vicinities of states\naround them. For example, the Kohn anomaly in the\nparticle-hole bubble is a result of such a local nesting\nfor the states scattering with the 2 kFmomentum trans-\nfer. The perfect local nesting corresponds to the Landau\ndamping of the particle-hole excitations with energy and\nmomentum around zero, in this case the scattered region\nin the particle-hole bubble is mapped onto itself.\nIt is also instructive to write down Eq. ( 61) for 2DEG\nand 3DEG explicitly:\nδΩ(B) =−g2|B|3\n3πv2\nF2π/integraldisplay\n0dφ\n2π/vextendsingle/vextendsingle/vextendsingle/vextendsingleβSO(φ)·b\nβSO(φ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n, D= 2,(62)\nδΩ(B) =g2B4\n3π2v3\nFln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2B\nΛ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n×/integraldisplay\nS2dn\n4π/vextendsingle/vextendsingle/vextendsingle/vextendsingleβSO(n)·b\nβSO(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle4\n, D= 3,(63)\nwhereb=B/Bis the unit vectoralong B. We neglected\nthe term ln |βSO(n)·b/βSO(n)|in Eq. (63) because it\njust slightly renormalizes the regular B4term. Here it\nis convenient to introduce the angular form-factor FD(b)\nwhich depends on the direction bof the magnetic field\nand on the SO splitting:\nFD(b) =/integraldisplay\nSD−1dn\nSD−1/vextendsingle/vextendsingle/vextendsingle/vextendsingleβSO(n)·b\nβSO(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingleD+1\n,(64)\nwhereSD−1is the area of a unit ( D−1)-dimensional10\nsphere.\nThe form-factors FD(b) can only be positive or zero,\nsee Eq. (64). If we demand FD(b) = 0 for any unit vector\nb, it is equivalent to say that βSO(n)·b= 0 for any b\nand also βSO(n)∝ne}ationslash= 0 from Eq. ( 56). As this is clearly\nimpossible, we conclude that FD(b) can never vanish for\nall unit vectors beven at arbitrary SO splitting βSO(n).\nTherefore, the non-analyticity with respect to Bcannot\nbe cut byanySO splittingneither in2DEGnorin 3DEG.\nNevertheless, the SO splitting is important because it\nleads to strong anisotropy of the non-analytic term, see\nEq. (61), which is described by the form-factor FD(b). If\nwe extrapolate this result to the vicinity of a FQPT, we\nconcludethatthedirectionofspontaneousmagnetization\nmustcoincidewiththemaximumof FD(b). Inparticular,\nwe predict a first-order Ising FQPT in electron gas with\na general SO splitting which breaks the spin rotational\nsymmetry down to Z2.\nAs an example, we consider a 2DEG with Rashba and\nDresselhaus SO splittings:\nβSO(φ)\n= ((αD+αR)kFsinφ,(αD−αR)kFcosφ,0),(65)where the xandyaxes correspond to the [110] and\n[110] crystallographic directions, αRandαDare the\nRashba and the Dresselhaus coupling constants, respec-\ntively. The qualitative picture of the SO-split Fermi sur-\nfaces is shown in Fig. 4(a). It is more convenient to in-\ntroduce the following SO couplings:\na±≡(αR±αD)kF. (66)\nThenwefindtheangularform-factor F2(b), seeEqs.( 62),\n(64):\nF2(b) =π/integraldisplay\n0dφ\nπ|a+bxsinφ−a−bycosφ|3\n/parenleftbig\na2\n+sin2φ+a2\n−cos2φ/parenrightbig3\n2,(67)\nwhereb=B/Bis the unit vector along B. We want to\nidentify the directions b∗whereF2(b∗) is maximal. It is\nclear that all such directions have b∗\nz= 0. Then b∗\nxand\nb∗\nycan be parametrized by a single angle Ψ:\nb∗\nx= cosΨ, b∗\ny= sinΨ. (68)\nThe integralin Eq. ( 67) is quite cumbersome but elemen-\ntary:\n2\n3F2(ζ,Ψ) =−ζcos(2Ψ)\nζ2−1+ζcosΨ\n(ζ2−1)3\n2/parenleftbig\nζ2cos2Ψ−3sin2Ψ/parenrightbig\narctan/parenleftBig/radicalbig\nζ2−1cosΨ/parenrightBig\n+sinΨ\n(ζ2−1)3\n2/parenleftbig\nsin2Ψ−3ζ2cos2Ψ/parenrightbig\nln/parenleftBigg/radicalbig\nζ2cos2Ψ+sin2Ψ/radicalbig\nζ2−1sinΨ+ ζ/parenrightBigg\n, ζ≡/vextendsingle/vextendsingle/vextendsingle/vextendsinglea+\na−/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingleαR+αD\nαR−αD/vextendsingle/vextendsingle/vextendsingle/vextendsingle>1. (69)\nWe added ζas additional argument of F2(b) for conve-\nnience. Equation ( 69) is true only if ζ >1. Ifζ <1, we\nuse the following identity:\nF2(ζ,Ψ) =F2/parenleftbigg1\nζ,π\n2−Ψ/parenrightbigg\n. (70)\nThe extremal values of the π-periodic function F2(ζ,Ψ)\ncorrespond to Ψ = 0 and Ψ = π/2:\nF2(ζ,0) =3\n2\nζ3arctan/parenleftBig/radicalbig\nζ2−1/parenrightBig\n(ζ2−1)3\n2−ζ\nζ2−1\n,(71)\nF2/parenleftBig\nζ,π\n2/parenrightBig\n=3\n2\nζ\nζ2−1−ln/parenleftBig\nζ+/radicalbig\nζ2−1/parenrightBig\n(ζ2−1)3\n2\n,(72)\nwhereζ >1. It is straightforward to see that at ζ >1\nthe maximum of F2(ζ,Ψ) correspondsto Ψ = 0. If ζ <1,\nwe use Eq. ( 70) and find that the maximum corresponds\nto Ψ = π/2. If these calculations are extrapolated to\nthe vicinity of the FQPT, we predict an Ising ferromag-\nnetism in 2DEG with Rashba and Dresselhaus SO split-\nting. The direction of spontaneous magnetization here\ncoincides with the spin quantization axis of the statesthat are maximally split by the SO coupling, namely,\nalong[110]([1 10]) ifαRandαDhavethe same (opposite)\nsigns. The case ζ= 1 is realized when either Rashba or\nDresselhaus SO splitting is zero, in this case all in-plane\ndirections are equivalent which corresponds to the easy-\nplane ferromagnet, the result predicted in Refs. [ 55,56].\nV. Strong interaction regime\nIn this section we consider the electron gas with strong\nelectron-electron interaction, i.e. g≫1 orkFaB≪1,\nsee Eq. ( 41). In this section we concentrate on the case\nof SO splitting with small magnetic field B≪β, where\nβis a characteristic SO splitting at the Fermi surface.\nIn this case we already know that only the scattering\nprocesses that are schematically shown in Fig. 4(b) con-\ntribute to the non-analytic correction in the regime of\nweak interaction. In preceding sections we already dis-\ncussed that the scattering processes are nearly collinear\nin order to support the local nesting , see the paragraph\nafter Eq. ( 61). Here we consider the effective interac-\ntion Hamiltonian whose matrix elements are only given\nby these processes, see Fig. 4(b). Note that the standard11\nbackscattering within the same band is forbidden due\nto Eq. (60). The momentum transfer in the diagram in\nFig.4(b) is about 2 kF, so we can use the approximation\nof contact interaction, see Eq. ( 32). The only difference\nfrom the previous sections is that here we consider the\nregime of strong electron-electron interactions g≫1, i.e.\nwe have to account for the processes in Fig. 4(b) fully\nself-consistently.\nThere is one more reason why we consider the case\nof finite SO splitting here. As we found earlier, the SO\nsplitting results in strong anisotropy of the non-analytic\nterms, see Eq. ( 61). For the case of general SO splitting\nwhich breaks the spin rotational symmetry down to Z2,\nthe magnetic order parameter is Ising, i.e. all fluctua-\ntion modes of the order parameter are gapped and can\nbe neglected even close to the FQPT. At the same time,\nthe resonant scattering processes shown in Fig. 4result\nin the negative non-analytic terms in Ω( B) and, thus,\ndestabilize the FQCP. We show that the non-analyticity\nis enhanced parametrically in the limit of strong interac-\ntiong≫1. This effect can be measured experimentally\nfromthestronglynon-analyticmagneticfielddependence\nof the spin susceptibility close to the FQPT.\nA. Self-consistent Born approximation\nNext, we apply the strategy that we used before in\nRef. [58] where we predicted non-Fermi-liquid phases\n(they correspond to certain magnetic quantum critical\npoints) in strongly interacting Fermi gases with multiple\nFermi surfaces. In fact, the case that we consider here is\nvery similar to a special case of Ref. [ 58]. One difference\nhere is that the Fermi surfaces are not spherical due to\nthe anisotropic spin splitting. Another difference here is\nthat we analyze the stability of a FQCP via considering\nthe non-analytic terms in the thermodynamic potential.\nHere we consider the effective interaction with non-\nzero matrix elements that are given only by the diagram\nin Fig.4(b) and its conjugate. The lowest order diagram\nto the self-energy coming from such an effective inter-\naction is shown in Fig. 5. If we apply the perturbation\ntheory and calculate the corresponding non-analytic cor-\nrection to Ω, we will restore Eq. ( 61). In this section we\nwant to go beyond perturbation theory. As a first step\ntowards this goal, we treat the diagram in Fig. 5self-\nconsistently, i.e. we dress all electron Green functions by\ncorresponding self-energies:\nGσ(iω,p)≡Gσ(iω,δp,n)\n=|σ,n∝an}bracketri}ht∝an}bracketle{tσ,n|\niω−vFδp−Σσ(iω,δp,n),(73)\nwherep=kσ+nδp, see Fig. 1(a),kσis the projectionof\nponto the Fermi surface FSσ,nis the outward normal\ntoFSσatkσ,δpis extended to the interval ( −∞,∞),ω\nis the fermionic Matsubara frequency, and |σ,n∝an}bracketri}htis given\nby Eq. ( 5),vFis the Fermi velocity at zero spin split-\nting. In Eq. ( 73) we use that the interband backscat-FIG. 5. The lowest order diagram for the self-energy that is\nconstructedfrom theeffectiveHamiltonianwith thematrixe l-\nements shown in Fig. 4(b). Here red (blue) color corresponds\ntoσ= +1 (σ=−1) Fermi surface. Inversion of the colors\nyields the self-energy for σ=−1 electrons. The wiggly lines\ncorrespond to the contact interaction defined in Eq. ( 32). We\nuse this diagram for the self-consistent Born approximatio n.\ntering does not alter the single-particle spinors because\nthe backscattering matrix elements equal unity in abso-\nlute value, |M+−(±n)|= 1, see Eq. ( 60). So far, we\nonly dress the electron Green functions leaving the ef-\nfective interaction as contact interaction, see Eq. ( 32),\nand also neglecting the interaction vertex corrections.\nThis kind of approximations are usually referred to as\nself-consistent Born approximations (SCBA). The SCBA\nself-energy shown in Fig. 5then reads:\nΣσ(z) =u2P−σσ(z)G−σ(z), (74)\nwherePσσ′(z) is the particle-hole bubble:\nPσσ′(z) =−Tr{Gσ(z)Gσ′(−z)}.(75)\nNote that the Green functions in Eqs. ( 74) and (75) are\ndressed by corresponding self-energies, see Eq. ( 73).\nHere we require the asymptotics of the dressed Green\nfunction Gσ(τ,r) atτ≫1/EFandr≫λF:\nGσ(τ,r)≈/parenleftbigg1\nλFr/parenrightbiggD−1\n2/bracketleftBig\nei(kσ(nr)r−ϑ)Gσ(τ,r,nr)\n+e−i(kσ(−nr)r−ϑ)Gσ(τ,−r,−nr)/bracketrightBig\n,nr=r\nr,(76)\nGσ(τ,x,n) =T/summationdisplay\nω∞/integraldisplay\n−∞dδp\n2πeiδpx−iωτGσ(iω,δp,n),(77)\nwhereϑis given by Eq. ( 19),λFis the Fermi wave-\nlength, and Gσ(τ,x,n) is the one-dimensional Fourier\ntransform of Gσ(iω,δp,n), see Eq. ( 73). The summation\nover Matsubara frequencies ωin Eq. (77) corresponds\nto finite temperatures T >0. If we neglect the self-\nenergyin Eq.( 73), then Eq. ( 76) transformsintoEq. ( 18)\nthat we derived for the free electron Green function. As12\nthe derivation of Eq. ( 76) is similar to the derivation of\nEq. (18) in many aspects, we refer to Appendix for de-\ntails.\nFirst, we separate the spinors using Eqs. ( 73) and (77):\nGσ(τ,x,n) =|σ,n∝an}bracketri}ht∝an}bracketle{tσ,n|gσ(τ,x,n),(78)\nwheregσ(τ,x,n) is a scalar 1D Green function:\ngσ(τ,x,n)=T/summationdisplay\nω∞/integraldisplay\n−∞dδp\n2πeiδpx−iωτ\niω−vFδp−Σσ(iω,δp,n).(79)\nUsing Eq. ( 76) in Eq. ( 75), we find the asymptotics of\nthe particle-hole bubble P−σσ(z):\nP−σσ(τ,r)≈ −/parenleftbigg1\nλFr/parenrightbiggD−1/bracketleftBig\ne−2iϑeir(k−σ(nr)+kσ(−nr))g−σ(τ,r,nr)gσ(−τ,r,−nr)\n+e2iϑe−ir(k−σ(−nr)+kσ(nr))g−σ(τ,−r,−nr)gσ(−τ,−r,nr)/bracketrightBig\n. (80)\nHere we used the matrix elements at B= 0, see Eq. ( 60),\nbecause we calculate the leading non-analytic contribu-\ntion. Then we substitute Eq. ( 80) into Eq. ( 74) and use\nEq. (76) for the Green function to represent the self-\nenergy in the following form:\nΣσ(τ,r)≈/parenleftbigg1\nλFr/parenrightbiggD−1\n2\n×/bracketleftBig\nei(kσ(nr)r−ϑ)Sσ(τ,r,nr)|σ,nr∝an}bracketri}ht∝an}bracketle{tσ,nr|\n+e−i(kσ(−nr)r−ϑ)Sσ(τ,−r,−nr)|σ,−nr∝an}bracketri}ht∝an}bracketle{tσ,−nr|/bracketrightBig\n,(81)\nwhereSσ(τ,x,n) is the following function:\nSσ(τ,x,n) =−u2e−iσ∆(n)x\n|λFx|D−1\n×g−σ(τ,x,n)g−σ(τ,−x,−n)gσ(−τ,x,−n),(82)\n∆(n) =k+(n)−k+(−n)+k−(−n)−k−(n)\n≈2\nvF(β(n)−β(−n))≈4\nvFβSO(n)·B\nβSO(n).(83)\nHere, we used Eqs. ( 9) and (59) to simplify ∆( n). Notice\nthat Eq. ( 81) has exactly the same form as Eq. ( 76) for\nthe Green function. This means that Sσ(τ,x,n) is just a\n1D Fourier transform of the self-energy:\nSσ(τ,x,n) =T/summationdisplay\nω∞/integraldisplay\n−∞dδp\n2πeiδpx−iωτΣσ(iω,δp,n).(84)Equation ( 82) provides the self-energy as a function of\ngσ(τ,x,n) which is itself connected to the self-energy via\nEq. (79).\nB. Limit of strong interaction\nWe expect a FQPT at large value of the dimensionless\ninteraction parameter g≫1. In this regime, the self-\nenergy dominates over the single-particle terms in the\nGreen function, so we can simplify Eq. ( 79):\ngσ(τ,x,n)≈T/summationdisplay\nω∞/integraldisplay\n−∞dδp\n2πeiδpx−iωτ\n−Σσ(iω,δp,n).(85)\nTaking the convolution of Sσ(τ,x,n) andgσ(τ,x,n)\ngiven by Eqs. ( 84) and (85), respectively, we find:\n−δ(τ)δ(x)\n=/integraldisplay\ndτ′dx′gσ(τ−τ′,x−x′,n)Sσ(τ′,x′,n).(86)\nSubstituting Eq. ( 82) for the self-energy in Eq. ( 86),\nwe find the self-consistent equation for the reduced\nGreen function gσ(τ,x,n) in the limit of strong electron-\nelectron interaction:\nδ(τ)δ(x) =u2/integraldisplay\ndτ′dx′gσ(τ−τ′,x−x′,n)gσ(−τ′,x′,−n)g−σ(τ′,x′,n)g−σ(τ′,−x′,−n)e−iσ∆(n)x′\n|λFx′|D−1.(87)\nC. Solutions at B= 0\nFirst, we analyze the solutions of Eq. ( 87) at zero mag-\nnetic field B= 0 when ∆( n) = 0, see Eq. ( 83). First ofall, we can drop the dependence of the reduced Green\nfunction on n, because there is no explicit dependence13\nonnin Eq. (87) if ∆(n) = 0:\ngσ(τ,x,n) =gσ(τ,x). (88)\nSecond, we observe that Eq. ( 87) is free from any energy\nor momentum scales which results in the scale invariance\n(conformal symmetry) of Eq. ( 87) with respect to inde-\npendent reparametrization of time and coordinate:\nτ→ℓ1τ, x→ℓ2x, (89)\nwhereℓ1andℓ2are arbitrary real numbers. Applying\nthis reparametrizationto Eq. ( 87), one can show that the\nGreen function with rescaled coordinates gσ(ℓ1τ,ℓ2x) is\njust proportional to gσ(τ,x):\ngσ(ℓ1τ,ℓ2x) =gσ(τ,x)\n|ℓ1|2d1|ℓ2|2d2, d1=1\n4, d2=3−D\n8,(90)\nwhered1andd2are the temporal and spatial scaling\ndimensions. Equation ( 90) implies the power-law scaling\nof the Green function gσ(τ,x) with respect to time and\ncoordinate:\ngσ(τ,x)∝1\n|τ|2d11\n|x|2d2. (91)\nThus, we see that we can separate the variables and rep-\nresent Eq. ( 87) at ∆(n) = 0 as the product of two inde-\npendent Dyson equations:\ngσ(τ,x) =g(τ)γσ(x), (92)\nδ(τ) =∞/integraldisplay\n−∞dτ′g(τ−τ′)g(−τ′)g2(τ′), (93)\nδ(x) =u2∞/integraldisplay\n−∞dx′γσ(x−x′)γσ(−x′)γ2\n−σ(x′)\n|λFx′|D−1.(94)\nEquation ( 93) is the well-known Dyson equation of the\nSachdev-Ye-Kitaev (SYK) model [ 65–67] which can be\napplied to describe the strange metal phase in cuprates\n[67,68]. In Ref. [ 58] we found that this model canemerge\nin strongly interacting electron systems with multiple\nFermi surfaces. The zero temperature solution of this\nequation is the following:\ng(τ) =1\n(4π)1\n4sgn(τ)\n|τ|1\n2. (95)\nIn fact, this solution can be easily generalized to the case\nof finite temperature T:\ng(τ) =sgn(τ)\n(4π)1\n4/vextendsingle/vextendsingle/vextendsingle/vextendsingleπT\nsin(πTτ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2\n, (96)\nwhereτ∈(−1/T,1/T). The factor sgn( τ) is needed\nhere to satisfy the antiperiodicity of the electron Green\nfunction:\nGσ/parenleftbigg\nτ+1\nT,r/parenrightbigg\n=−Gσ(τ,r). (97)\nEquation ( 94) is similar to the SYK equation ( 93) and\ncan be solved in terms of the power-law functions:\nγσ(x) =CDisgn(x)\n|x|2d2, d2=3−D\n8, (98)\nCD=/bracketleftbiggD+1\n8πu2λD−1\nFtan/parenleftbiggπ\n23−D\n4/parenrightbigg/bracketrightbigg1\n4\n.(99)Here we assume that the proportionality coefficient CD\nis the same for both spin-split bands, so γσ(x) is inde-\npendent of σin this case. This assumption is reasonable\nbecause the spin splitting is much smaller than the Fermi\nenergy.\nSubstituting Eqs. ( 96) and (98) into Eq. ( 76), we find\nthe asymptotic form of the SCBA Green function:\nGσ(τ,r)≈CD\nλD−1\n2\nFg(τ)\nrD+1\n4/bracketleftBig\nei(kσ(nr)r−ϑ+π\n2)|σ,nr∝an}bracketri}ht∝an}bracketle{tσ,nr|\n+e−i(kσ(−nr)r−ϑ+π\n2)|σ,−nr∝an}bracketri}ht∝an}bracketle{tσ,−nr|/bracketrightBig\n,(100)\nwhere the π/2 phase is due to the isgn(x) phase term\nin Eq. (98). Equation ( 100) has a certain phase freedom\nuponasmall translation r→r+a,a∼λF. Suchatrans-\nlation does not change the asymptotics of the power-law\ntailbecause r≫λFbut it resultsin thephaseshiftin the\noscillatory factors. In other words, the phase cannot be\ndetermined self-consistently from the long-range infrared\nlimit. Here we derive it from the particle-hole symmetry\nwhich is exact close to the Fermi surface, so we conclude\nthat the phase has to be just −ϑ. The true asymptotics\nofthe particle-hole symmetric MatsubaraGreen function\nwith positively defined spectral function then reads:\nGσ(τ,r)≈CD\nλD−1\n2\nFg(τ)\nrD+1\n4/bracketleftBig\nei(kσ(nr)r−ϑ)|σ,nr∝an}bracketri}ht∝an}bracketle{tσ,nr|\n+e−i(kσ(−nr)r−ϑ)|σ,−nr∝an}bracketri}ht∝an}bracketle{tσ,−nr|/bracketrightBig\n.(101)\nThe numerical coefficient CD>0 is given by Eq. ( 99).\nHere we would like to comment on the case D= 3\nbecauseCDvanishes in this case. In order to resolve this\nissue, we use the dimensional regularization:\nCD≈/radicalbigg\nλF\n4uδ1\n4, D= 3−δ, δ→+0.(102)\nSubstituting it into Eq. ( 98) and forgetting about the\nphase factor (see the paragraph above), we find:\nγσ(x) =/radicalbigg\nλF\n4u/parenleftbigg|x|δ\nδ/parenrightbigg−1\n4\n≈/radicalbigg\nλF\n4u/parenleftbigg1\nδ+ln|x|/parenrightbigg−1\n4\n.(103)\nHere 1/δcomes from the ultraviolet scale x∼λF, so we\ncan write 1 /δ= lnp0, wherep0∼kFcompensates the\ndimensionality of x:\nγσ(x) =/radicalbigg\nλF\n4u1\n(ln|p0x|)1\n4, p0∼kF.(104)\nAs a summary, we provide the Matsubara Green func-\ntionofstronglyinteracting2Dand3Delectrongaseswith\narbitrary SO splitting:14\nGσ(τ,r)≈/parenleftBigg\n3(√\n2−1)m2kF\nπg2/parenrightBigg1\n4ei(kσ(nr)r−π\n4)|σ,nr∝an}bracketri}ht∝an}bracketle{tσ,nr|+e−i(kσ(−nr)r−π\n4)|σ,−nr∝an}bracketri}ht∝an}bracketle{tσ,−nr|\n4πr3\n4\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingleπT\nsin(πTτ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2\nsgn(τ), D= 2,(105)\nGσ(τ,r)≈kF\n4π2/parenleftbigg4πm2\ng2/parenrightbigg1\n4ei(kσ(nr)r−π\n2)|σ,nr∝an}bracketri}ht∝an}bracketle{tσ,nr|+e−i(kσ(−nr)r−π\n2)|σ,−nr∝an}bracketri}ht∝an}bracketle{tσ,−nr|\nr|ln(p0r)|1\n4\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingleπT\nsin(πTτ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2\nsgn(τ), D= 3,(106)\nwheregis the dimensionless interaction coupling con-\nstant, see Eq. ( 41). We also provide the Fourier trans-\nforms:\nGσ(iω,δp,n) =/parenleftbigg3π\n2/parenrightbigg1\n4(√\n2−1)3\n4Γ/parenleftbig3\n4/parenrightbig\n2π√2gEFT\n×/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n4Γ/parenleftbigω\n2πT+1\n4/parenrightbig\nΓ/parenleftbigω\n2πT+3\n4/parenrightbig, D= 2,(107)\nGσ(iω,δp,n) =π1\n4\n16√gEFT\n×kF\n|δp|/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglep0\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg−5\n4Γ/parenleftbigω\n2πT+1\n4/parenrightbig\nΓ/parenleftbigω\n2πT+3\n4/parenrightbig, D= 3,(108)\nwhereω=πT(2n+1) is the Matsubara frequency, with\nnbeing an integer, and, again, p0∼kF.\nThese calculations have been performed within the\nSCBA. However, the emergent conformal symmetry\nmakes this approximation exact, see Ref. [ 58] for de-\ntails. In particular, the interaction vertex correction and\nthe dressing of the effective interaction only lead to the\nrenormalization of the interaction parameter g, while the\nscaling dimensions d1andd2remain the same. In this\npaper we consider gas a phenomenologicalparameter, so\nits renormalizations are not important for us. The only\ncondition that is implied here is that g≫1 close to the\nFQPT, so we can apply the limit of strong interaction.\nD. Solutions at finite B\nIfBis finite, then Eq. ( 87) contains an oscillatory\nterm making the self-energy irrelevant at large distance\nx≫1/∆(n), so we expect Fermi liquid behavior at such\ndistances. At small distances x≪1/∆(n) the oscilla-\ntory term is not important and the solutions, given by\nEqs. (105)–(106), are still valid. Following this reason-\ning, the Fourier transforms Eqs. ( 107)–(108) are valid if\nδp≫∆(n), where nhere labels the direction of p.\nAnother important limitation of Eqs. ( 105)–(108)\ncomes from the assumption that the self-energy\nΣσ(iω,δp,n) dominates over the single-particle terms\niω−vFδpwhich we neglected in Eq. ( 85). All in all,the non-Fermi liquid regime corresponds to the following\nconstrained region:\n∆(n)≪δp≪kF, w(δp)≪ω≪W(δp), (109)\nw(δp) =EF\ng/vextendsingle/vextendsingle/vextendsingle/vextendsingleδp\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2\n, W(δp) =gEF/vextendsingle/vextendsingle/vextendsingle/vextendsingleδp\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n2\n, D= 2,(110)\nw(δp) =EF\ng/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg−5\n2\n,\nW(δp) =gEF/parenleftbiggδp\nkF/parenrightbigg2/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg5\n2\n, D= 3,(111)\nwhere we used p0≈kFforD= 3. Here, it is required\nthatW(δp)≫w(δp) which further constrains δp:\nδp≫δp∗, (112)\nδp∗=kF\ng2, D= 2, (113)\nδp∗=kF\ng(lng)5\n2, D= 3. (114)\nAsg≫1 close to the FQPT, the scale δp∗is very small.\nHere we see that at the very deep infrared limit δp/lessorsimilarδp∗\nthe Fermi liquid is restored. This also means that at\n∆(n)≪δp∗the non-analyticity can be correctly esti-\nmated by the perturbative approach that we considered\nin the previous section. The most interesting regime\ncorresponds to ∆( n)≫δp∗where the dominant contri-\nbution to the thermodynamic potential comes from the\nquantum states that are strongly affected by the interac-\ntion. In the next sectionwecalculatethe thermodynamic\npotential Ω in this regime:\n∆(n)≫δp∗. (115)\nVI. Thermodynamic potential\nIn the preceding section we calculated the electron\nGreen function of 2DEG and 3DEG with arbitrary SO\nsplitting in the strongly interacting regime g≫1, see\nEqs. (105)–(108), and we also defined the crossover re-\ngion with the Fermi liquid, see Eqs. ( 109) and (115).\nThe Fermi liquid contribution to the non-analyticity is15\ncalculated in Sec. IV. Here we address the contribution\ncoming from the strongly correlated region of the phase\nspace, see Eqs. ( 109) and (115).\nIn order to calculate the thermodynamic potential Ω,\nwe consider the following energy functional Ω(Σ ,G):\nΩ(Σ,G) =−Trln/parenleftbig\nG−1\n0−Σ/parenrightbig\n−Tr(ΣG)−Φ(G),(116)\nwhere Tr is the trace over all indexes and Φ( G) the\nLuttinger-Ward functional. The saddle point of Ω(Σ ,G)\nyields the Dyson equation and the equation for the elec-\ntron self-energy:\n0 =δΩ(Σ,G)\nδΣσ=/parenleftbig\nG−1\n0−Σσ/parenrightbig−1−Gσ,(117)\n0 =δΩ(Σ,G)\nδGσ=−Σσ−δΦ(G)\nδGσ. (118)\nComparing Eq. ( 74) with Eq. ( 118), we reconstruct the\nLuttinger-Ward functional:\nΦ(G) =u2\n2/integraldisplay\ndz[Tr{G+(z)G−(−z)}]2.(119)\nThe thermodynamic potential Ω is given by the saddle-\npoint value of the functional Ω(Σ ,G):\nΩ = Trln G+3Φ(G), (120)\nwhereGis the Matsubara Green function. Here we used\nthat Σ satisfies Eq. ( 74), so that −Tr(ΣG) = 4Φ(G).\nA. The logarithmic term\nLet us first calculate Trln G:\nΩ1≡TrlnG=/summationdisplay\nσT/summationdisplay\nω/integraldisplaydp\n(2π)Dln(Gσ(iω,δp)).(121)\nHere we assume that Eq. ( 115) is satisfied, so the quan-\ntum critical regionis given by the constraintsEqs. ( 109)–\n(111).\nWe start from the low temperature regime:\nT≪W(∆(n)). (122)\nIn this case there are many Matsubara frequencies in the\nwindow ω∈(w(∆(n)),W(∆(n)), so we can approxi-\nmate the sum over frequencies by an integral. For the\nsame reason, we can also expand the gamma functions in\nEqs. (107)–(108) with respect to the large ratio ω/(2πT)\nsuch that ln Gσ(iω,δp) takes the following form:\nlnGσ(iω,δp) =3\n4ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n2ln|ω|+const, D= 2,(123)\nlnGσ(iω,δp) = ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle−5\n4lnln/vextendsingle/vextendsingle/vextendsingle/vextendsinglep0\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n−1\n2ln|ω|+const, D= 3.(124)The sum over σin Eq. (121) yields just a factor of 2. As\nlnGσ(iω,δp) is an even function of ωandδp, we can inte-\ngrate overthe positive intervals, i.e. δp∈(∆(n),kF) and\nω∈(0,W(δp)). Note that weextend the integrationover\nωall the way to zero frequency because W(δp)≫Tand\nW(δp)≫w(δp). This allows us to represent Eq. ( 121)\nin the following form:\nΩ1(B) = 22π/integraldisplay\n0dφ\n2π2kF/integraldisplay\n∆(φ)kFdδp\n2π2W(δp)/integraldisplay\n0dω\n2π\n×/parenleftbigg3\n4ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n2ln|ω|/parenrightbigg\n, D= 2,(125)\nΩ1(B) = 2/integraldisplaydn\n4π2kF/integraldisplay\n∆(n)k2\nFdδp\n2π22W(δp)/integraldisplay\n0dω\n2π\n×/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle−5\n4lnln/vextendsingle/vextendsingle/vextendsingle/vextendsinglep0\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n2ln|ω|/parenrightbigg\n, D= 3.(126)\nNext, we take the integral over ωwith logarithmic accu-\nracy. In particular, we neglect the lnln |δp|term in the\n3D case compared to the ln |δp|term:\nΩ1(B) =2k2\nF\nπ22π/integraldisplay\n0dφ\n2πkF/integraldisplay\n∆(φ)dδp\nkFW(δp)\n×/parenleftbigg3\n4ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n2ln|W(δp)|/parenrightbigg\n, D= 2,(127)\nΩ1(B) =2k3\nF\nπ3/integraldisplaydn\n4πkF/integraldisplay\n∆(n)dδp\nkFW(δp)\n×/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n2ln|W(δp)|/parenrightbigg\n, D= 3.(128)\nThen substituting W(δp), see Eqs. ( 110)–(111), we find:\nΩ1(B) =3k2\nFgEF\nπ22π/integraldisplay\n0dφ\n2π\n×kF/integraldisplay\n∆(φ)dδp\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingleδp\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\n2\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle, D= 2,(129)\nΩ1(B) =4k3\nFgEF\nπ3/integraldisplaydn\n4π\n×kF/integraldisplay\n∆(n)dδp\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingleδp\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\nδp/vextendsingle/vextendsingle/vextendsingle/vextendsingle7\n2\n, D= 3.(130)\nCalculating the integrals over δpwith the logarithmic\naccuracy and subtracting the part which is independent\nof ∆(n), we find the non-analytic contribution to the\nthermodynamic potential:\nΩ1(B) =−6k2\nFgEF\n5π216\n×2π/integraldisplay\n0dφ\n2π/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆(φ)\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle5\n2\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\n∆(φ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, D= 2,(131)\nΩ1(B) =−4k3\nFgEF\n3π3\n×/integraldisplaydn\n4π/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆(n)\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle3/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\n∆(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg7\n2\n, D= 3.(132)\nFinally, we have to calculate the Luttinger-Ward contri-\nbution to Ω.\nB. Luttinger-Ward contribution\nHere we calculate the second part of the thermody-\nnamic potential, see Eq. ( 120):\nΩ2(B) = 3Φ(G) =3u2\n2/integraldisplay\ndz[Tr{G+(z)G−(−z)}]2.(133)\nInthiscaseitismoreconvenienttousetherealspacerep-\nresentation of the Green function, see Eqs. ( 105)–(106).\nHere we will show that Ω 2(B) yields a subleading correc-\ntion compared to Ω 1(B).\nUsing Eqs. ( 105)–(106), we find:\n[Tr{G+(z)G−(−z)}]2≈3(√\n2−1)m2kF\n27π5g2\n×eir∆(nr)\nr3/vextendsingle/vextendsingle/vextendsingle/vextendsingleπT\nsin(πTτ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, D= 2,(134)\n[Tr{G+(z)G−(−z)}]2≈m2k4\nF\n25π7g2\n×eir∆(nr)\nr4ln(p0r)/vextendsingle/vextendsingle/vextendsingle/vextendsingleπT\nsin(πTτ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, D= 3,(135)\nFirst, we have to translate the conditions given by\nEqs. (109)–(111) to constraints on τandr. We do it\nvia the correspondence ω∼1/τ,δp∼1/r. Thus, in our\ncase the integration region is defined by the following\nconstraints:\nτ≫τ(r) =1\nW(r−1), r≪1\n∆(n).(136)\nHere we show only the most important conditions. As\nin the previous case, we consider small temperatures\nT≪W(∆(n)), so we can actually use the T= 0 ap-\nproximation. First, we integrate over τ:\n1\nT−τ(r)/integraldisplay\nτ(r)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingleπT\nsin(πTτ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≈2∞/integraldisplay\nτ(r)dτ\nτ2=2\nτ(r).(137)\nSubstituting this in Ω 2(B), we find:\nΩ2(B) =9(√\n2−1)\n16π2gEFk2\nF2π/integraldisplay\n0dφ\n2π×∞/integraldisplay\n0kFdr\n(kFr)7\n2eir∆(φ), D= 2,(138)\nΩ2(B) =3\n2π2gEFk3\nF/integraldisplaydn\n4π\n×∞/integraldisplay\n0kFdr\n(kFr)4(ln(kFr))3\n2eir∆(n), D= 3.(139)\nWe substituted p0≈kFforD= 3 case. Note that here\nwe do not even have to use the condition r≪1/∆(n)\nbecause it is effectively satisfied due to the oscillatory\ntermeir∆(n). The integral over kFrin Eq. (138) is cal-\nculated via Eq. ( 28). As ∆(φ+π) =−∆(φ), see Eq. ( 83),\nthen only the real part does not vanish after the integra-\ntion over φ. The integral in Eq. ( 139) can be calculated\nsimilarly. Here we calculate it with logarithmic accuracy:\n∞/integraldisplay\n0kFdr\n(kFr)4+δ(ln(kFr))3\n2eir∆(n)\n≈/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\n∆(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg3\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆(n)\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle3+δisgn(∆(n))+π\n2δ\n6δ,(140)\nwhere we introduced some small δ→0. As ∆( −n) =\n−∆(n), the divergent 1 /δpart cancels out after the inte-\ngration over n, so the physical contribution is finite. In\nconclusion, we find Ω 2(B):\nΩ2(B) =3(√\n2−1)\n5(2π)3\n2gEFk2\nF\n×2π/integraldisplay\n0dφ\n2π/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆(φ)\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle5\n2\n, D= 2,(141)\nΩ2(B) =gEFk3\nF\n8π\n×/integraldisplaydn\n4π/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆(n)\nkF/vextendsingle/vextendsingle/vextendsingle/vextendsingle3/parenleftbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglekF\n∆(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg3\n2\n, D= 3.(142)\nIndeed, compared to Ω 1(B), see Eqs. ( 131) and (132),\nΩ2(B) is only subleading, so we can neglect it.\nSumming over, the leading non-analyticity in Ω in the\nstrongly interacting regime g≫1 close to FQPT is given\nby the Ω 1contribution, see Eqs. ( 131) and (132). Here\nwe represent the answer using the angular form-factors\nFD(b)thataredifferentfromtheform-factors FD(b)that\nappear in the weak coupling regime, see Eq. ( 64):\nδΩ(B) =6k2\nFgEF\n5π2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2B\nEF/vextendsingle/vextendsingle/vextendsingle/vextendsingle5\n2\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2B\nEF/vextendsingle/vextendsingle/vextendsingle/vextendsingleF2(b), D= 2,(143)\nδΩ(B) =−32k3\nFg|B|3\n3π3E2\nF/vextendsingle/vextendsingle/vextendsingle/vextendsingleln/vextendsingle/vextendsingle/vextendsingle/vextendsingleEF\n2B/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle7\n2\nF3(b), D= 3,(144)\nFD(b) =/integraldisplay\nSD−1dn\nSD−1/vextendsingle/vextendsingle/vextendsingle/vextendsingleβSO(n)·b\nβSO(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingleD+3\n2\n, (145)17\nwhereb=B/B. First, we notice that the non-analytic\nterms are negative and that they are parametrically\nlargerthanthe Fermi liquidnon-analyticity, seeEqs.( 62)\nand (63). The angular form-factor FD(b) due to the SO\nsplitting is different from the one that we considered in\nthe weak coupling regime, see Eq. ( 64), but it still results\nin the same physical effect: if the SO splitting breaks the\nspin rotationalsymmetry down to Z2, then the ferromag-\nnetic ground state is Ising and the magnetization axis is\ndefined by the directionwherethe SOsplitting and, thus,\nFD(b) are maximal.\nEquations ( 143) and (144) are valid at very low tem-\nperatures when T≪W(∆(n)). This condition can be\nrewritten as follows:\nB≫B∗(T), (146)\nB∗(T) =EF/vextendsingle/vextendsingle/vextendsingle/vextendsingleT\ngEF/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n3\n, D= 2, (147)\nB∗(T) =EF\n(ln|gEF/T|)5\n4/radicalBigg\nT\ngEF, D= 3.(148)\nIfB≪B∗(T), then the electron self-energy at ω∼B\nandvFδp∼Bis irrelevant, so the regime B≪B∗(T)\ncorresponds to the Fermi liquid regime and the non-\nanalyticity in Ω is described by Eqs. ( 62)–(63). The\nregimeB≫B∗(T) is quantum critical and the non-\nanalyticities are described by Eqs. ( 143)–(144). The\nregimeB∼B∗(T) corresponds to the crossover between\nthe Fermi-liquid and the non-Fermi-liquid regimes.\nIn this section we treated the resonant backscatter-\ning processes shown in Fig. 4(b) non-perturbatively and\nfound that the non-analytic terms in Ω remain negative\nclosetotheFQPTandthattheyarestronglyenhancedin\nmagnitude compared to the regime of weak interaction.\nThis means that the FQCP separating paramagnetic and\nferromagnetic metals is intrinsically unstable even in the\npresence of arbitrarily complex SO splitting.\nVII. Spin susceptibility far away and close to\nFQPT\nThe non-analytic corrections destabilizing the FQCP\nseparating ferromagnetic and paramagnetic phases can\nbe measured experimentally via the magnetic field de-\npendence of the spin susceptibility χij(B) in the param-\nagnetic phase:\nχij(B) =−∂2Ω(B)\n∂Bi∂Bj. (149)\nDeep inside the paramagnetic phase where g≪1 or\nλF≪aB, see Eq. ( 41), the non-analytic correction\nis given by Eqs. ( 62)–(63), so the corresponding non-\nanalytic corrections to the spin susceptibility are the fol-\nlowing:\nδχij(B) =2g2|B|\nπv2\nFκ(2)\nij(b), D= 2, (150)δχij(B) =−4g2B2\nπ2v3\nFln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2B\nΛ/vextendsingle/vextendsingle/vextendsingle/vextendsingleκ(3)\nij(b), D= 3,(151)\nκ(D)\nij(b)\n=/integraldisplay\nSD−1dn\nSD−1/vextendsingle/vextendsingle/vextendsingle/vextendsingleβSO(n)·b\nβSO(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingleD−1βi\nSO(n)βj\nSO(n)\nβ2\nSO(n),(152)\nwhere Λ ∼EF,βi\nSO(n) is theithcomponent of the vec-\ntorβSO(n). These results are valid for B≫T. In the\nopposite regime B≪T, the non-analyticity is regular-\nized by the temperature. An important feature here is\nthe non-trivial angular dependence of the spin suscepti-\nbility on bdue to the SO splitting, see the angular tensor\nκ(D)\nij(b). Measuring κ(D)\nij(b) can resolve the angular de-\npendence of the SO splitting. However, it is not an easy\ntask to measure the non-analytic corrections to the spin\nsusceptibility in the weak coupling regime because they\nare proportional to g2, whereg≪1.\nIt is much more promising to measure the non-\nanalyticities in the spin susceptibility when the param-\nagnetic phase is close to the FQPT, such that g≫1 and\nwe can use the results of Sec. VI, see Eqs. ( 143)–(144):\nδχij(B)\n≈36√\n2\nπ2mgln/vextendsingle/vextendsingle/vextendsingle/vextendsingleEF\n2B/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalBigg\n|B|\nEFκ(2)\nij(b), D= 2,(153)\nδχij(B)\n≈256mg\nπ3vF/vextendsingle/vextendsingle/vextendsingle/vextendsingleln/parenleftbiggEF\n2B/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle7\n2\n|B|κ(3)\nij(b), D= 3,(154)\nκ(D)\nij(b)\n=/integraldisplay\nSD−1dn\nSD−1/vextendsingle/vextendsingle/vextendsingle/vextendsingleβSO(n)·b\nβSO(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingleD−1\n2βi\nSO(n)βj\nSO(n)\nβ2\nSO(n).(155)\nThese equations are valid if B≫B∗(T), where B∗(T) is\ngiven by Eqs. ( 147) and (148). IfB≪B∗(T), then the\nelectron self-energy is irrelevant on the scale ω∼Band\nthenon-analyticityisgivenbyEqs.( 150)and(151). Here\nwe see that at B≫B∗(T) the non-analytic corrections\nto the susceptibility are strongly enhanced. In particu-\nlar,δχij(B)∝/radicalbig\n|B|in a 2DEG and δχij(B)∝ |B|in\na 3DEG, which is much easier to detect experimentally\nthan the much weaker non-analyticities far away from\nthe FQPT, see Eqs. ( 150) and (151). To measure local\nspin susceptibility one can use, for example, ultrasensi-\ntive nitrogen-vacancy center based detectors [ 69].\nVIII. Conclusions\nIn this paper we revisited FQPT in strongly interact-\ning clean 2DEG and 3DEG with arbitrary SO splitting.\nFirst of all, we calculated the non-analytic corrections\nto the thermodynamic potential Ω with respect to arbi-\ntrary spin splitting in the limit of weak electron-electron18\ninteraction. So far, this has been done in the literature\nonly for very limited number of special cases [ 43,55–\n57]. Here we generalized the calculation for arbitrary\nspin splitting and arbitrary spatial dimension D >1, see\nEq. (44). The most important outcome of this calcula-\ntion is that even arbitrarily complex SO splitting is not\nable to cut the non-analyticity in Ω with respect to the\nmagnetic field, see Eq. ( 61). This is a direct consequence\nof the backscattering processes shown in Fig. 4. Such\nprocesses were not taken into account in Ref. [ 57] where\na complicated enough SO splitting is predicted to cut the\nnon-analyticity.\nThe weak coupling regime cannot be extended to the\nphase transition point where the electron-electron inter-\naction is very strong. In order to address this regime,\nwe treat the processes in Fig. 4non-perturbatively and\nfind that there is a sector in the phase space where the\nelectron Green function exhibits strong non-Fermi-liquid\nbehavior. We find that this non-Fermi-liquid sector dom-\ninates the non-analytic response at low temperatures.\nWe also find that the non-analyticity remains negative\nbut it is parametrically enhanced from δΩ∝ −|B|D+1\nin the weak interaction limit to δΩ∝ −|B|D+3\n2in the\nstrongcouplingregime. Inparticular,thenon-analyticity\nin interacting 3DEG is enhanced from the marginal ∝\nB4ln|B|term in the weak interaction regime to ∝ −|B|3\nclose to the FQPT where the interaction is strong. In ad-\ndition, SO splitting makes the non-analytic corrections\nstrongly anisotropic. In particular, if the SO splitting\nbreaks the spin symmetry down to Z2, then the ferro-\nmagnetic ground state is necessarily Ising-like. In this\ncase, all soft fluctuations of the magnetic order parame-\nter are gapped which validates the mean-field treatment\nof the magnetization. Nevertheless, there is still a gap-\nless fluctuation channel originating from the resonant\nbackscattering processes shown in Fig. 4(b) which con-\ntributes to the negative non-analytic terms in Ω destabi-\nlizing the FQCP. Basedon ouranalysis, we conclude that\nthe FQPT in clean metals is always first order regardless\noftheSOsplitting. Thenon-analytictermsleadingtothe\ninstability of the FQCP in clean metals can be measured\nviathe non-analyticdependence ofthe spinsusceptibility\non the magnetic field both far away, see Eqs. ( 150) and\n(151), or close to the FQPT, see Eqs. ( 153) and (154).\nThe candidate materials are the pressure-tuned 3D fer-\nromagnets ZrZn 2[59], UGe 2[60], and many others [ 62],\nor density-tuned 2D quantum wells [ 61].\nAcknowledgments\nThis work was supported by the Georg H. Endress\nFoundation, the Swiss National Science Foundation\n(SNSF), and NCCR SPIN. This project received funding\nfrom the European Union’s Horizon 2020 research and\ninnovation program (ERC Starting Grant, grant agree-\nment No 757725).A. ASYMPTOTICS OF THE GREEN\nFUNCTION\nWe start from the Fourier representation of the Green\nfunction:\nG(τ,r) =/integraldisplaydp\n(2π)Deip·rG(τ,p),(A1)\nwhereτis the imaginary time, raD-dimensional posi-\ntion vector, and paD-dimensional momentum vector.\nHere we do not indicate any band index because it is\nfixed. The asymptotics at large τ≫1/EFand large\nr=|r| ≫λFis dominated by the vicinity of the Fermi\nsurfaceFS, so we can expand the momentum pinto the\nmomentum kon the Fermi surface FSand the momen-\ntum along the outward normal n(k) atk, see Fig. 1(a):\np=k+n(k)δp,k∈ FS, (A2)\nwhereδpis the distance from pto the Fermi surface FS.\nNotice that δp >0 (δp <0) corresponds to the states\nabove (below) the Fermi surface. At large r≫1/kF,kF\nis the typical momentum scale on the Fermi surface, we\nhaveδp∼1/r≪kF, so we can approximate the integral\noverpby the integration over a thin layer around the\nFermi surface:\nG(τ,r)≈∞/integraldisplay\n−∞dδp\n2π/integraldisplay\nk∈FSdk\n(2π)D−1\n×eik·reiδpn(k)·rG(τ,δp,k), (A3)\nG(τ,δp,k)≡G(τ,k+n(k)δp). (A4)\nHere we extended the integral over δpto the interval\n(−∞,∞) because the convergence radius of this integral\nis very short at r→ ∞, namely δp∼1/r. Hence, we\napproximated the initial Fourier transform Eq. ( A1) by\nthe integral over the fiber bundle FS ×(−∞,∞).\nAsr→ ∞, we can use the stationary phase method\nto find the asymptotics. First, we evaluate the integral\nover the Fermi surface. The stationary condition for the\nphase of the rapidly oscillating factor eik·rin Eq. (A3)\nreads:\ndk·r= 0, (A5)\nwheredkis an arbitrary infinitesimal (but non-zero) el-\nement of the tangent space T(k) attached to the Fermi\nsurfaceFSat the point k. Hence, rhas to be orthogo-\nnaltothewholelinearspace T(k)whichhascodimension\none. This means that the stationary phase condition is\nsatisfied at the points k′∈ FSwhereris collinear with\nthe normals n(k′):\nn(k′) =s(k′,nr)nr,nr=r\nr, (A6)\nwheres(k′,nr) = +1 ( s(k′,nr) =−1) if the outward\nnormaln(k′) and the radius vector rare parallel (anti-\nparallel). We include all such points k′into a set P(nr):\nP(nr) ={k′∈ FS|n(k′) =±nr}.(A7)19\nIt is clear that P(−nr) =P(nr) ands(k′,−nr) =\n−s(k′,nr).\nAt this point we can take the integral over δpin\nEq. (A3):\n∞/integraldisplay\n−∞dδp\n2πeiδpn(k′)·rG(τ,δp,k′) =G(τ,s(k′,nr)r,k′),(A8)\nwhere we used that k′∈ P(nr) ands(k′,nr) =n(k′)·\nnr=±1, see Eq. ( A6). HereG(τ,x,k) is the 1D Fourier\ntransform of the Green function:\nG(τ,x,k) =∞/integraldisplay\n−∞dδp\n2πeiδpxG(τ,δp,k),k∈ FS.(A9)\nNote that such a 1D Fourier transform is generally de-\npendent on the point k∈ FS. Substituting this into\nEq. (A3), we find:\nG(τ,r)≈/summationdisplay\nk′∈P(nr)eik′·rJk′(r)G(τ,s(k′,nr)r,k′),(A10)\nJk′(r) =/integraldisplay\nk∈FSdk\n(2π)D−1ei(k−k′)·r. (A11)\nThe function Jk′(r) appears due to the integration over\na small vicinity of a point k′∈ P(nr).\nThe integral Jk′(r) converges due to the finite curva-\nture of the Fermi surface at k′. This is true even if the\ninteraction is very strong such that the Fermi surface be-\ncomes critical. In order to evaluate this integral, it is\nconvenient to introduce an auxiliary function ε(p) with\nthe following properties:\nε(p) = 0 ifp∈ FS, (A12)\nv(p) =∂ε(p)\n∂p∝ne}ationslash= 0 ifp∈ FS.(A13)\nWe notice here that there are infinitely many choices for\nsuch a function but we will show that the result is inde-\npendent of the choice. In case of a free electron gas the\nnatural choice for ε(p) is the electron dispersion. The\ncondition Eq. ( A13) is required to generate the outward\nnormaln(k) at each point k∈ FS:\nv(k) =v(k)n(k),k∈ FS. (A14)\nHerev(k)∝ne}ationslash= 0 for all k∈ FS.\nConsider two close points k∈ FSandk′∈ FS. Ac-\ncording to Eq. ( A12), we can write:\nε(k) =ε(k′) = 0. (A15)\nUsing the Taylor series expansion, we find:\n0 =ε(k)≈ε(k′)+(k−k′)·v(k′)\n+1\n2(k−k′)TR(k′)(k−k′), (A16)\nRij(p) =∂2ε(p)\n∂pi∂pj, (A17)where we used the matrix notations in Eq. ( A16), and\nthe superscriptTstands for transposition. Substituting\nEq. (A14) into Eq. ( A16), we find:\n(k−k′)·n(k′)≈ −(k−k′)TR(k′)\n2v(k′)(k−k′).(A18)\nIn Eq. (A11)k′∈ P(nr), son(k′) satisfies Eq. ( A6).\nThis allows us to write:\n(k−k′)·r≈ −s(k′,nr)r(k−k′)TR(k′)\n2v(k′)(k−k′).(A19)\nThe expression is quadratic with respect to the small\ndifference k−k′. At large rthe convergence radius of\nthe integral Jk′(r) scalesas |k−k′| ∝1/√r, so we indeed\nhave to integrate over a small vicinity of k′∈ P(nr) and\nthe Taylor expansion is valid. Equation ( A19) also shows\nthat the component of k−k′along the normal n(k′)\nis only quadratic with respect to k−k′, so at a given\naccuracy we can approximate k−k′on the right-hand\nside of Eq. ( A19) by its orthogonal projection onto the\ntangent space T(k′):\nJk′(r)≈/integraldisplay\nκ∈T(k′)dκ\n(2π)D−1e−irs(k′,nr)κTA(k′)κ,(A20)\nA(k′) =RT(k′)\n2v(k′), (A21)\nwhereRT(k′) is the restriction of the tensor R(k′) to\nthe tangent space T(k′). We reduced the initial integral\nJk′(r) to a standard Gaussian integral:\nJk′(r)≈/parenleftbigg1\n4πr/parenrightbiggD−1\n2e−iπ\n4s(k′,nr)S(A(k′))\n/radicalbig\n|detA(k′)|,(A22)\nS(A(k′)) =D−1/summationdisplay\ni=1sgn(ai), (A23)\nwhereaiare allD−1 eigenvalues of the symmetric ma-\ntrixA(k′). In all cases that we consider in this paper\ndetA(k)∝ne}ationslash= 0 at any point k∈ FS. In other words, we\nconsider only Fermi surfaces with non-zero Gauss curva-\nture at each point.\nLet us check that the matrix A(k),k∈ FS, is indeed\nindependent of the choice of ε(p). For this, we consider\nanother parametrization:\n˜ε(p) =f(p)ε(p), (A24)\nwheref(p) is an arbitrary smooth function such that\nf(p)∝ne}ationslash= 0. As f(p)∝ne}ationslash= 0, then ˜ ε(p) = 0 if and only if\nε(p) = 0, i.e. ˜ ε(p) = 0 defines the Fermi surface FS. Let\nus find the velocity ˜ v(k) whenk∈ FS:\n˜v(k)n(k) =∂˜ε(k)\n∂k=∂f(k)\n∂kε(k)+f(k)v(k)n(k),(A25)\nwherev(k) is defined in Eqs. ( A13) and (A14). Ask∈\nFS, thenε(k) = 0, so we find:\n˜v(k) =f(k)v(k),k∈ FS. (A26)20\nSimilarly, we can calculate the tensor ˜R(k),k∈ FS:\n˜Rij(k) =∂2˜ε(k)\n∂ki∂kj=f(k)Rij(k)\n+v(k)/parenleftbigg∂f\n∂kinj(k)+∂f\n∂kjni(k)/parenrightbigg\n,k∈ FS,(A27)\nwhereRij(k) is defined in Eq. ( A17). Second line in\nEq. (A27) contains a term which vanishes in all prod-\nuctsκT˜R(k)κwhereκ∈ T(k) i.e.κ·n(k) = 0. This\nmeans that the restriction to the tangent space T(k) is\nespecially simple:\n˜RT(k) =f(k)RT(k),k∈ FS.(A28)\nUsing Eqs. ( A26) and (A28), we indeed find that the op-\neratorA(k),k∈ FS, isinvariantwithrespecttodifferent\nchoices of ε(p):\n˜A(k) =˜RT(k)\n2˜v(k)=RT(k)\n2v(k)=A(k),k∈ FS.(A29)\nAll in all, the long range asymptotics of the Green\nfunction of an interacting Fermi gas is the following:\nG(τ,r)≈/summationdisplay\nk′∈P(nr)C(k′,nr)\n(4πr)D−1\n2eik′·r\n×G(τ,s(k′,nr)r,k′), (A30)\nC(k′,nr) =e−iπ\n4s(k′,nr)S(A(k′))\n/radicalbig\n|detA(k′)|.(A31)\nThis is the general result which is suitable for a Fermi\nsurface of arbitrary geometry. Next, we give examples\nfor spherical or nearly spherical Fermi surfaces.\n1. Spherical Fermi surface\nThe simplest example is the spherical Fermi surface\nwith the Fermi momentum kF. We considered this case\nin Ref. [58]. For an arbitrary direction nrthere are ex-\nactly two points on the Fermi surface whose normals are\ncollinear with nr:\nP(nr) ={±kFnr}. (A32)\nIn this case, the sum over k′in Eq. (A30) contains only\ntwo terms, namely, k′=±kFnr. In order to calculate\nthe matrix A(k′), we consider the function:\nε(p) =p2−k2\nF\n2. (A33)\nThe velocity v(p) and the tensor R(p) are then the fol-\nlowing:\nv(p) =p, (A34)\nRij(p) =δij. (A35)This allows us to identify the matrix A(k),|k|=kF:\nA(k) =I\n2kF,|k|=kF, (A36)\nwhereIis the (D−1)×(D−1) identity matrix on the\ntangent space T(k). Substituting this into Eq. ( A30), we\nfind the asymptotics of the Green function in case of the\nspherical Fermi surface:\nG(τ,r)≈/parenleftbiggkF\n2πr/parenrightbiggD−1\n2\n×/parenleftBig\nei(kFr−ϑ)G(τ,r)+e−i(kFr−ϑ)G(τ,−r)/parenrightBig\n,(A37)\nϑ=π\n4(D−1), (A38)\nG(τ,x) =∞/integraldisplay\n−∞dδp\n2πeiδpxG(τ,δp). (A39)\nHere we also used the spherical symmetry, i.e. G(τ,p) =\nG(τ,p), soG(τ,δp,k) is independent of k∈ FS.\n2. Nearly spherical Fermi surface\nHere we consider anotherexample when the Fermi sur-\nface is nearly spherical and can be modeled by the fol-\nlowing dispersion:\nε(p) =p2−k2\nF\n2m−β(p). (A40)\nWe denote points on the Fermi surface FSbyk, they\nsatisfy the equation ε(k) = 0:\nk2=k2\nF+2mβ(k). (A41)\nIn this partwe makethe followingassumptionsabout the\nsmooth function β(p):\n|β(k)| ≪EF,∇β(k)≡∂β(p)\n∂p/vextendsingle/vextendsingle/vextendsingle/vextendsingle\np=k≪vF,(A42)\nwherek∈ FS, 2EF=kFvFis the Fermi energy, and\nvF=kF/mthe Fermi velocity. Using the first condi-\ntion in Eq. ( A42), we find the approximate Fermi surface\nequation:\nk(e)≈kF+β(e)\nvF, k(e)e∈ FS,(A43)\nwhereeis an arbitrary unit vector and β(e) stands for\nβ(kFe).\nThe outwardnormal n(k) atk∈ FSis defined though\nthe gradient of ε(p) atp=k:\nv(k)n(k) =k\nm−∇β(k), (A44)\nv2(k) =k2\nm2−2k·∇β(k)\nm+(∇β(k))2,(A45)21\nwhere the second equation here is just the first one\nsquared. Here is where we use the second condition in\nEq. (A42). In linear order in β(k) we find:\nv(k)≈k(n)\nm−n·∇β(n), (A46)\nk·n(k)≈k(n) =kF+β(n)\nvF,(A47)\nwhereβ(n) stands for β(kFn).\nWe are only interested in the points k′∈ FSwith\nnormals n(k′) =snr,s=±1. Using Eq. ( A47), we find\nthe oscillating phase:\nk′·r=srk′·n(k′)≈sk(snr)r,(A48)\nwhere we used Eq. ( A47). Neglecting the weak depen-dence of the prefactor C(k′,nr) onβ(k′), see Eq. ( A31),\nwe find the asymptotic behavior of the Green function in\ncase of a nearly spherical Fermi surface:\nG(τ,r)≈/parenleftbiggkF\n2πr/parenrightbiggD−1\n2\n×/parenleftBig\nei(k(nr)r−ϑ)G(τ,r)+e−i(k(−nr)r−ϑ)G(τ,−r)/parenrightBig\n,(A49)\nwherek(n) is given by Eq. ( A43) andϑby Eq. (A38).\nHere,G(τ,x) is calculated at β(p) = 0, i.e. it coincides\nwith the sphericallysymmetric case. Importantly, the os-\ncillatoryfactors e±ik(±nr)rinEq.(A49)dependexplicitly\nonβ(±nr), which is crucial for the resonant scattering\nprocesses near the Fermi surface.\n[1] A. P. Ramirez, J. Phys. Condens. Matter 9, 8171 (1997).\n[2] J. M. D. Coey, M. Viret, and S. von Moln´ ar, Adv. Phys.\n48, 167 (1999).\n[3] K. Ghosh, C. J. Lobb, R. L. Greene, S. G. Karabashev,\nD. A. Shulyatev, A. A. Arsenov, and Y. Mukovskii, Phys.\nRev. Lett. 81, 4740 (1998).\n[4] D. Kim, B. Revaz, B. L. Zink, F. Hellman, J. J. Rhyne,\nand J. F. Mitchell, Phys. Rev. Lett. 89, 227202 (2002).\n[5] T. Dietl, H. Ohno, F. Matsukura, J. 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Leveraging the similarity between the dispersion properties of plate edge\nwaves and surface waves propagating in a semi-infinite medium, we use a setup consisting of a plate with a periodic\narrangement of bead-magnet resonators along one of its edges. Nonlinear contact between the ferromagnetic beads and\nmagnets is exploited to realize nonlinear local resonance effects. First, we experimentally demonstrate the nonlinear\nsoftening nature and amplitude-dependent dynamics of a single bead-magnet resonator on both rigid and compliant\nsubstrates. Next, the dispersion properties of the system in the linear regime are investigated. Finally, we demonstrate\nhow the interplay of nonlinear local resonances with plate edge waves gives rise to amplitude-dependent dispersion\nproperties. The findings will inform the design of more versatile surface acoustic wave devices that can passively adapt\nto loading conditions.\nSurface acoustic waves (SAWs) have broad applications in\nscience and engineering. At the micro and nano scales, these\nwaves are of interest in the design of radio-frequency filters\nfor wireless telecommunication systems1as well as biosen-\nsors for medical diagnostics2. At larger scales, the study of\nthese waves is essential for protecting the built environment\nfrom the damaging effects of seismic waves3,4. The advent of\nmetamaterials has realized unique engineering solutions for\nmanipulating these waves at vastly different frequencies. For\nexample, phononic crystals in the form of architected surface\nlayers have been used to design SAW filters, space-saving re-\nflective gratings and waveguides5. Periodic arrangements of\nlocal resonators have also been used to achieve subwavelength\nwave filtering and waveguiding6, as well as high-resolution\nimaging7,8.\nOnce fabricated, metamaterials for SAW control are usu-\nally bound to produce a desired effect at a specific frequency\nrange. Recently, efforts have been undertaken to increase\nthe versatility of these systems by making their response\ntunable9,10, or non-reciprocal11,12. Most previous work on\nSAW tunability has focused on using external stimuli (e.g.,\nthermal, magnetic, electrical) for tuning the wave-control ca-\npabilities of these systems11,12. However, the addition of ex-\nternal stimuli adds complexity to the system. A desirable al-\nternative is the design of self-tunable SAW devices, which can\npassively adapt to the loading conditions without the need for\nexternal stimuli. Incorporating nonlinearity in the design of\nmetamaterials provides an opportunity to explore amplitude-\ndependent self-tuning for SAWs.\nNonlinear metamaterials offer enhanced control over wave\ntransmission compared to their linear counterparts. Several\nexotic features have already been demonstrated in these sys-\ntems, including self-tunability13,14, nonreciprocity15–17, en-\nergy tunneling and localization18and, more recently, the\na)To whom correspondence should be addressed: daraio@caltech.eduemergence of subharmonic bandgaps19. Theoretical frame-\nworks have been developed for determining the dispersion\nproperties of nonlinear phononic lattices/crystals20,21. More\nrecent theoretical investigations have focused on the effects of\nmaterial or geometric nonlinearity in elastic metamaterials22.\nThe effects of nonlinear local resonators, on the other\nhand, have been mostly studied in the context of discrete\nsystems23,24. Limited works exist on wave propagation in\nsystems consisting of an array of nonlinear resonators embed-\nded in linear elastic continua25. Even though the interaction\nof SAWs with contact-based resonators, with an inherently\nnonlinear nature, has been studied previously, such studies\nare based on the assumption that the amplitude of the prop-\nagating waves is small and that the nonlinear stiffness can be\nlinearized26,27.\nAmplitude-dependent resonance is a well-documented phe-\nnomenon in nonlinear dynamics28. Several works have docu-\nmented this effect for Hertzian contact resonators29,30. A re-\ncent study on a cylindrical rod in contact with a bead provides\nexperimental proof that the nonlinear properties of the contact\nlead to amplitude-dependent resonance shifts31. A notable\nexperimental work had previously demonstrated how reso-\nnance shifts in a one-dimensional chain of beads connected\nwith nonlinear springs are intimately related to shifts in dis-\npersion curves for the overall system32. In a more recent and\nrelevant work, the propagation of Rayleigh waves in a half-\nspace coupled to nonlinear resonators was considered. The\nauthors provided a theoretical description of Rayleigh wave\ndispersion in the presence of hardening and softening interac-\ntion forces, and validated their findings using Finite Element\nsimulations33. However, experimental investigations of non-\nlinear dispersion shifts for SAWs have remained unexplored.\nIn this work, we leverage an experimental setup similar to\nthat of Ref.10and exploit the nonlinear dynamics of an ar-\nray of contact resonators to achieve amplitude-dependent dis-\npersion properties for plate edge waves. The compact table-\ntop experimental setup is shown in Fig. 1. It consists of an\nacrylic plate of dimensions 608 ×912×8 mm ( H×W×t),arXiv:2304.01494v2  [physics.app-ph]  5 Sep 20232\nFIG. 1: Configuration of the table-top experimental setup. (a)\nImage of the setup, and (b) its schematics showing the\ncomponents’ dimensions as well as the location of\nobservation points on the plate’s edge (red) and on the\nresonators (blue).\nYoung’s modulus E=5.5 GPa, Poisson ratio ν=0.35, and\ndensity ρ=1190 kgm−3. The plate is clamped to an opti-\ncal table at the bottom along the longer edge. A set of 41\ndisk magnets (K&J magnetics DH101; NdFeB, Grade N42)\nare glued at equal distances of d=15 mm on its top edge. The\nmagnets have a diameter of Dm=2.5 mm and a thickness of\ntm=0.8 mm. The Young’s modulus Emand Poisson ratio νm\nof the magnet are 190 GPa and 0.3, respectively34. Steel beads\n(McMaster-Carr 9642K49) with radius rb=4.8 mm and mass\nmb=3.5 g are placed on top of each magnet. The steel beads\nhave a Young’s modulus of Eb=210 GPa and a Poisson ra-\ntio of νb=0.334. The bead-magnet assemblies will serve as\nnonlinear mechanical oscillators. A vibration exciter (HBK\nType 4810) is glued to the plate at a distance of ls=168 mm\nfrom the first bead. A signal generator (Agilent 33220A) and\npower amplifier (HBK Type 2718) are used to drive the shaker\nand excite vertically-polarized edge waves along the edge of\nthe plate. A laser doppler vibrometer (LDV , Polytec OFV-\n5000) is mounted on a linear stage and a motor is used to move\nthe vibrometer and consecutively measure the vertical velocity\ncomponent at desired observation points. Measurement data\nis acquired using an oscilloscope (Tektronix DPO3034).\nWe use the analytical dispersion relation for thin semi-\ninfinite plates with stress-free boundary conditions35to pre-\ndict the phase velocity cRof edge waves in a pristine plate:\n\u0012\n2−c2\nR\nc2\nT\u00132\n−4s\u0012\n1−c2\nR\nc2\nT\u0013\u0012\n1−c2\nR\nv2\nP\u0013\n=0, (1)\nwhere vp=\u0002\nE/ρ(1−ν2)\u00031/2is the velocity of dilationalwaves in a thin plate and cT= [E/2ρ(1+ν)]1/2is the shear\nwave speed35. We note that this equation is similar to the one\ndescribing the dispersion relation of Rayleigh waves in a half-\nspace. Using this equation, we make a theoretical prediction\nofcR=1205 ms−1for the phase velocity of edge waves prop-\nagating on a pristine plate.\nPrior to investigating the dynamics of the overall system,\nwe characterize a single bead-magnet oscillator. We start by\nexperimentally investigating the resonance characteristics of\na single bead-magnet assembly on a rigid substrate. To do\nso, we attach the disk magnet to the surface of a piezoelec-\ntric transducer (Panametrics-NDT V1011) using cyanoacry-\nlate glue. The bead is then placed on top of the magnet (Fig. 2\n(a), (b)). Due to the importance of the contact surface in these\nFIG. 2: Single bead-magnet assembly on a rigid substrate.\n(a) Schematics of the problem and the forces exerted on the\nbead. The insert shows the static overlap δsbetween the\nsurfaces in contact at rest. FcandFmare the contact and\nmagnetic force, respectively. (b) Schematic of the\nexperimental setup. (c) Up-sweep (black) and down-sweep\n(red) experimental frequency response functions for the\nbead-magnet resonator at 8 mV-interval excitation\namplitudes.\nexperiments, we thoroughly clean the surface of the magnet\nas well as the steel bead before they get in contact. A Stan-\nford SR 860 analyzer is used for the excitation in a sine sweep\nmode from five to eight kHz, and the laser Doppler vibrom-\neter is used for measuring the velocity response of the bead.\nWe start at an amplitude of 8 mV and repeat the test by in-\ncreasing the excitation amplitude at 8 mV intervals. Fig. 2 (c)\nshows the frequency response plots of the bead-magnet res-\nonator. The steady-state amplitudes have been normalized by3\nthe static overlap between the bead and the magnet δs. The\nblack and red curves show results for sweep-up and sweep-\ndown tests, respectively.\nAt low excitation amplitudes, the bead-magnet assembly is\nexpected to behave as a linear oscillator. Thus, the frequency\nresponse curves from up and down sweeps coincide. The un-\nderlying linear natural frequency of the oscillator is approx-\nimately fr=7 kHz. The linearized normal stiffness of the\nresonator kNcan then be estimated as mb(2πfr)2. Assum-\ning a Hertzian contact law between the bead and the mag-\nnet, the linearized normal stiffness may also be written in\nterms of the static overlap δsaskN=2E∗r1/2\nbδ1/2\ns10, where\nE∗=\u0002\n(1−ν2\nb)/Eb+(1−ν2\nm)/Em\u0003−1. From here, the static\noverlap δsis approximately determined as 200 nm.\nAs the excitation amplitude is increased, nonlinearity bends\nthe frequency response and shifts the locus of the peak ampli-\ntude to lower frequencies. This is characteristic of a soften-\ning nonlinear response. In addition, the differences between\nup and down swept curves become increasingly stark. Emer-\ngence of jumps in the frequency response at an excitation am-\nplitude of 32 mV indicates the resonator’s loss of stability\nand is another evidence of the inherent nonlinear properties\nof contact resonance.\nNext, we study the dynamics of a single bead-magnet os-\ncillator on an acrylic plate. We present experimental evidence\nof higher harmonic generation and resonance frequency shifts\nin the contact resonator’s response. The schematic of the ex-\nperimental setup is similar to the one shown in Fig. 1, with\nthe difference that all bead magnet resonators, except the one\nclosest to the shaker, are removed. The linear resonance fre-\nquency of the oscillator is identified at roughly 5.15 kHz using\na broadband excitation. This shows a shift of approximately\n1.85 kHz in comparison to measurements on a rigid substrate\n(Fig. 2), which can be attributed to the substrate’s compli-\nance and its coupling to the rigid contact dynamics. A similar\neffect has been reported in previous work10. Based on the\ndetermined resonance frequency in the linear regime, we use\na narrow-band, slow (200 s−2) sweep-down excitation from\n6 kHz to 4 kHz to characterize the nonlinear response of the\noscillator. The voltage was set to 100 mV, and three differ-\nent excitation amplitudes (10, 20, and 30 dB) were chosen by\nchanging the gain on the amplifier. The response of the bead\nwas directly recorded by the LDV . Fig. 3 shows the time his-\ntory as well as frequency spectrum for the bead’s response. A\nmoving average filter was used to postprocess the response.\nTable I summarizes the main peaks in the frequency spectrum\nfor different levels of gain. The tabular data shows that the\nprimary resonance frequency f1shifts to lower frequencies as\nthe amplitude of the excitation increases. As discussed in the\nprevious section, this is characteristic of a softening nonlinear\nbehavior. Additionally, increasing the excitation amplitude\nleads to the generation of a higher harmonic f2at twice the\nresonance frequency. Another interesting feature is observed\nin the results by comparing the time history plots (top panel\nin Fig. 3). The velocity of the bead vhas been normalized\nby peak velocity vmaxin each case. At lower gains, the rise\nand fall of the amplitude at resonance is symmetric in shape.\nHowever, at 30 dB gain, the descent from resonance is abrupt,\nFIG. 3: Experimental results for the single bead-magnet\nresonator on the acrylic plate. Top panel shows the time\nhistory responses at the three gain levels, shifted for better\nvisualization. The bottom panel shows the frequency domain\nresponse of the single bead-magnet resonator.\nGain [dB] f1[kHz] f2[kHz] f2/f1\n10 5.15 - -\n20 5.12 10.2 1.99\n30 4.80 9.57 1.99\nTABLE I: Resonance frequencies of the single bead-magnet\nresonator on the acrylic plate.\nsuggesting loss of stability, a feature common to nonlinear res-\nonance phenomenon.\nWe now move on to determine the dispersion properties of\nedge waves for a plate with a periodic array of bead-magnet\nresonators. The experimental setup is shown in Fig. 1. Two\nprimary modes of excitation are utilized in this experiment: a\nwide-band sweep at low amplitudes that captures the linear re-\nsponse of the system, and a narrow-band slow sweep at higher\namplitudes that is used to investigate the nonlinear character-\nistics of the system. Dispersion reconstruction in the linear\nregime is carried out using a wide-band fast (590 s−2) sweep-\nup excitation from 100 Hz to 6 kHz. Due to evidence of soft-\nening nonlinearity in the response of the oscillator, the nonlin-\near system response is best characterized using a narrow-band\nslow (200 s−2) sweep-down excitation from 6 kHz to 4 kHz.\nFor all the above cases, we study the interaction of surface\nwaves with the array of bead-magnet resonators by recording\nthe vertical velocity at 42 stations on the edge of the plate,\nmarked as red circles in Fig. 1. The distance between adjacent\nobservation points is 15 mm.\nTo unveil the linear response of the system, which is used as\na baseline to understand the effects of nonlinearity, we recon-\nstruct the dispersion for the pristine plate, and the plate with an\narray of bead-magnet resonators. Fig. 4 (a) shows the exper-4\nFIG. 4: Dispersion reconstruction using experimental measurements: (a) the pristine plate in the linear regime, (b) the plate\nwith an array of bead-magnet resonators in the linear regime (10 dB gain), and (c) the plate with an array of bead-magnet\nresonators in the nonlinear regime (20 dB gain). The dashed orange line indicates the Rayleigh wave dispersion curve.\nimental dispersion curve for the pristine plate in a gray-scale\ncontour. The broadband chirp generated by the shaker travels\nalong the plate’s edge dispersionless, as expected. The dashed\norange curve shows the Rayleigh wave speed cR=1205 m /s\ngiven by Eq. 1. We note that gluing the array of magnets to the\nplate’s edge does not introduce any dispersion effects since the\nmagnets’ mass ( ≈0.03 g) is negligible. Fig. 4 (b) shows the\ndispersion curves for the plate with an array of bead-magnet\nresonators. Placing the contact resonators on the plate’s edge\nleads to hybridization between the traveling wave and the res-\nonance modes. The slow-propagating flat branch observed in\nthe dispersion plot is a result of SAW interaction with vertical\nresonances of the bead-magnet resonators10,27. The frequency\nwhere the branch flattens agrees well with the primary reso-\nnance frequency for a single bead on the acrylic plate, deter-\nmined previously.\nIn order to investigate the behavior of the system in the non-\nlinear regime, we use the slow narrow-band chirp. Three dif-\nferent excitation amplitudes were chosen by changing the gain\non the amplifier. In order to quantify confidence in the exper-\nimental results, three sets of measurements were done at each\ngain, leading to nine sets of data in total. After each mea-\nsurement, all beads were removed, cleaned and placed on the\nmagnets again. This was done to ensure that the results were\nnot significantly affected by the uncertainties associated with\nthe bead-magnet contact surface. Furthermore, the order in\nwhich the nine experiments were done was random. For each\nset of measurements at constant amplitude, recorded spatio-\ntemporal data on the plate’s edge was postprocessed using 2D\nFourier transforms. The average of normalized Fourier ampli-\ntudes over each three sets of measurements was then used to\nvisualize the system’s dispersion.\nFig. 4 (c) shows the reconstructed dispersion for the struc-ture at 20 dB gain. The gray-scale contour shows the full 2D\nvisualization of response in the wave number-frequency do-\nmain, with white showing the highest intensity. At each dis-\ncrete frequency value, the wave-number corresponding to the\nmaximum Fourier amplitude was identified. This gives the\noverlaid scattered plot in a gradient of red. The color of these\nmarkers at each point indicates the intensity of the normalized\nFourier amplitude, with white having the lowest intensity and\nred the highest. This approach will prove itself crucial later\nfor comparing the dispersion branches at different gain lev-\nels. It also helps highlight data points of greater significance.\nFor example, we can see that the data points lying outside the\nsound cone are of extremely low intensity and therefore of lit-\ntle significance. Thus, we can safely ignore them. The band\nstructure at the other two excitation amplitudes (10 and 30 dB)\nis constructed in a similar manner.\nFig. 5 (a) shows the dispersion branches reconstructed at the\nthree different amplitudes. The scattered plots are now shown\nin the form of error-bar plots; that is, at each frequency, the\nmarker indicates the mean and the horizontal bar shows the\nstandard deviation of the wavenumber corresponding to the\nmaximum Fourier amplitude for the three sets of measure-\nments. It is clear that in the regions where Fourier ampli-\ntude is high, standard deviation is extremely small. On the\ncontrary, as intensity approaches zero, the standard deviation\nbecomes very large. In the regions of high intensity and low\nstandard deviation, the figure shows that increasing the exci-\ntation amplitude shifts the dispersion curve to lower frequen-\ncies. In other words, with an increase in the excitation am-\nplitude, the wavenumber corresponding to a fixed frequency\nincreases. This is quantitatively shown in Table II for three\nselect frequencies.\nFig. 5 (b) shows the dispersion branches in a more limited5\nFIG. 5: Dispersion reconstruction in a limited frequency\nregion: (a) overlaid dispersion branches for the three\nexcitation levels. (b) same as (a), zoomed in a more limited\nregion to highlight the dispersion shift. The colored markers\nare graded according to the magnitude of the normalized\nFourier amplitude.\nWavenumber [1/m]\nFrequency [kHz] 10 dB 20 dB 30 dB\n4.2 4.65 4.69 4.76\n4.7 5.1 5.14 5.28\n5.2 6.63 6.76 7.28\nTABLE II: Increase of the wavenumber value with an\nincrease in excitation amplitude at sample low, medium and\nhigh frequencies.\nwave number-frequency region to highlight the amplitude-\ndependent dispersion shift. Just like in the softening the-\noretical prediction of Palermo et al.33, we observe that: i)\nan increase in excitation amplitude in the presence of soft-\nening nonlinearity shifts the dispersion branch down; and\nii) higher-amplitude branches tend to terminate early, i.e., at\nlower wavenumbers compared to their low-amplitude coun-\nterparts. The early termination may be explained by the on-\nset of instability for the surface resonators33. Comparing\nwavenumber-frequency pairs at a certain threshold of the nor-\nmalized Fourier amplitude intensity proves useful for quanti-\nfying the early termination of dispersion branches. For exam-\nple, a 0.3 normalized Fourier amplitude intensity corresponds\nto the point (7.0204 1/m, 5302.4 Hz) on the dispersion branch\nat 10 dB excitation amplitude. However, at the same intensity,\nwavenumber-frequency pairs at 20, and 30 dB are (6.901 1/m,\n5235.7 Hz) and (6.6515 1/m, 5092.6 Hz), respectively. As\nsuch, the termination wavenumber decreases by 1.7% from\n10 dB to 20 dB and 3.62% from 20 dB to 30 dB.\nIn conclusion, we have investigated the interaction of sur-\nface acoustic waves with nonlinear contact resonators and pro-\nvided experimental evidence of amplitude-dependent surface\nwave dispersion. Careful investigation of the bead-magnet’sdynamics revealed several features common to softening non-\nlinear oscillators, such as amplitude-dependent resonance fre-\nquency, higher harmonic generation and loss of stability.\nThese characteristics make the array of bead-magnet assem-\nblies suitable for use as nonlinear resonators in a compact\nsetup. In the current setup, the surface wave energy available\nfor interaction with the nonlinear resonators is limited due to\nthe maximum force rating of the shaker and the overall en-\nergy loss in the system. This prevents the realization of more\nsignificant shifts, such as those induced by external stimuli10.\nHowever, this proof-of-concept demonstration serves as a mo-\ntivation for other researchers to design novel solutions to in-\nduce more dramatic self-tuning effects for SAWs. These could\ninclude creating a waveguide close to the plate’s edge to max-\nimize the surface wave energy and exploring the dynamics of\nthe resonators in the vibroimpact region. Loss of contact non-\nlinearity for Hertzian contact resonators has been shown to\ninduce more significant resonance frequency shifts29,30.\nACKNOWLEDGMENTS\nWe thank Professor Alexander Vakakis and fellow re-\nsearchers Alireza Mojahed, Joaqin Garcia-Suarez, and Danilo\nKusanovic for the stimulating discussions. This research\nhas been supported by the US National Science Foundation\nGrant EFRI-1741565, the National Science Foundation Ca-\nreer Award No. 1753249, and the Graduate College Disserta-\ntion Completion Fellowship Award provided by the University\nof Illinois at Urbana-Champaign.\n1S. Benchabane and A. Reinhardt, “Elastic Metamaterials for Radiofre-\nquency Applications,” in Fundamentals and Applications of Acoustic Meta-\nmaterials (Wiley, 2019) pp. 207–262.\n2K. Länge, B. E. Rapp, and M. Rapp, “Surface acoustic wave biosensors: A\nreview,” (2008).\n3S. Brûlé, E. H. Javelaud, S. Enoch, and S. Guenneau, “Experiments on\nseismic metamaterials: Molding surface waves,” Physical Review Letters\n112, 133901 (2013).\n4Muhammad, C. W. Lim, and J. N. Reddy, “Built-up structural steel sections\nas seismic metamaterials for surface wave attenuation with low frequency\nwide bandgap in layered soil medium,” Engineering Structures 188, 440–\n451 (2019).\n5T. T. Wu, J. C. Hsu, J. H. Sun, and S. 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Gonella, “Doubly nonlinear waveguides with self-switching\nfunctionality selection capabilities,” Physical Review E 99, 042206 (2019).\n15A. Mojahed, J. Bunyan, S. Tawfick, and A. F. Vakakis, “Tunable Acoustic\nNonreciprocity in Strongly Nonlinear Waveguides with Asymmetry,” Phys-\nical Review Applied 12, 034033 (2019).\n16K. J. Moore, J. Bunyan, S. Tawfick, O. V . Gendelman, S. Li, M. Leamy,\nand A. F. Vakakis, “Nonreciprocity in the dynamics of coupled oscillators\nwith nonlinearity, asymmetry, and scale hierarchy,” Physical Review E 97,\n012219 (2018).\n17M. D. Fronk, S. Tawfick, C. Daraio, S. Li, A. Vakakis, and M. J. Leamy,\n“Acoustic non-reciprocity in lattices with nonlinearity, internal hierarchy,\nand asymmetry: Computational study,” Journal of Vibration and Acoustics,\nTransactions of the ASME 141(2019), 10.1115/1.4043783.\n18W. Jiao and S. Gonella, “Intermodal and Subwavelength Energy Trap-\nping in Nonlinear Metamaterial Waveguides,” Physical Review Applied 10,\n024006 (2018).\n19V . Zega, P. B. Silva, M. G. Geers, and V . G. Kouznetsova, “Experimen-\ntal proof of emergent subharmonic attenuation zones in a nonlinear locally\nresonant metamaterial,” Scientific Reports 10, 12041 (2020).\n20R. K. Narisetti, M. Ruzzene, and M. J. Leamy, “Study of wave propagation\nin strongly nonlinear periodic lattices using a harmonic balance approach,”\nWave Motion 49, 394–410 (2012).\n21K. Manktelow, R. K. Narisetti, M. J. Leamy, and M. Ruzzene, “Finite-\nelement based perturbation analysis of wave propagation in nonlinear pe-\nriodic structures,” Mechanical Systems and Signal Processing 39, 32–46\n(2013).\n22R. Khajehtourian and M. I. Hussein, “Dispersion characteristics of a non-\nlinear elastic metamaterial,” AIP Advances , 124308 (2014).23B. S. Lazarov and J. S. Jensen, “Low-frequency band gaps in chains with\nattached non-linear oscillators,” International Journal of Non-Linear Me-\nchanics 42, 1186–1193 (2007).\n24K. L. Manktelow, M. Ruzzene, and M. J. Leamy, “Wave Propagation in\nNonlinear Lattice Materials,” in Dynamics of Lattice Materials (John Wiley\n& Sons, Ltd, Chichester, UK, 2017) pp. 107–137.\n25X. Fang, J. Wen, D. Yu, G. Huang, and J. Yin, “Wave propagation in a non-\nlinear acoustic metamaterial beam considering third harmonic generation,”\nNew Journal of Physics 20, 123028 (2018).\n26N. Boechler, J. K. Eliason, A. Kumar, A. A. Maznev, K. A. Nelson, and\nN. Fang, “Interaction of a contact resonance of microspheres with surface\nacoustic waves,” Physical Review Letters 111, 036103 (2013), 1302.4698.\n27S. P. Wallen, J. Lee, D. Mei, C. Chong, P. G. Kevrekidis, and N. Boechler,\n“Discrete breathers in a mass-in-mass chain with Hertzian local resonators,”\nPhysical Review E 95, 022904 (2017), arXiv:1605.03377.\n28A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, 1995).\n29E. Rigaud and J. Perret-Liaudet, “Experiments and numerical results on\nnon-linear vibrations of an impacting Hertzian contact. Part 1: Harmonic\nexcitation,” Journal of Sound and Vibration 265, 289–307 (2003).\n30J. Perret-Liaudet and E. Rigaud, “Experiments and numerical results on\nnon-linear vibrations of an impacting Hertzian contact. Part 2: Random\nexcitation,” Journal of Sound and Vibration 265, 309–327 (2003).\n31A. Merkel, G. Theocharis, F. Allein, J. P. Groby, V . Gusev, and V . Tournat,\n“Testing a bead-rod contact with a nonlinear resonance method,” Journal of\nSound and Vibration 441, 84–95 (2019).\n32K. L. Manktelow, M. J. Leamy, and M. Ruzzene, “Analysis and ex-\nperimental estimation of nonlinear dispersion in a periodic string,” Jour-\nnal of Vibration and Acoustics, Transactions of the ASME 136 (2014),\n10.1115/1.4027137.\n33A. Palermo, B. Yousefzadeh, C. Daraio, and A. Marzani, “Rayleigh wave\npropagation in nonlinear metasurfaces,” Journal of Sound and Vibration\n520, 116599 (2022).\n34M. F. Ashby, “Materials selection in mechanical design,” (2018).\n35M. V . Wilde, M. V . Golub, and A. A. Eremin, “Experimental and theoret-\nical investigation of transient edge waves excited by a piezoelectric trans-\nducer bonded to the edge of a thick elastic plate,” Journal of Sound and\nVibration 441, 26–49 (2019)."    },    {        "title": "2210.00366v1.Nonlinear_features_of_the_superconductor__ferromagnet__superconductor___varphi_0__Josephson_junction_in_ferromagnetic_resonance_region.pdf",        "content": "Nonlinear features of the superconductor{ferromagnet{superconductor '0Josephson\njunction in ferromagnetic resonance region\nAliasghar Janalizadeh1, Ilhom R. Rahmonov2;3;4, Sara A.\nAbdelmoneim5, Yury M. Shukrinov2;3;4, and Mohammad R. Kolahchi1\n1Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45137-66731, Zanjan, Iran\n2BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n3Dubna State University, Dubna, 141980, Russia\n4Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Moscow Region, Russia\n5Physics department, Meno\fya University, Faculty of Science, 32511, Shebin Elkom,Egypt\n(Dated: October 4, 2022)\nWe demonstrate the manifestations of the nonlinear features in magnetic dynamics and IV-\ncharacteristics of the '0Josephson junction in the ferromagnetic resonance region. We show that\nat small values of system parameters, namely, damping, spin-orbit interaction, and Josephson to\nmagnetic energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar Du\u000eng os-\ncillator, driven by the Josephson oscillations. The role of increasing superconducting current in the\nresonance region is clari\fed. Shifting of the ferromagnetic resonant frequency and the reversal of\nits damping dependence due to nonlinearity are demonstrated by the full Landau-Lifshitz-Gilbert-\nJosephson system of equations, and in its di\u000berent approximations. Finally, we demonstrate the\nnegative di\u000berential resistance in the IV{characteristics, and its correlation with the foldover e\u000bect.\nI. I. INTRODUCTION\nThe coupling of superconducting phase di\u000berence with\nmagnetic moment of ferromagnet in the '0junction leads\nto a number of unique features important for supercon-\nducting spintronics, and modern information technology\n[1{5]. It allows to control the magnetization preces-\nsion by superconducting current and a\u000bects the current{\nvoltage (IV) characteristics by magnetic dynamics in the\nferromagnet, in particular, to create a DC component in\nthe superconducting current [6{8]. A remarkable mani-\nfestation of such coupling is the possibility to stimulate\na magnetization reversal in the ferromagnetic layer by\napplying current pulse through the '0-junction [3, 9{13].\nThere are two features of our Josephson junction that\ncome into play in our study. One is the broken inver-\nsion symmetry in the weak link of the Josephson junc-\ntion, when the link is magnetic, which introduces an ex-\ntra phase in the current|-phase relation, preventing it\nfrom being antisymmetric. Such Josephson junctions are\nnamed'0junctions [1], and examples exist such as MnSi\nand FeGe. Second is the nonlinear property of the system\nthat makes for an anomalous resonance behavior [14].\nWe couple such a Josephson junction to the model\nthat describes the magnetodynamics in thin \flms or\nheterostructure, to form the Landau-Lifshitz-Gilbert-\nJosephson model (LLGJ)[14{16]. It is shown that for\na particular set of parameters, the coupled equations\nreduce to the dynamics of a Du\u000eng oscillator [14].\nThe cubic nonlinearity in this oscillator has applications\nin describing several e\u000bects in other models too [17].\nOne being the resonance e\u000bects in the antiferromagnetic\nbimeron in response to an alternating current, which has\napplications in the detection of weak signals [15, 18, 19].\nThe Gilbert damping term is added phenomenologi-\ncally to the Landau|-Lifshitz model, to reproduce the\ndamping of the precessing magnetic moment. Gilbertdamping is important in modeling other resonance fea-\ntures too, as its temperature dependence a\u000bects them\n[20, 21], and in return in the superconducting correla-\ntions that a\u000bect it [22]. The magnetization precession\nin the ultra thin Co20Fe60B20layer stimulated by mi-\ncrowave voltage under a large angle, needs modeling by\nDu\u000eng oscillator too. This gets help from the so called\nfoldover features, again due to nonlinearity [16, 23, 24].\nThe consequences of the nonlinear nature of the cou-\npled set of LLGJ system of equations in the weak cou-\npling regime was demonstrated recently in Ref. [14]. We\nshowed in this regime, where the Josephson energy is\nsmall compared to the magnetic energy, the '0Joseph-\nson junction is equivalently described by a scalar non-\nlinear Du\u000eng equation. An anomalous dependence of\nthe ferromagnetic resonant frequency (FMR) with the\nincrease of the Gilbert damping was found. We showed\nthat the damped precession of the magnetic moment is\ndynamically driven by the Josephson supercurrent, and\nthe resonance behavior is given by the Du\u000eng spring.\nThe obtained results were based on the numerical simu-\nlations. The role of dc superconducting current, and the\nstate with negative di\u000berential resistance (NDR) in IV-\ncharacteristic were not clari\fed. Also, the e\u000bects of the\nJosephson to magnetic energy ratio and the spin-orbit\ncoupling (SOC) were not investigated at that time.\nIn the present paper, we study the nonlinear aspects\nof the magnetic dynamics and IV-characteristics of the\n'0Josephson junction in the ferromagnetic resonance re-\ngion. We compare description of the anomalous damp-\ning dependence (ADD) exhibited by full LLGJ system\nof equations with approximated equations and demon-\nstrate the Du\u000eng oscillator features in the small param-\neter regime. E\u000bects of the Josephson to magnetic energy\nratio, and the spin-orbit coupling on the ADD, referred\nto earlier as the \u000b-e\u000bect [14] are demonstrated. By de-\nriving the formula which couples the dc superconduct-arXiv:2210.00366v1  [cond-mat.supr-con]  1 Oct 20222\ning current and maximal amplitude of magnetization we\ndiscuss the correlation of superconducting current and\nthe negative di\u000berential resistance in the resonance re-\ngion. Finally, we discuss the experimentally important\nfeatures by emphasizing the details of the magnetization\ndynamics and the IV-characteristics of the '0junction.\nWe have shown that in the limit of small system pa-\nrameters; that is, the Josephson to magnetic energy ra-\ntioG, the damping \u000b, and the spin-orbit coupling r, the\ndynamics is given by the Du\u000eng spring [14]. We focus\non the shift in resonance and the e\u000bects of nonlinear in-\nteractions. We give semi-analytic models to explain our\nresults in various limits.\nThe paper is organized as follows. In Section II we\noutline the theoretical model and discuss the methods\nof calculations. The ferromagnetic resonance and ef-\nfects of system parameters on the anomalous damping\ndependence are considered in Subsection A of Section\nIII. In Subsection B we present analytical description of\nthe dynamics and IV-characteristics of the '0junction\nat small system parameters. Manifestation of the nega-\ntive di\u000berential resistance in IV-characteristics through\nthe foldover e\u000bect is discussed. We compare the de-\nscription of the anomalous damping dependence by full\nLLGJ system of equation with approximated equation,\nand show how the Du\u000eng oscillator captures the non-\nlinearities in the small parameter regime in Subsection\nC. We present results on the critical damping and de-\nrive the formula which couples the dc superconducting\ncurrent and maximal amplitude of magnetization in the\nferromagnetic layer. Finally, in Section IV we concludes\nthe paper.\nII. II. MODELS AND METHOD\nThe following section is closely related to our work\nin [13]. The '0junction [6, 12, 25] that we study is shown\nin Fig.1. The current-phase relation in varphi 0junction\nhas the form Is=Icsin ('\u0000'0), where'0=rMy=M0,\nMydenotes the component of magnetic moment in ^ ydi-\nrection,M0is the modulus of the magnetization. The\nphysics of'0Josephson juncton is determined by system\nof equations which consists of Landau-Lifshits-Gilbert\n(LLG), resistively capacitively shunted junction (RCSJ)\nmodel expression with current-phase relation ( Is) de-\nscribed above, and Josephson relation between phase dif-\nference and voltage.\nThe dynamics of the magnetic moment Mis described\nby the LLG equation [26]\ndM\ndt=\u0000\rM\u0002Heff+\u000b\nM0\u0012\nM\u0002dM\ndt\u0013\n; (1)\nwhere Mis the magnetization vector, \ris the gyromag-\nnetic relation, Heffis the e\u000bective magnetic \feld, \u000bis\nGilbert damping parameter, M0=jMj.\nFigure 1. Schematic view of SFS '0Josephson junction. The\nexternal current applied along x direction, ferromagnetic easy\naxis is along z direction.\nIn order to \fnd the expression for the e\u000bective mag-\nnetic \feld we have used the model developed in Ref.[6],\nwhere it is assumed that the gradient of the spin-orbit\npotential is along the easy axis of magnetization taken to\nbe along ^z. In this case the total energy of the system\ncan be written as\nEtot=\u0000\b0\n2\u0019'I+Es(';' 0) +EM('0); (2)\nwhere'is the phase di\u000berence between the supercon-\nductors across the junction, Iis the external current,\nEs(';' 0) =EJ[1\u0000cos ('\u0000'0)], andEJ= \b 0Ic=2\u0019\nis the Josephson energy. Here \b 0is the \rux quantum,\nIcis the critical current, r=l\u001dso=\u001dFl= 4hL=~\u001dF,L\nis the length of Flayer,his the exchange \feld of the\nFlayer,EM=\u0000KVM2\nz=(2M2\n0), the parameter \u001dso=\u001dF\ncharacterizes a relative strength of spin-orbit interaction,\nKis the anisotropic constant, and Vis the volume of the\nferromagnetic ( F) layer.\nThe e\u000bective \feld for LLG equation is determined by\nHe\u000b=\u00001\nV@Etot\n@M\n=\nF\n\r\u0014\nGrsin\u0012\n'\u0000rMy\nM0\u0013\nby+Mz\nM0bz\u0015\n(3)\nwhere \n F=\rK=M 0is frequency of ferromagnetic reso-\nnance andG=EJ=(KV) determines the ratio of Joseph-\nson energy to magnetic one.\nIn order to describe the full dynamics '0junction the\nLLG equations should be supplemented by the equation\nfor phase di\u000berence ', i.e. equation of RCSJ model for\nbias current and Josephson relation for voltage. Accord-\ning to the extended RCSJ model, which takes into ac-\ncount derivative of '0phase shift, the current \rowing\nthrough the system in underdamped case is determined\nby\nI=~C\n2ed2'\ndt2+~\n2eR\u0014d'\ndt\u0000r\nM0dMy\ndt\u0015\n(4)\n+Icsin\u0012\n'\u0000r\nM0My\u0013\n:\nwhereIis the bias current, CandRare the capacitance\nand resistance of Josephson junction respectively. The3\nJosephson relation for voltage is given by :\n~\n2ed'\ndt=V: (5)\nWe note that in the framework of RCSJ{model the\ndisplacement current is proportional to the \frst deriva-\ntive of voltage (or second derivative of phase di\u000berence).\nFrom the other hand, the magnetization dynamics plays\nrole of the external force and \frst order derivative of '0\nis a source of external current for JJ. This was demon-\nstrated in Ref.[25, 27] where the authors included the \frst\nderivative of '0as the source of the electromotive force.\nVoltage is determined by the phase di\u000berence, and does\nnot depend on '0. From this point of view, in the frame-\nwork of RCSJ model the external current source cannot\nmodify the expression for displacement current. That's\nwhy we do not include the second derivative of varphi 0\nin our model.\nUsing (1), (3), (4) and (5) we can write the system of\nequations, in normalised variables, which describes the\ndynamics of '0junction\n_mx=!F\n1 +\u000b2f\u0000mymz+Grm zsin('\u0000rmy)\n\u0000\u000b[mxm2\nz+Grm xmysin('\u0000rmy)]g;\n_my=!F\n1 +\u000b2fmxmz\n\u0000\u000b[mym2\nz\u0000Gr(m2\nz+m2\nx) sin('\u0000rmy)]g;\n_mz=!F\n1 +\u000b2f\u0000Grm xsin('\u0000rmy)\n\u0000\u000b[Grm ymzsin('\u0000rmy)\u0000mz(m2\nx+m2\ny)]g;\n_V=1\n\fc[I\u0000V+r_my\u0000sin('\u0000rmy)];\n_'=V(6)\nwheremx;y;z =Mx;y;z=M0and satisfy the constraintP\ni=x;y;zm2\ni(t) = 1,\fc= 2eIcCR2=~is McCumber pa-\nrameter. In order to use the same time scale in the\nLLG and RCSJ equations in this system of equations\nwe have normalized time to the !\u00001\nc, where!c=2eIcR\n~,\nand!F= \n F=!cis the normalized frequency of ferro-\nmagnetic resonance \n F=\rK=M 0. Bias current is nor-\nmalized to the critical current Icand voltage V{ to the\nVc=IcR. The system of equations (6), is solved numer-\nically using the fourth-order Runge-Kutta method(see\nRef.[14]).\nIII. III. RESULTS AND DISCUSSION\nA. A. E\u000bect of system parameters on the\nanomalous damping dependence\nADD of the FMR frequency with increasing \u000bwas dis-\ncussed in Ref. [14]. It was found that the resonance\ncurves demonstrate features of Du\u000eng oscillator, re-\n\recting the nonlinear nature of Landau-Lifshitz-Gilbert-\n 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14\n 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8mymax\nValpha=0.01\nalpha=0.02\nalpha=0.03\nalpha=0.04\nalpha=0.05\nalpha=0.06\nalpha=0.07\nalpha=0.08\nalpha=0.09\nalpha=0.1\n 0 0.1\n 0.5Figure 2. Maximal amplitude of magnetization\nmy\u0000component at each values of bias current and voltage\nalong IV-characteristics of the '0junction in the ferromag-\nnetic resonance region for various \u000b. Inset enlarges the main\nmaximum. Parameters: \fc= 25,G=0.05,r=0.05, !F= 0:5.\nJosephson (LLGJ) system of equations. There is a criti-\ncal damping value at which anomalous dependence comes\ninto play. This critical value depends on the system pa-\nrameters. Here we present the details of such transforma-\ntion from usual to anomalous dependence with variation\nin spin-orbit coupling and ratio of Josephson to magnetic\nenergy.\nTo investigate the e\u000bect of damping, we calculate\nthe maximal amplitude of magnetization component my\ntaken at each value of the bias current based on the\nLLGJ system of equations (6). In Fig.2 we show the\nvoltage dependence of maximal amplitude mmax\nyin the\nferromagnetic resonance region at di\u000berent damping pa-\nrameter and small values of Josephson to magnetic en-\nergy ratio G=0.05 and spin-orbit coupling r= 0:05. We\nfound that the ferromagnetic resonance curves demon-\nstrate the di\u000berent forms. An increase in damping shows\na nonuniform change in the resonant frequency: it is ap-\nproaching the !Finstead of moving away with increase\nin\u000b. We stress that this happens at small Gandr. We\nconsider that such behavior can be explained by the non-\nlinear nature of the LLGJ system of equations. There is\na manifestation of subharmonics of the FMR in Fig.2 at\n!= 1=2;1=3;1=4.\nWe usually expect the resonance peak to move away\nfrom resonance as the \u000bincreases. Figure 2 shows that\nthis normal e\u000bect is accompanied with an anomalous be-\nhaviour as can be seen in the inset to this \fgure, where\nthe resonance peak approaches !Fas\u000bincreases [14].\nThe manifestation of FMR in IV-characteristics of the\n'0junction at three values of damping parameter is\ndemonstrated in Fig. 3. The strong deviation of the\nIV-curve is observing at \u000b= 0:01, which is characteristic4\nFigure 3. Part of the IV characteristic of the '0junction\natG= 0:05;r= 0:05 and di\u000berent values of Gilbert damp-\ning. The numbers show \u000bvalue. Inset shows the total IV-\ncharacteristic and arrow indicates the resonance region\nvalue for many magnetic materials. This fact indicates\nthat ADD can be observed experimentally by measuring\nIV-characteristics in wide interval of the damping param-\neter.\nInteresting features of ADD appear by a variation of\nspin-orbit coupling. As it was demonstrated in Ref.[28],\nan increase in SOC leads to the essential change in IV-\ncharacteristics and magnetization precession in the fer-\nromagnetic resonance region. The nonlinearity is going\nstronger and the state with negative di\u000berential resis-\ntance appears at large SOC.\nFigure 4(a) demonstrates results of numerical simu-\nlations ofmmax\nydependence on \u000bat di\u000berent values of\nSOC parameter r. It shows two speci\fc features of ADD.\nFirst, with an increase in r, the critical value of Vpeakis\ndecreasing (the curve moves away from !F). The sec-\nond important feature is an increasing of \u000bcritwhich is\ndemonstrated in this \fgure by arrows.\nAnother model parameter which a\u000bects the phe-\nnomenon discussed in the present paper is the ratio G\nof Josephson to magnetic energies. Figure 4(b) demon-\nstrates the results of numerical simulations of mmax\nyde-\npendence on \u000bat di\u000berent values of G.\nSimilar to the e\u000bect of r, increasing Galso causes the\nvalue of\u000bcritto increase. By changing the volume of the\nferromagnetic layer, the ferromagnetic energy and con-\nsequently the value of G can be changed [6]. For small\nG, i.e. a situation where the magnetic energy is much\nlarger than the Josephson energy, the magnetic layer re-\nceives less energy, and its amplitude decreases in the y\ndirection, and also the maximum value of the oscillation\nfrequency is closer to the magnetic frequency, !F.\nVpeakα\n0.46 0.47 0.48 0.49 0.500.050.10.150.2\nαcrit\n1 23\nVpeakα\n0.46 0.47 0.48 0.49 0.500.050.10.150.2\nαcrit\n1 23Figure 4. (a) Demonstration of ADD at di\u000berent values of\nSOC parameter ratG= 0:05. Numbers indicate: 1 -\nr= 0:05; 2 -r= 0:1; 3 -r= 0:5; Arrows show critical \u000b\nvalue, corresponded to the reversal in the \u000bdependence (b)\nDemonstration of ADD at di\u000berent values of the Josephson\nto magnetic energy ratio Gatr= 0:05. Numbers indicate: 1\n-G= 0:01; 2 -G= 0:1; 3 -G= 1.\nB. B. Dynamics and IV-characteristics of the '0\njunction at small system parameters\nAs it was discussed in Refs.[6, 29, 30], in case of\nG;r;\u000b<< 1 andmz\u00191, \frst three equations of the sys-\ntem (6) can be simpli\fed. Taking into account '=!Jt\nand neglecting quadratic terms of mxandmy, we get\n(\n_mx=!F[\u0000my+Grsin(!Jt)\u0000\u000bmx]\n_my=!F[mx\u0000\u000bmy];(7)\nThis system of equations can be written as the second\norder di\u000berential equation with respect to my\nmy+ 2\u000b!F_my+!2\nFmy=!2\nFGrsin!Jt: (8)5\nCorresponding solution for myhas the form\nmy(t) =!+\u0000!\u0000\nrsin(!Jt)\u0000\r++\r\u0000\nrcos(!Jt);(9)\nwhere\n!\u0006=Gr2!F\n2!J\u0006!F\n\n\u0006; (10)\nand\n\r\u0006=Gr2!F\n2\u000b!J\n\n\u0006: (11)\nwith \n \u0006= (!J\u0006!F)2+ (\u000b!J)2(see Ref.[6] and corre-\nsponded Erratum[31]).\nWhen the Josephson frequency !Jis approaching the\nferromagnetic one !F,mydemonstrates the damped fer-\nromagnetic resonance. Di\u000berential resistance in the res-\nonance region is decreasing and it is manifested in the\nIV{characteristic as a resonance branch [7].\nTaking into account rmy<<1, we rewrite expression\nfor superconducting current as\nIs(t) = sin(!Jt\u0000rmy(t))\n= sin(!Jt)\u0000rmycos(!Jt) (12)\nUsing solution (9) we can obtain\nIs(t) = sin!Jt\u0000!+\u0000!\u0000\n2sin 2!Jt\n+\r++\r\u0000\n2cos 2!Jt+I0(\u000b) (13)\nwhere\nI0=\r++\r\u0000\n2: (14)\nThis superconducting current explains the appearance\nof the resonance branch in the IV{characteristic. The\ngenerated current I0can be expressed through the am-\nplitude ofmyand SOI parameter r\nI0=r\n2mmax\ny(!J); (15)\nwithmmax\ny(!J) being the frequency response of my.\nAt small model parameters \u000b<<Gr<< 1 of SFS'0\nJosephson junction, the states with a negative di\u000berential\nresistance appear in the IV-characteristics in the FMR\nregion. Due to the nonlinearity, the resonance peak is\nasymmetric. Increasing of nonlinearity leads to the bista-\nbility (foldover e\u000bect). A natural questions appears if the\nstates with a negative di\u000berential resistance are the origin\nof the foldover and ADD. In order to clarify this question,\nwe show in Fig.5 a part of IV{characteristics of '0junc-\ntion together with IV-characteristic of SIS junction in\nthe ferromagnetic resonance region and numerically cal-\nculated superconducting current through this junction.\nThe total IV{characteristics are demonstrated in the in-\nset to this \fgure.\nFigure 5. IV-characteristic of '0and SIS junctions and calcu-\nlated average superconducting current through the '0junc-\ntion.\nWe see the correlation of the foldover e\u000bect in super-\nconducting current (blue) with the NDR part of IV-curve.\nThe peak in the superconducting current and minimum\nof IV-curve are at the same voltage value. So, both ef-\nfects re\rect the nonlinear features of the ferromagnetic\nresonance in '0junction. But in contrast to the foldover\nand ADD e\u000bects, which start manifesting themselves at\nrelatively small deviations from linear case, the nonlin-\nearity in case of the NDR plays a more essential role.\nWe note that in the resonance region for considered\nlimit of model parameters the myamplitude are coupled\nto the value of superconducting current (see Eq.(15)). We\nstress the importance of the performed analysis demon-\nstrating the analytical coupling of time independent su-\nperconducting current and magnetization, re\recting the\nDu\u000eng oscillator features of the '0junction.\nAs it is well known, the states with negative di\u000beren-\ntial resistance appear in IV-characteristics of Josephson\nstructures in di\u000berent physical situations. In particular,\nthe nonlinear superconducting structures being driven\nfar from equilibrium exhibit NDR states [32]. The NDR\nstates plays an essential role in the applications related\nto the THz radiation emission [33]. A detailed expla-\nnation for di\u000berent types of negative di\u000berential resis-\ntance (NDR) in Josephson junction (i.e., N-shaped and\nS-shaped) is introduced in[34]. The authors emphasize\nthat the nonlinear behavior of the Josephson junction\nplays a key role in the NDR feature. In our case, the\nNDR states appear as a result of system's nonlinearity\nat small values of '0junction parameters, such as SOC,\nratio of Josephson to magnetic energy and Gilbert damp-\ning. We demonstrate these e\u000bects here by by presented\nresults of detailed investigation of the NDR state at dif-\nferent system parameters and discuss possibility of their\ncontrol near the ferromagnetic resonance.\nFigure 6 shows the e\u000bect of the spin-orbit coupling on\nthe IV-characteristic at G= 0:05 and\u000b= 0:01. We\nsee the NDR feature which is getting more pronounced\nwith increase in r. A further increase in rleads to the\njump down in voltage and then practically linear growth6\nIV\n0.49 0.495 0.5 0.505 0.510.460.470.480.490.09\n0.15\nIV\n0 0.5 100.20.40.60.811.2r =0.15\nFigure 6. Enlarged parts of the IV-curves in the resonance\nregion at di\u000berent values of SOC parameter rat\u000b= 0:01\nandG= 0:05. The numbers indicate the increasing of rfrom\n0.09 to 0.15 with an increment 0.01. Inset demonstrates total\nIV-characteristic ar r= 0:15\nof IV-characteristic.\nAn interesting question is concerning the e\u000bect of\nGilbert damping. Results of IV-characteristics simula-\ntions in the resonance region at a certain range of the\ndamping parameter \u000batG= 0:05 andr= 0:13 are\nshown in Fig.7(a). In this case, the most pronounced\ncharacteristic appears at \u000b= 0:01. At chosen set of pa-\nrametersG= 0:05 andr= 0:13, the range of \u000bwith\npronounced NDR features is 0 :016\u000b<0:014.\nThe maximal amplitude mmax\nyas a function of volt-\nage is shown in the Fig.7(b). Based on the results, pre-\nsented in Fig.7(a) and Fig.7(b), we came to the important\nconclusion that the foldover e\u000bect (bistability) and NDR\nstate have strong correlations and have the same origin\nrelated to the nonlinearity at small system parameters.\nBut anomalous damping dependence does not show a\none-to-one correlation with either negative di\u000berential re-\nsistance or foldover e\u000bect. The resonance peak positions\nofmmax\nyin bias current Ipeak, and in voltage Vpeakas the\nfunctions of \u000bare demonstrated in Fig.7(c). According\nto our results, we can divide \u000binterval into two regions\n(see Fig.7(c)). Region I introduces the values of \u000bwhere\nthe NDR feature is present, while in the Region II it dis-\nappears. In the Region II the foldover e\u000bect (bistability)\ndisappears as well, but ADD is realized.\nC. C. Du\u000eng oscillator features of '0junction and\ncritical damping\nThe system of equations (6) is nonlinear and very com-\nplex, so in order to provide analytical study of dynamics\nof the'0junction we need to derive an approximated\nequation for some limited values of model parameters.\nIn Ref. [14] was shown that the resonance curves demon-\nstrate features of Du\u000eng oscillator, re\recting the nonlin-\near nature of Landau-Lifshitz-Gilbert-Josephson (LLGJ)\nIV\n0.495 0.5 0.505 0.510.4650.470.4750.480.4850.49\n0.01(a)\n0.018\nIV\n0 0.5 100.20.40.60.811.2\nα= 0.01\nαIpeak, Vpeak\n0.01 0.012 0.014 0.016 0.018 0.020.470.480.490.5Ipeak\nVpeak(c)\nΙΙΙFigure 7. (a) Enlarged parts of the IV-curves at di\u000berent\nvalues of\u000b; (b) Voltage dependence of mmax\nyat di\u000berent \u000b;\n(c)\u000b-dependence of the resonance curve maximum in current\n(Ipeak) and voltage ( Vpeak). The numbers indicate the value\nof\u000bfrom 0.01 to 0.018 in (a) and from 0.01 to 0.02 in (b)\nby an increment 0.001. Results are obtained at r= 0:13, and\nG= 0:05.\nsystem of equations. In this section, we present analyt-\nical approach to describe the nonlinear dynamics of '0-\njunction and compare analytical results followed from ap-\nproximated Du\u000eng equation with numerical simulations\nof total system of equations (6). We show that in the\nlimit of\u000b << G;r << 1, we arrive to the Du\u000eng os-\ncillator. We start by \frst three equations of the (6) for7\nmagnetization components\n_mx\n!F=\u0000mymz+Grm zsin('\u0000rmy)\u0000\u000bmxm2\nz\n_my\n!F=mxmz\u0000\u000bmym2\nz (16)\n_mz\n!F=\u0000Grm xsin('\u0000rmy) +\u000bmz(m2\nx+m2\ny)\nSimplifying this system of equations by the same pro-\ncedure as it was done in Ref.[14] we can write equation\nformyas\nmy+2\u0018\u000b_my+\u00182(1+\u000b2)my\u0000\u00182(1+\u000b2\u0000\u000b4)m3\ny=\u00182\rsin':\n(17)\nFinally, by neglecting the \u000b2and\u000b4terms which are\nmuch smaller than 1, we come to well known Du\u000eng\nequation,\nmy+ 2!F\u000b_my+!2\nFmy\u0000!2\nFm3\ny=!2\nFGrsin': (18)\nIn the small parameter range, this Du\u000eng equation\ncan describe the dynamics of my. We will have the full\ndynamics, once we consider the coupling with the Joseph-\nson equation,\n'+1\n\fc[ _'\u0000r_my+ sin('\u0000rmy)] =1\n\fcI: (19)\nThe system of equations, (18) and (19) can replace the\nLLGJ equations in the limit of G;r<< 1 andG;r>>\u000b .\nTaking into account '=!Jtwe can right analytically\nobtained frequency response for equation (18)\n(mmax\ny)2=\u0000\nGr\u00012\n\u0002\n!2\u00001 +3\n4(mmaxy)2\u00032+\u0000\n2\u000b!\u00012\n(20)\nwhere!=!J=!F. From Eq. (20) we get\n(mmax\ny)6+8\n3(!2\u00001)(mmax\ny)4\n+\u00124\n3\u00132\u0014\n(!2\u00001)2+\u0000\n2\u000b!\u00012\u0015\n(mmax\ny)2\n\u0000\u00124\n3Gr\u00132\n= 0: (21)\nThis equation allows to determine analytically fre-\nquency dependence of the mmax\nyamplitude. To \fnd it\nwe solve the equation (21) by the Newton method. Re-\nsults of analytical calculations (blue dots) corresponded\nto (21) and numerical one (red doted line) corresponded\nto the full system of equation (6) are demonstrated in\nFig.8.\nFigure 8. Numerically (curve 1) and analytically (curve 2)\ncalculated amplitude dependence of my.\nFigure 9. Numerically calculated superconducting current for\nSFS junction (plot 1) and analytical I0(plot 2) and super-\nconducting current for SIS junction (plot 3).\nWe can see that they are close to each other which\nproves the correctness of the chosen approximation.\nBoth curves demonstrate an asymmetric resonance peak,\nwhich is common for Du\u000eng oscillator. When a role of\nthe cubic term is getting larger, we observe a bistability\nof the resonance curve, which is usually called a foldover\ne\u000bect. Note that the foldover e\u000bect can be also achieved\nby the damping decreasing; i.e., by the decreasing of dis-\nsipative term in (18), we can increase the in\ruence of the\ncubic term in this equation.\nThe comparison of analytically and numerically cal-\nculated superconducting current as a function of the\nJosephson frequency is demonstrated in Fig. 9. We note\nthat in our normalization V=!J. We can see the man-\nifestation of the asymmetric resonance peak in the fre-\nquency dependence of superconducting current. So, the\napproximated system of equations 7 re\rects one of the\nmain feature of Du\u000eng oscillator.\nFigure (10) compares anomalous damping dependence\nof the resonance peak of mmax\ny(V) calculated numeri-\ncally according to the full LLGJ system of equations (6)\nwith calculated numerically according to the generalized\nDu\u000eng model (equations (17, 19)). We see that in the\ndamping parameter interval [0.001 { 0.2] the coincidence8\nFigure 10. The \u000b-dependence of the resonance maximum of\nmmax\ny(V) in the damping parameter interval [0.001 { 0.12].\nGreen squares show results calculated numerically according\nto the full system of equations (6), blue circles show results\ncalculated numerically according to the generalized Du\u000eng\nand Josephson equations (17,19). The dashed line connects\nthe symbols to guide eyes. Solid line show analytical \u000b-\ndependence calculated according to the Eq. (22). All calcu-\nlation have been done at \fc= 25, G=0.05, r=0.05, !F= 0:5.\nof the dependences is enough good.\nUsing equation (18) with '=!Jt, we can \fnd (see\nSupplementary materials ??) a relation between posi-\ntion of the resonance peak in mmax\ny(V) dependence and\ndamping\n!peak=s\n1\u00003\u000b2\n2+1\n2r\n(1\u0000\u000b2)2\u000012(Gr\n4\u000b)2(22)\nwhere!peak=!J;peak\n!Fdetermines the position of the res-\nonance peak.\nEquation (22) allows to \fnd the formula for critical\ndamping\u000bcritwhich is an important parameter deter-\nmining the reversal point in damping dependence of the\nresonance peak of mmax\ny(V) .\nTaking into account equation (22) we can write equa-\ntion with respect of Gr=(4\u000b) (See supplementary mate-\nrials??).\n9\u0012Gr\n4\u000bcrit\u00134\n+ 3\u000b2\ncrit(10\u000b2\ncrit\u00001)\u0012Gr\n4\u000bcrit\u00132\n(23)\n\u00002\u000b4\ncrit(\u000b2\ncrit\u00001)2= 0\nUsing approximation 10 \u000b2\ncrit<<1 and\u000b2\ncrit<<1 it\ngives (see Supplementary Materials)\n\u000bcrit\u00191\n2sr\n3\n2Gr (24)\nFigure 11. Numerical calculations according to Eq. (6)\n(squares), analytical according to Eq. (23)(solid line) and\napproximated analytical according to Eq. (24) (dashed line).\nTable 1: A comparison between the numerical and an-\nalytical values of \u000bcrit:at di\u000berent values of Gandr.\nG r Gr\u000bcrit:;numerics \u000bcrit:;analytics\n0.01 0.05 0.0005 0.0100 0.0123\n0.05 0.05 0.0025 0.0300 0.0276\n0.05 0.10 0.0050 0.0400 0.0391\n0.05 0.30 0.0150 0.0700 0.0677\n0.05 0.50 0.0250 0.0900 0.0874\n0.10 0.05 0.0050 0.0391 0.0391\n0.60 0.05 0.0300 0.0950 0.0958\n0.70 0.05 0.0350 0.1000 0.1035\n1.00 0.05 0.0500 0.1200 0.1237\nFigure 11 presents comparison of numerical and ana-\nlytical results \u000bcritversusGr.\nAs we see, it shows a good agreement of numerical\nand analytical results of calculations at small product of\nJosephson to magnetic energy ratio and spin-orbit inter-\naction.\nIV. IV. CONCLUSIONS\nThe understanding of the nonlinear features of\nmagnetization dynamics in superconductor-ferromagnet-\nsuperconductor Josephson junction and their manifesta-\ntion in the IV-characteristics has implications for super-\nconductor spintronics, and modern information technol-\nogy. In'0junctions the nonlinear features can a\u000bect the\ncontrol of magnetization precession by superconducting\ncurrent and external electromagnetic radiation [28].\nHere, using numerical and analytic approaches, we\nhave demonstrated that at small system parameters,9\nnamely, the damping, spin-orbit interaction and Joseph-\nson to magnetic energy ratio in '0junction, magnetic dy-\nnamics is reduced to the dynamics of the scalar Du\u000eng\noscillator, driven by the Josephson oscillations. We have\nclari\fed the role of increasing superconducting current\nin the resonance region leading to the foldover e\u000bect in\nthe ferromagnet magnetization. We have demonstrated\nthe parameter dependence of the anomalous ferromag-\nnetic resonant shifting with anomalous damping depen-\ndence due to nonlinearity of the full LLGJ equation and\nin its di\u000berent approximations. We have derived the an-\nalytical expression for critical damping value. Also, we\ndemonstrated appearance of negative di\u000berential resis-\ntance in the IV-characteristics and the correlation with\noccurrence of the foldover e\u000bect in the magnetization of\nferromagnet.\nWe have stressed that the manifestation of negative\ndi\u000berential resistance is related to the nonlinear features\nof the system[34, 35]. It was demonstrated that in the\nsmall model parameters case the equation for magnetic\nsubsystem takes form of Du\u000eng equation where nonlin-\nearity manifest itself as the cubic term. We have shown\nthat the appearance of negative di\u000berential resistance in\nthe I-V curve is related to the appearance of foldover inthemmax\ny-Vcurve.\nWe believe that the experimentally measured IV-\ncharacteristics of '0junction with manifestations dis-\ncussed in detail in the present paper, would allow close\ninvestigations of its nonlinear features important for su-\nperconductor electronics and spintronics.\nV. SUPPLEMENTARY\nIn supplementary material are presented the details of\ncalculations for Eq.22 and Eq.24.\nVI. FUNDING\nNumerical simulations were funded by Project No. 18-\n71-10095 of the Russian Science Foundation. The pre-\nsented results concerning the calculations of DC super-\nconducting current in the section V are supported by the\nRussian Science Foundation in the framework of project\n22-42-04408. A.J. and M.R.K. are grateful to IASBS for\n\fnancial support.\n[1] Buzdin, A. Physical Review Letters 2008 ,101 (10),\n107005.\n[2] Linder, J., Robinson, J. W. Nature Physics 2015 ,11(4),\n307-315.\n[3] Bobkova, I., Bobkov, A., Silaev, M. Physical Review B\n2018 ,98(1), 014521.\n[4] Bobkova, I., Bobkov, A., Rahmonov, I., Mazanik, A.;\nSengupta, K., Shukrinov, Y. M. Physical Review B 2020 ,\n102(13), 134505.\n[5] Szombati, D., Nadj-Perge, S., Car, D., Plissard, S.,\nBakkers, E., Kouwenhoven, L. Nature Physics 2016 ,12\n(6), 568{572.\n[6] Konschelle, F., Buzdin, A. Physical Review Letters 2009 ,\n102(1), 017001.\n[7] Shukrinov, Y. M., Rahmonov, I., Sengupta, K. Physical\nReview B 2019 ,99(22), 224513.\n[8] Shukrinov, Y. M. Physics-Uspekhi 2022 ,65(4), 317.\n[9] Linder, J., Yokoyama, T. Physical Review B 2011 ,83\n(1), 012501.\n[10] Ho\u000bman, S., Blanter, Y. M., Tserkovnyak, Y. Physical\nReview B 2012 ,86(5), 054427.\n[11] Eschrig, M. Reports on Progress in Physics 2015 ,78\n(10), 104501.\n[12] Shukrinov, Y. M., Rahmonov, I., Sengupta, K.,\nBuzdin, A. Applied Physics Letters 2017 ,110 (18),\n182407.\n[13] Mazanik, A., Rahmonov, I., Botha, A., Shukrinov, Y. M.\nPhysical Review Applied 2020 ,14(1), 014003.\n[14] Shukrinov, Y. M., Rahmonov, I., Janalizadeh, A., Ko-\nlahchi, M. Physical Review B 2021 ,104(22), 224511.\n[15] Shen, L., Xia, J., Zhang, X., Ezawa, M., Tretiakov, O. A.,\nLiu, X., Zhao, G., Zhou, Y. Physical review letters 2020 ,\n124(3), 037202.[16] Azovtsev, A. V., Nikitchenko, A. I., Pertsev, N. A. Phys-\nical Review Materials 2021 ,5(5), 054601.\n[17] Zhu, J., Zhang, T., Yang, Y., Huang, R. Applied Physics\nReviews 2020 ,7(1), 011312.\n[18] Wang, G., Chen, D., Lin, J., Chen, X. IEEE Transactions\non industrial electronics 1999 ,46(2), 440{444.\n[19] Almog, R., Zaitsev, S., Shtempluck, O., Buks, E. Physical\nReview Letters 2007 ,98(7), 078103.\n[20] Zhao, Y., Song, Q., Yang, S.-H., Su, T., Yuan, W.,\nParkin, S. S., Shi, J., Han, W. Scienti\fc reports 2016 ,6\n(1), 1{8.\n[21] Yao, Y., Song, Q., Takamura, Y., Cascales, J. P.,\nYuan, W., Ma, Y., Yun, Y., Xie, X., Moodera, J. S.,\nHan, W. Physical Review B 2018 ,97(22), 224414.\n[22] Silaev, M. Physical Review B 2020 ,102(14), 144521.\n[23] Nayfeh, A., Mook, D. Nonlinear Oscillations. John Wil-\nley & Sons , 1979.\n[24] Chen, W., De Loubens, G., Beaujour, J.-M., Sun, J.,\nKent, A. Applied Physics Letters 2009 ,95(17), 172513.\n[25] Guarcello, C., Bergeret, F. Physical Review Applied\n2020 ,13(3), 034012.\n[26] Landau, L., Lifshitz, E. Phys. Rev 1955 ,100, 1243.\n[27] Rabinovich, D. S., Bobkova, I. V., Bobkov, A. M.,\nSilaev, M. A. Phys. Rev. Lett. 2019 ,123, 207001. doi:\n10.1103/PhysRevLett.123.207001 .\n[28] Abdelmoneim, S., Shukrinov, Y. M., Kulikov, K., ElSam-\nman, H., Nashaat, M. Phys. Rev. B 2022 ,106, 014505.\n[29] Shukrinov, Y. M., Rahmonov, I. Physics of Particles and\nNuclei 2020 ,51(4), 816{822.\n[30] Shukrinov, Y. M., Rahmonov, I., Botha, A. Low Temper-\nature Physics 2020 ,46(9), 932{938.\n[31] Konschelle, F., Buzdin, A. Physical Review Letters 2019 ,\n123(16), 169901.10\n[32] Pedersen, N. F., Filatrella, G., Pierro, V., S\u001crensen, M. P.\nPhysica C: Superconductivity and its Applications 2014 ,\n503, 178{182.\n[33] Kadowaki, K., Yamaguchi, H., Kawamata, K., Ya-\nmamoto, T., Minami, H., Kakeya, I., Welp, U.,\nOzyuzer, L., Koshelev, A., Kurter, C., Gray, K.,\nKwok, W.-K. Physica C: Superconductivity and its ap-\nplications 2008 ,468(7-10), 634{639.[34] Filatrella, G., Pierro, V., Pedersen, N. F., Sorensen, M. P.\nIEEE Transactions on Applied Superconductivity 2014 ,\n24(6), 1{7.\n[35] Nagel, J., Speer, D., Gaber, T., Sterck, A., Eichhorn, R.,\nReimann, P., Ilin, K., Siegel, M., Koelle, D., Kleiner, R.\nPhysical Review Letters 2008 ,100, 217001."    },    {        "title": "1402.3459v1.Extremely_high_resolution_measurements_of_microwave_magnetisation_dynamics_in_magnetic_thin_films_and_nanostructures.pdf",        "content": " 1 \nExtremely high-resolution measurements of microwave  magnetisation dynamics in \nmagnetic thin films and nanostructures \n \nEugene N. Ivanov and Mikhail Kostylev \nSchool of Physics, University of Western Australia,  Perth, WA 6009 \n \nAbstract:  In this work we discuss the use of interferometric  measurement technique to study \nmicrowave magnetization dynamics on ferromagnetic n anostructures. We demonstrate that in \nthis way one can resolve features which are impossi ble to resolve with broadband \nferromagnetic resonance and travelling spin wave sp ectroscopy otherwise. \n \nIntroduction \nThe concept of interferometric measurements at micr owave frequencies was first \nsuggested in the 50’s, but it took more than 40 yea rs before its high potential was fully realized. \nIn the mid 90’s, the synergy of the microwave inter ferometry and low-noise amplification \nenabled the “real-time” noise measurements with spe ctral resolution approaching the Standard \nThermal Noise Limit (STNL) [2]. This development br ought about experimental evidence of \nintrinsic fluctuations in the microwave components,  which had been earlier considered to be \n“noise free”. The further progress in the resolutio n of noise measurements went well beyond \nthe STNL [3, 4]. This was achieved via the “power r ecycling” technique when studying the \nnoise phenomena in the low-loss test samples. \nThe principles of microwave interferometry were als o behind the breakthrough in the \nphase noise performance of microwave oscillators [5 , 6]. Currently, the low-phase noise \nmicrowave oscillators with interferometric signal p rocessing are a key element of the advanced \nDoppler radars. Apart from military applications, l ow-phase noise signal sources play an \nimportant role in a range of physical experiments, such as generation of entangled states \nbetween macroscopic objects and microwave photons.  \nThe use of interferometric measurement techniques p roved to be essential for \nunderstanding the origin of the excess phase noise associated with demodulation of ultra-short \noptical pulses produced by the mode-locked lasers [ 7]. Unravelling the “mystery” of the excess \nnoise paved the way for generation of spectrally pu re microwave signals from the optical \nsources [8, 9]. \nIn this work we show the advantages of the interfer ometric measurements over \nconventional FMR and travelling spin wave spectrosc opy when studying microwave \nmagnetization dynamics of ferromagnetic micro- and nanostructures. We demonstrate that in \nthis way one can resolve features which are impossi ble to resolve with broadband FMR and \ntravelling spin wave spectroscopy otherwise. \n \nMeasurement Technique \nThe interferometric instrument may be tuned such th at it is sensitive to variation either \nin the amplitude or in the phase of the transmissio n or reflection coefficient of the device under  2 \ntest (DUT). This means that both characteristics ca n be measured with the same experimental \nsetup and with the same accuracy. The interferometr ic measurements are characterized by \ngreatly enhanced sensitivity, as compared to the co nventional techniques relying on the use of a \nphase bridge. This is due to the ability of the int erferometric systems to reconcile two \nseemingly contradictory requirements of having a hi gh power incident on the DUT with a \nsmall-signal operation of its microwave readout. Th e enhanced sensitivity of the interferometric \nmeasurements also stems from their relative immunit y to both amplitude and phase fluctuations \nof the output of the microwave source. This is, how ever, only true, if the DUT is linear and \nnon-dispersive. For non-linear and dispersive excit ations such as FMR and travelling spin \nwaves, to minimise the fluctuations it is preferabl e to use a low-noise microwave generator as a \nsource of microwave power.  \nAn analytical expression for the smallest detectabl e rms fluctuations of the \nphase/amplitude of DUT transmission or reflection c oefficient is given by   \n   \n0\nmin min 2 ( ) B i \ninc DUT k T T \nPδϕ δα α+=∼  , \nwhere kB is the Boltzmann constant, Ti is an operational characteristic of the interferom etric \ninstrument having sense of some characteristic nois e temperature, To is the ambient \ntemperature, Pinc  is the signal power at the input of the DUT and αDUT  is the insertion loss of \nthe DUT. For our instrument Ti=50 K. This leads to almost thermal noise limited s pectral \nresolution. Fig. 1 compares Single Sideband (SSB) p hase noise floors of the interferometric and \nconventional (phase bridge) measurement systems. Th e measurements were conducted at 10 \nGHz with a 3 dB broadband attenuator acting as a DU T. Agilent 8257C microwave frequency \nsynthesizer served as a pump source. As follows fro m the data in Fig. 1, “switching” from the \nphase bridge to interferometric system improves the  resolution of spectral measurements by 40 \ndB at Fourier frequency of 100 Hz. \nIt should be pointed out, that the phase noise floo r of the interferometric measurement \nsystem exhibits the 1/ f-dependence at low Fourier frequencies. The influen ce of this technical \nnoise can be avoided by transferring the measuremen ts to some intermediate frequency at \nwhich spectral density of technical fluctuations fa lls below the fundamental thermal noise \nbackground. \nAccordingly, to take FMR measurements of thin metal lic films and nanostructures [10] \nor to carry out travelling spin wave spectroscopy o f magnetic nano- and microwaveguides [11] \nthe measurement setup is configured as shown in Fig . 2. The interferometric instrument is a \nsingle-frequency device; therefore the measurements  are taken applied-field resolved. This can \nbe repeated at a number of frequencies within the t uneability range of the instrument \n(approximately from 6 to 17 GHz for our instrument. ) A microwave generator operational in \nc.w. regime is connected to the “GEN” input of the instrument. DUT is inserted between the \nports “DUT”. As a DUT a broadband FMR transducer in  the form of a coplanar or a microstrip \nline with a sample on top can be used [12, 13], as well as a microwave cavity for cavity FMR  3 \nor a microscopic magnetic stripe waveguide for trav elling spin waves with microscopic \ncoplanar antennas for travelling spin wave spectros copy measurements [11]. \nThe low-frequency output (DC OUT) of the instrument  is connected to the input of a \ndigital lock-in amplifier. The measurement data are  collected from the digital output of the \nlock-in via GPIB. Low-frequency (220Hz and 20.2KHz respectively for our microstrip and \ncavity FMR setups) modulation of the applied field is utilised, and the lock-in is locked to the \nmodulation frequency. To this end an additional mod ulation coil is fitted between the poles of \nthe electromagnet. Alternatively, amplitude modulat ion of the output of the microwave \ngenerator may be used (which may be useful for char acterisation of spin-torque nano-\noscillators). However, our measurements show that t he latter method is significantly less \nsensitive. Furthermore, the background (off-resonan ce) signal of non-magnetic nature is usually \nsignificant in this case, whereas it is naturally v anishing in the case of the field modulation.  \nTwo examples of the measurements taken with the ins trument are shown in Figs. 3 and \n4. In both cases the instrument was tuned such that  its sensitivity to variation in the amplitude \nof the signal from the output of DUT is maximised 1. In these conditions the instrument is \npractically insensitive to the variation in the sig nal phase. \nFig. 3 demonstrates a broadband-FMR absorption trac e taken with a 1.5mm-wide \nmicrostrip line on an array of parallel nanostripes  made from permalloy (Ni 80 Fe 20 ). The array \nwas fabricated by A.O.Adeyeye’s group at the Nation al University of Singapore [10]. The \nmacrosize of the array is 4x4 mm. The stripes are 3 00nm-wide, 30nm thick and 4mm long. The \nmagnetic field is applied along the stripes. The gr aph demonstrates a number of higher-order \nstanding spin waves across the stripe width. The am plitude of the outmost left-hand (negative) \npeak is just 20nV. From the inset one sees that no noise at all is seen in this low-applied field \nrange. Note that these are original raw data; no gr aphical smoothing was used to post-process \nthe registered trace.  \nThe field modulation frequency and the time of stab ilisation of a set magnetic field of \nthe electromagnet determines the time for a measure ment run. In our broadband FMR setup the \nmodulation coil sits on a pole piece of the electro magnet. Because of large inductance of a coil \nplaced on a pole piece we had to keep the frequency  of field modulation low: 220Hz.  \nTherefore the time constant of the lock-in was set to a relatively large value of 0.3 sec. \nAccordingly, the time of signal accumulation by the  lock-in was relatively large: 0.3s x 6 = \n1.8s. This resulted in 7 minutes for completing the  whole trace (200 points) 2.  \n                                                 \n1 It takes 3-5 minutes to tune the instrument to a p articular frequency. \n2 Note that if the coil is located well away from th e pole pieces, the modulation frequency can \nbe set much higher. Accordingly, the time for takin g one measurement point can be set \npractically equal to the time of stabilisation of t he set field of the electromagnet. This \nconfiguration is realised in our cavity FMR setup, where the coil is fixed on the wall of the \ncavity. In addition to the significant decrease in the measurement time, the increase in the \nmodulation frequency further decreases the 1/ f noise.  4 \nThe second example (Fig. 4) displays the results of  our measurements of higher-order \nmodes of a microscopic stripe waveguide of travelli ng spin waves 3. The Ni 80 Fe 20  stripe has \ncross-section 2 micron x 100nm. The spin waves are excited and received by microscopic \ncoplanar antennas with the total widths of 6 micron . Microscopic coplanar probes - \n“Picoprobes” from GGB Industries - are used to conn ect the antennas to the DUT ports of the \ninstrument. Other details of the experiment can be found in Ref.[11].  \nThe figure shows the record number of travelling sp in wave modes of the waveguide \ntaken with a fully-microwave method. In reflection from the input antenna seven modes are \nseen. With our simulation [14] they are identified as the lowest-order odd modes (from n=1 \n(fundamental) to n=13). In the signal from the output port of the spi n wave device (“transmitted \nsignal”) the 11 th  mode is easily noticeable, although the distance b etween the antennas is quite \nlarge: 12 micron. \n \nConclusion  \n We have demonstrated the possibility of extremely low-noise noise measurements of \nmicrowave magnetisation dynamics of magnetic nanost ructures. This was achieved by using \nthe principles of microwave circuit interferometry.   \n \nReferences:  \n1. A. Whitwell and N. Williams, \"A new microwave technique for determining noise sp ectra at \nfrequencies close to the carrier\" , Microwave J., 1959, pp. 27-32. \n2. E. Ivanov, M. Tobar and R. Woode, “Microwave interferometry: application to precision  \nmeasurements and noise reduction techniques” , IEEE Trans. on UFFC, 45 , no. 6, 1998, pp. \n1526-1537 \n3. E. Ivanov and M. Tobar, “Real time noise measurement system with sensitivit y exceeding the \nstandard thermal noise limit,”  IEEE Trans. On UFFC, 49 , no. 8, 2002, pp. 1160-1161. \n4. E. Ivanov and M. Tobar, ”Microwave phase detection at the level of 10  -11  rad”,  Rev. Sci. \nInstrum.  80 , 044701 (2009). \n5. E. Ivanov, M. Tobar and R. Woode, “Applications of interferometric signal processing to \nphase noise reduction in microwave oscillators”,  IEEE Trans. on MTT, 46 , no.10, 1998, \npp.1537-1545. \n6. E. Ivanov and M. Tobar, “Low Phase Noise Sapphire Crystal Microwave Oscilla tors: \nCurrent Status”,  IEEE Trans. on UFFC, 56 ,  no.2 , 2009, pp.263-269. \n7. L.-S. Ma, Z. Bi, Albrecht Bartels, L. Robertsson , M. Zucco, R. S. Windeler, G. Wilpers, C. \nOates, L. Hollberg, S. A. Diddams, “Optical Frequency Synthesis and Comparison with \nUncertainty at the 10 -19  Level” , SCIENCE, v. 303, pp. 1843-1845, 2004 \n8. J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wil pers, C. W. Oates, S. A. Diddams, L. \nHollberg, “Low-noise synthesis of microwave signals from an op tical source ”, Electronics \nLetters, vol. 41,  no. 11,  pp. 650-651, 2005. \n9. T. Fortier, M. Kirchner, F. Quinlan, J. Taylor, J. Bergquist, T. Rosenband, N. Lemke, A. \nLudlow, Y. Jiang, C. Oates and S. Diddams, “Generation of ultra-stable microwaves via \noptical frequency division” , Nature Photonics, v. 5, no. 7, pp. 425–429 (2011) . \n                                                 \n3 These measurements were taken by C. Chang .  5 \n10. G. Gubbiotti, S. Tacchi, M. Madami, G. Carlotti , A. O. Adeyeye and M. Kostylev, \n“Brillouin light scattering studies of planar metal lic magnonic crystals”, J. Phys. D: Appl. Phys. \n43, 264003 (2010). \n11. C. S. Chang, M. Kostylev, E. Ivanov, J. Ding,  and A. O. Adeyeye, “The phase \naccumulation and antenna near-field of microscopic propagating spin wave devices”, Appl. \nPhys. Lett., vol. 104 (2014) (in print). \n12. J.M. Shaw, H.T.Nembach, T.J. Silva, C.T. Boone,  “Precise determination of the \nspectroscopic g-factor by use of broadband ferromag netic resonance spectroscopy”, J. Appl. \nPhys., vol. 114, 243906 (2013). \n13. K. J. Kennewell, M. Kostylev, M. Ali, A. A. Sta shkevich, R. Magaraggia, D. Greig, B. J. \nHickey, and R. L. Stamps, “Microwave dynamic pinnin g at a Py/Co interface measured using \ninductive magnetometr y”, J. Appl. Phys. vol. 108, 073917 (2010). \n14. M. Kostylev, P. Schrader, R. L. Stamps, G. Gubb iotti, G. Carlotti, A. O. Adeyeye, S. Goolaup, N. \nSingh, “Partial frequency band gap in one-dimension al magnonic crystals”, Appl. Phys. Lett. vol. 92, \n132504 (2008).  6 \n \n \n \n \nFig. 1. SSB phase noise floors of a microwave inter ferometer and \nof an exemplary microwave phase bridge.  7 \n \n \n \n \nFig. 2. Diagram of the setup for measurements of mi crowave magnetisation \ndynamics. 1. Interferometric Instrument. 2. Device under test (DUT). 3. Microwave \ngenerator. 4. Modulation coil. 5. Lock-in amplifier . 6. Function generator to provide \nthe modulation coil with an ac current. 7. Pole pie ces of the electromagnet. “Sync” \ndenotes synchronisation signal to which the lock-in  is locked.  8 \n \n \n \nFig. 3. Broadband ferromagnetic resonance trace for  an array of \npermalloy nanostripes. Frequency is 15 GHz. Red sol id line (left-hand \naxis): measurement taken with standard sensitivity of broadband FMR. \nBlue dashed line (right-hand axis): measurement tak en with increased \nsensitivity of the lock-in amplifier to resolve the  signal of the low-\namplitude higher-order modes. Inset: zoom-in of the  lower applied field \nrange of the trace. The amplitude of the left-hand peak in the inset is \n100nv (200nV peak-to-peak).  9 \n \n \n \nFig. 4. Travelling spin waves on a Permaloy nanostr ipe with cross-\nsection 100nm x 2 micron. Distance between the inpu t and the output \nantennas is 12 micron. The field is applied in the stripe plane along \nthe 2 micron size. Frequency is 10GHz. Dashed line:  signal reflected \nfrom the input antenna. Solid line: signal from the  output antenna \n(transmitted signal). Vertical solid lines: theoret ical positions of the \nguided width modes for the nanostripe given here in  order to identify \nthe respective signals. "    },    {        "title": "1301.6544v1.Spin_filter_effect_at_room_temperature_in_GaN_GaMnN_ferromagnetic_resonant_tunneling_diode.pdf",        "content": "arXiv:1301.6544v1  [cond-mat.mes-hall]  28 Jan 2013Spin filter effect at room temperature in GaN/GaMnN ferromagn etic resonant\ntunnelling diode.\nP. W´ ojcik,∗J. Adamowski, M. Wo/suppress loszyn, and B.J. Spisak\nUniversity of Science and Technology, Faculty of Physics an d Applied Computer Science, Krak´ ow, Poland\nWe have investigated the spin current polarization without the external magnetic field in the\nresonant tunneling diode with the emitter and quantum well l ayers made from the ferromagnetic\nGaMnN. For this purpose we have applied the self-consistent Wigner-Poisson method and studied\nthe spin-polarizing effect of the parallel and antiparallel alignment of the magnetization in the\nferromagnetic layers. The results of our calculations show that the antiparallel magnetization is\nmuch more advantageous for the spin filter operation and lead s to the full spin current polarization\nat low temperatures and 35 % spin polarization of the current at room temperature.\nThe progress in homo- and heteroepitaxy of dilute\nmagnetic semiconductors1–8(DMS’s) during the past\ndecade allows to fabricate spintronic nanodevices, in\nwhich the spin polarization of the current can be con-\ntrolled by the magnetic or electric field. The spin filter\neffect in a resonant tunneling diode (RTD) with para-\nmagnetic quantum well embedded in II-VI DMS (Zn-\nMnSe) was studied theoretically9–11and experimentally\ndemonstrated by Slobodskyy et al.12For the paramag-\nnetic RTD, in the presence of the external magnetic field,\nthe exchange interaction between the conduction band\nelectrons and the Mn2+ions leads the giant Zeeman\nsplitting13of the quasi–bound state in the paramagnetic\nquantum-well. This splitting causes that the resonance\nconditions for the spin up and spin down electrons are\nsatisfied for different bias voltages leading to the separa-\ntion of the spin current components and consequently to\nthe spin polarization of the current. The spin filter effect\nin the paramagnetic RTD is limited to very low temper-\nature and requires the high external magnetic field.12,13\nThese restrictions cause that more interest is directed\ntowards the application of the ferromagnetic III-V semi-\nconductors, especially those with high Curie tempera-\nture, e.g. GaMnAs8,14–16or GaMnN.17,18Ohno et al.19\nexperimentally studied the ferromagnetic RTD based on\nGaMnAs in which the spin splitting was observed with-\nout external magnetic field but it still requires low tem-\nperature. Hovewer, the recent experiments reported that\nGaMnN can exhibit the ferromagnetic properties above\nroom temperature20–22at which the exchange splitting\nof the conduction band is about few tens of meV23and\nremains in the limit of thin layer of a few nanometer\nwidth.24Although the ferromagnetism in GaMnN is still\nunresolved theoretical problem, the spin filter effect in\nthe RTD’s based on GaMnN is a subject of research car-\nried out by many groups. Recently, Li et al.25have the-\noretically investigated the ferromagnetic RTD consisted\nof the InGaN quantum well between two GaMnN barri-\ners. In Ref. 25 the spin polarization of the current with-\nout magnetic field has been predicted at low temperature\nbut at room temperature it has been reduced to only 8 %.\nAnother way to obtain the spin polarization of the cur-\nrent at room temperature was proposed by Qui el at.26\nwho stated that the δdoping of the GaN quantum wellin the RTD with ferromagnetic (GaMnN) contacts en-\nhances the spin polarization of the current by two times.\nThe double enhancement of the spin polarization at room\ntemperature was also reported by Wang et al.27who in-\nvestigated the influence of the charge polarization at the\ninterface AlGaN/GaN in RTD with ferromagnetic con-\ntacts embedded in GaMnN.\nIn the present paper, we propose the RTD structure\nwith the ferromagnetic emitter and quantum well regions\nmade from GaMnN. Based on self-consistent Wigner-\nPoisson calculations we predict the full spin polarization\nof the current (i.e. P=±1) at low temperature for an-\ntiparallel magnetization of the magnetic layers. The spin\npolarization reduces to P=±0.35 at room temperature.\nTo the best of our knowledge, this is the highest value\nof the spin polarization predicted at room temperature\nin magnetic RTD. In particular it is about four times\nhigher than that reported by Li et al.25We also showe\nthat the proposed ferromagnetic RTD structure with the\nantiparallel alignment of magnetization can lead to the\nfairly large spin polarization of the current at room tem-\nperature.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s32/s115/s112/s105/s110/s32/s117/s112\n/s32/s115/s112/s105/s110/s32/s100/s111/s119/s110\n/s32/s32/s32\n/s32/s122/s32/s40/s110/s109/s41/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41\n/s32/s122/s32/s40/s110/s109/s41/s65/s108\n/s48/s46/s49/s71/s97\n/s48/s46/s57/s78/s71/s97/s77/s110/s78/s97/s41/s32/s112/s97/s114/s97/s108/s108/s101/s108/s32/s40/s80/s41/s32\n/s32/s69/s109/s105/s116/s101/s114\n/s71/s97/s77/s110/s78/s67/s111/s108/s101/s99/s116/s111/s114\n/s32/s32/s71/s97/s78/s32\n/s32\n/s32/s67/s111/s108/s101/s99/s116/s111/s114\n/s32/s32/s71/s97/s78/s32/s69/s109/s105/s116/s101/s114\n/s71/s97/s77/s110/s78\n/s71/s97/s77/s110/s78/s98/s41/s32/s97/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108/s32/s40/s65/s80/s41/s32\n/s65/s108\n/s48/s46/s49/s71/s97\n/s48/s46/s57/s78\nFIG. 1. Self-consistent potential energy profile for spin up\nand spin down electrons calculated for a) parallel (P) and\nb) antiparallel (AP) alignment of the magnetization of the\nemitter and quantum well layers.\nWe investigate the ferromagnetic RTD based on\nGaN/Al 1−xGaxN/GaMnN with the emitter and quan-\ntum well layer made from GaMnN (Fig. 1). The parallel2\n(P) and antiparallel (AP) alignments of the magnetiza-\ntion of the emitter and quantum well is considered. The\nconduction band profiles at zero bias for spin up and spin\ndown electrons are presented in Fig. 1. The calculations\nhave been performed with the following nanostructure\nparameters: the thickness of the GaMnN quantum well\nlayer is 6 nm, the barriers are assumed to be symmetric\nwith thickness 3 nm and x= 0.1 yields the barrier height\n130 meV.28We assume the conduction-band electrons of\nGaN, i.e., m/m 0= 0.228 and take on the relative electric\npermittivity ǫ= 8.6. In order to describe the pure spin\nfilter effect in the proposed ferromagnetic nanostructure\nwe neglect the charge polarization occurring at the inter-\nface AlGaN/GaN.28\nOur numerical calculations are based on the Wigner-\nPoisson approach, according to which the conduction\nband electrons are described by the spin dependent\nWigner distribution function (WDF). Assuming the\ntranslational invariance in the x−yplane and neglecting\nthe spin scattering (many transport experiments showed\nthat the spin scattering lenght is compared to the size\nof the RTD19,29) the time independent quantum trans-\nport equations can be reduced to the following one-\ndimensional form:30\n¯hk\nm∂ρW\nσ(z,k)\n∂z=i\n2π¯h+∞/integraldisplay\n−∞dk′Uσ(z,k−k′)ρW\nσ(z,k′),(1)\nwhereρW\nσ(z,k) is the spin-dependent WDF, kis thez-\ncomponent of the wave vector and σ= (↑,↓) is the spin\nindex.\nThe non-local potential Uσ(z,k−k′) in Eq. (1) is given\nby the formula\nUσ(z,k−k′) =+∞/integraldisplay\n−∞dz′/bracketleftbig\nUσ(z+z′\n2)−Uσ(z−z′\n2)/bracketrightbig\ne−i(k−k′)z′,\n(2)\nwhereUσ(z) is the spin-dependent potential energy pro-\nfile, which can be expressed as the sum of the three terms\nUσ(z) =U0\nσ(z)+Uel(z)+Uex\nσ(z), where consecutive terms\ndenote the spin-dependent conduction-band potential en-\nergy, the electrostatic potential energy calculated by solv-\ning the Poisson equation and the exchange energy.\nEquations (1) and the Poisson equation form the system\nof non-linear integro-differential equations that is solved\nby the self-consistent procedure.9After reaching conver-\ngence the spin dependent current density is calculated\nusing the formula\njσ=e\n2πLL/integraldisplay\n0dz+∞/integraldisplay\n−∞dk¯hk\nmρW\nσ(z,k), (3)\nwhereLis the length of the nanodevice.\nThe ferromagnetic properties of GaMnN causes that\nthe spin-degenerate quasi-bound state energy level in the\nquantum well splits into two levels for spin up and spin/s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48 /s48/s46/s49/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s48/s46/s48/s51 /s48/s46/s48/s54 /s48/s46/s48/s57 /s48/s46/s49/s50 /s48/s46/s49/s53/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s32 /s69/s61/s54/s32/s109/s101/s86/s44/s32/s115/s112/s105/s110/s32/s117/s112\n/s32 /s69/s61/s54/s32/s109/s101/s86/s44/s32/s115/s112/s105/s110/s32/s100/s111/s119/s110\n/s32 /s69/s61/s49/s48/s32/s109/s101/s86/s44/s32/s115/s112/s105/s110/s32/s117/s112\n/s32 /s69/s61/s49/s48/s32/s109/s101/s86/s44/s32/s115/s112/s105/s110/s32/s100/s111/s119/s110\n/s32 /s69/s61/s49/s52/s32/s109/s101/s86/s44/s32/s115/s112/s105/s110/s32/s117/s112\n/s32 /s69/s61/s49/s52/s109/s101/s86/s44/s32/s115/s112/s105/s110/s32/s100/s111/s119/s110/s99/s117/s114/s114/s101/s110/s116/s32/s100/s101/s110/s115/s105/s116/s121/s32/s40/s49/s48/s53\n/s32/s65/s99/s109/s45/s51\n/s41\n/s86\n/s98/s32/s40/s86/s41/s40/s98/s41/s32/s97/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108/s86\n/s98/s32/s40/s86/s41\n/s32/s32/s99/s117/s114/s114/s101/s110/s116/s32/s100/s101/s110/s115/s105/s116/s121/s32/s40/s49/s48/s53\n/s32/s65/s99/s109/s45/s51\n/s41/s40/s97/s41/s32/s112/s97/s114/s97/s108/s108/s101/s108\n/s32/s32\n/s32\nFIG. 2. Current-voltage characteristics for spin up (red) a nd\nspin down (blue) current components calculated for differen t\nvalues of the splitting energy ∆ Eand a) parallel, b) antipar-\nallel alignment of the magnetization of the emitter and the\nquantum well layers.\ndown electrons. Similarly, the conduction band in the\nferromagnetic emitter layer is splited into two subbands\nfor different spins. The spin splitting of the conduction\nbands in the ferromagnetic layers causes that the reso-\nnance transport conditions are different for the electrons\nwith different spins. In the present calculations the spin\nsplitting energy ∆ Eof of the conduction band is treated\nas an parameter of calculations that varies from 2 meV to\n15 meV (the reliable values reported in experiments17,18).\nFig. 2 shows the spin-dependent current-voltage charac-\nteristics calculated at temperature T= 4.2 K for (a) par-\nallel and (b) antiparallel alignment of the magnetization\nof the emitter and the quantum well layers. We see that\nfor the parallel magnetization of the layers the resonant\ncurrent peaks for spin up and spin down current compo-\nnent occur at almost the same bias Vb. If the splitting\nenergy ∆ Eincreases, the resonant current peak increases\nfor spin up and decreases for spin down current compo-\nnent. On the other hand for the antiparallel magnetiza-\ntion of the magnetic layers the resonant current peaks for\nboth the spin components behave in a different manner.\nNamely, the increasing ∆ Ecauses the separation of the\nresonant current peaks: the resonant peak for the spin\nup current component shifts towards the lower bias while\nthe resonant peak for the spin down current component3\n/s48/s46/s48/s52 /s48/s46/s48/s53 /s48/s46/s48/s54 /s48/s46/s48/s55 /s48/s46/s48/s56/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110\n/s32/s32\n/s40/s98/s41/s32/s97/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108\n/s86\n/s98/s40/s86/s41/s32 /s32/s69/s61/s54/s32/s109/s101/s86\n/s32 /s32/s69/s61/s49/s48/s32/s109/s101/s86\n/s32 /s32/s69/s61/s49/s52/s32/s109/s101/s86/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110\n/s86\n/s98/s40/s86/s41/s40/s97/s41/s32/s112/s97/s114/s97/s108/s108/s101/s108\n/s32\n/s32\n/s32\nFIG. 3. Spin polarization of current Pas a function of bias\nVbfor different value of the splitting energy ∆ Eand a) paral-\nlel and b) antiparallel alignment of the magnetization of th e\nemitter and the quantum well layers.\nshifts towards the higher bias. The separation of the spin\ncurrent components leads to the spin polarization of the\ncurrent defined as P= (j↑−j↓)/(j↑+j↓). In Fig. 3 we\npresent the spin polarization of the current as a function\nof the bias calculated for (a) parallel and (b) antiparallel\nalignment of the magnetization of the ferromagnetic lay-\ners. We see that for the parallel magnetization the spin\npolarization of the current is positive at the low bias and\ndecreases with increasing the bias. On the other hand for\nthe antiparallel magnetization of the ferromagnetic layers\nthe spin polarization of the current varies from P= +1\nfor the low bias to P=−1 for the high bias. This de-\npendence is observed for all values of the splitting energy\n∆E, however, for the higher ∆ Ethe transition between\nboth the fully polarized states occurs in a narrower bias\nrange.\nIn order to explain strongly polarizing effect of the fer-\nromagnetic RTD with the antiparallel magnetization of\nthe ferromagnetic layers we present the simple model of\nthe spin dependent electron transport through the RTD\nfor both alignments of the magnetization (Fig. 4). Fig. 4\nshows that only the antiparallel alignment of the magne-\ntization can lead to the full spin polarization of the cur-\nrent. Let us note that at room temparature the trans-\nport window in the magnetic emitter broadens in the\nnearest of the Fermi energy. This thermal effect causes\nFIG. 4. Schematic illustration of spin dependent resonance\ntunneling of electrons in ferromagnetic RTD. Only the an-\ntiparallel magnetization of the ferromagnetic layers lead s to\nthe full spin polarization of the current. E and C denotes\nemitter and colector region. The bias increases from top to\nbottom panels.\nthe broadenning of the resonant current peak for spin up\nand spin down current components (inset in Fig. 5). Our\ncalculations show that even for small but experimental\nrealiable value of the splitting energy ∆ E= 10 meV the\nspin polarization at room temperature is still quite large\nand achieves P= 0.35 (Fig. 5). This value is four times\nlarger than that reported in Ref. 25. Moreover we expect\n/s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48 /s48/s46/s49/s50 /s48/s46/s49/s52/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s84/s61/s51/s48/s48/s32/s75/s44/s32 /s32/s69/s61/s49/s48/s32/s109/s101/s86\n/s32/s32/s112/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110\n/s86\n/s98/s32/s40/s86/s41/s32/s115/s112/s105/s110/s32/s117/s112\n/s32/s115/s112/s105/s110/s32/s100/s111/s119/s110\n/s32/s32/s106/s32/s40/s65/s99/s109/s45/s50\n/s41\n/s32/s86\n/s98/s32/s40/s86/s41/s32\nFIG. 5. Spin polarization of current as a function of bias for\nantiparallel alignment of the magnetization between the em it-\nter and the quantum well at room temperature T= 300 K. In-\nset: current-voltage characteristics for spin up and spin d own\ncurrent components at room temperature.\nthat for the larger splitting energy ∆ E, the antiparallel\nmagnetization of the magnetic layers can lead to the full\nspin polarisation of the current at room temperature.\nIn conclusion, we have shown that the antiparallel4\nalignment of the magnetization in the ferromagnetic res-\nonant tunneling structure with the ferromagnetic emitter\nand quantum well can be used to obtain the full spin po-\nlarization of the current at room temperature. Our theo-\nretical calculations predicts that the spin polarization of\nthe current in the ferromagnetic RTD based on GaMnN\nachieves P= 0.35 at room temperature for experimen-\ntally reported splitting energy ∆ E= 10 meV in GaMnN.\nWe also argue that proposed nanostructure can allow toincrease the polarization up to |P|= 1 at room tem-\nperature for sufficiently large splitting energy ∆ E. 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Ser. 193, 012130 (2009)"    },    {        "title": "1806.09356v1.Wideband_and_on_chip_excitation_for_dynamical_spin_injection_into_graphene.pdf",        "content": "Wideband and on-chip excitation for dynamical spin injection into graphene\nDavid I. Indolese,1,\u0003Simon Zihlmann,1,\u0003P\u0013 eter Makk,1, 2,y\nChristian J unger,1Kishan Thodkar,1and Christian Sch onenberger1\n1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland\n2Department of Physics, Budapest University of Technology and Economics and Nanoelectronics 'Momentum'\nResearch Group of the Hungarian Academy of Sciences, Budafoki ut 8, 1111 Budapest, Hungary\n(Dated: June 26, 2018)\nGraphene is an ideal material for spin transport as very long spin relaxation times and lengths can\nbe achieved even at room temperature. However, electrical spin injection is challenging due to the\nconductivity mismatch problem. Spin pumping driven by ferromagnetic resonance is a neat way to\ncircumvent this problem as it produces a pure spin current in the absence of a charge current. Here,\nwe show spin pumping into single layer graphene in micron scale devices. A broadband on-chip\nRF current line is used to bring micron scale permalloy (Ni 80Fe20) pads to ferromagnetic resonance\nwith a magnetic \feld tunable resonance condition. At resonance, a spin current is emitted into\ngraphene, which is detected by the inverse spin hall voltage in a close-by platinum electrode. Clear\nspin current signals are detected down to a power of a few milliwatts over a frequency range of\n2 GHz to 8 GHz. This compact device scheme paves the way for more complex device structures\nand allows the investigation of novel materials.\nI. INTRODUCTION\nGraphene has proven to be an excellent material for\nspintronic applications1with spin relaxation times on the\norder of 10 ns2and spin relaxation lengths of 24 µm3at\nroom temperature. However, the conductivity mismatch\nproblem4poses severe challenges for e\u000ecient electrical\nspin injection into graphene as it requires a tunnel barrier\nbetween the ferromagnetic contact and graphene. Oxide\ntunnel barriers are the key ingredient for achieving large\nspin signals in magnetic tunnel junctions5,6. However,\nhigh quality oxide tunnel barriers are hard to grow on 2D\nmaterials (e.g. graphene)7,8. Insulating or semiconduct-\ning 2D materials have been investigated as possible tun-\nnel barriers9,10and hexagonal boron nitride has proven\nto be particularly useful11{16.\nSpin pumping driven by ferromagnetic resonance\n(FMR) is a another way to circumvent the conductiv-\nity mismatch as it produces a pure spin current in the\nabsence of a charge current17{19. The emission of a pure\nspin current goes along with an enhanced damping of the\nFMR17.\nAn enhanced damping of the FMR has been observed\nin metallic structures20,21as well as in graphene based\ndevices22. Compared to metallic structures, graphene\nhas the advantage that its properties are gate tunable.\nIt has been shown theoretically that this is also the case\nfor the spin mixing conductance23,24, which describes the\nspin pumping e\u000eciency.\nEven though \frst hints on spin pumping into graphene\nhave been observed, the detection of a spin current in\ngraphene was still missing until recently. Tang et al.25\nshowed spin pumping and the detection of a spin current\nby the inverse spin Hall e\u000bect in palladium25. However,\nthese experiments with macroscopic samples were per-\nformed in a RF cavity and therefore at a \fxed frequency\nand at high power levels (on the order of 100 mW).\nIn this letter, we show that the implementation of anon-chip RF current line to locally excite micron scale\npermalloy (Py, Ni 80Fe20) pads comes with the advantage\nof a compact sample design, which in addition allows for:\n1) broad frequency range of operation, 2) low RF power\nlevels. Additionally, we detect spin currents using the\ninverse spin Hall e\u000bect in a platinum electrode.\nII. WORKING PRINCIPLE\nA schematic drawing of the investigated samples is\nshown in Fig. 1 (a). On the left a spin current is dynam-\nically injected into the graphene spin transport channel,\nwhereas on the right (at a distance L) the spin current\nis converted into a charge current using a platinum elec-\ntrode employing the inverse spin-Hall e\u000bect (ISHE)26{28.\nAn external magnetic \feld H (red arrow along the y-\ndirection) de\fnes the equilibrium magnetization and a\nRF magnetic \feld h rf(black double arrow along the x-\ndirection) is used to resonantly drive the magnetization\nM(t) (black arrow) leading to spin pumping with the in-\njection of a spin current into graphene. The DC com-\nponent of the spin current js, with a spin orientation \u001b\nparallel to the external magnetic \feld H, is converted into\na charge current jcin a platinum electrode as a result of\nthe inverse spin Hall e\u000bect26{28.\nAll samples presented here were fabricated on intrinsic,\nhigh resistive silicon wafers (with 170 nm of SiO 2on top)\nto reduce RF losses. In a \frst step CVD graphene29was\ntransferred by a conventional wet transfer method using\nPMMA as a supporting layer and ammonium persulfate\nas the copper etchant. The graphene sheet was then pat-\nterned into an array of rectangles with width of 8 µm and\nlength of 12 µm by e-beam lithography and reactive ion\netching. Next, the platinum electrodes were deposited\neither by sputter deposition or by thermal evaporation.\nThe thickness of the Pt electrodes was kept at a maxi-\nmum ofdPt\u001910 nm to avoid shunting e\u000bects that occurarXiv:1806.09356v1  [cond-mat.mes-hall]  25 Jun 20182\nfor Pt electrodes much thicker than the spin relaxation\nlength within Pt ( \u0015Pt). Seven devices were connected in\nseries employing a meander structure of the Pt electrode\nto increase the total signal, see Fig.1 (b). For a clean\nfabrication of the ferromagnetic permalloy structures, a\nfabrication procedure based on ZEP resist was used30.\nPy pads of 8 µm\u00028µm were patterned on top of the\ngraphene, see also Fig. 1 (b) for a false-colour electron mi-\ncrograph of a sample at this fabrication stage. A thin Py\nlayer together with the negligible crystal anisotropy of Py\nfavours a homogeneous in-plane magnetization aligned\nwith the external magnetic \feld H31,32. The length of\nthe spin transport channel Lis de\fned by the Py and\nthe Pt electrode separation.\n(c) (d)Uxz y\ncurrent line\nSi/SiO2\nPy Pt graphene MgOxz\nyM(a)\n10 µm(b)\n10 µmIRF\nIRF\nH js\njcσM(t) hRFxyz\nU\nL\nFIG. 1. Device schematics: (a) shows a schematic draw-\ning of the device with the ferromagnetic Py pad on the left\n(brown), the graphene spin transport channel in the middle\n(grey) and the Pt strip on the right (blue) that is used to con-\nvert the spin current into a voltage using the inverse spin-Hall\ne\u000bect. (b)shows a false-colour scanning electron micrograph\nof seven devices. The respective Pt detection strips are con-\nnected in series. On top of the devices a RF current line is\nfabricated to drive the FMR of the Py pads. This is seen in a\nschematic cross cross section in (c)and as a top view in (d).\nA layer of MgO (75 nm thick) was used to insulate the\ndevice from the RF current line, which was fabricated di-\nrectly on top of the sample to maximize the amplitude of\nhRFdriving magnetic \feld, see Fig. 1 (c) for a cross sec-\ntion and Fig. 1 (d) for a false-colour electron micrograph.\nThe RF current line consists of 5 nm Ti and 100 nm of\nCu with 45 nm of Au on top that prevents the copper\nfrom oxidation.Here, a total of three devices are discussed. Two of\nwhich contain a graphene spin transport channel (sample\nA and B) whereas device C serves as a reference device\nwithout graphene.\nIII. RESULTS\nA vector network analyser was used to detect the FMR\nof the Py pads. This is achieved by sourcing a RF sig-\nnal that is sent into a 50 \n transmission line on a circuit\nboard that is terminated by the RF current line on the\nsample chip, which acts as a short. At the same time, the\nre\rection coe\u000ecient S 11is used to measure the resonance\ncondition. The voltage Uat the Pt electrode was mea-\nsured with a lock-in technique employing magnetic \feld\nmodulation ( \u00160dH\u00182:75 mT) at a frequency of 377 Hz.\nThis technique has the advantage that it is more sensitive\ncompared to DC measurements and it is not a\u000bected by\nthermal voltages that can drift over the long time scales\nof the measurements.\nA. Ferromagnetic resonance\nThe magnetization dynamics of a classical macro spin\n~Min an e\u000bective magnetic \feld ~Heffis described by the\nLandau-Lifshitz-Gilbert equation33,34\nd~M\ndt=\r~Heff\u0002~M+\u000b~M\u0002d~M\ndt; (1)\nwhere\r=g\u0016B=~is the gyromagnetic ratio with gthe\nLand\u0013 e g-factor and \u0016B=e~=(2me) the Bohr magneton,\nwhere ~is the reduced Planck constant, and \u000bthe Gilbert\ndamping constant. In the case of a thin \flm with the\nexternal magnetic \feld applied in-plane ( Heff\u0018Hext),\nthe resonance condition is given by the Kittel formula35:\nfres=\r\u00160\n2\u0019p\nHext(Hext+Ms); (2)\nwhere\u00160is the vacuum permeability and Msis the sat-\nuration magnetization.\nFig. 2 (a) shows S 11as a function of external mag-\nnetic \feld and frequency measured on sample A. In or-\nder to eliminate the standing wave background due to\nre\rections at each RF connector, a frequency dependent\nbackground was subtracted (100 mT trace). The remain-\ning vertical lines originate from a weak magnetic depen-\ndence of the standing wave background. At the FMR\ncondition RF power is absorbed by the precessing mag-\nnetization. This is seen in S 11, which is reduced at the\nresonance since not all of the RF signal is re\rected at the\nterminating short of the RF current line. The saturation\nmagnetization \u00160Ms= 0:96 T was extracted by \ftting\nthe resonance position with Eq. 2, while \fxing g= 2 to\nliterature values36. The extracted Msagrees well with\nliterature values36,37.3\n-50050µ0H (mT)\n8765432\n f (GHz) \n \n \n-15-10-50S11 (mdB) (a)\n-12-8-40 S11 (mdB)\n-50 -40 -30 -20 -10\nµ0H (mT) w/ Gr\n w/o Gr\n fit (ΔH = 8.1 mT)\n fit (ΔH = 5.5 mT) f = 4.8 GHz(b) \nFIG. 2. Ferromagnetic resonance condition: (a) shows\nS11as a function of magnetic \feld and frequency with a fre-\nquency dependent background subtracted. (b) shows S11as\na function of magnetic \feld at f = 4 :8 GHz for a device with-\nout graphene (orange) and for a device with graphene (red).\nLorentzian \fts are used to deduce \u0001 Has indicated in the\nlegend.\nThe line width of the FMR is given by the damping\nterm\u000bin Eq. 1. Cuts at f= 4:8 GHz are shown in\nFig. 2 (b) for sample A and for sample C (with and with-\nout graphene). The full width at half maximum (\u0001 H)\nwas extracted by \ftting a Lorentzian to the data. A\nsigni\fcant larger \u0001 Hwas observed for the sample with\ngraphene indicating an additional damping term. The\nlinewidth of the FMR, \u0001 H, can be related to the Gilbert\ndamping38\n\u0001H=4\u0019\u000b\n\rfres: (3)\nHere, inhomogeneous sample-dependent broadening of\nthe linewidth was neglected since it was shown22and also\nfound in our measurements to be negligible. This addi-\ntional damping term can be interpreted as spin pumping\ninto graphene17and the di\u000berence in linewidth of a sam-\nple with graphene (\u0001 H;Py=Gr ) and one without graphene\n(\u0001H;Py) can be used to estimate the real part of the ef-\nfective spin-mixing conductance38,39\ng\"#=MsdPy\n~f\u0000\n\u0001H;Py=Gr\u0000\u0001H;Py\u0001\n; (4)\nwheredFMis the thickness of the ferromagnetic Py layer.\nThe imaginary part of the spin mixing conductance can\nbe neglected since it is much smaller than the real partfor metallic ferromagnets39. The spin-mixing conduc-\ntance is a measure of the e\u000eciency of the spin injec-\ntion and was here estimated to be \u00181\u00021020m\u00002using\n\u00160Ms= 0:96 T as extracted above, dFM= 30 nm and\na literature value of g= 236. This values is roughly a\nfactor of two larger than previously reported in similar\nPy/graphene systems22.\nB. Spin current and inverse spin Hall voltage\nThe DC component of the spin current density \rowing\nacross the Py/graphene interface in z-direction due to\nspin pumping is given by\njs=P\u0001hf\n4\u0019g\"#\u0001sin2(\u0012); (5)\nwherePis a correction factor that accounts for a non-\ncircular precession mode and \u0012is the precession angle\nof the magnetization around the e\u000bective \feld Heff40.\nWe estimate the correction factor for the non-circular\nprecession at 5 GHz to be around 0.6 based on Ref. 38.\nEq. 5 describes the spin current density across the Py/Gr\ninterface. We would like to note that our device geometry\ndi\u000bers from the conventional metallic bilayer structures\nsince a spin current is detected laterally. Therefore, only\na region of length \u0015Pyat the Py interface to the lateral\ngraphene spin transport channel can contribute to a spin\ncurrent that is detected at a distance L.\nThe spin current due to spin pumping is detected with\na Pt electrode placed at a distance L(600 nm for de-\nvice A and 700 nm for device B) from the Py pad, see\nalso Fig. 1. The charge current jcdue to the inverse\nspin-Hall e\u000bect can be detected as a voltage in an open-\ncircuit con\fguration. This voltage changes sign if the\ndirection of the external magnetic \feld is reversed as the\nspin polarization \u001bis parallel to the external magnetic\n\feld (jc\u0018\u001b\u0002js26). The voltage due to the inverse\nspin-Hall e\u000bect follows the line shape of the FMR and is\ntherefore described by a Lorentzian.\nHere, the voltage Uat the Pt electrode was measured\nwith a lock-in technique employing magnetic \feld modu-\nlation. Therefore, we recorded dU=\u0016 0dHas a function of\nRF frequency and magnetic \feld as shown in Fig. 3 (a).\nThe signal follows the FMR condition, which is indicated\nby red dots. The slight discrepancy at larger frequen-\ncies can be explained by sample to sample variation as\nthe FMR condition was extracted from a di\u000berent sam-\nple but with nominally equal design. Fig. 3 (b) shows\ndU=\u0016 0dHas a function of magnetic \feld and reveals the\nexpected lineshape of a derivative Lorentzian. Since U\ndepends on the spin orientation \u001b, which itself depends\non the magnetic \feld H, it has opposite sign for HFMR\nand\u0000HFMR , whereHFMR is the magnetic \feld at which\nthe FMR occurs. As a consequence, dU=\u0016 0dHshows a\ndip-peak structure for negative values of Hand a peak-\ndip structure for positive values of H. This behaviour is\nseen for all frequencies investigated here, see Fig. 3 (a).4\nSimilar results were obtained for sample B, shown in\nFig. 3 (c).\n50\n0\n-50µ0H (mT)\n8765432\n f (GHz)\n-200-150-100-50050100150dU/µ0dH (µV/T)(a)sample A\n-50050dU/µ0dH (µV/T)(b) sample A, f = 4.1 GHz\n fitP = 7 dBm UISHE = 370 nV\nUAHE = 50 nV\n16\n12\n8\n4\n-40 -20 020 40\nµ0H (mT) sample B, f = 3.65 GHz P = 10 dBm\nFIG. 3. Inverse spin Hall voltage at Pt electrode: (a)\nshows the derivative of the voltage measured at the Pt elec-\ntrode with respect to the magnetic \feld as a function of mag-\nnetic \feld and frequency for sample A. The superimposed red\ndots mark the position of the FMR condition extracted from\na measurement of S 11of a di\u000berent sample. (b) shows the cut\nindicated in (a) and a cut from sample B. The signal clearly\nshows the mirror symmetry with respect to zero magnetic\n\feld. The red dashed line is a \ft with Eq. 6. In the case\nof sample B, the data points around zero magnetic \feld were\nremoved due to technical limitations.\nAs motivated above, a magnetic \feld modulation based\nmeasurement technique has its advantages when it comes\nto sensitivity and in\ruences by spurious e\u000bects. However,\nit can itself lead to a background signal. The small modu-\nlation of the magnetic \feld induces a voltage in the wires\nconnecting the sample to the voltage ampli\fers due to\nsimple inductive pick-up. This voltage is magnetic \feld\ndependent as it scales with the modulation amplitude\nthat itself depends on the magnetic \feld due to a non-\nlinear current to \feld conversion of the magnet set-up. In\norder to remove this background, the voltage at the Pt\nelectrode was once measured with the microwave source\nturned on and once with the microwave source turned\no\u000b. The di\u000berence of these two measurements is shown\nin Fig. 3 and is used in the following analysis.\nThe measurement set-up presented above is only sen-\nsitive to voltages that develop in x-direction. Contribu-\ntions due to the anomalous Hall e\u000bect (AHE) can there-\nfore be expected since RF eddy currents are induced by\nthe RF magnetic \feld in the Py pads. These currents\row in the yz-plane in the permalloy and in combination\nwith a varying magnetization in the xz-plane an anoma-\nlous Hall voltage can be expected to appear. This voltage\nwill consist of a component at twice the frequency and\nof a down mixed DC component along the x direction\nand therefore, we include the AHE into the analysis. In\norder to separate the ISHE from the AHE, the measured\ndU=\u0016 0dHwas \ftted with:\nU(H) =UISHE\u00012\nH\n(H\u0000HFMR )2+ \u00012\nH+\nUAHE\u00002\u0001H(H\u0000HFMR )\n(H\u0000HFMR )2+ \u00012\nH;(6)\nthat captures both contributions41. Here,UISHE and\nUAHE represent the amplitudes of the contribution of the\nISHE and the AHE. The ISHE contribution follows the\nLorentzian shape of the FMR condition and has its maxi-\nmum at the resonance frequency. On the other hand, the\nAHE contribution displays a di\u000berent line shape with a\ncontribution that changes sign across the resonance con-\nditions since M(t) phase shifts by \u0019at resonance. There-\nfore, the ISHE and the AHE contribution can be disen-\ntangled by their spectral shape41,42.\nC. Power dependence\nThe data and the \ft with the derivative of Eq. 6 is\nshown in Fig. 3 (b) for sample A and in the inset of\nFig. 4 for sample B. Power dependence was investigated\non sample B as shown in Fig. 4, where the ISHE and the\nAHE contribution are shown separately. The contribu-\ntion due to the ISHE is much larger than the contribution\ndue to the AHE for any microwave power investigated.\nUSHE linearly depends on the spin current density\njs\u0018sin2(\u0012). The cone angle \u0012\u00182hRF=\u0001Hitself is\nlinearly depending on the driving RF \feld hRF43. For\nsmall angles this leads to a linear power dependence of\nUSHE\u0018h2\nRF\u0018p\nP2\u0018PsincehRFscales with the\nsquare root of the applied power. The power dependence\nofUSHE shown in Fig. 4 is consistent with the linear\ndependence as indicated by the red solid line.\nUAHE depends linearly on the eddy currents that scale\nlinearly with hRFand it also depends linearly on the\nprecessing magnetization M(t), which also scales linearly\nwithhRF. Therefore, UAHE\u0018h2\nRF\u0018p\nP2\u0018Pand a\nlinear power dependence results. The AHE contribution\nindeed scales linearly with RF power as one can see in\nFig. 4.\nIV. DISCUSSION AND INTERPRETATION\nThe observed broadening of the FMR linewidth when\nthe FM is in contact with the graphene sheet shows the5\n15\n10\n5\n0U (nV)\n10 9 8 7 6 5 4 3\nP (mW)8\n6\n4\n2\n0dU/µ0dH (µ V/T)\n35 30 25 20 15 10\nµ0H (mT) P = 8.9 mW\n fit\n UISHE\n UAHEsample B\nf = 3.65 GHz\nFIG. 4. Power dependence of the voltage at the Pt\nelectrode: The contribution of the ISHE and the AHE to\nthe voltage at the Pt electrode are shown as a function of\nmicrowave power. The inset shows an actual measurement\nwith a \ft to Eq. 6. Solid lines are linear guides to the eye.\npresence of an additional damping channel that can be\nexplained by spin pumping into graphene. The two times\nlarger spin mixing conductance extracted here compared\nto the literature values22,25might be explained by the\ncleaner ZEP based fabrication protocol. Successful dy-\nnamical spin injection into graphene is further supported\nby the observation of a voltage at the Pt electrode that\nfollows a derivative Lorentzian lineshape that is expected\nfor the ISHE. This voltage follows the FMR condition\nover a broad frequency range and shows the sign change\nfor negative magnetic \feld. Power dependence of this\nvoltage reveals a linear scaling of the ISHE contribution\nas expected, with a minor contribution from the AHE.\nA. Quantitative analysis\nA quantitative analysis of the injected spin current\ncompared with the inverse spin-Hall voltage is given in\nthe following.\nThe injected spin current density at the Py/graphene\ninterface can be estimated from eq. 5. According to Guan\net al.43, the precession angle at resonance is given by\n\u0012\u00182hRF=\u0001H\u00188:4\u000e. The driving RF magnetic \feld\nhRFwas estimated to be 0 :6 mT using the Biot-Savart\nlaw where the RF current is assumed to \row homoge-\nneously within the RF current line and where the RF\ncurrent is given by the applied power of 7 dBm over a\n50 \n impedance. This is the maximum h RFthat can be\nexpected since neither RF losses in the cables nor re\rec-\ntions at the connectors are included.\nWe then estimate js\u00182:8\u000210\u00007J m\u00002at 4:1 GHz\nand 7 dBm. In order to get the lateral spin current (in y-\ndirection), one has to determine the area that contributes\nto the injected current. Since the Py is very well coupled\nto the graphene below, only a narrow strip of the size\nwGr=Py\u0002\u0015Pywill contribute as in the other parts the\nspins will have relaxed before reaching the spin trans-port channel. This then results in a lateral spin cur-\nrent density of jy\ns=js\u0001\u0015Py\u00181:2\u000210\u000015J m\u00001, using\n\u0015Py= 4:3 nm44. Similar spin current densities are com-\nmonly realized with electrical spin injection with tunnel\ncoupled ferromagnetic contacts.\nUsing this current density now we can calculate the\ninverse spin Hall voltage appearing on the Pt electrode\nand we can compare it with the measured values. The\nmeasured voltage U at the Pt electrode is given by:\nUISHE =2e\n~\u000bPt\u001aPtwgr=Ptjy\ns\nlPt\n\u0001\u0015Pt\ndPt1\u0000exp (\u0000dPt=\u0015Pt)\n1 + exp (\u0000dPt=\u0015Pt)\u0001\u0015Pt\nlPt;(7)\nwhere the \frst term describes the spin to charge con-\nversion via the inverse spin Hall e\u000bect, the second term\nincorporates spin relaxation in z-direction within the Pt\nelectrode following Ref. 28 and the last term is the ex-\ntension of that model considering also spin relaxation\nin y-direction and shunting due to the metallic elec-\ntrode. Note that lPt\u001d\u0015Ptand therefore the ex-\nponential corrections can be neglected in y-direction.\nWe used\u001aPt\u001846µ\n cm,wgr=Pt = 7\u00028µm = 56 µm\n(seven devices, each 8 µm),lPt=400 nm and dPt=10 nm\nthat were determined experimentally, whereas the spin\nHall angle \u000bPt\u00180.1528and\u0015Pt\u00185 nm26were\ntaken from literature. A lateral spin current density\njy\ns\u00181:2\u000210\u000015J m\u00001leads to an expected voltage\nUISHE\u0018170 nV, which is within a factor of two from\nthe experimentally determined value.\nThe di\u000berence between the measured voltage and\nexpected voltage at the platinum electrode can have\nserveral origins. First, the spin relaxation within the\ngraphene spin transport channel is neglected since spin\nrelaxations lengths of the order of 1 µm are commonly\nobtained also for low quality graphene devices. Next, we\nwould like to note that the cone angle is only a rough\nand upper estimate and an experimental determination\nwould reduce the uncertainty of that value. Moreover,\nseveral parameters used for the estimation of the voltage\nat the platinum electrode are not well known and espe-\ncially a large spread of the values for the spin Hall angle\nof Pt is found in literature26.\nV. CONCLUSION\nThe development of the compact, on-chip and broad-\nband excitation scheme for spin pumping into graphene\nand the detection of a spin current with a Pt electrode\npaves the way for future studies focussing on the extrac-\ntion of spin transport parameters. Hanle measurements,\nas shown in spin pumping experiments in silicon45, could\nbe performed in vector magnet set-ups, where an addi-\ntional magnetic \feld in the z-direction is available. An-\nother step forward would be the implementation of fer-\nromagnetic strips instead of squares, which would allow6\nspin pumping experiments at zero external in-plane mag-\nnetic \feld due to the non-zero remanent magnetization\nof nanomagnets36,46.\nIn conclusion, we demonstrated that FMR can be ob-\nserved in micronscale Py/graphene heterostructures with\non-chip and wideband microwave excitation in a simple\nre\rection measurement. The increased damping of the\nFMR in Py pads connected to graphene suggested the\npresence of spin-pumping, which is further supported by\nthe detection of a spin current at a Pt electrode employ-\ning the inverse spin Hall e\u000bect. This direct and compact\nway of spin pumping into graphene to power levels as\nlow as 3 mW paves the way for further studies on the\nspin dynamics in graphene and related heterostructures.\nFuture studies could investigate the spin-to-charge con-\nversion in graphene itself as recently reported by Mendes\net al.47. In addition, heterostrucutres of graphene and\ntransition metal dichalcogenides have shown a greatly\nenhanced spin-orbit coupling48with a dominating valley-\nZeeman term49. These systems are expected to show\nlarge spin-Hall angles50that would allow for an even more\ne\u000ecient spin-to-charge conversion. This is especially in-teresting and important since graphene is a promising\ncandidate for future building blocks in spintronic appli-\ncations (e.g. spin torque nano oscillators) considering\nthat it can withstand large current densities51and large\nspin accumulations15can be achieved.\nACKNOWLEDGMENTS\nThis work has received funding from the European\nUnions Horizon 2020 research and innovation programme\nunder grant agreement No 696656 (Graphene Flag-\nship), the Swiss National Science Foundation, the Swiss\nNanoscience Institute, the Swiss NCCR QSIT and ISpin-\nText FlagERA network OTKA PD-121052, OTKA FK-\n123894 and OTKA K112918. 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Bachtold,\nApplied Physics Letters 91, 163513 (2007),\nhttps://doi.org/10.1063/1.2789673."    },    {        "title": "1308.1232v1.Broadband_ferromagnetic_resonance_characterization_of_GaMnAs_thin_films.pdf",        "content": "Broadband ferromagnetic resonance characterization of GaMnAs thin films\nA. Ben Hamida,1S. Sievers,1K. Pierz,1and H.W. Schumacher1\nPhysikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig,\nGermany\nThe precessional magnetization dynamics of GaMnAs thin films are characterized by broadband network ana-\nlyzer ferromagnetic resonance (FMR) in a coplanar geometry at cryogenic temperatures. The FMR frequencies\nare characterized as function of in-plane field angle and field amplitude. Using an extended Kittel model of\nthe FMR dispersion the magnetic film parameters such as saturation magnetization and anisotropies are de-\nrived. The modification of the FMR behavior and of the magnetic parameters of the thin film upon annealing\nis analyzed.\nI. INTRODUCTION\nPrecessional magnetization dynamics of magnetic thin\nfilms and nanostructures are highly relevant for magnetic\ndevice applications. For example the minimum magneti-\nzation reversal times and hence the ultimate data rate\nof a magnetic memory devices is directly determined by\nthe precession frequency1–4. Ferromagnetic semicon-\nductors are a particularly promising class of magnetic\nmaterials as they could offer the combination of magnetic\nmemory and semiconductor logic functions in the same\nmaterial. Presently, GaMnAs can be considered the most\nprominent prototype of diluted ferromagnetic semicon-\nductors with well determined material parameters5. The\nprecessional magnetization dynamics of GaMnAs thin\nfilms and devices have been characterized by different\ntechniques. Time resolved precessional dynamics have\nbeen studied by all-optical pump-probe mageto optics\nusing fs lasers6–10or by time resolved magneto-optical\ncharacterization upon electrical excitation11,12. Further-\nmore low-temperature cavity ferromagnetic resonance\n(FMR) has been used to investigate anisotropies and\nlinewidths13,14as well as spin wave resonances15. In\naddition electrical measurements based on photovoltage\ndetection16or on spin-orbit ferromagnetic resonance17\nhave been tested.\nOver the last years broadband vector network an-\nalyzer based ferromagnetic resonance (VNA-FMR)18\nusing coplanar wave guides as inductive antennas\nhas become a versatile tool for the simple and fast all\nelectrical characterization of precessional dynamics of\nvarious magnetic thin films and multilayers19,20. In\nprinciple low-temperature VNA-FMR21should also be\nsuitable for the electrical characterization of ferromag-\nnetic semiconductors such as GaMnAs. However up\nto now VNA-FMR based measurements of the preces-\nsional magnetization dynamics of GaMnAs have proven\ndifficult to achieve. This is due to the low saturation\nmagnetization of GaMnAs in combination with strong\ncrystalline anisotropies. Both properties lead to a rather\nweak inductive signal which could be easily masked by\nthe noise of the high bandwidth measurement electronics.\nHere we present broadband coplanar VNA-FMR mea-surements of the precessional dynamics of GaMnAs thin\nfilms in a cryogenic environment. For sufficiently high ex-\ncitation powers a clear precessional signal is observed in\nthe VNA-FMR spectra. The field and angular dependence\nof the FMR peaks can be well described by a Kittel model\ntaking into account the different anisotropy components\nof GaMnAs. The precessional dynamics and material pa-\nrameters of an as-grown and an annealed thin film are\nanalyzed and compared.\nII. EXPERIMENTAL SETUP AND PROCEDURE\nGaMnAs layers of 100 nm thickness were grown\nin a low-temperature MBE environment on a 2 inch\nsemi-insulating GaAs(001) wafer at temperatures of Tg=\n240°C and 220°C, respectively. Details of the growth and\nannealing procedure can be found elsewhere22. Samples\nof 10 mm x 10 mm were cut from the wafers as well as\n5 mm x 5 mm pieces for superconducting quantum inter-\nference device (SQUID) magnetometry. For the as-grown\nsample of Tg= 240°C a saturation magnetization MS=\n30 mT is measured. The sample with Tg= 220°C was\nannealed at 200°C for 18 h in ambient air resulting in an\nincreased saturation magnetization of MS= 74 mT.\nThe setup for inductive FMR characterization of\nthe samples is described with respect to Fig. 1. The\nexperiments are carried out in a variable temperature\ninsert of a commercial He cryostat allowing to vary the\nsample temperature from TS= 1.5 . . . 250 K. All FMR\nmeasurements of this work were carried out at fixed sam-\nple temperature of TS= 10 K. The cryostat is equipped\nwith a three axial superconducting vector magnet allow-\ning application of static magnetic vector fields ¹0Hs=\n¹0(Hx,Hy,Hz) up to 1 T amplitude and arbitrary orien-\ntation. For FMR measurements the 5x5 mm2samples\nare placed on the center of a coplanar waveguide (CPW).\nDetails of the design and high frequency properties of the\nCPW can be found elsewhere23. Both ends of the CPW\nare connected to the two ports of a 24 GHz bandwidth\nVNA via 18 GHz bandwidth coaxial lines of about 1.5 m\nlength. Note that the rather long length of the coaxial\nlines is determined by the dimensions of the cryostat.\nInside the cryostat the CPW substrate is oriented in the\n1arXiv:1308.1232v1  [cond-mat.mes-hall]  6 Aug 2013xy-plane with the CPW line running along xand high\nfrequency (HF) field generation along y(HHF||Hy).\nIn the present experiments in-plane magnetic vector\nfields ¹0(Hx,Hy) up to 0.5 T amplitude are applied in\nthe sample plane ( Hz= 0). As sketched in Fig. 1(a) the\nGaMnAs samples are placed diagonally on the CPW with\nthe [010] crystalline axis oriented parallel to HHFand\nthe [100] axis along x.\nFor an FMR measurement at a given static vector\nfield ( Hx,Hy), the frequency output of the VNA is swept\nbetween 1 and 18 GHz and the forward scattering signal\nS21is measured. To maximize the weak inductive FMR\nsignal the maximum output power of 20 dBm is applied.\nThe HF output signal generates a HF excitation field\nHHFaround the CPW center conductor line and thus\nin the GaMnAs thin film sample. Under resonance con-\nditions the GaMnAs magnetization is excited into FMR\nprecession and the signal transmission is reduced leading\nto a Lorentzian absortion line in the VNA sweep. To\nenhance the visibility of the FMR signal a VNR reference\nmeasurement S21,refis carried out at a non-resonant\nfield. The normalized transmission signal T is then\ndeduced by subtracting the reference VNA sweep from\nthe actual measurement data. From the resonance peak\nthe FMR frequency fFMR and the absorption line width\n¢fFMR are derived. Note that for certain applied static\nfields only weak resonance peaks were found making a\nreliable linewidth analysis impossible. By variation of the\napplied static field the FMR properties were measured as\nfunction of the field vector ( Hx,Hy).\nIII. DATA ANALYSIS\nA detailed review on FMR in GaMnAs has been given\nby Liu and Furdyna24. In an FMR experiment, the mag-\nnetization ~MÆMs(sinµcosÁ,sinµsinÁ,cosµ) (see Fig. 2\nfor details of the angle nomenclature) of the film pre-\ncesses around its equilibrium position with the FMR fre-\nquency fFMR. Sweeping the value of the applied mi-\ncrowave frequency fHFat a fixed magnetic field ~HÆ\nH(sinµHcosÁH,sinµHsinÁH,cosµH) (cp. Fig. 2), the res-\nonance condition will be satisfied at fFMRÆfHF. The con-\ndition is given by:\n(2¼fHF\n°)2Æ1\n(2¼Mssinµ)2[@2F\n@µ2@2F\n@Á2¡(@2F\n@µ@Á)2] (1)\nwhere °Æ1.76¤1011denotes the gyromagnetic ratio and\nFis the free magnetic energy. The expression of Ffor a\nthin film with crystalline and uniaxial anisotropy is given\nby:\nFÆ ¡¹~M~HŹ\n2MsMef fcos2µ¡Ku1sin2µcos2(Á¡1)\nÅKc1\n4(sin2(2µ)Åsin4µsin2(2Á¡2c)) (2)\nFig. 1. (a) Experimental Setup: Sketch of the sample position on\nthe coplanar waveguide and connexions with the VNA. (b) Pho-\ntograph of the sample on top of the coplanar waveguide. (c) Ex-\nperimental FMR curve at H= 0.32 T. The absorption line has the\nshape of an asymmetric Lorentzian around fFMR = 12.6 GHz\nwith linewidth ¢f= 0.5 GHz.\nwhere ¹Æ4¼10¡7(SI units). Mef fis the effective mag-\nnetization of the thin film. Ku1is the in-plane uniaxial\nanisotropy constant corresponding to the easy axis ori-\nentation 1with respect to the crytallographic direction\n[100]. Kc1is the in-plane cubic anisotropy constant. c\nis the in-plane orientation of the cubic easy axis with\nrespect to [100].\nFig. 2. GaMnAs sample: Configuration of the magnetization ~M\nand the magnetic field with respect to the crytallographic direc-\ntions. ÁandÁHare the in-plane angles of ~Mand~H, respec-\ntively, as measured from the [100] orientation. µandµHare the\nnormal angles of ~Mand~H, respectively.\n2During our experiments, the external field ~His applied\nin-plane and the film magnetization ~Mis assumed to\nstay also in-plane ( µHƵÆ90°). At a given direction of\n~H, the resonance is then obtained by numerically solving\nthe above equation at the equilibrium position of ~M,\nfor@F\n@ÁÆ0. The anisotropy parameters of the GaMnAs\nthin films can then be derived for example from a fit\nto the measured angular dependence of the precession\nfrequency at fixed field amplitude H. Fig 3 shows fFMR\nas function of the in-plane field angle ÁHfor applied field\namplitude of ¹0H= 0.2 T (green) and 0.3 T (red). The\nsymbols represent the measured data whereas the lines\nare the fits according to the above FMR model. Two sets\nof data taken on the two different samples are shown. The\nupper panel (a) shows the data of the annealed sample\nwhereas in (b) the data of the as-grown sample are shown.\nFig. 3. Angular dependence of the precession frequency for two\nfield amplitudes of 0.2 T (green) and 0.3 T (red). The symbols\nare experimental data. The solid lines are the fit. (a) Annealed\nsample, (b) as-grown sample.\nThe data of both samples can be well described by a fit\nto the above model taking into account a thin film with\ncrystalline in-plane cubic and uniaxial anisotropy. The\nvalues of the derived anisotropies for the best fit to the\ndata are regrouped in Table 1. Note that in Table 1 the\nanisotropy fields HiÆ2Ki/MSare given instead of the\nanisotropy constants Ki.\nNote that the values of the saturation magnetization\nMSare based on the SQUID measurements and are not\nderived from fitting. Figure 3 shows that the model based\non the above parameters well describe the experimental\ndata both of the annealed and the as-grown sample.\nThe derived values of the magnetic parameters are very\nreasonable when compared to literature values of thin\nfilm anisotropies derived by conventional cavity FMR\nexperiments of GaMnAs thin films24. The comparisonof the two parameter sets shows the strong impack of\nannealing on all magnetic parameters from saturation\nmagnetization to the various anisotropy terms.\nThe broadband coplanar FMR setup also allows to\nderive experimental values of fFMR as function of field\namplitude Hfor a fixed in-plane angle ÁH. Such field\ndependent data is shown in Fig. 4 for selected field angles\nfor the (a) annealed and (b) as-grown samples. The\nsymbols again represent the data whereas the lines are\nthe model fit. The measured data is again well described\nby the model fit confirming the feasibility of the derived\nparameters.\nFig. 4. Field dependence of the precession frequency for different\nfield orientations ÁH. The points are experimental data. The\nsolid lines are the fit. (a) Annealed sample, (b) as-grown sample.\nAs mentioned above the linewidth could not be system-\natically analyzed from the VNA-FMR data. However for\nselected data points a sufficiently clear resonance peak al-\nlowed a reliable linewidth analysis. From this linewidth\ndata a Gilbert damping parameter of ®Æ0.018 was de-\nrived for the annealed sample. This value is in good agree-\nment with the literature values of annealed samples de-\nrived by X-Band ferromagnetic spectroscopy25. It is how-\never worth noting that this value is quite different from\nliterature data derived from time resolved optical pump\nprobe experiments6for an as-grown sample (the values of\n®ranged from 0.12 to 0.21). The reason of this strong de-\nviation of the literature values of the damping derived by\ndifferent methods can presently only be subject of spec-\nulation. However, it might be related to different ap-\nplied fields and hence to different contributions of extrin-\nsic line broadening in the experiments26. Furthermore\nthe difference could be related to inhomogeneous sample\nproperties12.\n3Sample MS Mef f Hu1 1 Hc1 c\nannealed 74 mT 30 mT 70 mT 45° 85 mT 0°\nas-grwon 30 mT 130 mT -20 mT 20° 100 mT -20°\nTable I. Experimental derived values of the magnetic parameters for annealed and as-grown 100 nm thick GaMnAs samples.\nIV. CONCLUSION\nConcluding we have demonstrated the suitability of\nbroadband network analyze based FMR for the charac-\nterization of the precessional dynamics of GaMnAs thin\nfilms. A coplanar inductive antenna was used to excite\nand detect the precessional signal of 100 nm thick an-\nnealed and as-grown GaMnAs thin films with saturation\nmagnetization down to 30 mT. The field and angular de-\npendence of the FMR frequency could be well described\nby a model taking into accound the different thin film\nanisotropy terms. The simple and yet powerful setup\ncould in the future allow investigations of more com-\nplex systems such as of the coupled dynamics of GaMnAs\nbased tunnel junctions and multilayers.\nV. ACKNOWLEDGMENTS\nThe work was supported by DFG SPP Semiconductor\nSpintronics and EMRP JRP IND08 MetMags and JRP\nEXL04 SpinCal. The EMRP is jointly funded by the\nEMRP participating countries within EURAMET and the\nEU.\nReferences\n1S. Kaka and S. E. Russek. Appl. Phys. Lett. , 80:2958,\n2002.\n2Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. Baer,\nand Th. Rasing. Nature , 418:509, 2002.\n3H. W. Schumacher, C. Chappert, P. Crozat, R. C. Sousa,\nP. P. 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App. Phys. , 95:5646, 2004.\n4"    },    {        "title": "2201.10722v1.Local_ferromagnetic_resonance_measurements_of_mesoscopically_patterned_ferromagnets_using_deterministically_placed_nanodiamonds.pdf",        "content": "Local ferromagnetic resonance measurements of mesoscopically\npatterned ferromagnets using deterministically placed\nnanodiamonds\nJe\u000brey Rable,1Benjamin Piazza,1,\u0003Jyotirmay Dwivedi,1and Nitin Samarth1,y\n1Department of Physics, Pennsylvania State University,\nUniversity Park, Pennsylvania 16802, USA\n(Dated: January 27, 2022)\n1arXiv:2201.10722v1  [cond-mat.mes-hall]  26 Jan 2022Abstract\nNitrogen-vacancy centers in diamond have recently been established as e\u000bective sensors of the\nmagnetization dynamics in vicinal ferromagnetic materials. We demonstrate sub-100 nm placement\naccuracy of nitrogen-vacancy-containing nanodiamonds and use these as local sensors that probe\noptically detected ferromagnetic resonance in mesoscopically patterned Permalloy islands. These\nmeasurements reveal variations in the ferromagnetic resonance signal at di\u000berent sites on these\nstructures with distinct behavior in the edge and the bulk of patterned features. These test\nmeasurements establish an easily implemented approach for spatially targeted measurements of\nspin dynamics in mesoscale ferromagnets. In principle, the methodology can also be extended to\nlocal studies of nanoscale ferromagnets such as single magnetic nanowires and nanoparticles.\nI. INTRODUCTION\nContemporary problems of interest in spintronics often require knowledge of the dy-\nnamical behavior of magnonic (spin wave) excitations in patterned ferromagnetic devices.\nFor example, this information is important in the creation and characterization of magnon\nquantum buses [1, 2] and other magnonic devices such as spin-based transistors [3]. The\nnitrogen-vacancy (NV) center in diamond has emerged as an e\u000bective non-perturbative local\nprobe for characterizing the magnetic properties of such systems [4]. NV center-based local\nmagnetometry has provided new insights into static spin con\fgurations of skyrmions [5, 6]\nand magnetic domain walls [7], as well as the dynamical behavior of magnons [8{11] and\nvortices [12]. In the latter context, it is important to develop techniques that allow local\nmeasurements of ferromagnetic resonance (FMR) at targeted locations in a ferromagnetic\nsample or device.\nContinuous wave NV center-based optically detected ferromagnetic resonance (ODFMR)\nmeasurements rely on the quenching of the NV center \ruorescence when a vicinal ferromag-\nnet meets the conditions for FMR [8]. This is attributed to an increased magnon density\nand an accompanying enhancement of the magnetic \feld noise sensed by the NV centers,\nthus leading to increased spin relaxation [9]. Local measurements of ODFMR have relied\n\u0003Current address: Network Science Institute, Northeastern University, Boston, MA 02115, USA\nynsamarth@psu.edu\n2on stochastic distribution of dropcast nanodiamonds [8, 13], stochastic distribution of NVs\nin a diamond \flm [12{15], proximate placement of a diamond nanobeam [9, 11], and chem-\nically patterned directed assembly of nanodiamonds [10]. In principle, scanned probe NV\ncenter magnetometry could provide a means of carrying out local ODFMR with imaging\ncapability. However, since NV detection of ODFMR rapidly decreases in sensitivity with\nincreasing sample-probe distance [15], NV scanning probe measurements of ODFMR may\nbe constrained by the tip-sample distances that are typically greater than about 100 nm\n[7, 16, 17]. Although one scanning NV technique allows for smaller tip-sample separation\n(\u001830 nm) [5], it would be technically challenging to engineer e\u000bective excitation of the FMR\nin a ferromagnetic sample in this geometry. This may account for the absence (as yet) of any\npublished reports of ODFMR using a scanning NV center probe. As an aside, we note that\nlocal magnetization dynamics of ferromagnets can also be e\u000bectively probed and imaged via\na completely di\u000berent method, namely scanning ferromagnetic resonance force microscopy\n(FMRFM) which uses a microscale force cantilever to detect FMR with \u0018100\u0000200 nm\nspatial resolution [18{21]. As with NV center based ODFMR, the FMRFM technique is\namenable to measurements at ambient temperature. However, in contrast with NV center\nODMFR, the FMRFM method requires more sophisticated instrumentation and it is more\ncumbersome to collect FMR data that densely spans frequency-magnetic \feld space. Thus,\na question of interest is whether one can develop a simpler approach for locally measuring\nFMR at targeted sites on a mesoscopic or nanoscale ferromagnetic structure without having\nto resort to the sophisticated instrumentation required by scanning microscopy techniques.\nIn this paper, we demonstrate the use of an atomic force microscope (AFM) to achieve\nwell-controlled positioning of NV-containing nanodiamonds as FMR sensors, with sizes be-\ntween 40 nm - 100 nm and with sub-100 nm accuracy. We use these deterministically placed\nnanodiamonds to perform local ODFMR measurements of mesoscale (5 \u000010\u0016m lateral size)\nfeatures patterned in ferromagnetic (Permalloy, Py) thin \flms. Although similar to a 'pick-\nand-place' technique previously reported for assembling nanodiamonds at desired locations\n[22, 23], this approach has not yet been exploited to probe the localized magnetization\ndynamics of magnetic materials. We note that chemically patterned directed assembly of\nnanodiamonds has been e\u000bectively used for probing ODFMR in YIG devices [10]; however,\nthe measured locations are constrained in advance by lithographic patterning and subject\nto overlay error. We seek a more \rexible approach that allows the measured locations to\n3FIG. 1. Demonstration of the nanodiamond placement process with a 40 nm diameter nanodi-\namond (1). After lifting the nanodiamond and con\frming it is no longer on the sample (2), we\nmove over to the deposition site, a 100 nm wide nanowire (3). Then, we can deposit via ramp or\nlift mode and con\frm placement via another scan (4).\nbe varied at will. The placement precision in our proof-of-concept demonstration can, in\nprinciple, also allow for the targeted measurement of smaller nanoscale structures, such as\nnanowires, or of localized modes in larger structures, such as edge modes and defect modes.\nII. METHODS\nA. Nanodiamond Placement and ODFMR\nWe \frst describe the experimental approach for deterministic placement. We begin with\na sample containing patterned permalloy structures in the center and drop cast an aqueous\nsolution of 100 nm diameter nanodiamonds with 3 ppm NV centers onto the edge of the\nsample. This is done to avoid directly depositing nanodiamonds onto the features. Next,\nwe scan the area where the nanodiamonds were drop cast using a Veeco Nanoscope IIIA\nMultimode AFM with a gold coated silicon tip, which increases the probability of pickup over\na standard silicon tip, likely because of stronger van der Waals forces or malleability of the\n4gold. When we \fnd a particle that matches the dimensions of the dispersed nanodiamonds,\nwe zoom in on the particle (Fig. 1, step 1) and enter ramp mode. Then, we ramp into the\nnanodiamond and re-scan the area where it was in the prior scan to con\frm pickup (Fig. 1,\nstep 2). If the nanodiamond remains at the site, we ramp in again until it is picked up or\nattempt pickup on a di\u000berent particle.\nIf the nanodiamond does not appear on the post-ramp scan, indicating that it was picked\nup, we can begin the placement process by moving the AFM tip to our patterned features.\nWith the nanodiamond still attached to the tip, we can scan the sample looking for the\nfeature that we want to deposit the particle on (Fig. 1, step 3). When we \fnd the device,\nwe either repeat the process used in pickup, ramping into the feature until the nanodiamond\ndislodges, or we scan across the sample at a constant height below the surface using the\nAFM's lift mode, scraping the nanodiamond-coated tip along until it dislodges. While this\nsecond method works more consistently, it risks scratching the sample if the tip is dug into\nthe feature. Finally, with the diamond dislodged, we perform a \fnal scan to con\frm that\nthe nanodiamond is in the desired location (Fig. 1, step 4). We caution that we do not\nyet have a systematic measure of the success rate of the method. The repeatability of the\ntechnique appears to depend on factors that are not completely understood and varies with\nthe details of the drop casting and the substrate.\nFor optical polarization and readout of the NV center \ruorescence, we used a 1 mW, 532\nnm continuous wave laser and an ID Quantique ID100 avalanche photodiode. A scanning\nmirror scans across the surface for imaging and allows focusing on a site for ODFMR mea-\nsurements. A static magnetic \feld is applied using a permanent N52 magnet mounted on\na highly repeatable stepper motor linear stage. The applied \feld is calibrated in the plane\nof the sample using a single crystal diamond \flm containing NV centers; this is achieved\nusing the known orientations and the Zeeman splitting of the NV electronic ground state\nspin transition. During the ODFMR measurements, a microwave magnetic \feld is applied\nvia a 25 µm diameter gold wire run across the sample. This microwave \feld both drives\nFMR in the magnetic features and the NV spin state transitions. Additionally, when FMR\nis driven in the sample, new, higher frequency magnons are generated via scattering and\nthermal mechanisms. The dipolar \feld noise generated by these incoherent magnons also\na\u000bects the NV spin state transitions; this e\u000bect is believed to be responsible for the detection\nof FMR via an optical contrast even though the FMR is driven at frequencies o\u000b-resonant\n5from those that drive the NV spin transitions [9].\nB. Micromagnetic Simulations\nPrior to our measurements, we performed micromagnetic simulations of the FMR modes\nin patterned Py features identical to those measured experimentally. The Py \flm thickness\nin all these simulations is 10 nm. The simulations were performed with the Mumax3 software\npackage, which uses the Landau-Lifshitz-Gilbert equation:\n@~M\n@t=\rLL1\n1 +\u000b2(~ m\u0002~Be\u000b+\u000b(~ m\u0002(~ m\u0002~Be\u000b)) (1)\nto calculate the evolution of the magnetization ~Mof \fnite ferromagnetic cells. In Eq. 1, \u000b\nis the Gilbert damping of the material, \rLLis the gyromagnetic ratio of the material, and\nBeffis the e\u000bective magnetic \feld at that cell, which includes contributions from external,\ndemagnetization, exchange, and anisotropy \felds [24].\nThe simulations were performed using 5 nm x 5 nm x 10 nm cells and the geometries\nconsisted of permalloy features with the parameters in table I.\nParameter Value\nMs 8 x 105A/m\nAex 1.3 x 10\u000011J/m\n\u000b 0.0063\nTABLE I. Permalloy material parameters used in micromagnetic simulations\nAfter de\fning the sample geometry, the system was given an initial magnetization point-\ning along the (0,0,1) direction out of the plane of the \flm and allowed to relax to the\nminimum energy state in the applied bias \feld, which ranged from 1 to 30 mT in our\nsimulations.\nTo excite the system, we applied a Gaussian pulse with a 20 ps full width half maximum\nand a 0.5 mT amplitude. The system was then allowed to freely evolve in time for 20 ns\nand average magnetization was sampled every 5 ps. Finally, we used a discrete fast Fourier\ntransform to analyze the data in the frequency domain. Using the above sampling rates and\nsimulation lengths, we obtain a resolution of 50 MHz.\n6FIG. 2. AFM images of the measured permalloy features with superimposed scanning confocal\n\ruorescence images showing where NV-containing nanodiamonds are located. (a) A 10 µm x 5\nµm x 10 nm permalloy rectangle with nanodiamonds located near the edges and in the middle.\n(b) Two 6 µm diameter, 10 nm thick permalloy circles connected at the edges. Nanodiamonds are\nlocated at the periphery of the left disk and near the junction between the disks.\nIII. RESULTS\nTo begin, we placed multiple nanodiamonds on the 10 µm x 5 µm x 10 nm rectangle in\nFig. 2 (a) to con\frm that we could replicate previous macroscale FMR measurements carried\nout on arrays of such rectangular islands. These studies showed the presence of an easy-axis\nand hard-axis resonance [25, 26] which can be modelled using a geometry-dependent form\nof the Kittel equation:\nf=\r\n2\u0019q\n(Happ\u0000(Nx\u0000Ny)4\u0019\u00160Ms)(Happ\u0000(Nx\u0000Nz)4\u0019\u00160Ms): (2)\nHere,\ris the gyromagnetic ratio of the material, Happis the applied \feld, Msis the\nsaturation magnetization of the material, and Niare the three geometric demagnetization\nfactors which sum to 1.[27] We use NxandNyas in-plane demagnetization factors and Nzas\nthe out-of-plane demagnetization factor. We can further simplify these three geometric pa-\nrameters and the saturation magnetization into two quantities that represent the anisotropy\n\felds of the features - BkandB?. This yields:\n7FIG. 3. Measurement of ODFMR using a nanodiamond in the bulk of a rectangular Py island\nwith the applied magnetic \feld oriented along (a) the longitudinal easy axis and (b) along the\ntransverse hard axis. The corresponding micromagnetic simulations of the FMR with the applied\nmagnetic \feld oriented along (c) the longitudinal easy axis and (d) along the transverse hard axis.\nf=\r\n2\u0019q\n(Happ+Bk)(Happ+B?): (3)\nBecause of Py's low intrinsic anisotropy, we can also assume that these anisotropy \felds\nsolely result from the shape anisotropy of our features. In the case of thin, rectangular\nfeatures like the one measured, NxandNyin eqn. 2 will be small and unequal, while Nz\nwill still be close to 1, its value in an in\fnite thin \flm. This results in B?being close to the\n4\u0019\u00160Msand inBkhaving a comparatively small magnitude. Furthermore, when rotated 90\u000e\nin plane,NxandNyswap positions in eqn. 2, leading to Bkretaining its magnitude while\nswitching signs in 3. This results in a positive Bkalong the easy axis, but negative Bkalong\nthe hard axis, leading to a divergence as Happapproaches it.\nFigure 3 (a) and (b) show the measurements of ODFMR obtained using a dim nanodia-\nmond located near the center of the rectangle. When a magnetic \feld is applied along the\n8easy axis (long edge of the rectangle), we detect a single mode that increases approximately\nlinearly starting at 2 GHz (Fig. 3 (a)), as expected from eqn. 3. Fitting eqn. 3 to this\nresult, we \fnd the in plane anisotropy \feld Bkof this feature to be 3.3 mT, and the out\nof plane anisotropy \feld B?to be 1.04 T, close to the accepted 1 T Msof Py as expected.\nWhen a magnetic \feld is applied along the hard axis (short edge of the rectangle), we see a\nV-shaped dispersion, which reaches a minimum at approximately 3 mT, near the divergence\npoint expected from our previously measured Bk(Fig. 3 (b)). The micromagnetic simula-\ntions shown in Fig. 3 (c) and (d) largely match these results, though the frequencies are\nhigher, most likely a result of our applied microwave power, which can result in NV contrast\nabove the FMR frequency [28].\nHowever, along the edges of the rectangle, we detected new features in addition to the\nones seen by the nanodiamond located in the middle (or bulk) of the ferromagnetic rectangle\n(Fig. 4). For a nanodiamond located on the long edge, when the magnetic \feld is applied\nalong the easy axis (long edge of the rectangle), we observe an incomplete switching with\na faint signal from the V-shaped resonance (Fig. 4 (a). We believe this to be the result of\na small magnetic \feld misalignment. When the \feld is applied along the hard axis (Fig.\n4 (b)), we see a faint easy-axis signal as well as faint traces of additional signals between\nthe easy and hard axis FMR signals. For a nanodiamond placed on the short edge of the\nrectangle and with the \feld applied along the easy axis Fig. 4 (c)), we observe a strong\neasy axis signal and a faint hard axis signal, similar to Fig. 4 (a). However, we can also\nsee faint traces of additional signals above the easy-axis resonance. We speculate that these\nadditional signals in Fig. 4(b) and (c) are higher order magnon modes.\nWe now discuss measurements on a feature composed of two connected 6 µm diameter, 10\nnm thick circular Py disks (Fig. 2 (b)). These features had a slight fabrication error in one of\nthe circles near the constriction where they meet, leading to a sharp edge slightly protruding,\nwhich could cause a local distortion of the ferromagnetic resonance. We positioned and\nmeasured nanodiamonds at three di\u000berent sites - one on the edge approximately 4 µm from\nthe constriction, one on the pristine side of the constriction approximately 2 µm from the\nfabrication error, and one on the misfabricated side of the constriction, approximately 750\nnm away from the error. Far away from the constriction, the data shows streaking at\napproximately 2 mT, and a signal that begins suddenly at approximately 4 mT (Fig. 5\n(a)). Near the constriction (Fig. 5 (b)), we see additional noise broadband noise at lower\n9FIG. 4. ODFMR measurements using a nanodiamond located on the long edge of a rectangular\nisland, with \feld oriented along (a) the hard axis (long edge), (b) along the easy axis (short edge).\n(c) ODFMR measurements using a nanodiamond located on the short edge of a rectangular island\nwith \feld oriented along the hard axis (long edge). We attribute the anomalous high \feld behavior\nof the NV center resonance lines to fringe \feld e\u000bects that occur at the edge of the patterned\nfeature.\n\felds (between 2 mT - 4 mT), but the signal largely matches that in Fig. 5 (a). On the\nopposite end of the constriction (Fig. 5 (c)), close to the misfabricated edge, the primary\nresonance detected at higher \felds matches the measurements at the other two sites, but\nan additional high frequency resonance emerges, leaving our detection range between 5 and\n10 mT. At this site, the signal up to 5 mT appears to be broadband noise, with greater\ncontrast as the \feld increases. The line width also increases dramatically, from 156 MHz at\n10 mT at the opposite, pristine side of the constriction to 278 MHz at 10 mT. Micromagnetic\nsimulations of FMR in the double circle feature (5 (d)) match our experimental results well\nfor \felds stronger than 4 mT. We speculate that the disagreement between measurements\n10FIG. 5. ODFMR measurements using a nanodiamond located at di\u000berent sites along the edge\nof a circular double disk feature where the disks meet to form a constriction. Measurements are\nmade at three distinct sites: (a) Far away from the constriction;(b) Near the constriction; (c) On\nthe opposite end of the constriction, close to a misfabricated edge. (d) Micromagnetic simulations\nof FMR in the double circle feature.\nand simulations at lower \felds is caused by di\u000berences in the magnetic texture of the material,\nas the applied \feld will not saturate the ferromagnet and our simulation setup procedure\ndoes not perfectly replicate the history of the features.\nIV. SUMMARY\nWe have demonstrated a straightforward method to deterministically place NV-containing\nnanodiamonds at desired locations on lithographically patterned ferromagnetic thin \flms.\nUsing this placement, we performed local ODFMR measurements on single mesoscopic Py\nislands, revealing position-dependent variations in spin dynamical behavior that cannot be\n11detected using conventional FMR measurements of ensembles of patterned islands. After\ncon\frming that this technique worked on a simple rectangular island, we then applied it to a\nmore complex feature composed of two circles with a subtle fabrication error near the point\nof closest approach. In the vicinity of this fabrication error, we measured both a larger line\nwidth and an additional signal that did not appear in the control measurement away from the\nerror or in our micromagnetic simulations of pristine patterns. This \fnding shows that local\nODMFR measurements using targeted placement of nanodiamonds can provide information\nabout the in\ruence of defects on the spin dynamical behavior of patterned ferromagnets.\nMoving forward, we see this technique being used as a more general method of measuring\nlocalized magnetization dynamics in various patterned ferromagnetic structures. For exam-\nple, the technique could provide new insights into the properties of spin wave edge modes\npreviously detected using scanning FMRFM [20]. It could also be used for probing the\nlocal magnetization dynamics of arti\fcial spin ice arrays [29, 30]. 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Lett. 127, 117203 (2021).\n15"    },    {        "title": "0806.3315v1.Penetration_Depth_of_Transverse_Spin_Current_in_Ferromagnetic_Metals.pdf",        "content": "arXiv:0806.3315v1  [cond-mat.mes-hall]  20 Jun 20081\nPenetration Depth of Transverse Spin Current in\nFerromagnetic Metals\nTomohiro Taniguchi1,2,Satoshi Yakata3,Hiroshi Imamura1,and Yasuo Ando4\n1Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology,\nTsukuba, Ibaraki 305-8568, Japan\n2Institute of Applied Physics, University of Tsukuba, Tsuku ba, Ibaraki 305-8573, Japan\n3Nanoelectronics Research Institute, National Institute o f Advanced Industrial Science and Technology,\nTsukuba, Ibaraki 305-8568, Japan\n4Department of Applied Physics, Graduate School of Engineer ing, Tohoku University, Sendai 980-8579, Japan\nAbstract—The line width of the ferromagnetic resonance\n(FMR) spectrum of Cu/CoFeB/Cu/Co/Cu is studied. Analyzing\ntheFMRspectrumbythetheoryof spinpumping,we determined\nthe penetration depth of the transverse spin current in the C o\nlayer. The obtained penetration depth of Co is 1.7 nm.\nIndex Terms —Ferromagnetic resonance, SpinPumping,Trans-\nverse Spin Current, Gilbert damping\nI. INTRODUCTION\nTHERE is great interest in the field of current-driven\nmagnetizationdynamics(CDMD)becauseofitspotential\napplications to non-volatile magnetic random access memor y\nand microwave devices. The concept of CDMD was first\nproposed by Slonczewski [1] and independently by Berger [2]\nin 1996. In the last decade, many experimental studies have\nshown the evidences of CDMD [3],[4].\nTheoretical studies of CDMD have also been developped\n[5],[6]. The origin of the CDMD has been understood as the\ntransferofspinangularmomentumoftheconductingelectro ns\nto the magnetization of the ferromagnetic metal. One of the\nmostimportantquantitiesinCDMD isthepenetrationdeptho f\nthetransverse(perpendiculartothemagnetization)spinc urrent\nλt, over which the transfer of spin angular momentum is\nachived. However, there is a controversial issue regarding the\npenetration depth of the transverse spin current. The balli stic\ntheory of electron transport argues that λtis on the order\nof the lattice constant in conventional ferromagnets such a s\nFe, Co and Ni, and their alloys [7],[8]. On the other hand,\nthe Boltzmann theory of electron transport argues that λtis\non the order of a few nm [9],[10],[11]. However, only a few\nexperimental measurements of the penetration depth has bee n\nreported [12],[13].\nIn our previous paper [14], we studied the line width\nof the ferromagnetic resonance (FMR) spectrum in a\nferromagnetic(F)/nonmagnetic(N) metal five-layer system\n(N1/F1/N2/F2/N3), and showed that the line width of the\nF1layer depends on the thickness of the F 2layer due to\nspin pumping [15],[16]. Analyzing the FMR spectrum, the\npenetration depth of the transverse spin current of NiFe,\nManuscript receivedFig. 1. Schematic illustration of a nonmagnetic/ferromagn etic metal five-\nlayer system. Li(i= 1,2,3)is the thickness of the i-th nonmagnetic layer\nanddk(k= 1,2)is the thickness of the k-th ferromagnetic layer. The\nmagnetization m1is in resonance and precess around the z-axis with the\nangleθ. The magnetization m2is fixed along the z-axis.Ipump\nsandIN→F\ns\nare pumped spin current and backflow of spin current, respect ively.\nCoFe and CoFeB were obtained [14]. Our result seems to\nsupport the Boltzmann theory of electron transport. Howeve r,\nwe cannot compare our results with the results of [9],[10],[ 11]\ndirectly since only the penetration depth of Co is studied in\n[9],[10],[11]. In this paper, we study the line width of FMR\nspectrum of Cu/CoFeB/Cu/Co/Cu five-layer system, and de-\ntermine the penetration depth of Co. The obtained penetrati on\ndepth of Co, 1.7 nm, has good agreement with the results of\n[9],[10],[11].\nII. THEORY\nSpin pumping [15],[16] is, in some sense, the reverse\nprocess of CDMD, where the precession of the magnetization\nin the ferromagnetic layer generates spin current flowing in to\nthe adjacent layers. In a ferromagnetic/nonmagnetic metal\nmulti-layer system the Gilbert damping constant of the fer-\nromagnetic layer is enhanced due to spin pumping. Analyzing\nthe dependence of the Gilbert damping on the thickness of\nthe nonmagnetic layer the spin diffusion length, i.e., the\npenetration depth of spin current in the nonmagnetic layer i s\ndetermined.\nThe penetration depth of the transverse spin current of\na ferromagnetic metal, λt, is also determined in a similar\nway. Let us consider N 1/F1/N2/F2/N3metal five-layer system\nshown in Fig. 1, where mk(k= 1,2) is the unit vector\nalong the magnetization of the k-th ferromagnetic layer. The\nmagnetization of the F 1layer (m1) is in resonance with\nthe oscillating magnetic field, and pumps spin current Ipump\ns2\nflowing into the other layers. The precession axis of m1is\nalong the direction of the magnetization of the F 2layer (m2).\nSince the magnetization vector of Ipump\nsis perpendicular to\nm1[14] and the precession angle θis very small (about 1\ndeg), the dominant component of the magnetization vector of\nspin current flowing into the F 2layer is perpendicular to m2,\ni.e., the dominant component of the spin current flowing into\nthe F2layer is the transverse spin current. Thus, analyzing\nthe dependence of the FMR spectrum of the F 1layer on\nthe thickness of the F 2layer, the penetration depth of the\ntransverse spin current of the F 2layer can be determined.\nHowever, the conventional theory of spin pumping assumes\nthat the penetration depth of the transverse spin current is\nzero. Thus, we need to extend the theory of spin pumping by\ntaking into account the finite penetration depth [14].\nThe spin current pumped from the F 1layer is given by [15]\nIpump\ns=/planckover2pi1\n4π/parenleftbigg\ng↑↓\nr(F1)m1×dm1\ndt+g↑↓\ni(F1)dm1\ndt/parenrightbigg\n,(1)\nwhere/planckover2pi1is the Dirac constant and g↑↓\nr(i)is the real (imaginary)\npart of the mixing conductance. The pumped spin current\ncreates spin accumulation in the other layers. The spin ac-\ncumulation is given by\nµ=/integraldisplay\nεFdεTr[ˆσˆf], (2)\nwhereˆσis the Pauli matrix and ˆfis the non-equilibrium\ndistribution matrix in spin space. In general, the distribu tion\nmatrix of a ferromagnetic layer, ˆfF, is given by ˆfF=f0ˆ1 +\nfzm·ˆσ+fxt1·ˆσ+fyt2·ˆσ, whereˆ1is the2×2unit matrix,\nf0= (f↑+f↓)/2is the non-equilibrium charge distribution\nandfz=(f↑−f↓)/2is the difference in non-equilibrium dis-\ntribution between spin-up ( f↑)and spin-down ( f↓) electrons.\n(t1,t2,m) is a set of orthogonal unit vectors in spin space\nwheremis theunitvectorparallelto themagnetizationvector.\nfxandfyarethenon-equilibriumdistributionofthetransverse\nspin components. Spin accumulation in a nonmagnetic layer\nis defined in a similar way.\nThe spin accumulation induces a backflow of spin current.\nThe backflow of spin current flowing from the N ilayer to the\nFklayer is expressed in terms of the spin accumulation as\nINi→Fks=1\n4π/bracketleftBigg\n2g↑↑\n(Fk)g↓↓\n(Fk)\ng↑↑\n(Fk)+g↓↓\n(Fk){mk·(µNi−µFk)}mk\n+g↑↓\nr(Fk)mk×(µNi×mk)+g↑↓\ni(Fk)µNi×mk\n−t↑↓\nr(Fk)mk×(µFk×mk)−t↑↓\ni(Fk)µFk×mk/bracketrightBig\n,\n(3)\nwhereµNiandµFkare the spin accumulation of the N i\nand the F klayer, respectively. g↑↑(↓↓)is the spin-up (spin-\ndown) conductance and t↑↓\nr(i)is the real (imaginary) part of the\ntransmisson mixing conductance defined at the F/N interface .\nIn the conventional theory of spin pumping, the penetration\ndepth of the transverse spin current is assumed to be zero,\nand the last two terms in (3) is neglected [8].\nThe spin current given by (1) and (3) satisfies the boundary\nconditions of the continuity of the spin current. In general , thecurrent operator in spin space is given by [9]\nˆj=1\neˆC∂V\n∂x−ˆD∂ˆn\n∂x, (4)\nwheree(>0)is the absolute value of electron charge and V\nis the applied voltage. Since we are interested in the FMR\nline width, we assume V= 0.ˆC,ˆDandˆnare the2×2\nmatrices representing the conductivity, the diffusion con stant\nand the density of the non-equilibrium electron, respectiv ely.\nThe conductivity and the diffusion constant are expressed a s\nˆC=C0(ˆ1 +βm·ˆσ)andˆD=D0(ˆ1 +β′m·ˆσ), where\nC0=(σ↑+σ↓)/2,D0=(D↑+D↓)/2,β=(σ↑−σ↓)/(σ↑+σ↓)\nandβ′= (D↑−D↓)/(D↑+D↓).σ↑(↓)andD↑(↓)are the\nconductivityand the diffusion constant of spin-up (spin-d own)\nelectrons,respectively. βandβ′arethepolarizationofthespin\ndependent conductivity and diffusion constant, respectiv ely.\nTheconductivityandthediffusionconstantsatisfytheEin stein\nrelationˆC=e2ˆNˆD, where ˆNis the density of states. For\nsimplicity,we assume that β=β′in this paper. The distribution\nˆfand the density ˆnare related with each other via\nTr[ˆσˆn] =/integraldisplay\nεFdεRe/bracketleftBig\nTr/bracketleftBig\nˆσˆNˆf/bracketrightBig/bracketrightBig\n. (5)\nThe spin current is given by Is=(/planckover2pi1S/2)Re[Tr[ ˆσˆj]], where\nSis the cross section area. Using (2), (4) and (5), the spin\ncurrentIsis expressed in terms of spin accumulation µ. The\nspin current in a nonmagnetic metal is expressed in a similar\nway, but β=β′=0.\nThe diffusion equation of the spin accumulation is obtained\nby the continuity of the charge and spin current. In a non-\nmagnetic metal, the spin accumulation µNobeys the diffusion\nequation given by [17]\n∂2\n∂x2µN=1\nλ2\nsd(N)µN, (6)\nwhereλsd(N)is the spin diffusion length of the nonmag-\nnetic metal. The spin accumulation can be expressed as a\nlinear combination of exp(±x/λsd(N)). The longitudinal spin\naccumulation in a ferromagnetic metal, µL\nF= (m·µF)m,\nalso obeys the diffusion equation, and is expressed as a\nlinear combination of exp(±x/λsd(FL)), whereλsd(FL)is the\nlongitudinal spin diffusion length.\nWe assume that the transverse spin accumulation in a\nferromagneticmetal, µT\nF=m×(µF×m),obeysthefollowing\nequation [9]:\n∂2\n∂x2µT\nF=1\nλ2\nJµT\nF×m+1\nλ2\nsd(FT)µT\nF, (7)\nwhereλJ=/radicalbig\n(D↑+D↓)/planckover2pi1/(2J)is the spin coherencelength\n[8] andλsd(FT)=λsd(FL)//radicalbig\n1−β2is the transverse spin\ndiffusion length. Jrepresents the strength of the exchange\nfield. The transversespin accumulationis expressedas a lin ear\ncombination of exp(±x/l+)andexp(±x/l−), where1/l±=/radicalBig\n(1/λ2\nsd(FT))∓(i/λ2\nJ). Therefore, we define the penetration\ndepth of the transverse spin current as\n1\nλt= Re/bracketleftbigg1\nl+/bracketrightbigg\n. (8)3\nReferences [10],[11] show that the order of λJis a few nm\nfor NiFe and Co. The exchange interaction, which determines\nλJ, does not give any contribution to λsd(FL), i.e., there’s no\nrelation between λJandλsd(FL). If the order of λsd(FL)is a\nfew nm, for example NiFe, λt≃λsd(FL). On the other hand,\nfor Coλsd(FL)≫λJ, and therefore λt≪λsd(FL).\nThe spin current at the F 2/N2(N3) interface is given by\nIN2→F2s(−IN3→F2s). Soving the diffusion equations of spin\naccumulations of the N 3and F2layers, (6) and (7), with these\nboundary conditions, the backflow at the N 2/F2interface can\nbe expressed as [13]\nIN2→F2\ns=1\n4π/bracketleftBig\n˜g∗\n(F2)(m2·µN2)m2\n+˜g↑↓\nr(F2)m2×(µN2×m2)+ ˜g↑↓\ni(F2)µN2×m2/bracketrightBig\n,(9)\nwhere the conductance ˜g∗\n(F2)depends on the ratio d2/λsd(FL),\nwhered2is the thickness of the F 2layer. Similarly, the\nrenormalized mixing conductances, ˜g↑↓\nr,i(F2), depend on the\nratiod2/l+(F2). If the thickness of the N 3layer is thin enough\ncomparedtoitsspindiffusionlength, ˜g∗\n(F2)isequalto g∗given\nin [14], and ˜g↑↓\nr,i(F2)are given by\n/parenleftBigg\n˜g↑↓\nr(F2)\n˜g↑↓\ni(F2)/parenrightBigg\n=1\n∆/parenleftbigg\nK1K2\n−K2K1/parenrightbigg/parenleftBigg\ng↑↓\nr(F2)\ng↑↓\ni(F2)/parenrightBigg\n,(10)\nwhere∆ =K2\n1+K2\n2.K1andK2are given by\nK1= 1+t↑↓\nrηr+t↑↓\niηi, (11)\nK2=−t↑↓\nrηi+t↑↓\niηr, (12)\nwhereηr(i)= Re(Im) ηandη={gttanh(d2/l+)}−1, where\ngt=hS/(2e2ρF2l+), andρF2is the resistivity of the F 2layer.\nThe mixing conductance of the F 1layer in (1) and (3) is also\nreplaced by the renormalized mixing conductance.\nThe spin pumping modifies the Landau-Lifshitz-Gilbert\n(LLG) equation of the magnetization of the F 1layer as\ndm1\ndt=−γm×Beff+τ+α0m1×dm1\ndt,(13)\nwhereBeffis the effective magnetic field, γis the gyromag-\nnetic ratio and α0is the intrinsic Gilbert damping constant. τ\nis the additional torque due to the spin pumping given by\nτ=γ\nMSd1m1×{(Ipump\ns−IN2→F1\ns)×m1},(14)\nwhereMis the saturated magnetization of the F 1layer and\nd1is the thickness of the F 1layer. We assume that the spin\nrelaxation in the N 2layer is so weak that the spin current in\nthe N2layer is conserved, i.e., Ipump\ns−IN2→F1s=IN2→F2s.\nThen the dynamics of the magnetization of the F 1layer is\naffected by the F 2layer. We notice that the effects of the N 1\nand N3layer are quite small because, as mentioned below, the\nthickness of these layers are thin enough compared to its spi n\ndiffusion length in our experiments. The LLG equation (13)\nis rewritten as [14],[15],[18]\ndm1\ndt=−γeffm1×Beff+γeff\nγ(α0+α′)m1×dm1\ndt,(15)Fig. 2. The dependence of derivative of the FMR spectrum of Co FeB on\nthe thickness of Co layer, d2. The center of the horizontal axis, 75 mT, is the\nresonance magnetic field of CoFeB.\nwhere (γeff/γ) andα′is the enhancementof the gyromagnetic\nratio and the Gilbert damping due to the spin pumping,\nrespectively. Assuming that g↑↓\nr≫g↑↓\ni, in the limit of θ→0,\nα′is reduced as\nα′≃γ/planckover2pi1\n4πMd1S˜g↑↓\nr(F1)˜g↑↓\nr(F2)\n˜g↑↓\nr(F1)+˜g↑↓\nr(F2), (16)\nand(γeff/γ)≃1. We should note that if we neglect the\npenetration depth of the transverse spin current in the ferr o-\nmagnetic layer the mixing conductances are not renormalize d,\nand that the enhancement of the Gilbert damping constant,\nα′, does not depend on the thickness of the F 2layer. This is\nbecause the dominant component of the pumped spin current\nis perpendicular to the magnetization of the F 2layer.\nIII. EXPERIMENT\nWe performed FMR experiments on Cu(5nm)/CoFeB(5nm)\n/Cu(5nm)/Co( d2)/Cu(10nm) five-layer system shown in Fig.\n1 [16], where CoFeB layer corresponds to the F 1layer and\nCo layer corresponds to the F 2layer. Figure 2 shows the\ndependence of the derivative of the FMR spectrum of CoFeB\non the thickness of Co, d2. The width of the peak to peak in\nFig. 2, namely the linewidth of the FMR spectrum ∆B, is a\nlinear function of the Gilbert damping constant [19]:\n∆B= ∆B0+4πf√\n3γ(α0+α′), (17)\nwherefis the frequencyof the oscillating magnetic field. The\nline width of CoFeB depends on the thickness of Co through\nα′, as shown in Fig. 2. Thus,we can determinethe penetration\ndepth of the transverse spin current of Co by the line width\nof CoFeB. The enhancement of the gyromagnetic ratio does\nnot give any contributions to the line width.\nThe sample was deposited on Corning 1737 glass substrates\nusing an rf magnetron sputtering system in an ultrahigh\nvacuum below 4×10−6Pa and cut to 5 nm2. The Ar pressure\nduring deposition was 0.077 Pa. The FMR measurement was\ncarried out using an X-band microwave source ( f=9.4[GHz])\nat room temperature. The microwave power, modulation fre-\nquency, and modulation field are 1 mW, 10 kHz, and 0.1 mT,\nrespectively. The precession angle of the magnetization of the\nF1layer was estimated to be 1 deg. The resistivity of CoFeB4\nFig. 3. The dependence of the FMR spectrum, ∆B, of CoFeB layer on\nthe thickness of Co layer, d2. The filled circles represent experimental data\nand solid line is fit to the experimental data according to the theory with the\nfinite penetration depth of the transverse spin current in Co ,λt. The dotted\nline represents the case of λt= 0.\nand Co are 1252 Ω·nm and 210 Ω·nm [20], respectively. The\nmagnetization ( 4πM) and the gyromagnetic ratio of CoFeB\nare 1.66 T and 1.846 ×1011Hz/T, respectively.\nA Cu layer typically shows an enhanced (111) orientation\nand the Co layer on it also shows an induced (111) texture.\nThus, the Co layer is considered to be (111) texture.\nIn Fig. 3 the measured line width of the FMR, ∆B, of\nCoFeB layer is plotted with full circles against the thickne ss\nof Co layer, d2. The solid line is a fit to the experimental data\naccording to the theory with the finite penetration depth of t he\ntransverse spin current λt. The dotted line is the calculated\nline width assuming λt= 0. If the Co film is not continuous\nbut consists of a Co islands, the thickness of the Co island\nis somewhat thicker than the nominal thickness. However,\nthe effect of the Co islands is not so significant because the\nimportant quantity in our analysis is the mean thickness whi ch\nis almost same as the nominal thickness.\nThe best fitting parameters are as follows. The real part of\nthe mixing conductances per unit area, g↑↓\nr/S, of CoFeB and\nCo are 128 nm−2and 20 nm−2, respectively. Although these\nvalues are determined by fitting, they have good agreement\nwith theab initio calculations [8]. For simplicity, we assume\nthatt↑↓\nr=t↑↓\ni, where the values of t↑↓\nr,i/Sof CoFeB and Co are\n0.8nm−2and6.0nm−2,respectively.Thespindiffusionlength\nof CoFeB and Co layer are 12 nm and 38 nm, respectively\n[20],[21]. The polarization of the conductance βare 0.56 for\nCoFeB and 0.31 for Co [20],[21]. We take g↑↓\ni/S=1.0nm−2\nand2g↑↑g↓↓/(g↑↑+g↓↓)S= 20nm−2both CoFeB and Co\n[15]; these are not important parameters for fitting. The spi n\ndiffusion length and resistivity of Cu are taken to be 500 nm\nand 21Ω·nm [22].\nThe obtained value of the penetration depth of Co is λt=\n1.7nm. References [9],[10],[11] estimate λt=√\n2λJ, and\npredict that λJof Co with (111) texture is 1.1 nm. Thus, we\nhave good agreement with [9],[10],[11].\nIV. CONCLUSION\nIn conclusion,we studythe line width of the FMR spectrum\nof Cu/CoFeB/Cu/Co/Cu five-layer system. The line width of\nthe CoFeB layer depends on the thickness of the Co layer due\nto spin pumping. 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Phys. , vol.91, pp.8105-8107,\n2002.\n[22] J. Bass and W. P. Pratt. Jr., ”Spin-diffusion length in m etals and alloys,\nand spin-flipping at metal/metal interfaces: an experiment alist’s review,”\nJ. Phys.: Condens. Matter , vol.19, pp.183201, 2007."    },    {        "title": "0905.2353v1.A_compact_apparatus_for_studies_of_element_and_phase_resolved_ferromagnetic_resonance.pdf",        "content": "arXiv:0905.2353v1  [cond-mat.mes-hall]  14 May 2009A Compact Apparatus for Studies of Element and Phase-Resolv ed Ferromagnetic\nResonance\nD.A. Arena∗, Y. Ding, and E. Vescovo\nNational Synchrotron Light Source, Brookhaven National La boratory, Upton, NY, U.S.A.\nS. Zohar, Y. Guan†, and W.E. Bailey\nMaterial Science Program, Department of Applied Physics, C olumbia University, New York, NY, U.S.A.\n(Dated: June 14, 2018)\nWe present a compact sample holder equipped with electromag nets and high frequency transmis-\nsion lines; the sample holder is intended for combined x-ray magnetic circular dichroism (XMCD)\nand ferromagnetic resonance measurements (FMR). Time-res olved measurements of resonant x-ray\ndetected FMR during forced precession are enabled by use of a rfexcitation that is phase-locked to\nthe storage ring bunch clock. Several applications of the co mbined XMCD + FMR technique are\npresented, demonstrating the flexibility of the experiment al design.\nI. INTRODUCTION\nAmong the most powerful methods developed to ex-\namine magnetic materials are ferromagnetic resonance\n(FMR) and x-ray magnetic circular dichroism (XMCD).\nFMR is certainly the more venerable technique and has\nbeen used to examine numerous phenomena ( e.g.gyro-\nmagnetic ratios, magnetic anisotropy energies, damping\nand relaxation processes [1, 2, 3], etc.). In thin films,\nthese capabilities have been extended to investigate is-\nsues suchas magneticreorientationtransitions, substrate\neffects in magneto-crystallineanisotropy, and coupling in\nmultilayer systems. FMR is a well-established approach\nand is supported by a firm theoretical foundation that\ncanbe applied in the interpretationofexperimentalspec-\ntra.\nBycontrast,XMCDisamuchmorerecently-developed\ntechnique [4, 5]. XMCD can also be applied to examine\na wide class of issues including overlapping topics such\nas magnetic anisotropies and spin-reorientation transi-\ntions. Moreover, because XMCD is based on core-level\nspectroscopy, it provides elemental sensitivity and can\nbe used, with appropriate care and via the application\nof sum-rules analyses, to determine absolute spin ( µs)\nand orbital ( µl) moments [6, 7, 8]. These characteris-\ntics make XMCD an indispensable tool for the investiga-\ntion of complex ferromagnetic systems such as alloys and\nother compounds and layered thin-film structures.\nThe combination of these two techniques has been a\ngoal of several research groups over the past few years\nand a few different implementations of XMCD + FMR\nhave been developed. The approaches used in these\nefforts typically fall into two groups: static, or time-\naveraged, measurements and time-resolved techniques.\nThe time-averaged methods generally operate with con-\n∗e-mail address: darena@bnl.gov\n†present address: SoloPower Inc., 5981 Optical Court, San Jo se,\nCA 95138tinuous wave ( cw) microwave excitations. While the de-\ntailsvary,inatypicalimplementationamicrowavefieldis\nused to excite uniform precession modes, and the change\nin the projection of an elemental moment ( ∆µZ) paral-\nlel to the incident x-ray wave vector ( ki) is monitored as\neither the microwave frequency or the magnetic field is\nswept through resonance [9, 10]. The absorption of the\ncwexcitation results in a change in ∆µZ. In general, in\nsuchtime-averagedmeasurementsthe rfexcitationbears\nno particular phase relation with the arrival of the x-rays\nand hence all frequencies are accessible.\nTime-resolved XMCD with continuous wave excita-\ntions, have been implemented to image sub-GHz gy-\nrotropic motion of vortex dynamics in mangetic nanos-\ntructures [11]. The time-resolved approach has been\nextended by our group [12, 13, 14, 15, 16]and other\nresearchers [17] to examine GHz-range FMR with\nelemental-specificity. An advantage of time-resolved\ntechniques is that the phase information between the cw\nexcitation and the precessing magnetic moments in the\nsample is preserved, permitting a flexible exploration of\nthe multi-dimensional phase space (excitation frequency,\nphase of the response, photon energy, and applied field)\ninherent in the experiment.\nWe present in this article an experimental apparatus\nthat allows for the combination of time-resolved XMCD\nand FMR. We designed a flexible measurement sys-\ntem, which is suitable for x-ray absorption spectroscopy\n(XAS), as well as conventional XMCD and in-situFMR.\nTime-resolved studies are enabled by the use of phase-\nlockedrfgeneration electronics. The hardware is com-\npact as it fits in the bore of a standard 4.5” UHV flange.\nThe measurements presented in this article were under-\ntaken at the soft x-raybeam line 4-ID-C at the Advanced\nPhoton Source (APS) at Argonne National Lab. The\nbeam line is equipped with an innovative hybrid elec-\ntromagnetic circularly polarizing undulator (CPU) [18],\nwhich can provide arbitrary x-ray polarization (linear\nhorizontal or vertical, right or left circular). However,\nwe stress that the apparatus and electronics described\nin this article are compact and easily portable to other2\nbeam lines and synchrotron facilities.\nII. EXPERIMENTAL SETUP\nA. Hardware: Sample Holder and Vector Magnet\nFMR in the linear regime implies small angular mo-\ntions. Therefore, regardless of the geometry of the mea-\nsurement ( e.g.transverse or longitudinal), the require-\nments on sample and beam stability are severe. Further-\nmore, near resonance, the phase of the magnetization\nmotion changes rapidly, and thus the field homogeneity\nand stability over the area sampled by the photon beam\nis critical. To address these restrictions, we designed a\ncustom sample holder to house the sample, rftransmis-\nsion lines and waveguides, and an in-plane vector magnet\nwhich provides vertical and horizontal fields. In addition\nto these restrictions, a goal of the design was compact-\nness as the entire assembly is intended to fit into the bore\nof ax,y,-ztranslation stage with a clear opening of 2.5”.\nFig. 1 presents a perspective view of the custom sam-\nple holder as well as a cross section through the plane\ncontaining the magnets and the sample. The coordinate\nsystem is chosen so the sample lies in the x-yplane, with\nthe sample normal along the zdirection, while the inci-\ndent synchrotron radiation is restricted to the x-zplane.\nThe angle of the photon beam with respect to the sam-\nple normal is varied by rotating the sample manipulator\nabout the y-axis.\nRF fields were delivered to the sample using a SMA\nend launch adapter to a grounded coplanar waveguide\n(CPW) transition, similar to those used in pulsed induc-\ntive microwave magnetometry[19, 20]. The orientation\nof the waveguide’s center conductor is along the y-axis\nand thus the rffield (Hrf) is in-plane and along the x-\ndirection. Theendlaunchadapter(SouthwestMicrowave\n292-06A-5 or similar), 1/2” in width, is screw-mounted\nto through-holes on the CPW, and has a coaxial cen-\nter pin diameter and dielectric diameter size-matched to\nthe CPW center conductor and ground shield to ground\nshield spacing, respectively. The coplanar waveguidewas\nformed from 10 mil thick, double-sided Au plated Rogers\nlaminate (R5880), with ǫr= 2.2, center conductor width\n21.4 mil, and ground-shield to ground shield spacing 36.5\nmil, with dielectric spacing on each side of the center\ntrace of 7.5 mil. The total length of the center trace is\nonly 327 mil.\nAn innovative feature for the transmission experiment\nwas the formation of a CPW transparent to x-rays, al-\nlowing x-rays to pass through the magnetic film and be\nmeasured at a photodiode behind the sample. We have\naccomplished this by drilling a 10 mil ( ≃250µm) di-\nameter pinhole through the center trace 50 mils above\nthe lower edge of the CPW, leaving ∼13 mils of center\ntrace on either side ofthe pinhole. Thin film samples, de-\nposited on Si 3N4membranes, are mounted, using epoxy,\nover the hole. The input rfpower is reflected by a dead\nFIG. 1: (Color Online) View of vector magnet and sample\nenvironment. Right Panel: perspective with coordinate axe s\nand indicating beam direction. Left panel: Cross section\nthrough the sample plane: 1: support tube; 2: high frequency\ncoaxial cable with SMA termination; 3: top and bottom ver-\ntical pole pieces; 4: top and bottom electromagnet coils; 5:\nend launcher; 6: sample; 7: yoke for horizontal magnetic fiel d;\n8: electromagnet coil for horizontal field.\nshort at the end of the CPW, formed by terminating\nthe dielectric spacing, or (equivalently) allowing the Au\nground shield and center conductor to continue for a 10\nmil width at the CPW edge opposite the end launch.\nThis places the optically accessible region of the sample\nat a voltage node and rffield maximum. An upper limit\nfor therffield can be given by the DC field from a planar\nconductor,\nBrms\nrf=µ0\n2W/radicalbigg\nP\nZ0(1)\nwhereW= 563µmis the CPW center conductor\nwidth, and Z0≃50Ω is the characteristic impedance of\nthewaveguide,wheretheexpressionassumesfilmspacing\nfrom the CPW much less than the CPW width, reason-\nably well justified in the present case. For the maximum\ninput power of 30 dBm (1 W), and present center con-\nductor width, this is Brms\nrf≤163.6µT, orHpp\nrf≤2.3 Oe.\nIn the vicinity of the hole, the rffield will likely be less,\nalthough more detailed simulation would be necessary to\nestimate its value.\nThe sample environment contains two electromagnets\noriented alongthe y-direction. A high bandwidth coaxial\nrfcablepassesaxiallythroughaholeinthetoppolepiece\nand up to a rfvacuum feedthrough (not shown). These\ntwo electromagnets provide a magnetic field referred to3\nas the vertical bias field ( HB), which is normal to both\ntoHrfand the incident photon beam. Hrfforces the\nmagnetization in the sample to precess about HB(or,\nmore accurately, about Heff, which is the sum of HB\nand the anisotropy and dipolar fields). At a fixed driving\nfrequency, the precession of Mcan be tuned by varying\nHB. A third electromagnet is connected to a C-shaped\nyoke and provides an in-plane magnetic field along the\nx-direction ( i.e.horizontal and orthogonal to HB). This\nfield (Hx)is used to acquire conventional XMCD spectra\nand element-specific hysteresis curves.\nThe pole pieces for all three electromagnets were ma-\nchined from conventional low carbon-content steel. To\nreduce the internal stress caused by the facbrication pro-\ncesses, the pole pieces were annealed in an inert atmo-\nsphere at 960◦C for 8 hours, followed by a slow cool\ndown. This procedure produced relatively “soft” mag-\nnetic cores with low remanent magnetization. In the lin-\near region of the magnetization curve, the slope of the\nHvs.applied current curve is 32 Oe / amp and 38 Oe\n/ amp for the vertical and horizontal electromagnets, re-\nspectively. Ohmic heating of the coils, combined with\nthe lack of conductive and convective cooling in vacuum,\nlimits the sustainable current to ∼3 amps. However,\nthe coils are equiped with cooling strips made of thin\nCu foil. Future modifications will add in-situcooling to\nthe three electromagnets, as well as sample cooling for\ntemperature-dependent measurements.\nA significant advantage of the experimental system\npresented is the simplicity of sample preparation. The\nsamples are planar ferromagnetic thin films, deposited\nonto commercially available, 100 nm thick Si 3N4mem-\nbranes. In the spectral range of interest ( Ledges of\n2ndrow transition metals and Medges of rare earth el-\nements) such membranes have a high degree of trans-\nmission ( >∼75%). The membranes are supported by Si\nframes (500 µm thickness) and these are attached to the\nwaveguide using an insulating adhesive.\nSpectroscopic x-ray absorption data and timing scans\nare acquired in transmission mode. Assuming an appro-\npriate selection of materials and sample thickness (re-\nquiredtoavoidsaturationeffects), transmissionmeasure-\nments have a significant advantage over electron or fluo-\nrescence yield techniques in improved signal-to-noise ra-\ntio. A related benefit is simplified detection utilizing a\nstandard soft x-ray photodiode operated in current am-\nplification mode. Also, in contrast to pulse counting,\nnoise reduction via lock-in amplification is implemented\nin a straightforward fashion.\nB. Microwave Excitation and Timing Electronics\nIn our measurements, as Hrfdrives the precession of\nM, the projection of Malong the photon beam direc-\ntion (Mproj(t)) is sampled stroboscopically by the x-ray\npulses. This requires an rfsignal in the microwave re-\ngion of the spectrum (in our case, ∼1 to 4.5 GHz) that/X74/X69/X6D/X65 /X42/X75/X6E/X63/X68/X20/X43/X6C/X6F/X63/X6B \n/X7E/X38/X38/X20/X4D/X48/X7A \n/X66/X72/X65/X71/X2E /X4C/X50/X4E/X20/X43/X6F/X6D/X62 \n/X66/X72/X65/X71/X2E /X42/X50/X46 \n/X28/X47/X48/X7A/X29\n/X30\n/X31\n/X2D/X31 /X30\n/X31\n/X2D/X31 \n/X52/X65/X66 /X49/X6E /X4F/X75/X74 /X52/X46/X20/X53/X77/X69/X74/X63/X68/X20/X2F \n/X50/X6F/X77/X65/X72/X20/X4D/X6F/X64/X75/X6C/X61/X74/X69/X6F/X6E \n/X50/X6F/X77/X65/X72 \n/X41/X6D/X70/X6C/X69/X66/X69/X65/X72 /X44/X69/X72/X65/X63/X74/X69/X6F/X6E/X61/X6C \n/X43/X6F/X75/X70/X6C/X65/X72 \n/X54/X6F/X20/X52/X46/X20/X50/X6F/X77/X65/X72 \n/X4D/X65/X61/X73/X75/X72/X65/X6D/X65/X6E/X74 \n/X58/X2D/X52/X61/X79 \n/X70/X68/X6F/X74/X6F/X2D \n/X64/X69/X6F/X64/X65 /X49/X6E/X63/X69/X64/X65/X6E/X74 \n/X42/X65/X61/X6D \n/X54/X54/X4C \n/X54/X72/X69/X67/X67/X65/X72 \n/X28/X31/X20/X6B/X48/X7A/X29 \n/X4C/X6F/X63/X6B/X2D/X49/X6E \n/X41/X6D/X70/X6C/X69/X66/X65/X72 /X54/X6F/X20/X44/X41/X51 \n/X43/X6F/X6D/X70/X75/X74/X65/X72 /X54/X72/X61/X6E/X73/X2E \n/X42/X65/X61/X6D \n/X53/X61/X6D/X70/X6C/X65 \n/X6F/X6E/X20/X43/X50/X57 /X44/X69/X67/X69/X74/X61/X6C \n/X44/X65/X6C/X61/X79 \nFIG. 2: Simplified schematics of rfand signal detection elec-\ntronics. The generation of the phase-locked rfexcitation fol-\nlows heavy solid lines while the detection of the x-ray signa l\nproceeds along the light dashed lines. Note that the lock-in\ndetection branch may be bypassed bysetting the TTL trigger\nto “high.”\nmaintains a well-defined phase relationship with the pho-\nton bunch clock. Such phase-locking imposes a further\nrestriction that the rfexcitation frequency must be a\nharmonic of the bunch clock frequency. Fig. 2 presents\nasimplifiedschematicoftheelectronicsdevelopedtogen-\nerate this rfsignal. The path of the microwave genera-\ntion is indicated by the heavy solid line while the x-ray\nphotodiode and lock-inoutput and controlsignalsarede-\npicted with a light dashed line. For clarity, intermediate\nattenuation and amplification conditioning stages have\nbeen ommitted.\nTherfgeneration sequence starts with the photon\nbunch clock. For our time-resolved measurements, we\nutilize special operating mode 4 (SOM-4) at the APS,\nwhich is available several times a year. In SOM-4, every\nfourthrfbucket in the storage ring is populated with\nelectrons, which results in an x-ray pulse repetition fre-\nquency of ∼88 MHz. The bunch clock signal is fed into\na local digital delay generator. After the delay stage,\na high-bandwidth switch, triggered by a periodic TTL\nsignal that also serves as the reference for the lock-in\namplifier of the detection circuit (see below), modulates\nthe signal on and off. An amplified version of the delayed\nbunch clocksignalis directedintoa lowphasenoisecomb\ngenerator (LPN comb), which generates the harmonic\nspectrum of the 88 MHz input signal. A tunable narrow\nband pass filter (BPF) is used to select the desired har-\nmonic of the input signal. After the final amplification\nstage in the circuit, a directional coupler is directs the4\nsignal to the CPW. The reflected signal is detected with\na microwave diode or rfpower meter at the auxiliary\nport of the coupler.\nFor fixed-frequency, conventional FMR measurements,\nthereflected rfpowerismonitoredatthedirectionalcou-\npler while sweeping HB. For the XMCD measurements,\nthe signal from the photodiode is amplified by use of a\ncurrent preamplifier (not shown). To improve the signal\nto noise ratio in the small-amplitude XMCD+FMR mea-\nsurements, the power of the cwexcitation is modulated\nat∼1 kHz by the rfswitch, located between the bunch\nclock and the LPN comb. The signal from the current\npre-amplifier is directed to a lock-in amplifier to improve\nthe signal to noise ratio. Note that the rfpower mod-\nulation and noise suppression via lock-in amplification\ncan be bypassed by setting the the TTL control signal to\ntherfswitch to“high.” In this configuration, the signal\nlevelfromthephotodiodeduringdrivenprecessioncanbe\ncompared readily to hysteresis measurements, providing\na simple angular calibration for signal levels.\nIII. APPLICATIONS\nA. Conventional XMCD Spectroscopy and\nElement-Specific Magnetometry\nAs mentioned, the design of the apparatus depicted in\nFig. 1 is quite flexible. By using the horizontal electro-\nmanget (item 8 in Fig. 1), Hxcan be used to provide an\nalternating saturating field for soft ferromagneticmateri-\nals, permitting measurement of standard XMCD spectra\nin field-switching mode with fixed x-ray helicity. An ex-\nample of such spectroscopy is presented in Fig. 3, repro-\nduced from ref. [13]. The figure shows the XMCD spec-\ntra, acquiredin transmissionmode, ofamagnetictrilayer\n[(Si3N4substrate/Ni 81Fe19(25 nm)/Cu(20 nm)/Co 93Zr7\n(25 nm)/Cu cap(5 nm)], similar to technologically im-\nportant giant magneto-resitance (GMR) stacks found in\nmany modern magnetic field sensors. The spectra were\nacquired in a single sweep with a reasonable dwell time\nper point ( ∼1 s). High-quality XMCD spectra, clearly\nshowing the spin-orbit split L3(2p3/2) andL2(2p1/2)\ncore levels, are acquired from all three magnetic elements\nin the multilayer structure. The low noise in the data in-\ndicates that the vibration of the sample holder relative\nto the photon beam and the photodiode is low.\nThe insets to Fig. 3 show element-specific hysteresis\ncurves acquired by settting the photon energy to the L3\nabsorption edges of Fe, Ni or Co and then sweeping Hx.\nOne important benefit of employing a core-level spec-\ntroscopic technique is immediately apparent in compar-\ning the hysteresis loops of the Fe and Ni edges with the\nloop measured at Co L3edge: the Co 93Zr7clearly has a\nlarger coercive field than the Ni 81Fe19layer. More rele-\nvant to the element-resolved FMR scans (presented be-\nlow) is the angular calibration provided by the hysteresis\nmeasurements; the total magnitude of the photodiode/X2D/X32/X30 /X30/X46/X65/X20/X58/X4D/X43/X44/X20 /X28/X61/X2E/X75/X2E/X29 \n/X37/X35/X30 /X37/X34/X30 /X37/X33/X30 /X37/X32/X30 /X37/X31/X30 /X37/X30/X30 \n/X50/X68/X6F/X74/X6F/X6E/X20 /X45/X6E/X65/X72/X67/X79/X20 /X28/X65/X56/X29 /X2D/X32/X30 /X30/X32/X30 /X4E/X69/X20/X58/X4D/X43/X44/X20 /X28/X61/X2E/X75/X2E/X29 \n/X39/X30/X30 /X38/X39/X30 /X38/X38/X30 /X38/X37/X30 /X38/X36/X30 /X38/X35/X30 /X2D/X32/X30 /X30/X32/X30 /X43/X6F/X20/X58/X4D/X43/X44/X20 /X28/X61/X2E/X75/X2E/X29 \n/X38/X33/X30 /X38/X32/X30 /X38/X31/X30 /X38/X30/X30 /X37/X39/X30 /X37/X38/X30 /X37/X37/X30 \n/X31/X30 \n/X30\n/X2D/X31/X30 /X46/X65/X20/X4C/X33/X20/X53/X69/X67/X6E/X61/X6C/X20/X28/X61/X2E/X75/X2E/X29 \n/X2D/X32/X30 /X30 /X32/X30 /X32/X30 \n/X31/X30 \n/X30\n/X2D/X31/X30 \n/X2D/X32/X30 /X4E/X69/X20/X4C/X33/X20/X53/X69/X67/X6E/X61/X6C/X20/X28/X61/X2E/X75/X2E/X29 \n/X2D/X32/X30 /X30 /X32/X30 /X2D/X32/X30 /X2D/X31/X30 /X30/X31/X30 /X32/X30 /X43/X6F/X20/X4C/X33/X20/X53/X69/X67/X6E/X61/X6C/X20/X28/X61/X2E/X75/X2E/X29 \n/X2D/X32/X30 /X30 /X32/X30 \nFIG.3: (Color online)XMCDspectraandelement-specichys-\nteresis measurements of the Ni 81Fe19/Cu/Co 93Zr7trilayer.\nThe hysteresis curves were measured at the photon energies\nindicated by the arrows (707.5eV for Fe, 851.5 eV for Ni and\n778 eV for Co). The XMCD signal levels at saturation in\nthe hysteresis curves provide an angular calibration for ti me-\nresolved XMCD measurements. Reproduced from ref. [13]\nsignal in sweeping the field from ±Hx,maxcorresponds\nto a change of the direction magnetization by 180◦(as-\nsuming the magnetization of the sample is saturated at\nthe extremal values of the applied field).\nB.In-Situ FMR Spectroscopy\nThe electronics and hardware described above can be\nusedinastraightforwardfashiontomeasureconventional\nFMR spectra. In these experiments, a sinusoidal rfsig-\nnal in the frequency range of ∼1-4.5 GHz is introduced\nalong the high bandwidth coaxial transmission line. The\nreflected rfpowerismonitoredas HBisvaried. Measure-\nments aretypically conducted using a lock-in amplifierto\nimprove the signal-to-noise ratio. With this set-up, the\nresonance fields and FMR linewidths can be determined\nbefore proceeding with the time-resolved measurements.\nExamples of such in-situmeasurements can be found in\nreferences [13, 15]. The conventional FMR spectra pre-\nsented in these references are static measurements which5\nintegrate the absorbed power throughout the sample.\nAdditional details on the motion of elemental moments\nis revealed by utilizing XMCD and time-resolution.\nC. Measurement of Precession Orbits\nWith use of an rfexcitation that is phase-locked with\nthe photon bunch clock at the synchrotron, the projec-\ntion of the precessing elemental moments can be sam-\npled stroboscopically at a fixed delay. The time-varying\nprojection of the precession orbit is recorded by sweep-\ning the variable delay of the rfrelative to the photon\nbunch clock. An example is presented in Fig. 4 (adapted\nfrom ref. [13]). The data were acquired from the same\nNi81Fe19/Cu/Co 93Zr7trilayer sample, at HB= 40 Oe,\nnear the main resonance for the Ni 81Fe19layer at the\nexcitation frequency of 2.3 GHz.\nThe variation in the signal measured in the element-\nspecific hysteresis loops provides an angular calibration\nfor the oscillatory signal. As the functional form of the\nprecession orbit, projected onto the photon beam direc-\ntion, is a simple sinusoid, a fit to the data results in high-\nprecision determination for the amplitude and phase of\nthe orbit. The amplitude of the oscillation has been mea-\nsured from ∼1.5◦down to∼0.1◦, with an estimated error\nas low as ∼0.1◦. The estimated error on the phase of the\noscillation (relative to the photon bunch clock) can be as\nlow as 2◦of the full oscillation period, which corresponds\ntoanoveralltime resolutionof2psatthe2.3GHzexcita-\ntion frequency used in this example. This is considerably\nsmaller than the ∼70 ps bunch length (FWHM) of the\nphoton bunches from the storage ring.\nD. Phase-Resolved XMCD Spectroscopy\nBy selecting a specific time delay between the rfex-\ncitation and the photon bunch clock, the relative phase\nof the oscillations, and hence Mproj(t), can be held con-\nstant, permitting variation of other external parameters.\nFig. 5 presents one option, where the photon energy is\nswept through relevant absorption edges of the sample\nwhileHBis held constant. In this example, the sam-\nple consists of a single ferromagnetic layer, Ni 81Fe19[25\nnm], with a 5 nm Cu cap. Panel (A) of Fig. 5 (top)\nshows a representative timing scan acquired at the Fe L3\nedgenearthe resonanceconditionforthe Ni 81Fe19film at\nf=2GHz. The arrowsin the figurepoint to specific delay\nvalues for the energy scans presented in panel (B). The\npoints are selected to span more than half a cycle of the\nprecession orbit, and thus are assured to cover positive\nandnegativeextrema(nearpoints 4and12, respectively)\nand an antinode (near point 8).\nPanel(B)inFig. 5(bottom) presentsthe energyscans\nacrosstheFe L3andL2edges. Thespectrawereacquired\nat fixed helicity of the CPU and with a point spacing of\n0.5 eV and in a single scan of ∼10 minutes. The data1.5\n1.0\n0.5\n0.0Cone Angle (deg.)\n50 40 30 20\nHB (Oe)(b)\n Fe\n Co\n (a)\n200\n150\n100\n50\n0Phase (deg.)\n50 40 30 20\nHB (Oe)(c)(b)4\n3\n2\n1\n0Cone Angle (deg.)2.0 1.5 1.0 0.5 0.0\nDelay (ns)(a) Ni Fe  Co\nFIG. 4: (Color online) Panel (a): representative time-reso lved\nXMCD data of a Ni 81Fe19/Cu/Co 93Zr7trilayer undergoing\nforced precession at 2.3 GHz. The solid lines are fits to a\nsinusoidal function. Panel (b): extracted values for the am -\nplitude of the forced precession orbits, resolved to the Fe i n\nthe Ni 81Fe19layer or the Co in the Co 93Zr7layer. Panel (c):\nthecorrespondingphaseoftheprecession orbits. Inpanels (b)\nand (c), the dashed lines are simulations based on indepen-\ndent layers while the solid lines are simulations which assu me\nweak coupling. See reference [13] for details.\nclearly show the change in the sign of the XMCD signal\nbetween the L3andL2edges, and initially the dichro-\nism of the L3edge is positive. The overall intensity of\nthe dichroism starts off rather low, and increases in in-\ntensity as the delay value approaches point 4, close to\nthe positive maximum of the oscillating Mproj(t). Af-\nter passing through this maximum, the intensity of the\nXMCD signal decreases and disappears at around point\n8, where Mproj(t) = 0. Upon increasing the delay delay\nbetween the rfand the photon bunch clock, the dichro-\nism signal remerges, but now, as Mproj(t)<0, the sign\nof the dichroism is reversed. With a further increase of\nthe delay, the intensity again increases, and the magni-\ntude reaches another maximum at the negative extrema,\nnear point 12 in Panel (A) of Fig 5.\nThe data presented in Fig. 5 clearly show that phase-\nresolvedXMCD spectra can be acquiredefficiently. With\nno significant changes to the apparatus, high-quality\nXMCD spectra can be acquired at specific points of the\nprecessionorbit. In the future, by use ofsum-rulesanaly-\nses on these spectra, the spin-orbit ratio ofthe precessing\nelemental moments can be monitored with the same level\nof phase precision ( ∼2◦) as has been demonstrated in the\ndata in Fig. 4.6\nE. Element-Resolved Complex Susceptibility ( χ)\nPanel (A) of Fig. 6 presents the same type of delay\nscan as is found in Panel (A) of Fig. 5. However, panel\n(B) of Fig. 6 presents a different variation of the experi-\nmental parameters. Instead of keeping the relative phase\nandHBfixed and varying the photon energy, in Fig. 6\nthe photon energy is held constant at the Fe L3edge\nwhileHBis swept from 0 through the resonance condi-\ntion. The HBscans are acquired at specific delay points,\nindicated by the arrows in panel (A) of Fig. 6 . As the\ndelay values define a specific relative phase between the\nrfand the photon bunch clock, the HBscans can be\ninterpreted as the phase and element-resolved complex\nmagneticsusceptibility ( χ=χ′+iχ′′), in this case χ(Fe)\nin the Ni 81Fe19film. By varying the phase, the in-phase\ncomponent of the susceptibility ( χ′) or the out-of-phase\ncomponent ( χ′′) can be measured in an element-specific\n0.63\n0.62\n0.61\n0.60\n0.59\n0.58\n0.57\n0.56Lock-In Signal (arb. units)\n730 720 710 700\nPhoton energy (eV)1.16\n1.15\n1.14\n1.13\n1.12Lock-In Signal (arb. units)1.0 0.8 0.6 0.4 0.2 0.0Delay (ns)\n14\n812\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12(A)\n(B)\nFIG. 5: Panel (A): Timing delay scan acquired at 707 eV (Fe\nL3edge) from a Ni 81Fe19(25 nm) sample at an excitation\nfrequency of 2 GHz. Panel (B): Photon energy scans at the\ndelay points indicated in the top panel. Note the variation i n\nintensity of the dichroic signal, as well as the reversal of t he\ndichroism on the L3andL2edges.12\n10\n8\n6\n4Lock-In Signal (arb. units)\n1.2 0.8 0.4 0.0\nControl Voltage ( ∝ HB )20\n16\n12\n8Lock-In Signal (arb. units)1.0 0.8 0.6 0.4 0.2 0.0Delay (ns)\n13579111315171921(A)\n(B)1715\n21\nFIG. 6: Panel (A): Timing delay scan acquired at 707 eV\n(FeL3edge) from a a Ni 81Fe19(25 nm) sample. Panel (B):\nMagnetic field sweep scans at varying timing delays, corre-\nsponding to the point labeled in panel (A), with the photon\nenergy fixed at the Fe L3edge.\nfashion.\nAs was the case with the energy scans, the HBscans\nshow a dramatic variation with the relative phase be-\ntween the rfand the photon bunch clock. Panel (B) in\nFig. 6 presents the data (open circles) and also a fit to a\ncomplex Lorentzianfunction (thin solid line). At point 1,\nχ(Fe) is asymmetric, with both real and imaginary con-\ntributions to the susceptibility. At approximately point\n3,theresponsetosweeping HBisasymmetricandpurely\nimaginary Lorentzian; at this condition, the response of\nthe system is absorptive and 90◦out-of-phase with the rf\ndrivingfield. Bypoint12,atthenodalpointof Mproj(t),\nthe response is purely real and in-phase the rf. Beyond\nthis nodal point, the response is again asymmetric until\naround point 19, where χ(Fe) is again symmetric, al-\nthough the reversal of the sign of Mproj(t) causes the\nLorentzian function to be inverted.7\nIV. OUTLOOK AND CONCLUSIONS\nAs can be seen in the various applications presented\nabove, the combination of a compact vectormagnet com-\nbined with an rfexcitation sourcethat is phase-lockedto\nthe photon bunch clock produces a powerful tool to ex-\namine issues the dynamic response of scientifically- and\ntechnologically-relevant systems and materials. Static\nmeasurements of materials properties permit the identifi-\ncation of elements and even chemical species in a sample\nwhile the element-specific magnetometry can be used to\nestablishmagneticanisotropies. Theresonancefieldsand\nlinewidth, averaged across the sample, are easily mea-\nsured via in-situFMR scans. Extremely high-precision\nassessments of the precession orbit amplitude and phase\nare determined by timing delay scans at selected values\nofHB. Finally, we have recently upgraded these capa-\nbilities by use of rfpower modulation and lock-in am-\nplification. These advances have in turn permitted im-\nplementation of time-resolved XMCD spectroscopy and\nmeasurements of element-resolved complex susceptibility\n(χ).\nThere are numerous issues that can be investigated\nby use of the experimental apparatus as presented ( e.g.\norigins of intrinsic[21, 22] and impurity damping[23], in-\nterfacial effects in multilayer structures[24], spin-transfer\nand spin-pumping effects[25, 26], etc.). These capabili-\nties can be extended with relatively minor modifications,\nsuch as incorporation of sample cooling and implemen-\ntation of a full return path for the rfexcitation. Theseenhancementswillfacilitateinvestigationsofanextended\nset of issues such as the origins of viscous damping in fer-\nromagnets.\nThe use of synchrotron radiation for these measure-\nments is critical, of course. However, the properties of\nmany 3rdgeneration storage ring sources sets an upper\nboundaryon the frequencyrangeaccessiblein these mea-\nsurements. The photon bunch length must be less than\n1/2 of the oscillation period[27]. For our measurements,\nthe photon bunch length is ∼70 ps (FWHM), which sets\nan upper boundary of about 7 GHz. For many systems,\nit would be desirable to extend this frequency range up-\nwards. Fortunately, shorter pulse lengths are available\nin special operating modes at some storage rings. For\nexample, the momentum compaction (or low α) mode\navailable at certain synchrotrons [28] may permit mea-\nsurements in the tens of GHz range.\nSupport and Acknowledgements: The technical assis-\ntance of Gary Nintzel and Tony Lenhard is gratefully\nacknowledged, as is the beam line support provided by\nDavid Keavney and colleagues at the Advanced Photon\nSource. This work was partially supported by the Army\nResearchOfficewithGrantNo. ARO-43986-MS-YIPand\nthe National Science Foundation with Grant No. NSF-\nDMR-02-39724. Use of the Advanced Photon Source was\nsupported by the U.S. Department of Energy Office of\nScience, Office of Basic Energy Sciences, under Contract\nNo. W-31-109-Eng-38. The support of the NSLS under\nDOE contract No. DE-AC02-98CH10886 also is grate-\nfully acknowledged.\n[1] M. Farle, Reports On Progress In Physics 61, 755 (1998).\n[2] J. Lindner and K. Baberschke, Journal of Physics: Con-\ndensed Matter 15, R193 (2003).\n[3] E. Simanek and B. Heinrich, Physical Review B (Con-\ndensed Matter and Materials Physics) 67, 144418 (2003).\n[4] J. Erskine and E. Stern, Physical Review B 12, 5016\n(1975).\n[5] C.T. Chen, F. Sette, Y. Ma, and S. Modesti, Phys Rev\nLett42, 7262 (1990).\n[6] B. Thole, P. Carra, P, F. Sette, F., and G. Van der Laan,\nPhysical Review Letters 68, 1943 (1992).\n[7] P. Carra, B. Thole, M. Altarelli, and X. Wang, Physical\nReview Letters 70, 694 (1993).\n[8] C. T. Chen, Y. U. Idzerda, H. J. Lin, N. V. Smith,\nG. Meigs, E. Chaban, G. H. Ho, E. Pellegrin, and\nF. Sette, Physical Review Letters 75, 152 (1995).\n[9] G. Boero, S. Rusponi, P. Bencok, R. S. Popovic,\nH. Brune, and P. Gambardella, Applied Physics Letters\n87, 152503 (2005).\n[10] J. Goulon, A. Rogalev, F. Wilhelm, N. Jaouen,\nC. Goulon-Ginet, G. Goujon, J. B. Youssef, and M. V.\nIndenbom, JETP Lett 82, 969 (2005).\n[11] A. Puzic, B. Van Waeyenberge, C. Kang Wei, P. Fischer,\nH. Stoll, G. Schutz, T. Tyliszczak, K. Rott, H. Bruckl,\nG. Reiss, et al., Journal of Applied Physics 97, 10E704\n(2005).[12] W. E. Bailey, L. Cheng, E. Vescovo, C.-C. Kao, and D.A.\nArena, Physical Review B (Condensed Matter and Ma-\nterials Physics) 70, 172403 (2004).\n[13] D. A. Arena, E. Vescovo, C.-C. Kao, Y. Guan, and W. E.\nBailey, Physical Review B (Condensed Matter and Ma-\nterials Physics) 74, 064409 (2006).\n[14] Y. Guan, W. E. Bailey, C.-C. Kao, E. Vescovo, and D. A.\nArena, Journal of Applied Physics 99, 08J305 (2006).\n[15] Y. Guan, W. E. Bailey, E. Vescovo, C.-C. Kao, and D. A.\nArena, Journal of Magnetism and Magnetic Materials\n312, 374 (2007).\n[16] D. A. Arena, E. Vescovo, C.-C. Kao, Y. Guan, and W. E.\nBailey, Journal of Applied Physics 101, 09C109 (2007).\n[17] T. Martin, G. Woltersdorf, C. Stamm, H. A. Duerr,\nR. Mattheis, C. H. Back, and G. Bayreuther, Journal\nof Applied Physics 103(2008).\n[18] J. W. Freeland, J. C. Lang, G. Srajer, R. Winarski,\nD. Shu, and D. M. Mills, Reviewof Scientific Instruments\n73, 1408 (2002).\n[19] A. Kos, T. Silva, and P. Kabos, Review of Scientific In-\nstruments 73, 3563 (2002).\n[20] S. Reidy, L. Cheng, and W. Bailey, Applied Physics Let-\nters82, 1254 (2003).\n[21] V. Kambersky, Physical Review B 76(2007).\n[22] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Physical\nReview Letters 99(2007).8\n[23] W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE\nTransactions on Magnetics 37, 1749 (2001).\n[24] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys-\nical Review Letters 88, 117601/1 (2002).\n[25] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Jour-\nnal of Magnetism and Magnetic Materials 272-276 , 1981\n(2004).[26] G. Woltersdorf, M. Buess, B. Heinrich, and C. H. Back,\nPhysical Review Letters 95, 037401/1 (2005).\n[27] P. Horowitz and W. Hill, The Art of Electronics, 2nd\nedition(Cambridge University Press, 1989).\n[28] J. Feikes, K. Holldack, P. Kuske, and G. Wustefeld, Pro-\nceedings of EPAC2004 (2004)."    },    {        "title": "1104.4246v1.__59_Co_Nuclear_Quadrupole_Resonance_and_Nuclear_Magnetic_Resonance_studies_on_YCoGe_____Comparison_between_YCoGe_and_UCoGe____.pdf",        "content": "arXiv:1104.4246v1  [cond-mat.str-el]  21 Apr 2011Typeset withjpsj3.cls <ver.1.1> FullPaper\n59Co-NuclearQuadrupole Resonanceand NuclearMagnetic Reso nancestudies onYCoGe\n—Comparisonbetween YCoGeandUCoGe—\nKosukeKarube1,∗, TaisukeHattori1, YoshihikoIhara1,+,YusukeNakai1,2,Kenji Ishida1,2,†, NobuyukiTamura3,Kazuhiko\nDeguchi3,NoriakiK.Sato3,andHisatomoHarima4\n1Department of Physics,Graduate School of Science, KyotoUn iversity, Kyoto606-8502, Japan\n2Transformative Research ProjectonIronPnicides (TRIP),J apan Science and Technology Agency (JST),\nChiyoda, Tokyo102-0075, Japan\n3Department of Physics,Graduate School of Science, Nagoya U niversity, Nagoya 464-8602, Japan\n4Department of Physics,Graduate School of Science, Kobe Uni versity, Kobe 657-8501, Japan\nWehaveperformed59Co-nuclearquadrupole resonance(NQR)andnuclearmagneti cresonance (NMR)\nstudies on YCoGe, which is a reference compound of ferromagn etic superconductor UCoGe, in order\nto investigate the magnetic properties at the Co site. Magne tic and superconducting transitions were not\nobserved down to 0.3 K, but a conventional metallic behavior was found in YCoGe, although its crystal\nstructure is similar to that of UCoGe. From the comparison be tween experimental results of two com-\npounds, theferromagnetismandsuperconductivity observe dinUCoGeoriginatefromtheU-5 felectrons.\nKEYWORDS: YCoGe, UCoGe, NMR, nuclear quadrupole resonance\n1. Introduction\nThe discovery of superconductivity in ferromagnet UGe 2\nunder pressure gave a great impact for researchers study-\ning superconductivity,1)because ferromagnetism and super-\nconductivity have been thought to be mutually exclusive. In\n2007, a similar superconductivity was discovered in ferro-\nmagnet UCoGe at ambient pressure by Huy et al.2)Ferro-\nmagnetic (FM) and superconducting (SC) transition temper-\natures (TCurieandTSC) of UCoGe were reported to be 3 and\n0.8 K, respectively.3)In addition,µSR and59Co-NQR mea-\nsurementssuggestthatferromagnetismandsuperconductiv ity\nmicroscopicallycoexist, and that the SC gapis formedin the\nFM region.4,5)In UCoGe, it has been believed that the U-5 f\nelectrons are responsible for ferromagnetism and supercon -\nductivityfromthe analogyof UGe 2. However,there is a pos-\nsibilitythatferromagnetisminUCoGeoriginatesfromCo-3 d,\nnotfromU-5 felectrons,becauseit iswellknownthatCo-3 d\nelectronsgiverisetomagnetisminsomeCo compounds,and\nthereisareportindicatingthattheCo-3 delectronscontribute\ntothemagnetisminUCoGein ahighfield(12T).6)\nTo clarify the role of Co-3 delectrons to ferromagnetism\nin UCoGe, we point out that YCoGe would be a good ref-\nerence compound for UCoGe, because Y has no felectrons\nand YCoGe has the similar TiNiSi-type crystal structure and\nlattice constants7)to UCoGe,8)as shown in Fig. 1. In addi-\ntion, the band calculation suggests that the contribution o f\nCo-3delectrons to the density of states is quite similar both\nin UCoGe and YCoGe.9)Although the crystal structure of\nYCoGe studied by X-ray di ffraction analysis was reported\nin the literature,7)its physical properties have not been re-\nportedso far. Inthis paper,we report59Co-NMRand nuclear\nquadrupole resonance (NQR) results as well as electrical re -\nsistivity and specific-heat results in YCoGe measured down\nto0.3K.Theresultsstronglysuggestthattheferromagneti sm\nandsuperconductivityobservedinUCoGe originatefromthe\n∗karube@scphys.kyoto-u.ac.jp\n†kishida@scphys.kyoto-u.ac.jp\nFig. 1. (Color online) TiNiSi-type orthorhombic crystal st ructures of\nUCoGeandYCoGe.Thedi fferencebetweenthetwostructuresisthealign-\nmentof Co-Ge.\nU-5felectrons.\n2. ExperimentalProcedure\nPolycrystalline and single-crystal samplesof YCoGe were\nprepared by arc-melting and Czochralski-pulling methods\nwith a tetra-arc furnace, respectively. From X-ray di ffraction\nmeasurements,small peaksassigned to impurityphaseswere\nobservedinapolycrystallinesample,butwerenotinasingl e-\ncrystal sample. This implies that the quality of the single-\ncrystal sample is higher than that of the polycrystalline sa m-\nple. Tiny single crystals were crushed into fine powder and\npacked in a sample case made from a straw of 5 mm diame-\nter.ThepowderwasmixedwithGEvarnish,stirred,andfixed\nto a randomorientationin orderto avoida preferentialorie n-\ntationundermagneticfields.\n3. ExperimentalResults\nBeforeshowingNMRandNQRresults,bulkpropertieson\nYCoGe are overviewed.As shown in Fig. 2, the resistivity in\nthe single-crystal sample exhibits a typical metallic beha vior\nwithout magnetic and superconducting anomalies. The inset\nindicatesthatlow-temperatureresistivityisproportion altoT2\n1/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s48 /s50/s48/s48/s48 /s52/s48/s48/s48 /s54/s48/s48/s48 /s56/s48/s48/s48/s48/s49/s50/s51/s52/s53\n/s84/s32 /s32/s40/s75/s41/s32/s32/s40 /s32/s99/s109/s41/s89/s67/s111/s71/s101/s32/s32/s32/s40 /s32/s99/s109/s41\n/s84 /s32/s50\n/s32/s32/s40/s75/s50\n/s41/s32\n/s32\n/s32\nFig. 2. Temperature dependence of the resistivity of YCoGe d own to 3.4\nK.Theinset displays the resistivity as a function of temper ature squared.\n/s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50 /s49/s51 /s49/s52/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s50/s48/s50/s52/s54/s56\n/s124 /s32/s32/s55/s47/s50 /s62\n/s124 /s32/s32/s49/s47/s50 /s62/s124 /s32/s32/s51/s47/s50 /s62/s89/s67/s111/s71/s101/s32/s32\n/s53/s57\n/s67/s111/s45/s78/s81/s82\n/s84 /s32/s61/s32/s52/s46/s50/s32/s75/s53/s57\n/s67/s111/s45/s78/s81/s82/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s32/s40/s77/s72/s122/s41/s54/s46/s50/s51/s32/s77/s72/s122/s57/s46/s56/s52/s32/s77/s72/s122\n/s54/s46/s50/s50/s32/s77/s72/s122/s50/s49/s51/s69/s110/s101/s114/s103/s121/s32/s108/s101/s118/s101/s108/s32/s32/s40\n/s81/s41\n/s124 /s32/s32/s53/s47/s50 /s62/s51\n/s50\n/s49/s32/s61/s32/s48/s46/s53/s57\n/s49/s50/s46/s52/s53/s32/s77/s72/s122/s49/s43\n/s50\n/s105/s109/s112/s117/s114/s105/s116/s121/s32/s63/s32/s32\n/s32/s43/s45/s43/s45/s43/s45/s43/s45/s43/s45/s43/s45\n/s43/s45/s43/s45\nFig. 3.59Co-NQRspectrum ofYCoGeat4.2K.Intensities arenormalize d\nat each peak. The inset exhibits nuclear spin energies for I=7/2 (in non-\naxial EFG) as a function of η. The analysis yields ν1=6.23 MHz,ν2=\n6.22 MHz andν3=9.84 MHz, and the quadrupole parameters νQ=3.40\nMHzandη=0.59.\nbelow100K,whichischaracteristicoftheFermi-liquidsta te.\nThespecificheat( C)wasmeasuredonthesingle-crystalsam-\nple down to 0.3 K. Below 10 K, the experimentaldata Cwas\nwellfittedby C=γT+βT3,andtheelectronic( γ)andphonon\n(β) coefficients were evaluated as γ=6.6 (mJ/mol·K2) andβ\n=0.134(mJ/mol·K4),respectively.\nWe performed59Co nuclear quadrupole resonance (NQR)\nonYCoGe,andsearchedNQRsignalsoverawidefrequency\nrange from 3 to 20 MHz. Three sharp peaks, displayed in\nFig. 3, were observed,whose frequenciesare 6.23, 9.84, and\n12.45MHz at 4.2K. TheNQRHamiltonianisprovidedas,\nHQ=/planckover2pi1νQ\n6/braceleftbigg\n3(I2\nz−I2)+η\n2(I2\n++I2\n−)/bracerightbigg\n, (1)\nwhereνQisthefrequencyalongtheprincipalaxisoftheelec-\ntric field gradient (EFG) and ηis the asymmetry parameter,\ndefined asη≡(Vxx−Vyy)/Vzz. Here,Vijis the componentofTable I. Wave functions of four energy levels of the Co nuclea r spin in\nYCoGe.\n|±7/2/angbracketright |±5/2/angbracketright |±3/2/angbracketright |±1/2/angbracketright\nΨ±7/2-0.996 -0.003 -0.090 -0.012\nΨ±5/2-0.012 0.972 0.075 0.221\nΨ±3/20.085 0.166 -0.886 -0.425\nΨ±1/20.031 -0.164 -0.449 0.878\n/s54/s46/s48 /s54/s46/s53 /s55/s46/s48 /s55/s46/s53 /s56/s46/s48/s32/s61/s32/s57/s48/s32/s32/s61/s32/s52/s50\n/s32/s61/s32/s57/s48/s32/s53/s57\n/s75 /s32/s61/s32/s48\n/s32/s53/s57\n/s67/s111/s45/s78/s77/s82/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s48/s72 /s32/s32/s40/s84/s41/s89/s67/s111/s71/s101\n/s53/s57\n/s67/s111/s45/s78/s77/s82\n/s84 /s32/s61/s32/s53/s32/s75\n/s102/s32/s61/s32/s55/s50/s46/s49/s32/s77/s72/s122\n/s54/s51\n/s67/s117/s32/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110\nFig. 4. (Color online) Field-swept59Co-NMR spectrum at 5 K. The black\nsolid line shows the experimental spectrum and the red dotte d line shows\nthe random powder simulation result.59K=0 shows the field where the\nisotropic Knight shift of59Co is zero. The two central peaks marked by\narrowsarederived from θ=90and42◦.Thesmallpeak arising from63Cu\nin the coil is also marked by an arrow.\nthe EFG tensor. When59Co with nuclear spin I=7/2 is in\nthe presenceof EFG, the degenerateeight nuclear-spinstat es\n|m/angbracketright(m=7/2,···,−7/2) are split into four energy levels by\nelectric quadrupole interaction, yielding three resonanc e fre-\nquencies,ν1,ν2andν3, as shown in the inset of Fig. 3. One\nmayconsiderthatthesignalsobservedat6.23and12.45MHz\narisefromtheν1andν2(ν2andν3)signalswithη=0,butthis\npossibilityisexcludedsincethe ν3(ν1)signalisnotobserved\nat 18.7MHz (3.1MHz).Theobservedthreefrequenciescan-\nnot be interpreted by the ν1,ν2andν3transitions; alterna-\ntively, the observed NQR peaks are assigned as follows. ν1\nandν2are almost overlapped and observed at 6.23 and 6.22\nMHz, andν3is observed at 9.84 MHz. The 12.45 MHz peak\nisassignedtoν1+ν2=12.45MHz,whichiscausedbyhybrid\nstatesduetononzero η.Fromtheassignment, νQ=3.40MHz\nandη=0.59 are derived, and the wave functions of the four\nenergylevelsare expressed,as shownin Table I. In this case ,\nthe resonance peak corresponding to ν2+ν3=16.06 MHz\nis expected, but the transition probability between Ψ±7/2and\nΨ±3/2is one magnitude smaller than that between Ψ±5/2and\nΨ±1/2; thus, we could not observesignals around16 MHz. A\ntinysignalwasobservedataround8.7MHz,whichisthought\nto arisefromimpurityphases.\nTo confirm the validity of the NQR-parameter identifi-\ncation, we performed59Co-NMR measurements. The field-\nswept59Co-NMR spectrum, shown in Fig. 4, was obtained\nfrom the powdered single-crystal sample at 5 K. The redTable II. NQR and Knight-shift parameters as well as an elect ronic coeffi-\ncient in the specific-heat measurements in YCoGe and UCoGe.\nYCoGe UCoGe\nνQ(exp.) [MHz] 3.40 2.85\nη(exp.) 0.59 0.52\nνQ(band) [MHz] 3.91 3.11\nη(band) 0.33 0.47\nKiso[%] 1.88 (5 K) 1.58 (200 K)\nKaniso[%] -0.28 (5 K) 0.02 (200 K)\nγ[mJ/molK2]6.6 65\ndotted line in Fig. 4 shows the result of the simulation,10)\nin which the principal axis of EFG is randomly distributed\nagainst the external field, and νQ=3.40 MHz andη=0.59,\nwhicharedeterminedbytheabovementionedNQRmeasure-\nment, and the isotropic Knight shift Kiso=1.88% and the\nanisotropic Knight shift Kaniso=−0.28% are used. The sim-\nulation result is in good agreement with the observed NMR\nspectrum, indicating that the NQR parameters can also be\nused to interpret the NMR spectrum.The Knight-shiftvalues\nin YCoGe are comparable to those in UCoGe at higher tem-\nperatures. The NQR and Knight-shift values of YCoGe and\nUCoGearesummarizedinTableII.\nThe nuclear spin-lattice relaxation rate 1 /T1of Co was\nmeasured with the NMR and NQR methods in order to in-\nvestigate the anisotropy of magnetic fluctuations. In gener al,\n1/T1isdeterminedwiththefluctuationsofthehyperfinefields\nperpendicularto the quantumaxis. 1 /T1was measured at the\nν3peakinthe59Co-NQRspectrum,whose1 /T1detectsmag-\nnetic fluctuationsalongthe b-andc-axisdirections,sincethe\nEFG principal axis is considered to be almost parallel to the\na-axis from the analogy of UCoGe.11)1/T1was derived by\nfittingtherecoverycurves R(t)=1−m(t)/m(∞)withthefol-\nlowingtheoreticalNQRrecoverycurvefor ν3,12)\nR(t)=0.163exp/parenleftBigg−2.74t\nT1/parenrightBigg\n+0.675exp/parenleftBigg−9.22t\nT1/parenrightBigg\n+0.162exp/parenleftBigg−17.9t\nT1/parenrightBigg\n. (2)\nHere,m(t) is the nuclear magnetization measured at a time t\nafter a saturation pulse. 1 /T1was also measured at the NMR\ncentralpeakscorrespondingto θ=90and42◦shownwithar-\nrowsinFig.4,where θistheanglebetweenthe externalfield\nandtheEFGprincipalaxis.The1 /T1measuredattheθ=90◦\npeakdetectsthe magneticfluctuationsincludingin the a-axis\ndirection, since the EFG principal axis is almost a-axis. The\n1/T1measuredattheNMRpeaksisderivedbyfittingwiththe\ntheoreticalrecoverycurveforthecentraltransition.13)Theex-\nperimental R(t)andtheoreticalcurvesfortheNQRandNMR\nmeasurementsareshowninFig.5,and1 /T1wasderivedfrom\nthe reasonable results of fitting. The 1 /T1results are shown\nin Fig. 6. We foundthat 1 /T1is isotropic and proportionalto\na temperature above 1.5 K. The isotropic T1T=const. (so-\ncalled“Korringa”)behaviorindicatesthatYCoGeisinacon -\nventional metal state without notable magnetic fluctuation s.\nThe 1/T1result is consistent with the resistivity and specific-\nheat results. By assuming the Korringa relationship betwee n\ntheconstant T1TandthespinpartoftheKnightshift( Ks),Ks/s48 /s49/s48 /s50/s48/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s89/s67/s111/s71/s101/s32/s32/s32/s53/s57\n/s67/s111/s45/s78/s81/s82\n/s51/s32/s32/s112/s101/s97/s107/s32/s32/s32/s32/s32/s32\n/s84 /s32/s61/s32/s51/s48/s32/s75\n/s89/s67/s111/s71/s101/s32/s32/s32/s53/s57\n/s67/s111/s45/s78/s77/s82\n/s99/s101/s110/s116/s101/s114/s32/s112/s101/s97/s107/s32/s32 /s32/s61/s32/s57/s48\n/s84 /s32/s61/s32/s51/s48/s32/s75/s82 /s32/s40 /s116/s41\n/s84/s105/s109/s101/s32/s32/s40/s109/s115/s41/s48 /s49/s48 /s50/s48/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s82 /s32/s40 /s116/s41\n/s84/s105/s109/s101/s32/s32/s40/s109/s115/s41\n/s32/s32\nFig. 5. (Color online) Recovery curves R(t) of the nuclear magnetization\nm(t) atatime tafter asaturation pulse with the theoretical curves for eva l-\nuating 1/T1.(a)R(t) measured atν3in Fig. 3,and (b) R(t) measured atthe\nθ=90◦peak in the NMR spectrum of Fig. 4(see in text).\nisdescribedby\nKs=/radicalbigg\n1\nT1T/planckover2pi1\n4πkB/parenleftBiggγe\nγn/parenrightBigg\n, (3)\nwhereγeandγnaregyromagneticratioswith respectto elec-\ntrons and nuclei, respectively. Using the experimental val ues\nof 1/T1T(=2.03 s−1K−1),Ksis estimated as Ks=0.31%,\nwhich is much smaller than the experimental value of the\nKnight shift ( Kiso∼1.88%); thus, the orbital part of the\nKnight shift Korb∼1.57% is derived, since the experimental\nvalueoftheKnightshiftisthesumof KsandKorb.Theexper-\nimentalvalueof Korbisareasonablevalueof Korbreportedin\nseveral Co compoundssuch as a Co metal ( Korb=1.7 %)14)\nandnonmagneticNaCoO 2(Korb=1.9%).15)\n4. Discussion\nHere, we compare our results in YCoGe with those in\nUCoGe.5)First, YCoGe has a similar59Co-NQR spectrum\nto UCoGe, although ν1andν2are overlapped, indicative of\nthe low symmetry of the crystal structure. As shown in Ta-\nble II, quadrupole parameters in YCoGe are slightly greater\nthanthoseinUCoGe.Thedi fferencecanbeinterpretedbythe\ndifference in the Co-Ge alignment along the a-axis. The Co-\nGe alignment is almost straight along the a-axis in UCoGe,\nwhereas it is zigzag in YCoGe (see Fig 1). Therefore,the lo-\ncal symmetry of Co atoms in YCoGe is lower than that in\nUCoGe, providing larger quadrupole parameters. However,\ntheasymmetricparameter ηinYCoGecalculatedbybandcal-\nculation is smaller than that in UCoGe, which is inconsisten t\nwiththeexperimentalobservation(see TableII).Thiscont ra-\ndictionseemstobecausedbytheelectronicstateoftheYsit e\nin thebandcalculation./s49 /s49/s48 /s49/s48/s48/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52\n/s84\n/s67/s117/s114/s105/s101/s85/s67/s111/s71/s101/s32/s32/s32/s53/s57\n/s67/s111/s45/s78/s81/s82\n/s89/s67/s111/s71/s101/s32/s32/s112/s111/s119/s100/s101/s114\n/s32/s53/s57\n/s67/s111/s45/s78/s81/s82/s49/s47 /s84\n/s49/s32/s32/s40/s115/s45/s49\n/s41\n/s84 /s32/s32/s40/s75/s41/s126 /s84/s32/s112/s111/s108/s121/s99/s114/s121/s115/s116/s97/s108/s108/s105/s110/s101\n/s32/s115/s105/s110/s103/s108/s101/s45/s99/s114/s121/s115/s116/s97/s108\n/s32/s32/s32/s53/s57\n/s67/s111/s45/s78/s77/s82\n/s32 /s32/s61/s32/s57/s48\n/s32 /s32/s61/s32/s52/s50\nFig. 6. (Color online) Temperature dependence of59Co-NQR 1/T1(red\nclosed square) and59Co-NMR 1/T1forθ=90◦(blue closed circle) and θ\n=42◦(green closed triangle) in YCoGe, compared with59Co-NQR 1/T1\nin polycrystalline (open star) andsingle-crystal (open di amond) UCoGe.5)\nTheferromagnetic transition temperature in UCoGemarked b yTCuriewas\nreported to be2.5K.\nAs for magnetic fluctuations, the 1 /T1in UCoGe is much\ngreater than that in YCoGe in the entire temperature range\nand displays a prominent large peak due to FM ordering at\naroundTCurie∼2.5 K, as seen in Fig. 6. In addition, recent\ndirection-dependentCo-NMRmeasurementsinsingle-cryst al\nUCoGe revealed that the Knight shift and 1 /T1show a sig-\nnificant anisotropic behavior, indicating the presence of t he\nIsing-type FM fluctuations along the magnetic easy axis ( c-\naxis) at low temperatures below 10 K.11)It is also shown\nthatsuchcharacteristicFMfluctuationsareintimatelyrel ated\nto the unconventional superconductivity in UCoGe.11)Such\nmagnetic fluctuations were not observed at all in YCoGe.\nHowever, it should be noted that the experimental values of\nKand1/T1at 230K in UCoGeareisotropicandcomparable\nto those in YCoGe.11)This implies that the high-temperature\nelectronic state in UCoGe is similar to that in YCoGe, since\nthehybridizationbetweenconductionelectronsandU-5 flo-\ncal moments is weak at high temperatures and the electronic\nstate in UCoGe is governedby the conductionelectron; thus,\nYCoGe is a good reference compound for understandingthe\nelectronic state without U-5 felectrons. Therefore, we con-\ncludethatthestrongIsing-typeFMfluctuationsaccompanie d\nbytheweakFMorderingandunconventionalsuperconductiv-\nity as well as the heavy-fermioncharacter are ascribed to th e\nU-5felectrons. However, the field-induced magnetism ob-\nservedinUCoGe,6)whichisrelatedtoa5 f-3dhybridization,\nis an important issue in the understanding of the high mag-\nnetic state in UCoGe. High-field59Co-NMR measurements\nareexpectedtoprovidecrucialinformationonthisissue.5. Conclusion\nFrom59Co-NQR/NMR measurements, we determined\nNQR parameters in YCoGe, which are similar to those in\nUCoGe. The temperature dependence of 1 /T1shows that\nYCoGe is in a conventional metallic state without notable\nmagneticfluctuationsdownto1.5K,whichisconsistentwith\nresistivityandspecific-heatmeasurements.Fromthecompa r-\nison between the experimental results in YCoGe and those\nin UCoGe, we conclude that U-5 felectrons simultaneously\ncarry ferromagnetism and unconventional superconductivi ty\nin UCoGe.\nAcknowledgments\nTheauthorsthankD.AokiandJ.Flouquetforvaluabledis-\ncussions,andD.C.Peets,S.Yonezawa,andY.Maenoforex-\nperimental support and valuable discussions. This work was\npartially supported by Kyoto Univ. LTM Centre, the “Heavy\nElectrons” Grant-in-Aid for Scientific Research on Innova-\ntiveAreas(No.20102006,No.21102510,andNo.20102008)\nfromtheMinistryofEducation,Culture,Sports,Science,a nd\nTechnology(MEXT) of Japan, a Grant-in-Aidfor the Global\nCOE Program “The Next Generation of Physics, Spun from\nUniversality and Emergence” from MEXT of Japan, and a\nGrant-in-Aid for Scientific Research from the Japan Soci-\nety for Promotion of Science (JSPS), KAKENHI (S) (No.\n20224015).\n+presentaddress:DepartmentofPhysics,FacultyofScience ,\nHokkaidoUniversity,Nishi8,Kita10,Kita-ku,Sapporo060 -\n0810,Japan\n1) S.S.Saxena,P.Agarwal,K.Ahilan,F.M.Grosche,R.K.W.H aselwim-\nmer, M.J.Steiner, E.Pugh, I.R. Walker, S.R. Julian, P.Mont houx, G.\nG. Lonzarich, A. Huxley, I. Sheikin, D. Braithwaite, and J. F louquet:\nNature (London) 406(2000) 587.\n2) N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P. Klaa sse,\nT. Gortenmulder, A. de Visser, A. Hamann, T. G¨ orlach, and H. v.\nL¨ ohneysen: Phys.Rev. Lett. 99(2007) 067006.\n3) N.T.Huy,D.E.deNijs,Y.K.Huang,andA.deVisser:Phys.R ev.Lett.\n100(2008) 077002.\n4) A. de Visser, N. T. Huy, A. Gasparini, D. E. de Nijs, D. Andre ica, C.\nBaines, and A.Amato: Phys.Rev. Lett. 102(2009) 167003.\n5) T. Ohta, T. Hattori, K. Ishida. Y. Nakai, E. Osaki, K. Deguc hi, N. K.\nSato, and I.Satoh: J.Phys.Soc. Jpn. 79(2010) 023707.\n6) K. Proke, A. de Visser, Y. K. Huang, B. Fak, and E. Ressouche : Phys.\nRev. B81(2010) 180407(R).\n7) A.E.Dwight, P.P.Vaishnava, C.W.Kimball, and J.L.Matyk iewicz: J.\nLess-Common. Met. 119(1986) 319.\n8) F. Canepa, P.Manfrinetti, M. Pani, and A. Palenzona: J. Al loys Comd.\n234(1996) 225.\n9) H.Harima: private communication.\n10) G. C. Carter, L. H. Bennett, and D. J. Kahan: Metallic Shift in NMR\n(Pergamon, New York,1977).\n11) Y. Ihara, T. Hattori, K. Ishida, Y. Nakai, E. Osaki, K. Deg uchi, N. K.\nSato, and I.Satoh: Phys.Rev. Lett. 105(2010) 206403.\n12) J.Chepin and J.H. Ross,Jr:J.Phys.Condens. Matter. 3(1991) 8103.\n13) A.Narath: Phys.Rev. 162(1967) 320.\n14) U. El-Hanany and W. W. Warren, Jr.: Bull. Am. Phys. Soc. 19(1974)\n202.\n15) G.Lang,J.Bobro ff,H.Alloul,P.Mendels,N.Blanchard, andG.Collin:\nPhys.Rev. B 72(2005) 094404."    },    {        "title": "0904.1780v1.Coupled_Superconducting_Phase_and_Ferromagnetic_Order_Parameter_Dynamics.pdf",        "content": "arXiv:0904.1780v1  [cond-mat.supr-con]  11 Apr 2009Coupled Superconducting Phase and Ferromagnetic Order Par ameter Dynamics\nI. Petkovi´ c1∗, M. Aprili1, S. E. Barnes2,3, F. Beuneu4, and S. Maekawa5,6\n1Laboratoire de Physique des Solides, Universit´ e Paris-Su d, CNRS,\nUMR 8502, 91405 Orsay, France.2Theory of Condensed Matter Group,\nCavendish Laboratory, Madingley Road, Cambridge CB3 0HE,\nUnited Kingdom.3Physics Department, University of Miami, Coral Gables,\nFL 33124, USA.4Laboratoire des Solides Irradi´ es, CNRS, ´Ecole Polytechnique,\nF-91128 Palaiseau, France.5Institute for Materials Research,\nTohoku University, Sendai 980-8577, Japan.6CREST,\nJapan Science and Technology Agency, Sanbancho, Tokyo 102- 0075, Japan.∗\n(Dated: November 13, 2018)\nVia a direct coupling between the magnetic order parameter a nd the singlet Josephson supercur-\nrent, we detect spin-wave resonances, and their dispersion , in ferromagnetic Josephson junctions in\nwhich the usual insulating or metallic barrier is replaced w ith a weak ferromagnet. The coupling\narises within the Fraunhofer interferential description o f the Josephson effect, because the magnetic\nlayer acts as a time dependent phase plate. A spin-wave reson ance at a frequency ωsimplies a\ndissipation that is reflected as a depression in the current- voltage curve of the Josephson junction\nwhen ¯hωs= 2eV. We have thereby performed a resonance experiment on only 107Ni atoms.\nThe coupled dynamics of the electromagnetic field and\na Josephson junction has a number of manifestations\nand is very well understood [1, 2, 3, 4]. When the\nusual insulating or metallic barrier is replaced with a\nweak ferromagnet there is a coupling to another field,\nnamely the spontaneous magnetisation of the ferromag-\nnet. Spin-waves are elementary spin excitations which\ncan be viewed as both spatial and time dependent vari-\nations of the magnetisation. In a ferromagnet the lowest\nenergy excitation, the Ferromagnetic Resonance (FMR),\ncorresponds to the uniform precession of the magnetisa-\ntion around an externally applied magnetic field at the\nfrequency ωs. This mode can be resonantly excited by\nan alternative (ac) magnetic field that couples directly to\nthe magnetisation, as described by the Landau-Lifshitz\nequations [5]. The Josephson phase difference φbetween\nthe two superconductors has its own dynamics. A bias\nvoltageV0causesφto become time-dependent so that\nφ=φ0+ωJt, where ωJ= (2e/¯h)V0andφ0is arbi-\ntrary. Corresponding to the ac Josephson effect [1], for\nour junctions, to a good approximation, the resulting ac\nJosephson current density is Js=Jcsin(φ0+ωJt), where\nJcis the critical current density.\nIn analogy with the A-phase [6] of3He, coupled mag-\nnetic and phase oscillations should exist in ferromagnetic\nsuperconductorswithtripletpairing,buthaveneverbeen\nobserved. We show here that a similar coupling for sin-\nglet superconductorscan be realised in a Josephsonjunc-\ntion with a ferromagnetic barrier. The dynamical cou-\nplingstemsfromthespatialinterferenceoftheAharonov-\nBohm phase caused by M(t), resulting in the spatial de-\npendence of φ(r,t). The ac Josephson current produces\nan oscillating magnetic field H(t) and when the Joseph-\nson frequency matches the spin wavefrequency, ωJ≈ωs,\nthis resonantly excites M(t). Due to the nonlinearity of\nthe Josephson effect, there is a rectification of current\nacross the junction, resulting in a dip in the average dccomponentof Jsatvoltage Vs= (¯h/2e)ωs. Theprincipal\nresult reported here is the observation of these coupled\ndynamics.\nMagnetised Josephson junctions [7] require weak ferro-\nmagnetic materials and nanosized junction area to keep\nthe overallmagnetic flux in the junction smaller than the\nflux quantumΦ 0. An electronmicroscopeimageofa typ-\nical ferromagnetic junction used in this study is shown in\nFig. 1(a), while Fig. 1(b) is a schematic representation of\nthedifferentlayers. Thesuperconductingelectrodescom-\nprise50nmofNb ( Tc= 7.6K),while thebarrieris20nm\nof Pd0.9Ni0.1(TCurie= 150 K). The current-voltage (IV)\ncharacteristics are measured using current bias and are\nreported, as function of the applied in-plane field, in the\nright insert of Fig. 2. The IV characteristics are not hys-\nteretic, and overall they correspond closely to those ex-\npected for a junction with a conductive barrier [8, 9].\nThe junction normal resistance is Rn≈0.8Ω, and the\nJosephson coupling is IcRn≈5µV, as expected for ferro-\nmagneticjunctions ofthis thickness[7], yielding the criti-\ncalsupercurrentof Ic≈7µA. Thehostile natureofeven a\nweak ferromagnetic environment for singlet Cooper pairs\nis illustrated by a similar junction with 70 nm of non-\nmagnetic Pd which, despite the almost four times larger\nthickness, has a larger critical current Ic≈44µA.\nFor a square junction of side L, the total supercurrent\nis given by the integral [10]\nIs=Jc/integraldisplayL/2\n−L/2dx/integraldisplayL/2\n−L/2dysinφ(x,y,t) (1)\nwith\nφ(r,t) =φ0+ωJt−2e\n¯h/integraldisplay\nA·dr, (2)\nwhere the last term is the Aharonov–Bohm phase [11],\ninvolving the vector potential A. We use a gauge where2\nFIG. 1: (color online). (a) SEM photo of the ferromagnetic\nJosephson junction used in the experiment. Junction area is\n500nm×500 nm. (b) Schematic cross-section of the Joseph-\nson junction. The ac Josephson current I(t)flows through the\njunction creating an rf magnetic field H(t), causing the mag-\nnetisation precession M(t), which in turn resonantly couples\nwith the Josephson phase at frequency ωJ. Layers are respec-\ntively from the bottom: Nb (50 nm), Pd 0.9Ni0.1(20 nm) and\nNb (50 nm). (c) The equivalent RSJ model and the sketch of\nthe effect of the FMR on the current-voltage characteristics .\nThe ferromagnetic layer is modeled as a series LCR 0oscilla-\ntor, in parallel with the Josephson junction. Resistance R0is\nproportional to the imaginary part of the susceptibility χ′′.\nA=A(r,t)ˆ z, the direction ˆ zbeing perpendicular to the\njunction surface [see Fig. 1]. Therefore φ(r,t) =φ0+\nkx+ωJt+φm, whereφm= (4ae/¯h)Amzreflects time\ndependent fields and k= (4ed/¯h)µ0H+(4ea/¯h)µ0M0y.\nHereM0yis theycomponent of the static magnetisation\nM0, the applied field His in the ydirection, 2 aand\n2d=2(a+λ) are the actual and magnetic thickness of\nbarrier and λthe London penetration depth. Equations\n(1) and (2) are used to describe both the statics and the\ndynamics of our junctions.\nIn the absence of a bias ( V0= 0), we are dealing\nwith static fields, and the Equations (1) and (2) lead\nto the Fraunhofer pattern Is=JcL/integraltextL/2\n−L/2dxsinkx[11].\nThe magnetisation M0of the barrier has the same effect\nas inserting a wedge shaped phase plate in front of the\nslit, it displaces the diffraction pattern. Experimentally,\nthe diffraction pattern is shifted to the right for increas-\ning (positive M0y), and the left for decreasing (negative\nM0y), fields. This illustrates the linear nature of the\ncoupling to M. In Fig. 2, the dotted curves are a fit us-\ning Eqs. (1) and (2), along with the magnetisation data\nmeasured on a trilayer with the same cross section as the\njunction [see the left insert of Fig. 2]. The periodicity\nand the asymptotic behaviour of the measured diffrac-\ntion pattern attest to the high quality of our junctions.\nThey confirm the close-to-uniform current distribution\nand single-harmonic current-phase relation, while the re-\nFIG. 2: (color online) Dependence of the Josephson critical\ncurrent on the external magnetic field in plane. The solid\ncurve represents the normalized experimental data taken at\n35 mK and the dotted curve is the Fraunhofer pattern ex-\npected for our junction parameters including the magnetisa -\ntion. Left insert: The magnetic hysteresis taken at 10 K on\na millimetric trilayer with the same cross section as the jun c-\ntion, with field applied in plane (dotted curve)and perpendi c-\nular to the plane (solid curve). Right insert: Current-volt age\ncurves for different field values, measured at 35 mK.\nproduction of the shift with the two sweep directions,\nusing experimental magnetostatic data, confirms the va-\nlidity of our description.\nThe dynamical coupling reflects a similar phase con-\ntribution ϕ(r,t) due to M(r,t), but which now has both\na temporal and a spatial dependence, the equivalent of\na phase plate in the optical analog with a similarly de-\npendent refractive index n(r,t). The dynamics of the\nmagnetisation (timescale of 1 ns) is much slower than\nthe diffusion time through the ferromagnetic layer (0 .5\nps). The Josephson coupling is thus adiabatic with re-\nspect to the magnetisation dynamics. This assumption\nis implicit in Eqs. (1) and (2). The signal is seen for\nV > I cR, implying Eq. (1) can be linearized. The dc\nmagnetic signal then corresponds to [10]\nIm=4ae\n¯h/integraldisplayL/2\n−L/2dx/integraldisplayL/2\n−L/2dyJccos(kx+ωJt)φm,(3)\nwhere the bar denotes a time average. Substituting for\nφm= (4ae/¯h)Amzand using J=∇×H, following both\na time and space integration by parts, the dc signal re-\nflecting the magnetic resonance is\nIm=1\nV0/integraldisplay\ndrH·dM\ndt, (4)\nwithMi(r,t) =/integraltext\ndt′χi(t−t′)Hi(r,t′),i=x,y,z, where\nχi(t) isthe dynamic susceptibility. Thishasan appealing\ninterpretation in terms of magnetic losses. Here, as illus-\ntrated by Fig. 1(b), H(r,t) is the magnetic field which3\ncirculates inside the junction by virtue of the ac Joseph-\nson current. The junction lateral size Lis smaller than\nbothλand the skin depth for the frequencies involved.\nThe displacement current is therefore negligible and all\nthat is needed is to integrate Amp` ere’s law in order to\ndetermine H(r,t). More details of these calculations are\ngiven elsewhere [12]. The current due to M(r,t) is\nIm= 2πIc(0)Φrf\nΦ0/bracketleftbig\nFxχ′′\nx(ωJ)+Fyχ′′\ny(ωJ)/bracketrightbig\n,(5)\nwhere Φ rf= (2aL)Brf= (2aL)µ0Ic(0)/Lis the flux due\nto the radio frequency field and Fx= (1/12)(Ic(B0)/Ic)2\nandFy= (2/θ2\nL)[1 + sin( θL/2)cos(θL/2) + ((13 /12)−\n(4/θ2\nL))sin2(θL/2)];θL=kL, reflect the geometrical\nstructure of the coupling. As the equilibrium magnetisa-\ntion is along the zaxis, the magnetic resonance signal is\ncontained in χ′′\nx(ωJ) andχ′′\ny(ωJ), the Fourier transforms\nofthe imaginarypartof the susceptibility. Therefore, the\ntotal dc current within the Resistively Shunted Joseph-\nson junction (RSJ) model [8, 9] is\nI=V0\nR(0)+Ic2(B0)\n2V0R(ωJ)−Im, (6)\nwhereR(0) andR(ωJ)≈Rare the junction resistances\nfor dc and frequency ωJ. A simple physical argument\ncan account for the three terms in Eq. (6). The average\npower dissipated in the junction is IV0and so the first\nterm,V02/R, corresponds to the Ohmic loss at dc, while\n1\n2Ic2R(ωJ) is the similar loss at ωJ. The key third term\nrepresents a self-inductance L(M), stemming from the\nferromagnet, in parallel with the junction and modeled\nas an LCR 0oscillator [see Fig 1(c)], where R0reflects\nthe magnetic damping. At the magnetic resonance fre-\nquency, energy is absorbed by the ferromagnet, causing\nthe oscillator to be lossy. This actually reduces the effec-\ntive junction resistance, leading to a dip in I(V). In this\nmanner, the Josephson junction rectifies the self-induced\nmagnetic resonance.\nThis coupling to the magnetic system is evident in the\nmeasured dynamical resistance dV/dIcurves reported in\nFig. 3. We measure the dynamical resistance rather than\nthe IV characteristics to improve amplitude resolution.\nThe mode labeled FMR (Ferromagnetic Resonance) is\nseen only for ferromagnetic junctions. There is good\nagreementbetweenthe experiment, solidcurves, andthe-\nory, dotted curves. The magnetic resonance mode ob-\nserved in our experiments reflects the properties of a thin\nfilm of the ferromagnet Pd 0.9Ni0.1. Magnetisation curves\nM(H), measured directly for a large area Nb /PdNi/Nb\ntrilayer with the same cross section as the junction, are\nshown in the insert of Fig. 2. They indicate that Mis\nperpendicular to the junction plane, a conclusion rein-\nforced by earlier anomalous Hall effect measurements on\nsimilar thin films [13]. The FMR mode, shown in Fig. 3,\noccurs at V0= 23µV. This is unambiguously identified\nFIG. 3: (color online). Dynamical resistance of the fer-\nromagnetic Josephson junction (solid curve, SFS, bottom\nand left axis) shows resonances compared to a similar non-\nferromagnetic junction (solid curve, SNS, top and right axi s).\nDottedcurveis afittotheory[Eq. (6)]. The mode at 23 µVis\nthe ferromagnetic resonance (FMR). Bottom insert: Conven-\ntional cavity ferromagnetic resonance on a macroscopic tri -\nlayer. Top insert: Comparison between the field dependence\nof the FMR in the Josephson junction (solid square) with the\ncavity measurement on a macroscopic trilayer (open square) .\nDotted curve is a parameter free fit of the FMR using the\nKittel formula [Eq. (7)].\nas such, since the frequency ωs= 2eV0/¯hagrees, without\nfitting parameters, with the Kittel formula [14]\nωs=γe/radicalbig\n(HK−4πMS)2−H2 (7)\nfor the in-plane magnetic field dependence of the uni-\nform FMR mode when the anisotropy field is perpendic-\nular to the plane. The anisotropy field HK= 4900 G\nand the magnetisation at saturation MS= 930 G are\nboth determined directly from the static magnetisation\ndata, and γe=µB/¯h, where µBis the Bohr magne-\nton. For comparison, the ferromagnetic resonance of a\nmacroscopic Nb/PdNi/Nb trilayer has been measured in\na conventional 9 .5 GHz cavity spectrometer at 10 K with\nfield applied parallel to the substrate. The cavity FMR,\nshown in the bottom insert of Fig. 3, occurs at 2160 G,\nagain exactly as predicted by Eq. (7). Displayed in the\ntop insert of Fig. 3 is the comparison of the resonant\nmode in the Josephson junction (solid square) and in the\nmacroscopic trilayer (open square). The dotted curve\nshows the frequency of the FMR mode calculated from\nthe Kittel formula [Eq. (7)]. The spectra presented in\nthe main part of Fig. 3 contain an extrinsic broaden-\ning caused by a lock-in modulation voltage of ∼1µV.\nFor the ac modulation voltage of 0.5 µV, the junction\nresonance width saturates at 0.5 µV, which corresponds\nto the conventional resonance width (150 G). The sig-\nnal amplitude corresponds to a resonant susceptibility of4\napproximately 10, consistent with the FMR mode mea-\nsured in a microwave cavity and reported in the bottom\ninsert of Fig. 3.\nFIG. 4: (color online). (a) The dynamical resistance of the\nferromagnetic Josephson junction with applied external mi -\ncrowaves. Pronounced dip is the Shapiro resonance, and ar-\nrows indicate sideband resonances at the same frequency as\nthen=2modefrom Fig. 3. Theexternalmicrowave frequency\nis 17.35 GHz and the temperature 35 mK. (The solid curves\nare the measured derivative of the dynamical resistance fro m\nFig. 3, and dotted curves are a fit [Eq. (6)]. The spin wave\nfrequency increases with field. Insert: The field dispersion\nrelation of the modes. The solid curve is Eq. (7) when the\nspatial dependence of the FMR modes is taken into account.\nIn order to demonstrate that the magnetic system is\ncoupled to the super- but not to the normal current,\nwe have performed Shapiro step [2] measurements, re-\nported as the dynamical resistance dV/dIin Fig. 4(a).\nThe junction is irradiated with microwaves of frequency\nν=17.35 GHz at 35 mK. The Shapiro steps arise from\nthe mixing ofthe microwavesignal with the ac Josephson\neffect and are smaller replicas of the zero-voltage current\nstep displaced from zero voltage by Vn(¯h/2e)ω, where\nω=2πν, andnis an integer. We do not observe half in-\nteger Shapiro steps, indicating negligible higher harmon-\nics in the current-phase relation. However, as expected\nwithin the RSJ model, the ferromagnetic resonance can\nbe exited at voltage Vns= (¯h/2e)ωs/n[4]. The sub-\nharmonic for n=2 is visible in the spectrum in Fig. 3.\nAs shown in Fig. 4(a), it is reproduced as a side-band to\neach regular step when V=(¯h/2e)(nω±ωs/2). Exper-\nimentally, we do not have available a high enough fre-\nquency to separate similar side-bands for the main FMR\nmode at ωs.\nFinally, the field dependence of the resonance at ωs/2\nhas been studied in more detail in the second deriva-\ntive,d2V/dI2[Fig. 4(b)], where the minima correspond\ntoV2s= (2e/¯h)ωs/2. Measurementswere limited in field\ndue to the rapid decrease of the critical current above\n800 G.\nIn the insert of Fig. 4(b), we show (1 /2π)ωs/2 as a\nfunction of the applied magnetic field. The error barsare due to the drift of the amplifier. The solid curve\nis Eq. (7), without fitting parameters, with the spatial\ndependence of FMR taken into account. The spatial de-\npendence of the spin-waves leads to an additional term\nto Eq. (7) given by ak2, wherek=(πd/Φ0)His the spin\nwave momentum and a=Eexb2, where Eex= 50 meV\nis the PdNi exchange energy and b=0.1 nm the lattice\nconstant. Since the width of the junction is only about\n500 nm, this leads to a small but finite correction to the\nuniform FMR energy which is larger than the direct ef-\nfect of the applied dc field. Illustrated in this manner\nis the direct determination of spin-wave dispersion using\nthe present technique.\nIn conclusion, we have demonstrated the dynamical\ncoupling of the superconducting phase with the spin\nwaves in a ferromagnet and measured their dispersion.\nWe have performed a photon free FMR experiment on\nabout 107Ni atoms, which would be infeasible with\nstandard FMR techniques, and have illustrated a new\nmethodology for the study of spin dynamics. There are\ndirect and implied applications to spintronics and nano-\nmagnetism [15].\nWe thank J. Gabelli, B. Reulet, D. Feinberg, R. Melin,\nZ. Radovi´ c, I. Martin and M. Houzet for stimulating dis-\ncussions. M.A. is indebted to H. Bouchiat for an illu-\nminating conversation and to A. Thiaville and H. Hur-\ndequint for many tutorials about spin dynamics. This\nwork was in part supported by CREST of JST, and EP-\nSRC(UK).\n∗petkovic@lps.u-psud.fr\n[1] B.D. Josephson, Physics Letters 1, 251 (1962).\n[2] S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).\n[3] D.D. Coon, and M.D. Fiske, Phys. Rev. 138, A744\n(1965).\n[4] K.K. Likharev, Dynamics of Josephson Junctions and\nCircuits (Gordon and Breach, New York, 1986).\n[5] C. Kittel, Phys. Rev. 115, 1587 (1959).\n[6] A.J. Leggett, Phys. Rev. Lett. 29, 1227 (1972).\n[7] T. Kontos, M. Aprili, J. Lesueur, F. Genet, B. Stephani-\ndis, and R. Boursier, Phys. Rev. Lett. 89, 137007 (2002).\n[8] W.C. Stewart, Appl. Phys. Lett. 10, 277 (1968).\n[9] D.E. McCumber, J. Appl. Phys. 39, 3113 (1968).\n[10] A. Barone and G. Paterno, Physics and Applications of\nthe Josephson Effect (Wiley, New York, 1982).\n[11] P.W. Anderson and J.M. Rowell, Phys. Rev. Lett. 10,\n230 (1963).\n[12] S.E. Barnes et al., in preparation.\n[13] T. Kontos, M. Aprili, J. Lesueur, and X. Grison, Phys.\nRev. Lett. 86, 304 (2001).\n[14] C. Kittel, Rev. Mod. Phys. 21, 541 (1949).\n[15]Concepts in Spin Electronics , Ed. S. Maekawa (Oxford\nUniversity Press, 2006)."    },    {        "title": "1308.3069v1.Electrical_Detection_of_Spin_Wave_Resonance_in_a_Permalloy_Thin_Strip.pdf",        "content": "The 3rd International Symposium on Advanced Magnetic Materials and Applications (ISAMMA2013)  1 \n  \nAbstract —We investigated the microwave -induced DC \nresponse of spin wave resonance (SWR) in a permalloy thin strip  \nvia electrical detection . Our exper imental results  obtained by \nsweeping the external field reveal that: 1. the amplitude of SWR \nsignals depend on t he direction of external field and, 2. unlike the \nDC response of ferromagnetic resonance,  SWR spectra are always \nanti-symmetrical. The spin dyna mics are discussed based on these \nunusual signals  in resonant condition.   \n \nI. INTRODUCTION  \nLECTRICAL detection of direct c urrent (DC) response  of \nferromagnetic resonance (FMR)  under microwave  \nradiation in very thin sheets of ferromagnet ic conductors  has \nbeen intensively studied since 1960s [1]. The \nmicrowave -induced DC response is a non -zero time -averaged \nsignal, which is generated by the microwave induced  current \nand the time -varying anisotropic magnetoresistance (AMR) \noriginates from magnetization precess ion. This DC response is \nvery informative to further understand the magnetization \nprecessional motion  and spin dynamics, e.g., magnetic \nanisotropy [2], electromagnetic relative phase (the phase \ndifference between electrical field component and magnetic \nfield component of electromagnetic wave)  [3] and damping [4].  \nAs a result, DC response detection bears an important role for \nboth scientific points of view and technological perspectives.  \nThe motion s of spin and magnetization are  well described by \nLandau -Lifshitz-Gilbert (LLG) equation [ 4, 5]. This equation \nindicates that , excited by time -varying field of microwave, \nelectron spin will precess about  its average direction fixed by a \nstatic magnetic field.  Under suitable conditions, the amplitude \nof precession st rongly enhan ced and is named as “resonance”.   \nFMR  is a typical resonance for  uniform precession mode,  while \nunlike FMR, the spin precession in SWR exhibits as a  \nnon-uniform mode in terms of exchange or dipole interaction  \nbetween spins . SWR usually performs  as a mode  of sta nding \n \nThis work was supported in part by the State Key Program for Basic \nResearch of China grants (2007CB613206), the National Natural Science \nFoundation of China grants (10725418, 10734090, 10990104, and 60976092).  \nZiqian Wang is wi th National Laboratory for Infrared Physics, Shanghai \nInstitute of Technical Physics, Chinese Academy of Sciences, Shanghai, \n200083, China. (email: ziqian86@mail.sitp.ac.cn ). \nWei Lu is with National Laboratory for Infrared Physics, Shanghai Institute \nof Technical Physics, Chinese Academy of Sciences, Shanghai, 200083, China. (email: luwei@mail.sitp.ac.cn).  \nBogen Wang is with Department of Physics, Nanjing University, Nanjing, \nJiangsu Pr ovince, 210093, China. (email: bgwang@nju.edu.cn).  \n wave  by taking b oundary condition into account , and, from a \nmacroscopic view, might result  in magnetization precession . \nSWR is possible to be electrical detected since the \ntime-dependent AMR  [6]: \n( )2\n0 sin cosAA RR R R t θ θ ωψ = + − +∆ + \n \nis generat ed by precessional magnetization . Here R0 represents \nthe resistance while the magnetization M is perpendicular to the \nmicrowave induced current , RA is the decrement resistance, and \nθ is the angle of magnetization  M with respect to current . ∆θ  \nrefers to the amplitude of precession angle, ω is the precession \nfrequency and Ψ is used to identify the phase of magnetization \nprecession. Generally, in the electrical detection, the induced \ncurrent is originated from microwave and is given by \n() cos jj t ω = . As a res ult, the voltage response is expressed \nby Ohm’s law as:  \n()\n( )()\n()( )\n()0Re\ncos\n1sin 2 cos 22\n1sin 2 cos2A\nA\nAU jR\njR R t\njR t\njRω\nθ θ ωψ\nθ θψ= ×\n= +\n−∆ +\n−∆\n                                         (1)  \nThe last time -independent item of (1) is the DC response and \ncan be measured via electrical detection . \n In this article, w e are focusing on the DC response of \nexchange -dominated SWR in a permalloy thin strip. The wave \nvector of this kind of SWR is perpendicular to this strip’s plane \nand is signif icantly larger than other spin waves , e.g. dipole \ninteraction dominated spin waves . Our purpose is to research \nthe dynamics of this spin wave through electrical detection, and \nprovide a general picture to show how the spin wave evolves in \nresonant conditio n. \nII. EXPERIMENT : SAMPLE PREPARATION AND MEASUREMENT  \nThis work is performed on a Ni 80Fe20 (permalloy, Py) t hin \nstrip with polycrystalline structure . The dimensions of this \nspecimen are: length=2400μm, width=200μm and \nthickness=49nm (prepared as described elsewhere [ 7]). The Py  \nthin strip is bonded between two poles . A rectangular \nwaveguide  is applied  to transmit microwave  and ensure them  \nnormally propagate into the strip . An electromagnet  is \nemployed to provide an external static magnetic field µ0Hex \nwith maximum amplitude  of 1.5T. The signal is extracted by \nusing a lock- in amplifier  (SR830, Stanford)  with the \nmodulation frequency at 8.33 kHz. All data were obtained at Electrical Detection of Spin Wave Resonance in \na Permalloy Thin Strip  \nZiqian Wang, Wei Lu and Bogen Wang  \nE The 3rd International Symposium on Advanced Magnetic Materials and Applications (ISAMMA2013)  2 \nroom temperature. H ex is in-plane with  an angle θ to the strip’s \nlong axis . For simplicity, a  coordinate system is  selected,  as \nshown in Fig.1.  \nIII. RESULTS AND DISCUSSION  \nA. Anti- symmetrical SWR  DC-response  Spectra  \nThe direct comparison between the DC response spectra of \nSWR and FMR is s hown in Fig.2(a). In the previous research \n[7], researchers  found that the  FMR DC response  represents as \na combination of symmetric and anti -symmetric line shapes , \nshown in Fig.2(b). However, the DC response spectra of SWR \nare always anti -symmetrical.  The ratio of symmetric \ncomponent and anti -symmetric component reveals the phase \ndifference between precession and microwave -induced AC \ncurrent. Since the phase difference between precession and \nmagnetic  component of microwave can be obtained by solving \nLLG  equation, we can get the information of the relative phase \nbetween microwave’s electric component and magnetic component. As a result , the electromagnetic relative phase \ndifference cannot be obtained via SWR spectra because it does \nnot consist of symmetric li ne shape.  \nThe DC -responses measured at different microwave \nfrequencies  via sweeping H\nex are shown in Fig.2( c). The \nresonance s are associated with FMR and SWR.  In spite of the \ncomplex physical mechanism of SWR, its anti -symmetrical \nspect rum is empirically c oncluded as:  \n( )\n( )2 2ex r\nDC\nex rHH HU\nHH H∆−\n− +∆ ,                                                       (2) \nhere H∆ represents the line width of SWR signal  and rHis the \nintensity of external field at resonance. A peak  and a nadir exist \nat each side of ex rHH= .  \nThe anti -symmetrical shape of SWR signal brings facilities \nfor us to find H r and the signal’s amplitude. In this article, we \ndefine the amplitude of SWR as the difference between the \nstrengths  of signals at peak and nadir , as illustrated in Fig.2( d). \nAccording to (1), the amplitude also depends on  θ, and is proportional to sin(2 θ ), see in Fig.2( e). \nB. The spin dynamics in SWR  \nThe most intrigue issue for SWR is: why the SWR spectra \nare different from those of FMR? Before this question is \nanswered, we will begin our discuss ion with the origin of  \n“exchange interaction equivalent  magnetic field”, or H exchange . \nIn our experiment, H exchange  is directly obtained as \nµ0Hexchange =1100Gauss by recording the  dispersion curves for \nSWR and FMR, see in Fig.3(a) . Here we begin the following \ndiscussion with calculating the origin of H exchange . As noticed in  \nthe introduction, the exchange  interaction dominated spin wave \nin thin strip  exhibits as a standing  spin wave mode (SSW) due \nto the pinning condition  at the sample ’s surfaces , accordingly \nwe may draw a picture for SSW , shown in Fig. 3(b) and \nFig.3(c) . \n \nFor the nst spin in a SSW chain, considering the exchange \ninteraction, the LLG  equation is modified as [ 8]:  \nFig. 1.  The schematic diagram of the experimental setup and the selected \ncoordinate system. Here we set the direction of external field Hex as z-axis, and \nthe direction perpendicular to our sample’s plane , or xz -plane  as y-axis. Hex \nencloses the angle θ with respect to the length, or the long -axis of our sample.  \n \n \nFig. 2. (a). The comparison between SWR and FMR DC -responses. The \nfrequency of assisted microwave is fixed at 11.5GHz while θ  is 45 °. (b). The \nFMR spectra in (a) exhibits as a typical signal curve , which  consists of \nsymmetrical and anti -symmetrical line shapes. (c). The experimental (black  \ndots and lines ) and fitted (red line s) of SWR DC -response at different \nmicrowave frequen cies while θ  is fixed at 45 °. (d). The definitions of resonant \nexternal field Hr and amplitude U swr for SWR signals  in this article . (e). The \nexperimental (grey dots) and fitted (red line) of θ  dependence of U swr at fixed \nmicrowave frequency of 10.8GHz.  \nThe 3rd International Symposium on Advanced Magnetic Materials and Applications (ISAMMA2013)  3 \n( ) 11 n n nn n n nMH M MMM M MMαγ γλ−++× =×− ×+        (3) \nhere Mn represents the nst spin’s magnetic momentum, γ is the \nelectron’s gyromagnetic ratio, α represents the damping \ncoefficient  and λ is the exchange coefficient. Only the \nexchange interactions of nearby spins are considered.  For \nsimplicity, we treat the amplitude of each spin’s precessional \nmotion follows sinusoidal distribution, and  the precession for \nthe spin at antinode of SSW is  \n2\n__,,itit\nsw sw x sw y M em e m Mπωω−\n=  \nfor small m sw_x and msw_y, and  Mn is written by : \n2\n() ()\n2\n__,,\nsin , sin ,itit\nn xn yn\nitit\nsw x sw yM em e m M\nnnem e m MNNπωω\nπωω ππ−\n−\n=\n\n=\n \nAnd we obtain:  \n( ) 11\n2\n__2 sin cos 1\n, ,0nn n\nitit\nsw y sw xnMM MNN\ne m M em Mπωωππγλ γλ−+\n−+ ×= − \n\n× \n                 \nSo the exchange interaction of nearby spins acts on the nst spin \nis equivalent to a static field  Hexchange  and is given by:  \n0, 0, 2 1 cosexchangeHMNπλ = −                                         (4) \n(4) implies that, for series of spins with same phases, precession \ntraces and different amplitudes, the exchange interaction \namong them can be equalized as a static magnetic field with the \nsame direction as H ex. If the precessional motion s for each spin \nin SSW follow  other distributions, H exchange  for each spin might \ndifferent . However, Hexchange  is also equivalent to a static field  \nwithout time -varying component  if each spin precesses at the \nsame phase.  \nNonetheless , it is indicated by  solving LLG equation that , for \ntwo spins, while the damping coefficient s, the static magnetic \nfields (including H exchange  and Hex) and the exciting magnetic \nfields hmw are the same , the precessional amplitude ∆θ and \nphase Ψ for them  are also the same . Furthermore, if the static \nmagnetic fields are different while α and the assisted time \nvarying magnetic field h mw are the same, their precessional  \nphases will be different. In non -resonant condition, the phase of \nspin precession with respect to h mw is approximately  0 (for Hex \nlarger than H r) or 180°  (for Hex smaller than H r ), and ∆ θ is very \nsmall. That is to say, in non-resonant condition, the \npresumption, each spin in a SSW  precesses coherently, is \ncorrect . While Hex is swept at resonant region, with the \nincreased amplitude of spin precession, the non -uniformity of \nspin pre cession inside the ferromagnetic sheet is strengthened, \naccompany with the emergence of  phase difference between \ntwo nearby spins  along y -axis. Consequently, from a \nmacroscopic point of view, the time -varying component of \nmagnetoresistance is offset by the phase differences between \nspins, and equals zero . That is why the SWR signals are 0 at the \nexpected resonant  field Hswr.  If the DC response of SWR for H ex smaller than H r is written \nby \n( ) ()1sin 2 cos2DC ex r A U H H jR θ θψ <= − ∆ ,  \nin the field -swept mode, the signal fo r Hex larger than H r should \nbe expressed as:  \n( ) ()( )\n()\n( )1sin 2 cos2\n1sin 2 cos2DC ex r A\nA\nDC ex rU H H jR\njR\nUH Hθ θ ψπ\nθ θψ>= − ∆ −\n= ∆\n= −<                      (5)  \nThat is to say, the DC response of SWR is anti -symmetrical \nwith respect to H ex=Hr. \nIV. CONCLUSIONS  \nWe have demonstrated the anti -symmetrical SWR DC \nresponse spect ra in a P y thin strip through electrical detection . \nThe amplitude of SWR DC response is found to be influenced \nby the direction of H ex. We have also obtained H exchange  via \nmeasuring the dispersion curves of FMR and SWR. The origin of H\nexchange  is discussed. Furthermore, we have found that i n the \nresonant region, it is impossible for spins inside our sample precess as a standing wave mode with the same phase due to the \nrequirement of LLG equ ation, and the DC response thereby \nvanishes at H\nex=Hr. In the  non-resonant condition, the \nprecessional phase of spin wave changes 180°  from Hex<Hr to  \nFig. 3. (a). The experimental (colored dots) and fitted (colored lines) of SWR \nand FMR dispersion curves. Hexchange  is indicated. (b). The amplitude of the \nprecessional motions for each spin follows sinusoidal distribution, while the  \nmagnetic momentum of spin at the antinode of SSW (consists of N spins) is \nrecorded as Msw. (c). Only nearby spins, Mn-1 and Mn+1 are taken into account \nwhile calculating Hexchange  for a certain Mn. \n \nThe 3rd International Symposium on Advanced Magnetic Materials and Applications (ISAMMA2013)  4 \nHex>Hr, and finally results in the anti -symmetrical line shape of \nSWR DC response.  \n \nREFERENCES  \n[1] H. J. Juretschke, J. Appl. Phys. 31, 1401 (1960).  \n[2] W. Kwiatkowski , M. Stabrowski , and S.  Tumanski,  IEEE Trans. Magn.  \nvol. 19, no. 6, pp. 2502 -2505 (1983).  \n[3] J. D. Jackson, Classical Electrodynamics  (John Wiley & Sons, New \nYork, 1975), 2nd edition.  \n[4] T. L. Gilbert, IEEE Trans. Magn . 40, 3443 (2004 ). \n[5] L. Landau and L. Liftshitz, Phys. Z. Sowjetunion . 8, 153 ( 1935 ). \n[6] R. C. O’Handley, Modern Magnetic Materials: Principles and \nApplications  (Qiley, New York, 2000).  \n[7] N. Mecking, Y. S. Gui, and C.- M. Hu, Phys. Rev. B . 76, 224430 \n(2007).  \n[8] John W. Hartwell, Proceedings of the IEEE, vol. 56, no. 1, pp. 23 -31 \n(1968).  "    },    {        "title": "0802.1755v2.FMR_induced_Josephson_Current_in_a_Superconductor_Ferromagnet_Superconductor_Junction.pdf",        "content": "arXiv:0802.1755v2  [cond-mat.mes-hall]  12 Sep 2008FMR induced Josephson Current in a\nSuperconductor/Ferromagnet/Superconductor Junction\nS. Hikino1, M. Mori1, S. Takahashi1,2, and S. Maekawa1,2\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2CREST, Japan Science and Technology Agency, Kawaguchi 433- 0012, Japan\nAbstract\nWe propose the phase dynamics induced by spin waves in a super conduc-\ntor/ferromagnet/superconductor (SC/FM/SC) Josephson ju nction. The resistively shunted\njunction (RSJ) model, which describes the dynamics of super conducting phase difference, is\nextended to include the spin wave excitation by ferromagnet ic resonance (FMR) using the\ngauge invariant phase difference between two s-wave superconductors. The current-voltage\ncharacteristics show step structures when the magnetizati on in FM is driven by tuning the\nmicrowave frequency to FMR in the SC/FM/SC junction. The res ult presents a new route to\nobserve the spin wave excitation using the Josephson effect.\nPACS numbers: 74.50.+r, 76.50.+g\n1The Josephson effect is characterized by a zero voltage current t hrough a thin insu-\nlating barrier sandwiched by two superconductors[1]. This effect is a macroscopic quan-\ntum phenomenon involving phase coherence between two supercon ductors. When a finite\nvoltage-drop ( V) appears in the junction, the phase difference ( θ) evolves according to\n∂θ/∂t= 2eV//planckover2pi1. As a result, an alternating current flows due to the time dependen ce ofθ.\nMoreover, when an alternating electric field due to microwave irradia tion is applied to the\njunction, thecurrent-voltage( I-V)characteristicsshowShapirosteps[2]. Similarphenomena\nare observed in a normal metal weak link due to the proximity effect[3 , 4]. The Josephson\njunctions are useful devices. For instance, the ac Josephson cu rrent is utilized to detect the\nelectron-spin resonance in the Josephson junction doped with mag netic impurities[5]. The\ndynamics of the Josephson junction which are reflected in the I-Vcharacteristics are de-\nscribed by the resistively shunted junction (RSJ) model, which is use d to analyze Josephson\ndynamics[6, 7].\nTheJosephson effect insuperconductor/ferromagnet/superc onductor (SC/FM/SC) junc-\ntions has been of considerable interest in recent years[8, 9, 10, 1 1, 12, 13, 14, 15]. One of\nthe most interesting phenomena is the oscillation of the Josephson c ritical current ( Ic) in\nSC/FM/SC junctions. The origin of the oscillation is the exchange split ting of the conduc-\ntion band in the FM, in which Cooper pairs penetrate into the FM with a fi nite center of\nmass momentum. If the thickness of the FM is about half of the perio d of the oscillation,\nthe current-phase relation is shifted by πfrom that of a conventional Josephson junction\n(0-junction). This is called a π-junction, which has a potential application as a quantum\nbit[16]. The π-junctionis also used to measure experimentally a nonsinusoidal cur rent-phase\nrelation[17]. Most of the studies on SC/FM/SC junctions have so far been focused on the\neffect of the exchange splitting on Cooper pairs penetrating into th e FM. Ferromagnetic\nmaterials possess dynamic properties such as spin waves, which may be excited by using\nferromagnetic resonance (FMR). Therefore, the interaction be tween Cooper pairs and spin\nwaves in the FM is expected to play an important role in the transport . Up to now, the\nspin dynamics in the FM has been disregarded in SC/FM/SC junctions.\nIn this Letter, we study the effect of the spin dynamics on the Jose phson effect in a\nSC/FM/SC junction. The RSJ model is extended to include the effect of spin wave excita-\ntions using the gauge invariant phase difference between supercon ducting leads. We adopt a\nmodel in which the magnetization of FM exhibits a precessional motion . TheI-Vcharacter-\n2istic is calculated using the extended RSJ model. It is found that this c haracteristic exhibits\nstep structures and the dc Josephson current is induced due to t he spin wave excitation by\nFMR. The step structures appear whenever the relation ωJ= 2ℓΩ is satisfied with ℓbeing\nan integer, where Ω is the FMR frequency, and ωJ= 2eV//planckover2pi1. We propose a new route to\nobserve spin waves using the Josephson current in SC/FM/SC junc tions.\nThe system considered is a Josephson junction with a ferromagnet sandwiched by two\ns-wave superconductors. We choose a coordinate system such th at the electrode surfaces\nare parallel to the yzandxzplanes as shown in Fig. 1, and the current flows along x. The\nJosephson junction is characterized by the phase difference θ(y,z,t) between SCs, which is\ndescribed by the RSJ model as follows\ni=i0\ncsinθ(y,z,t)+1\nRΦ0\n2πdθ(y,z,t)\ndt+CΦ0\n2πd2θ(y,z,t)\ndt2, (1)\nwhereiis anexternal current density and i0\ncis thecritical current density. The flux quantum\nis denoted by Φ 0. The resistance, R, and the capacitance, C, in the Josephson junction are\nnormalized by the junction area, Syz, asR=R0/Syz,C=C0/Syz.R0andC0are the\nresistance and the capacitance in the Josephson junction, respe ctively.\nWeconsider thesituation, inwhichtheFMisexposedtoacircularlypola rizedmicrowave.\nThe microwave causes the precessional motion of the magnetizatio n, which corresponds to\nthe excitation of the uniform mode of a spin wave. The motion of magn etization due to the\nmicrowave radiation is described by the Landau-Lifshitz-Gilbert (LL G) equation[19]\ndM\ndt=−γM×Heff+α\nM/bracketleftbigg\nM×dM\ndt/bracketrightbigg\n, (2)\nwhereM(t)isthemagnetizationoftheFM, γisthegyromagneticratio, and αistheGilbert\ndamping. The effective field, to which M(t) responds, is given by Heff=H0+hac(t),\nwhereH0is an uniaxial magnetic anisotropic field, which is parallel to zaxis and hac=\n(haccosΩt,hacsinΩt,0) is the microwave driving field with frequency Ω. In Eq. (2), we\nneglect an anisotropic precession of the magnetization due to the d emagnetization field for\nsimplicity[20, 21]. The linearized solution of Eq. (2) is given by\nM±(t) =γMzhac\nΩ−Ω0∓iαΩe±iΩt, (3)\nwhich describes the precession of M±(t) around H0with frequency Ω and has a resonance\nwith Ω 0=γH0, whereM±(t) =Mx(t)±iMy(t). The rotating magnetic field is induced in\n3the FM through the magnetic flux density B(t) = 4πM(t). Due to the magnetic field, the\nsuperconducting phase is not gauge-invariant. Therefore, we de rive the equations for the\ngauge invariant phase difference between SCs. In the SC/FM/SC ju nction, the magnetic\nfield in the FM is equal to 4 πM(t), since we assume that the two superconductors separated\nby the FM of the thickness dare thick compared with the London’s penetration depth and\nthe magnetic field produced by the Josephson current is negligible. T herefore, the flux ∆Φ y\nparallel to the yaxis enclosed by a contour C in Fig. 1 is ∆Φ y= 4πMy(t)d/Φ0∆z. In a\nsimilar way, the flux ∆Φ zparallel to the zaxis is ∆Φ z= 4πMd/Φ0∆y. Combining these\nrelations with a gauge invariant phase difference, we obtain the differ ential equations with\nrespect toθ(y,z,t)[22, 23],\n∇y,zθ(y,z,t) =−4πd\nΦ0M(t)×n, (4)\nwhere▽y,z= (0,∂/∂y,∂/∂z ),nis the unit vector perpendicular to the yzplane. Equation\n(4) is satisfied by the following solution,\nθ(y,z,t) =θ(t)−4πMzd\nΦ0y+4πMy(t)d\nΦ0z, (5)\nwhere we adopt the gauge, such that the superconducting phase difference is equal to θ(t)\nwithout magnetic field. Introducing Eq. (5) in Eq. (1) and integratin g over the junction\narea, we obtain the extended RSJ model that includes the spin dyna mics in the FM,\nI/Ic=sin/parenleftBig\nΦs˜My(τ)/parenrightBig\nΦs˜My(τ)sin(θ(τ))+dθ(τ)\ndτ+βcd2θ(τ)\ndτ2, (6)\nwith\nΦs=Lzd4π2\nΦ0γMzhac\nΩ0, (7)\n˜My(t) =(Ω/Ω0−1)sin(ΩτJτ)\n(Ω/Ω0−1)2+(αΩ/Ω0)2+αΩ/Ω0cos(ΩτJτ)\n(Ω/Ω0−1)2+(αΩ/Ω0)2, (8)\nwhereIisanexternalcurrentand Ic=I0\nc(πΦ0/Φz)sin(πΦz/Φ0).I0\ncistheJosephsoncritical\ncurrent, andΦ zisthetotalmagneticfluxparalleltothe z-axisintheFM.Theothervariables\nare given by, τ=t/τJ,τJ= Φ0/(2πIcR), andβc=RC/τJ. When Φ z= 0 and Φ s˜My(τ) = 0,\nEq. (6) reduces to a conventional RSJ model. The extended RSJ mo del is applicable to both\n0- andπ-junctions. Below, we solve numerically Eq. (6) for βc= 0 (overdamped junction)\n4and calculate the I-Vcharacteristic using the relation, /angb∇acketleft∂θ/∂t/angb∇acket∇ight= 2e/angb∇acketleftV/angb∇acket∇ight//planckover2pi1, where/angb∇acketleft···/angb∇acket∇ight\ndenotes a time average. For the numerical calculation, αis taken to be 0.01[19].\nFirst, we consider thesituationthatthemagnetic fieldcomponent o fmicrowave is applied\nto the junction and the FMR occurs in the FM. This condition is realized by using a\nmicrowave cavity[24]. The I-Vcharacteristic is calculated using Eq. (6) and the result\nis shown in the solid and dashed lines in Fig. 2(a). The vertical axis is the normalized\ncurrent,I/Ic, and the horizontal axis is the normalized voltage, V/IcR. As shown by the\ndashed line in Fig. 2(a), when the microwave frequency (Ω) deviates from the ferromagnetic\nresonance frequency (Ω 0), theI-Vcharacteristic is in agreement with that of a conventional\nJosephson junction. On the other hand, when the microwave freq uency is in the condition\n|1−Ω/Ω0|≤αat which FMR occurs in the FM, the I-Vcharacteristic shows the step\nstructure under the FMR as shown by the solid line in Fig. 2(a).\nSecond, we consider the case that the electromagnetic field due to microwave irradiation\nis applied to the junction. For the electric field component, we use th e ac current source\nmodel in which Iis replaced by I+Iacsin(Ωt) in Eq. (6), where Iacis the amplitude of ac\ncurrent[25]. Figure 2(b) shows the I-Vcharacteristic which reflects the electromagnetic field\nirradiation. In the dashed line in Fig. 2(b), when Ω deviates from Ω 0, theI-Vcharacteristic\nshows Shapiro steps due to the ac current similarly to conventional Josephson junctions.\nWhen Ωisequal to thatofthealternating Josephson current, the se steps ariseat V=n1\n2e/planckover2pi1Ω\nwithnbeing an integer. When the microwave frequency is in the condition |1−Ω/Ω0|<\nα, the step structure dramatically changes as shown by the solid line in Fig. 2(b). The\namplitude of the second step is much larger than that of the first st ep. This behavior is\ndifferent from that of the Shapiro step for which the amplitude of th e steps decreases with\nincreasing the voltage. In addition, the step structures appear in the higher voltage region\n(V/IcR>1.5) unlike the case of the Shapiro steps.\nTo elucidate the origin of the step structure in the I-Vcharacteristic due to the spin\nwave excitations, we analyze the first term of the Josephson curr entIJ(t) in Eq. (6). Using\nthe generating function of Bessel functions and the standard ma thematical expansion of the\nsine function, we have\n5IJ(t) =−Ic\nζs∞/summationdisplay\nn=0∞/summationdisplay\nm=−∞J2m+1(ζs)ei[θ(t)+2(n+m+1)Ωt+2(n+m+1)ψ]\n+Ic\nζs∞/summationdisplay\nn=0∞/summationdisplay\nm=−∞J2m+1(ζs)e−i[θ(t)−2(n+m+1)Ωt+2(n+m+1)ψ], (9)\nwhereJm(ζs) is the Bessel function of the first kind, ζs= Φs/(Ω−Ω0), andψ=\narctan/parenleftBig\nαΩ\nΩ−Ω0/parenrightBig\n. In Eq. (9), when θ(t) =±2(n+m+1)Ωt, the time averaged IJ(t) has\na nonzero component of the dc Josephson current. As a result, t heI-Vcharacteristics rep-\nresent the step structure without changing voltage. Using the re lation,∂θ(t)/∂t=ωJ, we\nobtain the voltage at which the step structures occur,\nωJ= 2(n+m+1)Ω, (10)\nwherenandmareintegers. Figure 3 shows the amplitude of the step structure a s a function\nofthefrequency2( n+m+1)Ωinthe I-Vcharacteristic. Theverticalaxisisthenormalized\namplitude of the step. It is seen that the step structures only app ear at even number unlike\nthe case of the Shapiro step.\nFinally, we discuss the condition 2( n+m+ 1)Ω occuring step structures in the I-V\ncharacteristic. We convert Eq. (10) into V= 2(n+m+1)1\n2e/planckover2pi1Ω, which is quite similar to the\nconditionV=n1\n2e/planckover2pi1Ω of the Shapiro step, where the voltage is proportional to the num ber\nof photonnand its energy /planckover2pi1Ω[2, 25]. In an analogous way, we interpret the above results\nas follows. In the present situation, a uniform mode of spin wave is ex cited in the FM by\nthe FMR due to the microwave irradiation. Therefore, 2( n+m+1) and /planckover2pi1Ω in Eq. (10)\ncorrespond to the number of quantized spin waves (magnons) and their energy, respectively.\nIf Cooper pairs penetrating into the FM are scattered by the odd n umber of magnons,\nthe Josephson coupling vanishes between the s-wave SCs because of the formation of a\nspin triplet state[26], resulting in no Josephson current. On the oth er hand, the Josephson\ncoupling occurs when Cooper pairs penetrating into the FM compose the spin singlet state\nbecause of the even number of magnon scattering, so that the Jo sephson current flows.\nTherefore, the even number of magnons couples to the Josephso n current.\nIn conclusion, we have studied the effect of the spin dynamics on the ac Josephson effect\nin the superconductor/ferromagnet/superconductor (SC/FM /SC) junction. The resistively\nshunted junction (RSJ) model is extended to include the spin wave e xcitations induced\n6by the ferromagnetic resonance (FMR) and the equations for the gauge invariant phase\ndifference between SCs have been derived. We find that the curren t-voltage characteristics\nshow step structures when the relation ωJ= 2ℓΩ withℓbeing an integer is satisfied, where\nωJ= 2eV//planckover2pi1and Ω the FMR frequency. The results provide a new route to obser ve the spin\nwaves using the Josephson current in SC/FM/SC junctions.\nThe authors thank M. Aprili, I. Petkovic and S. E. Barnes for valuab le discussions. This\nworkissupportedbyaGrantinAidforScientific ResearchfromMEXT , thenextGeneration\nSupercomputer Project of MEXT. The authors thank the Superc omputer Center, Institute\nfor Solid State Physics, University of Tokyo for the use of the facilit ies.\n[1] B.D. Josephson, Phys. Lett. 1, 251 (1962).\n[2] S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).\n[3] P. G. de Gennes, Rev. Mod. Phys. 36, 225 (1964).\n[4] K. K. Likharev, Rev. Mod. Phys. 51, 101 (1964).\n[5] S. E. Barnes and F. Mehran, Phys. Rev. B 34, 4537 (1986)\n[6] W.C. Stewart, Appl. Phys. Lett. 12, 277 (1968).\n[7] D.E. McCumber, J. Appl. Phys. 39, 3113 (1968).\n[8] A.I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n[9] A. A. Golubov, M. Yu. Kupriyanov, and E. Ilichev, Rev. Mod . Phys.76, 411 (2004).\n[10] V. V. Ryazanov, V. A. Oboznov, A.Yu. Rusanov, A.V. Veret ennikov, A. A. Golubov, and J.\nAarts, Phys. Rev. Lett. 86, 2427 (2001).\n[11] T. Kontos M. Aprili, J. Lesueur, F. Genˆ et, B. Stephanid is, and R. Boursier, Phys. Rev. Lett.\n89, 137007 (2002).\n[12] V. A. Oboznov, V. V. Bolginov, A. K. Feofanov, V. V. Ryaza nov, and A. I. Buzdin, Phys.\nRev. Lett. 96, 197003 (2006).\n[13] J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, and M. G. Blamire, Phys. Rev. Lett. 97,\n177003 (2006).\n[14] M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D. Koelle , R. Kleiner, and E. Goldobin,\nPhys. Rev. Lett. 97, 247001 (2006).\n[15] M. Mori, S. Hikino, S. Takahashi, and S. Maekawa, J. Phys . Soc. Jpn. 76, 054705 (2007).\n7[16] Y. Yamashita, K. Tanikawa, S. Takahashi, and S. Maekawa , Phys. Rev. Lett. 95, 097001\n(2005).\n[17] H. Sellier, C. Baraduc, F. Lefloch, and R. Calemczuk Phys . Rev. Lett. 92, 257005 (2004).\n[18] G. Tkachov, E. McCann, and V. I. Falko, Phys. Rev. B 65, 024519 (2001).\n[19] B. Hillebrands andK.Ounadjela, Spin dynamics in confined magnetic structures II (Springer-\nVerlag Berlin Heidelberg, New York, 2003)\n[20] C. Kittel, Introduction to Solid State Physics (Eighth Edition) , (Wiley, New York, 2005)\n[21] S. Chikazumi, Physics of Ferromagnetism , (Oxford University Press, New York, 1997).\n[22] M. Tinkham, Introduction to superconductivity (Second Edition) (Dover, New York, 2004).\n[23] R. D. Parks, Superconductivity, Vol. 2 (Marcel Dekker, New York, 1969).\n[24] H. Y. Inoue, K. Harii, K. Ando, K. Sasage, and E. Saitoh, J . Appl. Phys. 102, 083915 (2007).\n[25] A. Barone and G. Patern` o Physics and applications of the Josephson effect , (John Wiley &\nSons, New York, 1982)\n[26] S. Takahashi, S. Hikino, M. Mori, and S. Maekawa, Phys. R ev. Lett. 99, 057003 (2007).\n8SC FM SC yz\nxC\nΔz\nIac h\nFIG. 1: Josephson junction with FM between two s-wave SCs. A microwave radiation haconto\nFM causes the precession of the magnetization of FM. Coordin ate system is chosen such that the\nelectrodes surfaces are parallel to yz-plane. C is a contour in xz-plane.\n9(a) \n(b) 0 1 2 30123\nV/IcRI/Ic Ω/Ω0=0.9 \n Ω/Ω0=1 \n0 1 2012\n Ω/Ω0=0.9 \n Ω/Ω0=1 \nV/IcRI/Ic\nFIG. 2: Current-voltage characteristics. The solid and das hed lines show the I-Vcharacteristics\nmicrowave frequency for Ω /Ω0= 0.9 and Ω /Ω0= 1, respectively. We choose Φ s= 0.03. (a)\nIac/Ic= 0. (b) Iac/Ic= 0.6.Iacis ac current across the junction due to the electromagnetic field.\n10FIG. 3: The amplitude of the step in the I-Vcharacteristics vs the normalized Josephson fre-\nquency. The positions of step lie on the even number. ∆ I2ℓ/Icis the amplitude of step in the I-V\ncharacteristics, and ℓis an integer.\n11"    },    {        "title": "0801.1019v1.Magnetization_reversal_driven_by_spin_injection___a_mesoscopic_spin_transfer_effect.pdf",        "content": "arXiv:0801.1019v1  [cond-mat.mes-hall]  7 Jan 2008Magnetization reversal driven by spin-injection :\na mesoscopic spin-transfer effect\nJ.-E. Wegrowe∗and S. M. Santos, M.-C. Ciornei, H.-J. Drouhin\nLaboratoire des Solides Irradi´ es, Ecole Polytechnique, 9 1128 Palaiseau Cedex, France.\nJ. M. Rub´ ı\nDepartement de Fisica Fonamental, Universitat de Barcelon a,\nDiagonal 647, Barcelona 08028, Spain.\n(Dated: November 2, 2018)\nAbstract\nA mesoscopic description of spin-transfer effect is proposed , based on the spin-injection mecha-\nnism occurring at the junction with a ferromagnet. The effect o f spin-injection is to modify locally,\nin the ferromagnetic configuration space, the density of mag netic moments. The corresponding\ngradient leads to a current-dependent diffusion process of th e magnetization. In order to describe\nthis effect, the dynamics of the magnetization of a ferromagne tic single domain is reconsidered in\nthe framework of the thermokinetic theory of mesoscopic sys tems. Assuming an Onsager cross-\ncoefficient that couples the currents, it is shown that spin-d ependent electric transport leads to\na correction of the Landau-Lifshitz-Gilbert equation of th e ferromagnetic order parameter with\nsupplementary diffusion terms. The consequence of spin-inje ction in terms of activation process of\nthe ferromagnet is deduced, and the expressions of the effecti ve energy barrier and of the critical\ncurrent are derived. Magnetic fluctuations are calculated: the correction to the fluctuations is\nsimilar to that predicted for the activation. These predict ions are consistent with the measure-\nments of spin-transfer obtained in the activation regime an d for ferromagnetic resonance under\nspin-injection.\nPACS numbers: 75.40.Gb,72.25.Hg,75.47.De\n∗Electronic address: jean-eric.wegrowe@polytechnique.edu\n1In the context of spintronics, spin transfer is a generic term that describes the magnetiza-\ntion reversal or magnetization excitations of a ferromagnetic laye r provoked by the injection\nof a spin-polarized electric current. This effect has now been obser ved while measuring the\nmagnetization of many different systems [1]. It has also been implemen ted into the last\ngeneration of magnetic random access memories [2].\nSpintronics immerged first with the discovery of spin-injection and g iant magnetoresis-\ntance (GMR) [3]. Effects of spin-injection at a junction of a ferroma gnet are successfully\ndescribed within the two spin-channel model [4, 5, 6, 7, 8]. In this de scription, GMR, or\nspin-accumulation, isduetothespins oftheconduction electrons t hataredriven out-ofequi-\nlibrium by an interface: the difference of the electro-chemical pote ntials ∆µbetween the two\nspin-channels leads to a redistribution of the spin populations throu gh spin-flip mechanisms.\nDepending on the magnetic state of the ferromagnet at the junct ion, this redistribution of\nspin-dependent electronic populations modifies theresistance. Fo r this reason, theGMR can\nbe used as a probe (e.g. read heads for hard disk drive) in order to m easure precisely the\nposition of a nanoscopic ferromagnetic moment in the correspondin g configuration space.\nInversely, from the point of view of the ferromagnetic properties , it is natural to expect\nthat the spin-flip mechanism at the interface modifies locally the dens ity of the magnetic\nmoments in this configuration space. This is a consequence of both t he spin-orbit coupling\n(or s-d relaxation), and the redistribution of spins in the magnetic c onfiguration space. The\nspin-injection would then be responsible for a gradient of the densit y in the magnetic con-\nfiguration space, that contributes to the diffusion processes of t he ferromagnetic moment.\nThe aim of this work is to investigate the consequences of this redist ribution mechanism\nfor the ferromagnetic moments. This task is performed in the fram ework of the nonequilib-\nrium theory of mesoscopic systems [9, 10]. In this context, the cou pling between the spin of\nconduction electrons and the ferromagnetic order parameter is d ue to the introduction of a\nrelevant Onsager cross-coefficient that links the spin-polarized ele ctric currents (i.e. trans-\nport of mass, spins, and electric charges in the normal space) to t he ferromagnetic current\n(i.e. a massless transport of ferromagnetic moments in the corres ponding space). Physically,\nthis spin-transfer cross-coefficient accounts for the fact that the spins of the conduction elec-\ntrons also contribute to the transport of the ferromagnetic ord er parameter in the space of\nmagnetic moments, in analogy with the thermoelectric Peltier-Seebe ck cross-coefficient that\naccounts for the fact that charge carriers contribute also to th e transport of heat.\n2The motivation for a classical mesoscopic analysis was the need to ac count for the fol-\nlowing highly specific properties observed in spin-transfer experime nts that can hardly be\naccounted for in direct microscopic approaches [11, 12, 13, 14]. Fir st, in single domain\nferromagnets, the reversible part of the hysteresis loop is not sig nificantly modified under\ncurrent injection, while the irreversible jump is drastically modified [14 ]. Second, the am-\nplitude of spin-transfer is proportional to the giant magnetoresis tance and to the amplitude\nof the current [15, 16, 17]. Third, the N´ eel-Brown activation law is still valid under current\ninjection, with a correction of the barrier height that is quasi-symm etric under both the\npermutation of the magnetic configuration (parallel to anti-paralle l and inversely) and the\nchange of the direction of the current [15, 18, 19, 20]. ”Quasi”-sy mmetric means here that a\nquantitative shift (a factor 2 to 4 in general) is systematically obser ved in the amplitude of\nspin transfer for both transitions in spin-valve structures. Four th, this last quasi-symmetry\nisalso observed inthecontext offerromagnetic resonanceunder current injectionclose tothe\nequilibrium states (i.e. with an weak spin-injection) [21]. In order to co mpare the results of\nthe model with these observations, after calculating the diffusive c orrection of the Landau-\nLifshitz-Gilbert equation (LLG) due to spin-injection in the first sec tion, the correction to\nactivation law is derived in a second section, and the correction to th e fluctuations is derived\nin the third section.\nA. Spin-injection correction to the ferromagnetic Landau-Lifshitz-Gilbert Equa-\ntion\nIntheframeworkofthetwoconductingchannelapproximation, t hesystemisdescribedby\ntwo electronic populations. A first conducting channel is carrying t he conduction electrons\nof spin up ↑with the electric conductivity σ↑and the other channel is carrying the electron\nconduction of spin down ↓with the electric conductivity σ↓. The quantification axis is\ndefined by the direction of the magnetization of the ferromagnet. The Ohm’s law is valid\nfor each channel: in a 1D ferromagnetic wire described with a single sp ace coordinate z, the\nelectric current Je\n↑(resp.Je\n↓) is related to the localelectric field E/a∇∇owbothv=−1\ne∂µe\n/arrowbothv\n∂z[7] through the\nconductivity: Je\n/a∇∇owbothv=σ/a∇∇owbothvE/a∇∇owbothv, whereµe\n↑(resp.µe\n↓) is the electrochemical potential of the channel\nof spins↑(resp.↓).\nIn the case of an interface between a normal metal and a ferroma gnet (or between two\n3ferromagnets), the spin-dependent relaxation between both ele ctronic populations leads to\na redistribution of spins within the two channels, that is driven by the difference between\nthe two electro-chemical potentials ∆ µe(z) =µe\n↑−µe\n↓(the ”spin-neutral” electrochemical\npotential is defined, in turns, by µe\n0=µe\n↑+µe\n↓). The parameter ∆ µeaccounts for the\nspin-injection, or spin accumulation, mechanism (see Fig. 1). The eq uilibrium state of the\nspin system is recovered in the bulk where, by definition, ∆ µe(∞) = 0. This corresponds to\na distance of some few times the spin-diffusion length (some tens of n anometers in usual Co\nor Ni layers). The description can be generalized to multichannel mo del that includes the\nspins of the conduction electron of the sband and of the spins of the conduction electrons\nof thedband, with the corresponding interband relaxation [14, 22]. For con venience, a\ntwo-channel approximation is used in the following, in which we define Je\n0=Je\n↑+Je\n↓as the\n”spin-neutral” electrical current, and δJe=Je\n↑−Je\n↓is the spin-polarized electrical current.\nThe system under consideration is composed not only by the microsc opic spins up and\ndown carried by the conduction electrons of different nature (ban dsord), but also by\na ferromagnet of length v(vis also the volume in the normal space for a section unity)\ndescribed by a ferromagnetic order parameter /vectorM. The last variable is defined in the space\nof the ferromagnetic moments ( Fig. 2) of constant modulus : /vectorM=Ms/vector ur, calledγ- space\nin the following [23]. This space can be defined on the unit sphere (see F ig. 2) with the two\nanglesθandϕ, the radial unit vector /vector ur, the azimuth unit vector /vector uθand the zenith unit\nvector/vector uϕ. The magnetization is then described statistically in the configuratio n space by\nthe density ρF(θ,ϕ) of ferromagnetic moments oriented at a given direction γ={θ,ϕ}, and\nalso by the ferromagnetic potential energy VF(θ,ϕ) that contains at least the contributions\ndue to the external magnetic field /vectorHand the anisotropy energy. Typically, for a uniaxial\nanisotropy with anisotropy constant K, the ferromagnetic potential writes: VF(θ,ϕ) =\nKsin2(θ)−MsHcos(θ−φ) whereφgives the direction of the applied field. This potential\nenergy has the form of a double well potential (Fig. 2).\nIn the absence of spin-injection, the magnetization is a conserved variable, so that the\nconservation law writes∂ρF\n0\n∂t=−div(/vectorjF\n0), where /vectorjF\n0=jFθ\n0/vector uθ+jFϕ\n0/vector uϕis the ferromagnetic\ncurrent density andthe operator divis thedivergence defined onthe surface of a unit sphere.\nThis is no longer the case under spin-injection at an interface: due t o the redistribution of\nspins in the different channels (especially from sband todband) spins are transferred from\n4one sub-system to the other, and the ferromagnetic sub-syste m becomes an open system .\nIn order to work in a larger system that is closed, i.e. that does not e xchange magnetic\nmoments with the environment, the total ferromagnetic density ρF\ntotand total ferromagnetic\ncurrent/vectorjF\ntotare defined in what follows.\nIn the total system, that includes both the ferromagnetic layer a nd the spin-polarized\ncurrent, the entropy production d S/dt (per unit of solid angle and per unit of length) is\ngiven by\ndS\ndt=−1\nT/parenleftbigg\n/vectorjF\ntot./vector∇µF−δJe∂∆µe\ne∂z−Je\n0∂µe\n0\ne∂z/parenrightbigg\n(1)\nwhereTis the temperature assumed uniform, /vector∇is the gradient defined on the surface of\nthe unit sphere, µFis the total ferromagnetic chemical potential, and ethe charge of the\nelectron. The last term in the right hand side is the Joule heating, the second term is the\ndissipation related to the giant magnetoresistance, and the first t erm is the ferromagnetic\ndissipation that definesthe total ferromagnetic current /vectorjF\ntotin the internal space of magnetic\nmoments [24].\nFrom the expression of the entropy production Eq. (1) and the se cond law of thermo-\ndynamics d S/dt≥0, the flux involved in the system are related to the generalized forc es\nthrough the matrix of the Onsager transport coefficients [25]\n\njϕF\njθF\nδJe\nJe\n0\n=−\nLϕϕLϕθ0 0\nLθϕLθθl0\n0˜l σ0βσ0\n0 0βσ0σ0\n\n1\nsin(θ)∂µF\n∂ϕ\n∂µF\n∂θ\n1\ne∂∆µe\n∂z\n1\ne∂µe\n0\n∂z\n(2)\nAll coefficients are known, except the new cross-coefficients land˜l, introduced in this\nmodel, and related to the experimental parameters at the end of t he next section. The\nelectric conductivity σ0is given, in the two channel approximation, by σ0=σ↑+σ↓\n2, and\nthe conductivity asymmetry βis given by β=σ↑−σ↓\nσ0. The four ferromagnetic transport\ncoefficients are {Lθθ,Lθϕ,Lϕθ,Lϕϕ}. However, the Onsager-Casimir reciprocity relations\ngiveLθϕ=−Lϕθ, and the symmetry imposes that Lθθ=Lϕϕ[14]. Furthermore, the two\ntransport coefficients left are not independent since they are bot h related to the Gilbert\ndamping coefficient ηand the gyromagnetic factor Γ . The following relation holds [14]:\n5FIG. 1: Illustration of the spin-accumulation occurring in a nanoscopic ferromagnetic layer with\nits two non-ferromagnetic contacts: the profile of the elect rochemical potential difference ∆ µeis\nplotted as a function the spatial coordinate z.\nLθθ=−αLθϕ=ρF\n0αΓ\nvMs(1+α2)(3)\nwhereα=ηΓMsis the normalized Gilbert damping coefficient.\nIntheabsence ofspin-injection ∆ µe= 0 andthereis nocoupling between the currents. In\nthat case, the well-known LLG equation of the ferromagnetic layer of volume vis recovered\nby inserting the ferromagnetic chemical potential [26]\nµF=kBTln(ρF\n0)+vVF(4)\ninto the expression of the ferromagnetic current density /vectorjF\n0=−¯L/vector∇µFwhere¯Lis the 2x2\nmatrix of components {Lθθ,Lθϕ,Lϕθ,Lϕϕ}=Lθθ/α({α,−1,1,α}). We have the expression:\n/vectorjF\n0v=−ρF\n0Γ\nMs(1+α2)/braceleftigg\n/vector ur×/parenleftigg\n/vector∇VF+kBT\nv/vector∇ρF\n0\nρF\n0/parenrightigg\n−α/vector ur×/bracketleftigg\n/vector ur×/parenleftigg\n/vector∇VF+kBT\nv/vector∇ρF\n0\nρF\n0/parenrightigg/bracketrightigg/bracerightigg\n(5)\nThe LLG equation (that includes diffusion terms) is deduced immediate ly by dividing\nEq. (5) with the density ρF\n0, thanks to the relations ρF\n0d/vector ur\ndt=/vectorJF\n0and/vectorM=Ms/vector ur[14].\n6FIG. 2: The configuration space of the ferromagnetic order pa rameter is represented by a sphere\nof radius unity. The double well shown is a projection over th e plane that contains the two\nequilibrium states at θ1andθ2, and the top of the barrier at θ0. The step approximation for\nthe chemical potential µ(θ) is plotted with the thermal fluctuations sketched by the das hed area\naround the two minima.\nIn the presence of spin injection, ∆ µ∝ne}ationslash= 0 and a correction to the above LLG equation\nis expected, due to the phenomenological transport spin-transf er coefficient l, introduced in\nthe Onsager matrix of Eq. (2). For the sake of simplicity, we only tre at the coupling of\nlongitudinal spin relaxation along the vector /vector uθ.\nAssuming lconstant, the longitudinal component of the total ferromagnet ic current in\nthe layer /vectorJF(θ,ϕ,v) =/integraltext\nv/vectorjF(θ,ϕ,z)dzhas the form,\nJθF(θ,ϕ,v) =jθF\n0(θ,ϕ)v−l/integraldisplay\nv∂∆µe\n∂z(θ,z)dz (6)\nThe quantity β/integraltext\nv∂∆µe\n∂zdzis proportional to the giant magnetoresistance RGMRgenerated\nat the interface [6, 14] (see Fig. 1 and [29]):\nβ/integraldisplay\nv∂∆µe\n∂zdz= 8eIRGMR(θ) (7)\nwhereJe\n0=Iis the electric current injected in the junction of section unity. Sinc e the\ngiant magnetoresistance RGMR(θ) depends on the angle between the incident spin-polarized\ncurrent (e.g. defined by the magnetization state of a second magn etic layer in a spin-valve\n7structure) and the ferromagnetic layer, it depends on the state of the ferromagnetic layer,\ni.e. the position in the γ-space. Eq. (6) shows that the total ferromagnetic current is s imply\nrelated to the gradient of a total chemical potential µF\ntot:\n/vectorJF=−¯L/vector∇µF\ntot (8)\nThetotal chemical potential µF\ntotwrites:\nµF\ntot(θ) =µF(θ)+l\nLθθVe(θ) (9)\nwhere the electrospin chemical potential Ve(θ) is given by an integration over the length\nof the ferromagnet in the direction z, and over the angle θ:\nVe(θ) =1\ne/integraldisplay\nθ/integraldisplay\nv∂∆µe\n∂z(z,θ′)dzdθ′=8eI\nβ/integraldisplay\nθRGMR(θ′)dθ′(10)\nInserting the ferromagnetic chemical potential Eq. (4) into Eq. (9) yields\nµF\ntot(θ) =kT ln(ρF\n0(θ))+l\nLθθVe(θ)+vVF(θ) (11)\nIn order to deal with the density of the total system ρF\ntot, the contribution of the spin-\ndependent scattering is included in the logarithm :\nµF\ntot=kT ln(ρF\ntot)+vVFwhere\nρF\ntot(θ) =ρF\n0(θ).el\nkTLθθVe(θ)(12)\nConsequently, the expression of the ferromagnetic-electroche mical potential takes the same\nform as in the case of a ferromagnetic chemical potential without s pin-injection, but with a\nmodified density (that depends on the spin dependent scattering p rocess).\nThe total ferromagnetic current, given by Eq. (8), writes now:\n/vectorJF=−ρF\ntotΓ\nMs(1+α2)/braceleftigg\n/vector ur×/parenleftigg\n/vector∇VF+kBT\nv/vector∇ρF\ntot\nρF\ntot/parenrightigg\n−α/vector ur×/bracketleftigg\n/vector ur×/parenleftigg\n/vector∇VF+kBT\nv/vector∇ρF\ntot\nρF\ntot/parenrightigg/bracketrightigg/bracerightigg\n(13)\nOnce again, the corresponding generalized LLG equation is directly obtained from the\nexpressiond/vector ur\ndt=/vectorJF\ntot\nρF\ntot. The equation canberewritten with introducing the ”pseudo” effec tive\nfield/vector˜Heff=−/vector∇VF+kBT\nv/vector∇ρF\ntot\nρF\ntot, so that the Generalized LLG equation takes the usual form:\n8d/vector ur\ndt=Γ\nMs(1+α2)/braceleftig\n/vector ur×/vector˜Heff−α/vector ur×/bracketleftig\n/vector ur×/vector˜Heff/bracketrightig/bracerightig\n(14)\nThis equation has the same form than that of the LLG equation witho ut spin-injection\n(note that it is mainly a consequence of the approximation of longitud inal coupling only,\ndisregarding precessional coupling), but the diffusion part of the e ffective field is modified\nthrough the correction of the density ρF\ntot=ρF\n0el\nkBTLθθVe:\nkBT/parenleftigg/vector∇ρF\ntot\nρF\ntot−/vector∇ρF\n0\nρF\n0/parenrightigg\n=l\nLθθ∂Ve\n∂θ/vector uθ (15)\nthe correction due to spin-injection8le\nβkBTLθθvRGMRIis consequently a correction to the\ndiffusion term that is proportional to the GMR and to the injected cu rrent. However, this\ndiffusiontermaccountsforfluctuations[27]. Theeffectofthediffus iontermscannotbetaken\ninto account in the quasi-static hysteresis loop (i.e. for vanishing te mperature or infinite\nmeasurement times) as a deterministic effective field (this point is disc ussed in the references\n[14, 18]). A deterministic correction to the reversible part of the hy steresis (the quasi-static\nstates) is hence not expected here. In contrast, the effect of t his correction is considerable in\nthe activation regime or in ferromagnetic resonance near equilibrium , described in the next\nsections.\nB. Spin-injection correction to the activation process\nWhat is the consequence of the diffusion correction Eq. (15) in the a ctivation regime of\nmagnetization reversal? This question is investigated below for large time scales (beyond\nnanosecond time scales, or ”high barriers”), with the correspond ing activation process, i.e.\ntheso-calledN´ eel-Brownrelaxation[30]. Intheactivationregime, theeffect oftheprecession\ncan be neglected: the gradient /vector∇and the divergence operator can be reduced to the scalar\nderivative∂\n∂θ.\nTheferromagneticpotential VFhasadoublewell structure (Fig. 2), withthetwo minima\nθ1andθ2, and a maximum at θ0. The description of the activation process is based on the\nhigh barrier approximation under which the ferromagnetic current becomes a step function\nin theθspace:\n9JF(θ,t) =JF(t)[Θ(θ−θ1)−Θ(θ−θ2)] (16)\nwhere Θ( θ) is the Heaviside step function.\nThe chemical potential can also be approximated by a step function that takes the value\nof the equilibrium states in the left ( θ1) or in the right ( θ2) side of the potential barrier in the\nferromagneticconfigurationspace(Fig. 2): µF(θ,t) =µF(θ1,t)Θ(θ0−θ)+µF(θ2,t)Θ(θ−θ0).\nUsing now this expression, the density function ρF\ntot(θ) of the ferromagnet in configuration\nspace writes:\nρF\ntot(θ,t) =ρF\ntot(θ1,t)e−v(VF(θ)−VF(θ1))\nkBTΘ(θ0−θ)+ρF\ntot(θ2,t)e−v(VF(θ)−VF(θ2))\nkBTΘ(θ−θ0) (17)\nThe ferromagnetic current is related to the gradient of the gener alized chemical potential\nthrough our fundamental relation Eq. (8) : JF(θ,t) =−Lθθ∂µF\ntot(θ,t)\n∂θ. This equation can be\nwritten into the more convenient form\nJF(θ,t) =−DF\ntot(θ)e−vVF(θ)\nkBT∂eµF\ntot(θ,t)\nkBT\n∂θ(18)\nwith the diffusion coefficient\nDF\ntot(θ)≡kTLθθ\nρF\ntot(θ)=DF\n0.e−lVe(θ)\nkBTLθθ (19)\nwherethesecondequalityisdeduced fromEq. (12)andtheparame terDF\n0=Lθθ.kBT/ρF\n0\n(dimension of angle per unit of time) is the usual ferromagnetic diffus ion coefficient that is\nconstant.\nThe activation process is described by a rate equation (see Eq. (23 ) below), e. i. a\ncontracted description that is obtained by performing a reduction of the continuous internal\nvariableθover the equilibrium states θi(i={1,2}).\nThe total flow has a zero divergence current density divJF= 0. The system is quasi\nstationary and the total current is I= 2πsin(θ)J(t). Eqs. (16) and Eqs. (18) can be\nintegrated over the measure exp(vV/kBT)dθto give:\nI/integraldisplayevVF(θ)\nkBT\n2πsinθ[Θ(θ1−θ)−Θ(θ2−θ)]dθ=−/integraldisplay\nDF\ntot(θ)e−vVF(θ)\nkBT∂eµF(θ)\nkBT\n∂θdθ(20)\n10so that the total current writes [32]:\nI=DF\ntot(θ0)eµ(θ2)/kBT−eµ(θ1)/kBT\n/integraltextθ2\nθ1evVF(θ)/kBT\n2πsin(θ)dθ(21)\nDefining the number of representative points near equilibrium by n(θi) =\n/integraltextθi+ǫ\n−ǫρ(θ′)2πsin(θ′)dθ′, the density in the double well potential Eq. (17) leads to the expre s-\nsions:\neµ(θ1)/kBT=ntot(θ1)\n2π/integraltextθ0\n−ǫe−vVF(θ)/kBTsin(θ)dθ\neµ(θ2)/kBT=ntot(θ2)\n2π/integraltextθ2+ǫ\nθ0e−vVF(θ)/kBTsin(θ)dθ(22)\nwhereǫis a real number. Inserting the above equation into Eq. (21) leads t o the\ngeneralized rate equation:\nI= ˙ntot(θ1) =−˙ntot(θ2) =ntot(θ1)\n˜τ1→2−ntot(θ2)\n˜τ2→1(23)\nUsing the steepest descents approximation [33] for the three inte grals present in Eq. (21)\nafter inserting Eqs. (22), the total relaxation times write:\n˜τ−1\ni→i±1=Dtot(θ0)sin(θ0)\nsin(θi)v/radicalbig\n|(VF)′′(θ0)||(VF)′′(θi)|\n2πkTevVF(θi)−vVF(θ0)\nkBT (24)\nMore explicitly, this generalized rate equation as the form of the usu al N´ eel -Brown\nrelaxation rates τi→i±1, with an exponential correction expressed in terms of the electrospin\nchemical potential Ve(θi):\n˙ntot(θ1) =−˙ntot(θ2) =n(θ1)\nτ1→2e−l(Ve(θ1)−Ve(θ0))\nkBT Lθθ−n(θ2)\nτ2→1e−l(Ve(θ2)−Ve(θ0))\nkBT Lθθ (25)\nwhereVe(θ) = 8eI/integraltextθ\n0RGMR(θ′)dθ′/β. Accordingly, the N´ eel-Brown activation law is still\nvalid under current injection, with a correction that can be added t o the potential energy\nbarrier. The total potential energy including the contribution of t he spin-injection writes :\nVtot(θ) =VF(θ)+8leI\nβLθθ/integraldisplay\nθRGMR(θ′)dθ′(26)\nThe correction to the energy barrier is proportional to the curre nt I and to the GMR\nintegrated over the magnetization states corresponding to the e nergy barrier height. Eq.\n11(25) shows that the process follows the N´ eel-Brown activation law with the relaxation times:\n˜τ=τ0e−∆Vtot\nkBTwhere ∆Vtot=Vtot(θi)−Vtot(θ0),θi={θ1,θ2}andτ0is the usual waiting time\n(i.e. the prefactor in Eq.(24) ).\nIn order to compare this analysis with experimental results perfor med on nanopillars,\nlet us assume a spin-valve structure with two ferromagnetic layers composed of identical\nmaterials with β≥0, in which only two states along the anisotropy axes are allowed. One\nlayer is fixed (the pinned layer) and the states of the other (the ”f ree” layer) areinvestigated.\nThe magnetization states of the free layer are θ1= 0 for the parallel configuration (P) and\nθ2=πfor the ant parallel configuration (AP). The GMR for the P state RGMR(0) = 0\ncorresponds to the reference configuration (no spin-flip). The A P configuration RGMR(π) =\n∆Rcorresponds to the maximum GMR. If we take the most simple form fo r the angular\ndependence of the GMR [31] RGMR(θ) = ∆R(1−cos(θ)), we have Ve(0)− Ve(π/2) =\n−4I∆R(π−2)\nβandVe(π)−Ve(π/2) = +4I∆R(π+2)\nβ.\nNote that the GMR parameter ∆ Ris a function of β: expressed in terms of the spin\ndiffusion length lsfit writes [14] ∆ R/R=β2\n1−βlsf\nv(wherevis the length of the layer).\nIn conclusion, the injection of the current leads to suppress one t ransition and to accel-\nerate the other: the current provokes the magnetization rever sal from one configuration to\nthe other, and the transition depends on the current direction. I n the exemple ebove, the\ncurrent provokes the magnetization reversal from P to AP config uration for positive current,\nand provokes the magnetization reversal from AP to P configurat ion for negative current.\nThis is a sufficient condition in order to accounts for the hysteresis lo op of the magnetization\ndriven by the current. In the general case, both transitions are defined with the relaxation\nrate\nτi=τ0e−∆VF±cil∆R\nβLθθI\n2kBT (27)\nwith a asymmetry factor ci≡/integraltextθ0\nθiRGMR(θ)dθ/∆R, and the coefficient Lθ,θis defined in Eq.\n(3)\n(cP= (π−2)/2 andcAP= (π+ 2)/2 in the simple exemple given above). The quasi-\nsymmetry under both the permutation of the magnetic configurat ions and the change of the\ncurrent direction observed experimentally is hence contained in the result expressed in Eq.\n(25).\nFrom an empirical point of view, the expression often used in order t o fit the data in-\n12troduces the critical current Ic(measured at zero external field and extrapolated at zero\nKelvin), such that: ˜ τ=τ0e−Ea\nkT(1−I\nIc)whereEais the anisotropy energy of the ferromagnetic\nlayer under consideration. Result Eq. (25) shows that the critical current is given by the\nexpression:\nIc=−ΓαEa\nvMs(1+α2)ci/parenleftbiggρF\n0\nel/parenrightbiggβ\n8∆R(28)\nTheN´ eel-Brownlawundercurrent injectionismeasuredinreferen ces[15, 18, 19, 20], with\nthetypicalasymmetrybetween thetwotransitionsofafactor2t o4. Therelation Ic∝1/∆R\nis verified in reference [15]. The proportionality with βis observed through the change of the\nsign while changing the scattering anisotropy [28]. Note that the phe nomenological results\npresented here can be generalized to tunnel junctions (see e.g. t he work of Schmidt and et\nal. in terms of spin injection in magnetic semiconductors [34]). In the c ase of tunnel barrier\na factor ten is typically gained in the magnetoresistance ∆ R, so that the critical currents in\nEq. (28) are also decreased by a factor ten [35].\nWhat is the value of the spin-transfer coefficient l? In order to compare with the fer-\nromagnetic transport coefficient Lθθ/ρF\n0= Γα/(Ms(1 +α2)) - expressed as the inverse\nof an action J−1.s−1- the phenomenological coefficient is compared in the same units:\nl\neρF\n0=−Eaβ\ne8∆RIcci/parenleftig\nLθθ\nρF\n0/parenrightig\n.\nExperiments performed on typical pseudo spin-valve systems sho w that it is possible to\nswitch the magnetization at zero external field in both directions (A P to P or P to AP in\nthe previous exempla) for currents of the order of ±1 mA. For such currents, the energy\ntransferred is of the order of 10 meV [15]: the measured quantity is the slope of the points\n1/Icplotted as a function of ∆ Rfor the two transitions. The anisotropy energy Eais of the\norder of 0.1 eV. The spin-transfer coefficient l/eis consequently of the order of 10−1Lθθto\n10−2Lθθ.\nOn the other hand, the activation experiments under current inje ction allow to access\ndirectly to the spin-transfer coefficient through the N´ eel-Brown law, without the need to\nmeasure the activation energy Ea. The quantity measured is the slope s=∂(ln(τ/τ0))/∂I.\nEq. (26) and Eq. (27) show that:l\neLθθ=kTsβ/parenleftig\n8e/integraltextθ0\nθeqRGMR(θ′)dθ′/parenrightig−1\n≈0.1. The order of\nmagnitude 10−1is confirmed experimentally in references [15, 18], together with the factor\n2 to 4 in the asymmetry.\n13C. Spin-injection correction to the fluctuations\nDue to its diffusive nature, the correction produced by the spin-inj ection can hardly be\nobserved onthe reversible state ofthe hysteresis. The above se ction shows thatthe activated\nirreversible jump of the magnetization is in contrast strongly modifie d by the spin-injection.\nBeyond the activation process, the presence of supplementary f erromagnetic diffusion pro-\ncesses strongly affects another experimentally accessible parame ter: the linear response of\nthe ferromagnetic moment to magnetic field, spin-injection, and th ermal excitations. The\nresponse is then proportional to the fluctuations.\nThe fluctuations occurring near the quasi-static states in the dou ble-well potential can\nbe analyzed from the general fluctuation-dissipation theorem (FD T) in the γ-space [10].\nThe density ˜ ρFis subjected to random fluctuations that are introduced through arandom\ncurrentJF\nr, which satisfies FDT:\n∝an}b∇acketle{tJF\nr(θ,t)Jr(θ′,t′)∝an}b∇acket∇i}ht= 2DF\ntot(θ)∝an}b∇acketle{tρF\ntot(θ,t)∝an}b∇acket∇i}htδ(θ−θ′)δ(t−t′) (29)\nThe variation of density is now corrected by the presence of the flu ctuation current:\n∂\n∂tρtot(θ,t) =−JF(θ,t)−JF\nr(θ,t) (30)\nApplying step by step the method described above for the rate equ ation (Eqs. (20) to (23))\nto the Eq. (30) , the following expression of the fluctuations is obta ined (see reference [10]):\n∝an}b∇acketle{tIr(t)Ir(t′)∝an}b∇acket∇i}ht=∝an}b∇acketle{tnF\ntot∝an}b∇acket∇i}ht(θ1)\n˜τ1→2−∝an}b∇acketle{tnF\ntot∝an}b∇acket∇i}ht(θ2)\n˜τ2→1(31)\nwhere the relaxation times ˜ τare that previously defined. This expression has not the\nusual form of a FDT which means that this theorem, strictly valid whe n fluctuations take\nplace around equilibrium states, is not fulfilled. The theorem is restor ed near an equilibrium\nstate∝an}b∇acketle{tnF\ntot∝an}b∇acket∇i}hteq=∝an}b∇acketle{tnF\ntot∝an}b∇acket∇i}ht(θ1) or∝an}b∇acketle{tnF\ntot∝an}b∇acket∇i}hteq=∝an}b∇acketle{tnF\ntot∝an}b∇acket∇i}ht(θ2) because transitions from one equilibrium\nstate to the other are neglected. We obtain from Eq. (31) ∝an}b∇acketle{tIr(t)Ir(t′)∝an}b∇acket∇i}hteq=/angb∇acketleftnF\ntot/angb∇acket∇ighteq\n˜τeqδ(t−t′).\nThis last expression is valid in the cases of linear ferromagnetic reson ance experiments, i.e.\ninasituationwhere the current is well below the critical current Icdefined inEq. (28)[21]. A\nmorecomplicatedbehavior(highlynon-linear)shouldbeexpectedfo rstrongexcitationsnear\nor beyond the critical current Icin order to interpret the non-linear resonance experiments\n[36].\n14In conclusion, in the case of linear ferromagnetic resonance (FMR) measured below Ic\nand observed close to one equilibrium state ( θ1orθ2), a correction to the response of the\nferromagnet is expected, that takes the same form as that calcu lated for the activation\nprocess:\n∝an}b∇acketle{tIr(t)Ir(t′)∝an}b∇acket∇i}hteq= 2DF\n0∝an}b∇acketle{tnF\n0∝an}b∇acket∇i}ht\nτeqexp/parenleftbigg\n−l(Ve(θi)−Ve(θ0))\nkBT Lθθ/parenrightbigg\nδ(t−t′) (32)\nThe behavior expected is then surprisingly similar to that predicted in the case of the\nactivation process, except that it holds for the amplitude of the line ar response and not for\nthe transition rates. For β≥0, we expect an exponential increase (resp. suppression) of the\nresponseintheAPstatewithapositive(resp. negative)current, andanexponential increase\n(resp. suppression) of the response in the P state with a negative (resp. positive) current.\nThis highly specific characteristic is in agreement with that observed experimentally in the\ncontext of FMR measurements under spin-injection below critical c urrentIc, i.e. a situation\nin which the magnetization is close enough to equilibrium states (see re sults presented in\nreference [21]).\nD. Link with microscopic theories\nBefore concluding, a last question must be invoked about the relatio n between the model\npresented here and the microscopic theories of spin-transfer to rque [11, 12, 13, 37]. The\nphenomenological transport coefficient l(and of course the known transport coefficient Lθθ\nandβσ0) could formally be defined from the relevant Hamiltonian expression w ith the help\nof projection-operator formalisms [38], or any other techniques [3 9, 40] that lead to the\ncoupled stochastic transport equations of the spin-polarized cur rent and the ferromagnetic\norder parameter in the corresponding configuration space. The d ifficulty is to manipulate\non an equal footing a microscopic degree of freedom, the spin of co nduction electrons, and\na collective variable, the magnetic order parameter. This task is far beyond the present\nreport, but it is possible to gain some insight with dimension considerat ions. The physical\nmechanism proposed here for spin-transfer is based onthe spin-in jection only, that is respon-\nsible for the supplementary diffusion effect of the magnetization thr ough the modification\nof the local densities of magnetic moments (redistribution of spins) . This redistribution of\nspins between the electric sub-system and the magnetic sub-syst em is governed by specific\n15spin-flip scattering mechanisms (or spin dependent creation-anhila tion mechanisms). The\nmicroscopic approach would define the relevant mechanism and dedu ce the typical spin-\ntransfer relaxation time τtr. The relation between mesoscopic and microscopic approaches\ncan consequently be invoked through the relation between the cor rection of the diffusion\nconstant δDF\ntot(expressed in dimension of angle per unit of time) and the relaxation t ime:\nδDF\ntot∝δn/τtr, whereδnis the amount of spins transferred from the electric sub-system t o\nthe ferromagnet.\nIn the approach proposed by Berger [12] in a pioneering work, a spin -transfer process\nat the interface is described at the electronic level by a typical spin relaxation time τsd\ncalculated from the s-d exchange Hamiltonian. If we assuming that t he relevant spin-flip\nrelaxation is governed by this mechanism τtr=τsd, we would have δDF\ntot∝1/τsdunder\nthe relevant hypotheses. In that case, the spin-transfer desc ribed here in terms of diffusion\nprocess would be a consequence, in parallel to GMR effects, of the s -d exchange interaction\noccuring at the microscopic level.\nE. Conclusion\nA description of spin-transfer has been proposed at the mesosco pic level, based uniquely\non the spin-injection mechanism occuring at the junction with a ferr omagnet. The spin-\naccumulation at the interface leads to a local change of the density of magnetic moments in\nthe corresponding configuration space. The gradient of density g enerates a diffusion process\nof the ferromagnetic order parameter, which is responsible for th e magnetization reversal.\nThe spin-injection has been described at the interface of a ferrom agnet by means of the\nusualtwo-conductionchannelmodelwhichsimplifiesthespin-polar izedcurrentandthegiant\nmagnetoresistance analyses. The dynamics of the ferromagnet is treated by means of anout-\nof-equilibrium mesoscopic model with a ferromagnetic current defin ed in the configuration\nspace of uniform magnetic moments. The coupling between the two c urrents is introduced\nthrough a new phenomenological Onsager cross-coefficient lthat accounts for the fact that\nthe spins carried by the electric charges in the normal space contr ibute also to the transport\nof ferromagnetic moments in the ferromagnetic configuration spa ce. It as been shown that\nthe correction to the LLG equation that governs the dynamic of th e magnetization comes\nfrom a diffusion term. We have found that the N´ eel-Brown activatio n law is still valid,\n16with a correction to the barrier height that is proportional to the in tegral of the giant\nmagnetoresistances over the ferromagnetic states, from the e quilibrium to the top of the\nbarrier. The expression of the critical current Icis given as a function of GMR, the damping\nfactor and the new spin-transfer cross-coefficient l. Furthermore, the correction to the\nfluctuations is shown to be analogous to that of the activation and is also expressed as an\nexponential term. 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Brown Jr.,\nPhys. Rev. 130, 677 (1963)\n18[28] S. Urazhdin, N. O. Birge, W. P Pratt, J. Bass, Appl. Phys. Lett.84, 1516 (2004).\n[29] In the framework of the two channel approximation (with channels ↑and↓), an interface\nbetween two layers of length vandv′, conductance σandσ′, and spin-diffusion length lsf\nandl′\nsfgives rise to the resistance: RGMR=1\n8e/parenleftigσ↑−σ↓\nσ0−σ′\n↑−σ′\n↓\nσ′\n0/parenrightig/parenleftigσ↑\nσ0−σ′\n↓\nσ0/parenrightig/parenleftig\n1\nvlsf+v′l′\nsf/parenrightig\n, as\nderived in Wegrowe et al. J. Phys.: Condens. Matter 19, 165213 (2007) .\n[30] This approach cannot be identified, without further con siderations, to the derivation of an\nactivation process under spin-transfer-torque, provoked by adeterministic spin-torque term\nadded to the Landau-Lifshitz-Gilbert equation (in our case , the derivation is based on the\ndiffusion term). Such studies with spin-transfer torque can b e found in Z. Li and S. Zhang,\nPhys. Rev.B 69134416 (2004), D. M. 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B\n62, R4790 (2000).\n[35] Yiming Huai et al. Appl. Phys. Lett. 84, 3118 (2004), K. Yagami et al. Appl. Phys. Lett.\n85, 5634 (2004), Zhitao Diao et al. Appl/ Phys. Lett. 87, 232502 (2005), Jun Hayakawa et\nal. Japanese. J. Appl. Phys. 44, L1267 (2005), Zhitao Diao et al. J. Appl. Phys. 99, 08G510\n(2006), Hitoshi Kubota et al. Appl. Phys. Lett. 89, 032505 (2006), T. Inokuchi et al. Appl.\nPhys. Lett. 89, 102502 (2006), G. D. Fuchs et al. Phys. Rev. Lett. 96, 186603 (2006) Z. T.\nDiao et al. J. Phys.: Condens. Matter 19, 195209 (2007), Masatoshi Yoshikawa et al. J. Appl.\nPhys.101, 09A511 (2007).\n[36] S. I. Kiselev et al. Nature425, 380 (2003)., M. R. Pufall et al. Phys. Rev. B 69, 214409\n19(2004), M. Covingtonet al. Phys. Rev. B, 69, 184406 (2004), I.N. Krivorotovet al. Science 307\n28 (2005), T. Devolderet al. Phys. Rev. B 71, 184401 (2005).\n[37] Ya. B. Bazaliy, B.A. Jones, and S.-C. Zhang, Phys. Rev. B ,57, R3213 (1998), X. Waintal , E.\nB. Myers, P. W. Brouwer, and D. C. 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Saradzhev, F. C. Khanna, Sang Pyo Kim, M. de Montign y, Phys. Rev. B 75, 024406\n(2007).\n20"    },    {        "title": "1208.6573v1.On_the_interpretation_of_the_angular_dependence_of_the_FMR_spectrum_in_heterogeneous_ferromagnetic_thin_films.pdf",        "content": "On the interpretation of the angular dependence\nof the FMR spectrum in heterogeneous ferromagnetic thin \flms\nMaciej Kasperski and Henryk Puszkarski\nSurface Physics Division, Faculty of Physics,\nAdam Mickiewicz University\n61-614 Pozna\u0013 n, Umultowska 85, Poland\nWe demonstrate that a multi-peak FMR spectrum, with lines corresponding to resonance in\ndi\u000berent ferromagnetic regions of a heterogeneous thin-\flm sample, can collapse to a single-peak\nspectrum if there exists a particular \feld con\fguration, or the con\fguration of the external magnetic\n\feld with respect to the \flm surface, in which dHres=dM e\u000b= 0 within the region magnetically\ndominating in the sample.\nI. INTRODUCTION\nThe ferromagnetic resonance (FMR) spectrum of thin \flms is known to evolve with the angular con-\n\fguration of the applied magnetic \feld with respect to the \flm surface. In this evolution the multi-peak\nFMR spectrum often becomes a single-peak one in a certain angular con\fguration, to regain its multi-peak\ncharacter beyond it. A careful analysis of this e\u000bect, in which a multi-peak FMR spectrum 'collapses' to\na single-peak one, leads to the observation of two types of behavior of the collapsing spectrum. In one\ntype the intensity of all the peaks except the \frst one diminishes progressively to vanish completely in\nthe collapse con\fguration. Beyond this con\fguration angle the 'satellite' resonance lines emerge again in\nan unchanged order. The other type of collapse involves resonance positions rather than intensities: in\nthe collapse con\fguration the lines shift to the position of the main peak, to reemerge in a reversed order\nbeyond this con\fguration.\nThe \frst type of collapse has been long known in the literature to be a surface e\u000bect due to the changes\nof the surface magnetic anisotropy (responsible for the surface spin pinning) with the con\fguration of\nthe external \feld with respect to the surface of the thin \flm. In this pattern a complete collapse of the\nresonance spectrum occurs in a speci\fc angular con\fguration (referred to as critical angle of the surface\nanisotropy) in which the surface anisotropy has no e\u000bect on the surface spin pinning, and the resonance\nprecession of the all spins is homogeneous throughout the sample. In contrast, the other type of collapse is\na bulk e\u000bect that occurs as a result of the ferromagnetic resonance in separate regions of slightly di\u000berent\nmagnetic character in a heterogeneous thin-\flm sample. A heterogeneous magnetic structure is known to\nresult in heterogeneous conditions of ferromagnetic resonance in the sample. Thus, in this interpretation\neach line in the multi-peak resonance spectrum corresponds to a resonance in a di\u000berent region of the\nsample, and the collapse con\fguration of the external \feld is a particular con\fguration in which all the\nmagnetically di\u000berent regions participate in the resonance for resonance \felds that di\u000ber only negligibly.\nThe collapse of the FMR spectrum in this speci\fc angular con\fguration of the external \feld in which\nthe resonance heterogeneity of the sample is eliminated has not been thoroughly analyzed in the literature\nso far. The aim of this paper is to elucidate in detail the theoretical grounds of this e\u000bect.\nThe paper is organized as follows. In Section II we recall the derivation of the universal Smit-Beljers-\nSuhl formula expressing the condition for ferromagnetic resonance to occur in a homogeneous bulk fer-\nromagnetic sample. On the basis of this formula, in Section III we derive the con\fguration condition\nof resonance in thin \flms. In Section IV we use this condition for analyzing the con\fguration evolution\nof the resonance \feld in thin-\flm samples with a perpendicular uniaxial anisotropy. We formulate a\nuniversal condition for a multi-peak FMR spectrum to collapse into a single-peak one in such samples,\nand (in Section V) demonstrate the bulk character of this e\u000bect.\nII. SIMPLIFIED DERIVATION OF THE SMIT-BELJERS-SUHL RESONANCE FORMULA\nLet us consider the dynamics of a body with an angular momentum Jand a magnetic moment m=\rL\ncollinear to it ( \ris the gyromagnetic ratio) in a magnetic \feld H. The \feld acts on the dipole to producearXiv:1208.6573v1  [cond-mat.mtrl-sci]  31 Aug 20122\na torquem\u0002Hthat sets the body in motion according to the equation:\n_m=\rm\u0002H: (1)\nLet us represent the vectors in the above equation in the local basis of orthonormal spherical vectors\n^r;^\u0012;^\u001edetermined by the vector m(m;\u0012;\u001e ) (obviously, m=m^r, see Fig.1 and comments in [1]). Thus:\n_m=\r(m^r)\u0002(Hr^r+H\u0012^\u0012+H\u001e^\u001e): (2)\nNow, let us determine the components H\u0012andH\u001eof the magnetic \feld Hby considering the change in\nthe energy of the dipole with an in\fnitesimal rotation (we only consider rotation by the polar angle \u0012).\nThe energy of a magnetic moment min a magnetic \feld His expressed by the equation:\nE(m) =\u0000m\u0001H; (3)\nwhich, represented in the basis ^r;^\u0012;^\u001e, becomes:\nE(m) =\u0000(m^r)\u0001(Hr^r+H\u0012^\u0012+H\u001e^\u001e): (4)\nAs a result of the rotation of mby a small angle \u0001 \u0012(m\u0001\u0012!m0=m^r0) the energy changes to:\nE(m0) =\u0000(m^r0)\u0001(Hr^r+H\u0012^\u0012+H\u001e^\u001e): (5)\nThe vector ^r0can be expressed as ^r+ \u0001\u0012^\u0012, so the above equation will become:\nE(m0) =\u0000m(^r+ \u0001\u0012^\u0012)\u0001(Hr^r+H\u0012^\u0012+H\u001e^\u001e): (6)\nThe following relations are seen to occur:\n\u0001E=\u0000mH\u0012\u0001\u0012)H\u0012=\u00001\nm@E\n@\u0012\f\f\f\f\n\u0012;\u001e; (7)\nH\u001e=\u00001\nmsin\u0012@E\n@\u001e\f\f\f\f\n\u0012;\u001e: (8)\nNow, we can get back to the equation of motion (1) and write the cross product as the determinant:\n2\n4_m\nm_\u0012\nm_\u001esin\u00123\n5=\r\f\f\f\f\f\f\f^r ^\u0012 ^\u001e\nm 0 0\nHr\u00001\nm@E\n@\u0012\f\f\n\u0012;\u001e\u00001\nmsin\u0012@E\n@\u001e\f\f\f\n\u0012;\u001e\f\f\f\f\f\f\f: (9)\nHence we obtain the system of equations:\n8\n>>>>>><\n>>>>>>:_m= 0;\nm\n\r_\u0012sin\u0012=@E\n@\u001e\f\f\f\f\n\u0012;\u001e;\nm\n\r_\u001esin\u0012=\u0000@E\n@\u0012\f\f\f\f\n\u0012;\u001e:(10)\nThis system of equations is to be solved in four steps. We (i) determine the equilibrium angles \u00120\nand\u001e0as the solution of the system of equations @E=@\u0012 =@E=@\u001e = 0; (ii) expand Eand sin\u0012into\na Taylor series at \u00120and\u001e0, respectively; (iii) assume that the time dependence of the angles has the3\nformei!t, and (iv) exclude all the terms except the lowest-order ones (see [2]). This procedure leads to\nthe Smit-Beljers-Suhl formula [3, 4] that represents the ferromagnetic resonance condition:\n!\n\r=1\nmsin\u00120vuut@2E\n@\u00122\f\f\f\f\neq@2E\n@\u001e2\f\f\f\f\neq\u0000 \n@2E\n@\u0012@\u001e\f\f\f\f\neq!2\n; \r =g\u0016B\n~; (11)\nwhere the values of the derivatives correspond to the equilibrium angles \u00120and\u001e0. (Note by the way\nthat the radicand is the Hessian of the function Eat the equilibrium point; its positive value is indicative\nof the existence of a minimum of the function at the point at which the \frst partial derivatives vanish.)\nSince we shall henceforth consider samples with a magnetization M, the formulas derived above, with M\nin place ofm, will apply to the description of the magnetization dynamics in the studied case.\nIII. RESONANCE CONDITION IN THIN FILMS WITH UNIAXIAL ANISOTROPY\nLet us consider a sample in which the free energy density Eis the sum of the Zeeman energy, the\ndemagnetization energy and the uniaxial anisotropy energy:\nE=\u0000M\u0001H+ 2\u0019(M\u0001n)2\u0000K(M\u0001u=M)2; (12)\nwherenis a unit vector normal to the surface of the sample, and uis a unit vector oriented along the easy\nmagnetization axis. The applied magnetic \feld and the magnetization of the sample will be henceforth\nrepresented as (see Fig.2a):\nM=M(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012)\nH=H(0;sin\u0012H;cos\u0012H):(13)\nWe shall consider three special cases corresponding to three di\u000berent choices of the easy axis ([100] ;[010]\nor [001], see Fig.2b). In each case we shall investigate the angular con\fguration dependence Hres(\u0012H)\nimplied by the assumed form of the free energy density (12).\nA. Perpendicular uniaxial anisotropy, u= [001]\nIn this case the unit vector nnormal to the surface of the sample and the unit vector uoriented along\nthe easy axis are identical:\nn=u= [001]: (14)\nThus, the free energy density reads:\nE=\u0000HM (sin\u001esin\u0012sin\u0012H+ cos\u0012cos\u0012H) + (2\u0019M2\u0000K) cos2\u0012: (15)\nLet us determine the equilibrium conditions of the system. The \frst partial derivatives with respect to\nthe angles\u0012and\u001ehave the form:\n8\n>><\n>>:@E\n@\u0012=\u0000MH (sin\u001ecos\u0012sin\u0012H\u0000sin\u0012cos\u0012H)\u0000(2\u0019M2\u0000K) sin 2\u0012\n@E\n@\u001e=\u0000MH cos\u001esin\u0012sin\u0012H:(16)\nFrom the condition of vanishing of these derivatives we obtain the equations that determine the coordi-\nnates\u0012Mand\u001eMof the equilibrium point (Fig.2b):\n2Hsin(\u0012M\u0000\u0012H) = 4\u0019Me\u000bsin 2\u0012M (17a)\ncos\u001eM= 0(\u001eM=\u0019=2; (17b)4\nwhereMe\u000bis the e\u000bective magnetization, de\fned as:\n4\u0019Me\u000b= 4\u0019M\u00002K=M: (18)\nIn this case the Smit-Beljers-Suhl formula becomes:\n!2\n\r2=\u0002\nHcos(\u0012M\u0000\u0012H)\u00004\u0019Me\u000bcos2\u0012M\u0003\n[Hcos(\u0012M\u0000\u0012H)\u00004\u0019Me\u000bcos 2\u0012M]: (19)\nIn order to plot the corresponding con\fguration dependence Hres(\u0012H) of the resonance \feld we must\nsolve the system of two nonlinear equations (17a) and (19).\nB. In-plane uniaxial anisotropy, u= [010]\nIn this case the unit vector nnormal to the surface and the unit vector uoriented along the anisotropy\ndirection are:\nn= [001] and u= [010]: (20)\nThe free energy density is:\nE=\u0000HM (sin\u001esin\u0012sin\u0012H+ cos\u0012cos\u0012H) + 2\u0019M2cos2\u0012\u0000Ksin2\u0012sin2\u001e: (21)\nThe \frst partial derivatives with respect to the angles \u0012and\u001eread:\n8\n>>>><\n>>>>:@E\n@\u0012=\u0000MH (sin\u001ecos\u0012sin\u0012H\u0000sin\u0012cos\u0012H)\n\u00004\u0019M2sin\u0012cos\u0012\u00002Ksin2\u001esin\u0012cos\u0012\n@E\n@\u001e=\u0000MH cos\u001esin\u0012sin\u0012H\u00002Ksin\u001ecos\u001esin2\u0012:(22)\nThus, the equilibrium direction of magnetization is determined by the equations:\n8\n>><\n>>:0 = 2H(sin\u001eMcos\u0012Msin\u0012H\u0000sin\u0012Mcos\u0012H) + (4\u0019M+2K\nMsin2\u001eM) sin 2\u0012M\n0 = cos\u001eMsin\u0012M\u0012\nHsin\u0012H+2K\nMsin\u001eMsin\u0012M\u0013\n:(23)\nThe Smit-Beljers-Suhl formula for this case is:\n!2\n\r2=\u0014\nHcos(\u0012M\u0000\u0012H)\u0000\u0012\n4\u0019M+2K\nM\u0013\ncos 2\u0012M\u0015\u0014\nHsin\u0012H\nsin\u0012M+2K\nM\u0015\n; (24)\nor, equivalently:\n!2\n\r2=\u0014\nHcos(\u0012M\u0000\u0012H)\u0000\u0012\n4\u0019M+2K\nM\u0013\ncos 2\u0012M\u0015\n\u0002\u0014\nHcos(\u0012M\u0000\u0012H)\u0000\u0012\n4\u0019M+2K\nM\u0013\ncos2\u0012M+2K\nM\u0015\n:(25)\nNote that this condition does not involve the azimuth angle \u001e, since the equilibrium condition (23)\nrequires again that \u001eM=\u0019=2.5\nC. In-plane uniaxial anisotropy, u= [100]\nIn this case the unit vector nnormal to the sample surface and the unit vector uparallel to the\nanisotropy direction are:\nn= [001] and u= [100]: (26)\nThe free energy density reads:\nE=\u0000HM (sin\u001esin\u0012sin\u0012H+ cos\u0012cos\u0012H) + 2\u0019M2cos2\u0012\u0000Ksin2\u0012cos2\u001e: (27)\nThe \frst partial derivatives of the energy density with respect to the angles \u0012and\u001ehave the form:\n8\n>>>><\n>>>>:@E\n@\u0012=\u0000HM(sin\u001ecos\u0012sin\u0012H\u0000sin\u0012cos\u0012H)\n\u00004\u0019M2sin\u0012cos\u0012\u00002Kcos2\u001esin\u0012cos\u0012\n@E\n@\u001e=\u0000HM cos\u001esin\u0012sin\u0012H+ 2Ksin\u001ecos\u001esin2\u0012:(28)\nThus, the equilibrium direction of magnetization is determined by the angles \u0012Mand\u001eMthat ful\fll the\nequations:\n8\n><\n>:0 =\u0000HM(sin\u001eMcos\u0012Msin\u0012H\u0000sin\u0012Mcos\u0012H)\n\u00004\u0019M2sin\u0012Mcos\u0012M\u00002Kcos2\u001eMsin\u0012Mcos\u0012M\n0 =\u0000HM cos\u001eMsin\u0012Msin\u0012H+ 2Ksin\u001eMcos\u001eMsin2\u0012M:(29)\nThe Smit-Beljers-Suhl formula for this case has the form:\n!2\n\r2=\u0014\nH(sin\u001eMsin\u0012Msin\u0012H+ cos\u0012Mcos\u0012H)\u0000\u0012\n4\u0019M+2K\nMcos2\u001eM\u0013\ncos 2\u0012M\u0015\n\u0002\u0014Hsin\u001eMsin\u0012H\nsin\u0012M+2K\nMcos 2\u001eM\u0015\n\u0000\u0012Hcos\u001eMcos\u0012Msin\u0012H\nsin\u0012M\u00002K\nMsin 2\u001eMcos\u0012M\u00132\n(30)\nor, equivalently:\n!2\n\r2=\u0014\nH(sin\u001eMsin\u0012Msin\u0012H+ cos\u0012Mcos\u0012H)\u0000\u0012\n4\u0019M+2K\nMcos2\u001eM\u0013\ncos 2\u0012M\u0015\n\u0002\u0014\nH(sin\u001eMsin\u0012Msin\u0012H+ cos\u0012Mcos\u0012H)\u0000\u0012\n4\u0019M+2K\nMcos2\u001eM\u0013\ncos2\u0012M+2K\nMcos 2\u001eM\u0015\n\u0000\u00122K\nM\u00132\nsin2\u001eMcos2\u001eMcos2\u0012M:(31)\nIV. THE EXISTENCE OF A RESONANCE INTERSECTION POINT IN\nHETEROGENEOUS THIN FILMS\nIn this Section we shall only consider the simplest of the three cases mentioned above, i.e. a thin \flm\nwith perpendicular uniaxial anisotropy (case [001]); we leave the remaining two cases to be similarly\ndiscussed in a separate paper. The condition (17a) allows to determine the equilibrium angle \u0012Mof\nthe magnetization vector corresponding to a speci\fc con\fguration of the external \feld \u0012H. Figure 3\nshows the consequent con\fguration dependence \u0012M(\u0012H) for two values of 4 \u0019Me\u000b. The deviation of the\nmagnetization angle \u0012Mfrom the \feld angle \u0012His seen to be the largest in the middle of the interval\n[0;\u0019=2] , and to grow with increasing value of 4 \u0019Me\u000b; in this range the deviation is of ca. 10\u000eto 15\u000e, far\nfrom being negligible!6\nNow, let us assume that the thin \flm under consideration is heterogeneous and planarly strati\fed into\nregions with two di\u000berent values of e\u000bective magnetization 4 \u0019Me\u000b. The resonance condition (19) { with\nthe equilibrium condition (17a) used { leads to two respective con\fguration resonance curves Hres(\u0012H),\nsee Fig.4a. This means that for a con\fguration \u0012Heach of the ferromagnetic strata resonates for a\ndi\u000berent value of the external \feld. This applies to the whole range of \u0012Hwith the exception of one\nparticular con\fguration corresponding to the point of intersection of the two curves. In this particular\ncon\fguration both regions { in spite of their signi\fcantly di\u000berent magnetic parameters! { produce\na resonance simultaneously for the same magnitude of the external \feld. This implies homogeneous\nprecession throughout the sample, as if its heterogeneity were completely eliminated in this con\fguration!\nThus, for this particular angle \u0012Hthe sample can be regarded as dynamically homogeneous . The dynamic\nhomogeneity (DH) is evidenced by the collapse of the two-peak resonance spectrum to a single resonance\nline. Note that past this particular angle the two peaks reemerge, but the resonance sequence is reversed.\nThe occurrence of a dynamic homogeneity angle (DHA) is an intrinsic characteristic of the Smit-Beljers-\nSuhl (SBS) resonance formula. Thus, we have grounds to ask whether the SBS equation can provide the\nbasis for an analytical condition that would allow to predict the DHA angle for a ferromagnetic thin-\n\flm sample. We think we have managed to formulate such a condition a posteriori on the basis of the\nnumerical results shown in Fig.4b, presenting the con\fguration dependence of the derivative @H=@M e\u000b\ndetermined for each of the ferromagnetic regions shown in Fig.4a. The DHA angle is seen to be in\nthe range delimited by the extreme con\fgurations for which the considered derivative vanishes in each\nstratum. The range shrinks with decreasing di\u000berence between the extreme values of 4 \u0019Me\u000bin the two\nregions. Thus, we can expect that the DHA angle will be determined with a good approximation by the\ncondition of vanishing of the derivative:\n@Hres(\u0012H)\n@Me\u000b= 0 (32)\nin the region the magnetic properties of which predominate in the sample.\nV. CONCLUSION\nIt is worthy of notice that the above-discussed collapse of the FMR spectrum in a particular \feld\ncon\fguration de\fned by the DHA angle is a dynamic volume e\u000bect, since only bulk quantities \fgure in\nthe adopted expression for free energy density, and the presence of the surface is only manifested in the\nshape anisotropy of the sample. Thus, the energy of surface anisotropy was not taken into account in\nour considerations. This is an important statement, since also the surface anisotropy by itself can cause\na multipeak SWR spectrum to collapse to a single resonance line in a certain con\fguration, referred to\nascritical angle [5], of the applied \feld with respect to the \flm surface. However, there is a substantial\ndi\u000berence between the collapse of an FMR spectrum and that of an SWR spectrum. In the former the\nresonance line positions merge to form a single line at the DHA angle, while the collapse of an SWR\nspectrum consists in the suppression of the intensity of all the lines except the main one at the critical\nangle. This is due to the di\u000berent nature of the two e\u000bects: the essence of the FMR collapse is the\nattainment of homogeneous resonance dynamics throughout the volume of the sample, whereas the SWR\ne\u000bect consists in the elimination, in the critical \feld con\fguration, of the impact of the surface anisotropy\non the spin dynamics in the thin \flm. Therefore, we shall refer to the latter con\fguration as surface\ncritical angle (SCA). The SCA is sensitive to the conditions on the surface of the sample and can vary\nwith them, while the DHA, only determined by the bulk characteristics that enter the condition (32), is\na \fxed parameter of the sample. It should be emphasized that the above-mentioned di\u000berences between\nthe two e\u000bects can be used in practice as a criterion that allows to determine whether a collapse of a\nmultipeak FMR spectrum observed in a thin \flm is of surface or bulk character.\nVI. ACKNOWLEDGMENTS\nThis study was supported by the Polish Ministry of Science and Higher Education, Grant No. N N202\n1945 33.7\n \nm\nFig.1\nFIG. 1. Local basis of spherical vectors ^r;^\u0012;^\u001erelated to vector m.\nH\nM\nθH\nΦ\nθ\nxyz(a) (b)\n[010]M\nΦM\nθM\n[100][001]\nFig.2\nFIG. 2. De\fnition of the angular con\fguration of the magnetization Mand the applied magnetic \feld Hwith\nrespect to the \flm surface.\n0102030405060708090EQUILIBRIUM MAGNETIZATION ANGLE,M\n0 10 20 30 40 50 60 70 80 90\nEXTERNALFIELD ANGLE,HθM=θH4πM eff= 6280Gs\n4πM eff= 3760Gs\nFIG. 3. Equilibrium magnetization angle \u0012Mvs. external \feld angle \u0012Hdetermined for a thin \flm with per-\npendicular uniaxial anisotropy (f=33.5 GHz); two curves corresponding to two e\u000bective magnetization values are\nplotted for comparison.8\n-4-2024681012DERIVATIVE,dHres/dMeff\n0 10 20 30 40 50 60 70 80 90\nCONFIGURATIONANGLE,H910111213141516171819FMRFIELD, Hres[kOe](a)\n(b)4πM eff= 6280Gs\n4πM eff= 3760Gs\nf= 33.5GHz\nFIG. 4. Con\fguration dependence of (a) the resonance \feld Hres(\u0012H) and (b) its derivative dHres=dM e\u000bin a thin\n\flm with perpendicular uniaxial anisotropy. (a) Note the intersection of the resonance curves plotted for the two\nregions of di\u000berent e\u000bective magnetization magnitude. The intersection point corresponds to the particular \feld\ncon\fguration in which the sample produces a homogeneous resonance for the same \feld magnitude throughout\nits volume. (b) Note that the homogeneous resonance con\fguration lies in the range determined by the vanishing\nof the derivative dHres=dM e\u000bin each resonating region.\n[1] David J. Gri\u000eths, Introduction to Electrodynamics (Prentice-Hall, New Jersey, 1999) pp. 38-43\n[2] Allan H. Morrish, Physical Principles of Magnetism (John Wiley and Sons, New York, 1965)\n[3] J. Smit, H. G. Beljers, Philips Res. Repts. 10(1955) 113\n[4] H. Suhl, Physical Review 97(1955) 555\n[5] H. Puszkarski, Progress in Surface Science 9(1979) 191"    },    {        "title": "1407.3029v1.Persistent_ferromagnetism_and_topological_phase_transition_at_the_interface_of_a_superconductor_and_a_topological_insulator.pdf",        "content": "Persistent ferromagnetism and topological phase transition at the interface of a\nsuperconductor and a topological insulator\nWei Qin and Zhenyu Zhang\nInternational Center for Quantum Design of Functional Materials (ICQD),\nHefei National Laboratory for Physical Sciences at Microscale (HFNL),\nand Synergetic Innovation Center of Quantum Information and Quantum Physics,\nUniversity of Science and Technology of China, Hefei, Anhui, 230026, China\n(Dated: June 3, 2022)\nAt the interface of an s-wave superconductor and a three-dimensional topological insulator, Ma-\njorana zero modes and Majorana helical states have been proposed to exist respectively around\nmagnetic vortices and geometrical edges. Here we \frst show that a single magnetic impurity at such\nan interface splits each resonance state of a given spin channel outside the superconducting gap, and\nalso induces two new symmetric impurity states inside the gap. Next we \fnd that an increase in\nthe superconducting gap suppresses both the oscillation magnitude and period of the RKKY inter-\naction between two interface magnetic impurities mediated by BCS quasi-particles. Within a mean\n\feld approximation, the ferromagnetic Curie temperature is found to be essentially independent\nof the superconducting gap, an intriguing phenomenon due to a compensation e\u000bect between the\nshort-range ferromagnetic and long-range anti-ferromagnetic interactions. The existence of persis-\ntent ferromagnetism at the interface allows realization of a novel topological phase transition from\na non-chiral to a chiral superconducting state at su\u000eciently low temperatures, providing a new\nplatform for topological quantum computation.\nPACS numbers: 73.20.-r, 75.30.Hx, 74.45.+c, 03.67.Lx\nIntroduction .|Non-Abelian fermions have attracted\nmuch attention because of their potential applications\nin topological quantum computation (TQC) [1, 2]. One\ncommon physical entity obeying non-Abelian braiding\nstatistics is the zero-energy Majorana fermion [3], which\nis its own anti-particle described by \r=\ry. In con-\ndensed matter physics, a chiral topological superconduc-\ntor (TSC) [4] is characterized by the existence of two\ntypes of Majorana fermions, chiral Majorana edge modes\nand a single Majorana zero mode surrounding a magnetic\nvortex, the latter can be manipulated for realization of\nTQC [5{8]. The simplest chiral TSC is a spinless px+ipy\nsuperconductor or super\ruid [9]; however, it is di\u000ecult\nto quench the spin degrees of freedom in order to realize\nspinless superconductors.\nRecently, Fu and Kane proposed that the proximity-\ninduced superconductivity on the surface of a topolog-\nical insulator (TI) deposited on a conventional s-wave\nsuperconductor possesses a px+ipypairing feature [11].\nThe non-chiral nature of such a spinfull superconductor is\ncharacterized by the existence of Majorana helical edge\nstates and a pair of Majorana zero modes surrounding\na magnetic vortex. To convert such a TSC into a chi-\nral one, time reversal symmetry (TRS) must be broken.\nTwo schemes have been proposed to break TRS, both\nrelying on the e\u000bect of a Zeeman \feld. The \frst con-\nsists of a superconductor-TI-magnet junction [11]; in the\nsecond scheme, the TI can further be replaced by a tradi-\ntional semiconducting thin \flm with strong Rashba spin-\norbit coupling (SOC) [12, 13]. These intriguing propos-\nals have motivated extensive experimental e\u000borts for the\ndetection of Majorana fermions [14{16], but so far de\fni-tive proofs of their existence remain controversial. Here\nwe note that both schemes face the inherent challenge\nthat the proximity-induced Zeeman \feld decays rapidly\nthrough the TI or semiconductor thin \flm.\nIn this Letter, we introduce an alternative and con-\nceptually new scheme to realize a chiral TSC within a\nsimpler structure, achieved by doping magnetic impuri-\nties directly at a superconductor-TI interface. We \frst\nshow that a single magnetic impurity at such an interface\nsplits each resonance state of a given spin channel outside\nthe superconducting gap, and also induces two new sym-\nmetric impurity states inside the gap. Next we \fnd that\nan increase in the superconducting gap suppresses both\nthe oscillation magnitude and period of the Ruderman-\nKittel-Kasuya-Yosida (RKKY) interaction between two\nmagnetic impurities mediated by BCS quasi-particles.\nThe ferromagnetic Curie temperature is found to be es-\nsentially independent of the superconducting gap, due\nto a compensation e\u000bect between the short-range ferro-\nmagnetic and long-range anti-ferromagnetic interactions.\nThe existence of persistent ferromagnetism at the inter-\nface provides a strong and uniform Zeeman \feld for the\nrealization of a chiral TSC. In particular, by investigating\nthe edge states and the corresponding \frst Chern num-\nber [17] , we reveal a topological phase transition from a\nnon-chiral to chiral TSC at su\u000eciently low temperatures.\nThese \fndings in principle provide a new and more ap-\npealing platform for TQC.\nTheoretical model .|The surface states of strong TIs\nare described by the time reversal invariant Hamilto-\nnianH0=P\nk y\nk(\u0017F~ \u001b\u0001~k) k. Here y\nk= (cy\nk\";cy\nk#),\n~ \u001b= (\u001bx;\u001by) are the Pauli spin matrices, \u0016is the chem-arXiv:1407.3029v1  [cond-mat.mtrl-sci]  11 Jul 20142\nical potential, and \u0017Fis Fermi velocity, given by 4.08\neV\u0001\u0017A for Bi 2Se3[18] and 3.70 eV\u0001\u0017A for Sb 2Te3[19]. By\ndepositing an s-wave superconductor on the surface of a\nTI, the proximity-induced pairing Hamiltonian is given\nasHp=P\nk(\u0001cy\nk\"cy\n\u0000k#+h:c:). Here \u0001 = \u0001 0ei\u001eis the\nsuperconducting gap with phase \u001e. The states at the\nsuperconductor-TI interface can then be described by [11]\nH0=1\n2X\nk\ty\nkH(~k)\tk;\nH(~k) = (\u0017F~ \u001b\u0001~k\u0000\u0016)\u001cz\u0000\u00010(\u001cxcos\u001e\u0000\u001cysin\u001e);(1)\nwhere \ty\nk= (cy\nk\";cy\nk#;c\u0000k#;\u0000c\u0000k\") are 4-dimensional\n\feld operators in Nambu spinor basis. TRS and particle-\nhole symmetry are expressed as \u0002 = i\u001byKand \u0004 =\n\u001by\u001cyK, which satisfy [\u0002 ;H] = 0 andf\u0004;Hg= 0 at \u0000\npoint of the Brillouin zone, respectively, where Kis the\ncomplex conjugate operator.\nAt the microscopic level, we treat the s-dinterac-\ntion between a magnetic impurity located at ~Riand\nthe electrons at the superconductor-TI interface to be\nisotropic, described by Hi\nsd=\u0000J(~ \u001b\u0001~S)\u000e(~ r\u0000~Ri), where\n~ \u001b= (\u001bx;\u001by;\u001bz) is the real electron spin, and ~Sis the\nspin of the magnetic impurity. In Nambu notations, the\ninteraction Hamiltonian can be rewritten as\nHsd=\u0000J\n2X\nkk0\ty\nk(~S\u0001~ \u001b)\u001c0\tk0; (2)\nwhereJdenotes the s-dexchange coupling strength at\nthe interface, estimated to be 0 :1\u00000:5 eV [19{21]. Hamil-\ntonian (2) describes the interaction between the magnetic\nimpurities and BCS quasi-particles, which, together with\nHamiltonian (1) de\fne our theoretical model and the\nstarting point of this study.\nSingle magnetic impurity .|We \frst study a single\nmagnetic impurity at the superconductor-TI interface.\nThe matrix form of the retarded Green's function of\nEq. (1) reads\nGret\n0(~k;!) =1\n!\u0000H(~k) +i\u000e: (3)\nIn order to study the e\u000bect of a single magnetic impurity,\nwe investigate the local density of states (LDOS) using\ntheT-matrix technique [22], which can be expressed us-\ning the Lippmann-Schwinger equation:\n^T(!) =^U+^UGret\n0(!;0)^T(!); (4)\nwhereGret\n0(!;0) is the retarded Green's function in real\nspace and ^U=\u0000J\n2(~S\u0001~ \u001b)\u001c0in our system. The algebra is\nsimplest for \u0016= 0, where the retarded Green's function\nin real space is given by the Fourier transformation of\nEq. (3). Forj!j>\u00010the result is\nGret\n0(!;~ r) =f1(!)H(1)\n0(rp\n!2\u0000\u00012\n0\n\u0017F) sgn (!)\n+f2(^r;!)H(1)\n1(rp\n!2\u0000\u00012\n0\n\u0017F);(5)\nFIG. 1. (Color online) (a) LDOS as a function of the electron\nenergy Eat the location r= 3 nm away from a magnetic\nimpurity. The solid and dashed lines are the spin-resolved\nLDOS for \u0001 0= 15 meV and 0 meV, respectively, with each\nof the spin-down (blue) and spin-up (red) resonance states\nsplit by the superconducting gap. (b) Spatial distribution\nof the spin-resolved LDOS at E= 0:2 eV, with the arrow\nand color indicate the in-plane and z-direction projections,\nrespectively. Here in order to highlight the resonance e\u000bect,\nwe take a large value of J= 8:0 eV [23].\nwheref1(!) =\u0000iv\n4\u00172\nF[!\u0000\u00010(\u001cxcos\u001e\u0000\u001cysin\u001e)] and\nf2(^r;!) =v\n4\u00172\nF(~ \u001b\u0001^r)\u001czp\n!2\u0000\u00012\n0,H(1)\n0;1are the Hankel\nfunctions,vis the volume of the lattice primitive cell,\nand ^ris the unit vector. For ~ r!0, The Green's function\ntakes the following asymptotic form:\nGret\n0(!;0) =f1(!)fsgn (!) +i2\n\u0019[ln(p\n!2\u0000\u00012\n0\n2W) +\u001f]g;\n(6)\nwhereWis a large band cuto\u000b, \u001fis the Euler-Mascheroni\nconstant, and sgn ( x) is the sign function. From the alge-\nbraic relations of Eqs. (4) and (6), we can calculate the\nT-matrix, and further obtain the full retarded Green's\nfunction as:\nGret(!;~ r) =Gret\n0(!;~ r) +Gret\n0(!;~ r)^T(!)Gret\n0(!;\u0000~ r):\n(7)\nThe spin-resolved LDOS in direction iis given as\n\u001a\u0006\ni(!;~ r) =\u00001\n2\u0019ImfTr[Gret(!;~ r)(1\u0006\u001bi)(1 +\u001cz)]g:(8)\nIt was shown previously that a strong magnetic impu-\nrity will induce a pair of low-energy resonance states on\nthe surface of a 3D TI [23]. In the present study, such\nLDOS resonances are derived from the minima of the de-\nnominators of the T-matrix. As illustrated in Fig. 1(a),\nthese low-energy resonance states are further shown to be\nrobust when the surface state of the 3D TI is proximity-\ncoupled to the superconductor. Furthermore, each of the\nspin-resolved resonance states outside the superconduct-\ning gap will further be split due to the appearance of the\nsuperconducting gap around the Fermi level. Such reso-\nnance state splittings could be directly observed exper-\nimentally. The spatial distribution of the spin-resolved3\nLDOS at a given energy is shown in Fig. 1(b). Similar\nto the case of TI surface states [19], a magnetic impurity\npolarized along the vertical zdirection will induce spin\ntextures both perpendicular and parallel to the interface\ndue to the presence of SOC.\nIn order to study the emergent electronic proper-\nties inside the superconducting gap induced by the\npresence of the magnetic impurity, we calculate the\nretarded Green's function for j!j<\u00010:Gret\n0(!;~ r) =\n2\n\u0019[\u0000if1(!)K(1)\n0(rp\n\u00012\n0\u0000!2\n\u0017F) +f2(^r;!)K(1)\n1(rp\n\u00012\n0\u0000!2\n\u0017F)];\nwhereK(1)\n0;1are the modi\fed Bessel functions. For\n~ r!0, the above retarded Green's function takes\nthe asymptotic form Gret\n0(!;0) =i2\n\u0019f1(!)c(!) with\nc(!) = ln (p\n\u00012\n0\u0000!2=2W) +\u001f, which is a real function\nof!. From Eq. (4), we \fnd two poles of the T-matrix,\ngiving rise to two impurity states inside the supercon-\nducting gap (not shown in Fig. 1(a)). For J > 0, the\nimpurity states can be obtained from the self-consistent\nrelation:!=\u0006[\u00010+\u0019\nJSc(!)]. The symmetric nature\nof the two spin-up and spin-down impurity states stems\nfrom the particle-hole symmetry.\nMultiple magnetic impurities .|In this part, we fo-\ncus on the electronic and magnetic properties of the\nsuperconductor-TI interface doped with randomly dis-\ntributed magnetic impurities. In order to study the col-\nlective magnetic behavior of such a system, we \frst con-\nsider the RKKY interaction between two magnetic impu-\nrities mediated by the BCS qusi-particles. Hamiltonian\n(1) can be mapped into a two-band spinless px+ipy\nHamiltonian as\nH0=X\nkm\u0018km\u000by\nkm\u000bkm\u00001\n2(m\u0001ei\u0012k\u000by\nkm\u000by\n\u0000km+h:c:);(9)\nwhere\u0018km=m\u0017Fk\u0000\u0016are the Dirac electron spectra,\nm=\u00061 are the band indices, and \u000bkm= (mei\u0012kck\"+\nck#)=p\n2. Using the same basis set, Hamiltonian (2) can\nbe rewritten as\nHi\nsd=\u0000JX\nmm0kk0ei(~k0\u0000~k)\u0001~Ri(~Si\u0001~ \u001bkm;k0m0)\u000by\nkm\u000bk0m0;\n(10)\nwhere~ \u001bkm;k0m0are the spin matrices.\nIn the following, we treat the many-body prob-\nlem using perturbation theory. The corrected ground\nstate energy due to s-dhybridization is E=\nh\njTH0S(\u00001;1)j\ni, whereTis the time-order op-\nerator,j\niis the ground state of the BCS Hamilto-\nnian (9), and the S-Matrix is de\fned as S(t;t0) =\nTexp [\u0000iRt\nt0dt1^Hsd(t1)]. The normalized ground state\nof Hamiltonian (9) can be written as:\nj\ni=Y\nkm0(ukm+\u0017km\u000by\nkm\u000by\n\u0000km)j0i; (11)\nwhere the sign0indicates lack of double counting of\nelectron pairs,j0iis the vacuum state, and ukmand\nFIG. 2. (Color online) RKKY interaction between two mag-\nnetic impurities as a function of the separation Rand the\nFermi energy Efcalculated with J= 0:5 eV, Ef= 100 meV\nfor (a) and R= 10 nm for (b). The insert in (a) shows the\ncase for the Fermi surface located at the Dirac point.\n\u0017kmare determined by the Bogoliubov transformation.\nThe normalization condition h\nj\ni= 1 is ensured by\njukmj2+j\u0017kmj2= 1. By expanding the S-matrix to the\nsecond order in Hsd, and only considering the loop ap-\nproximation between two di\u000berent magnetic impurities i\nandj, the RKKY interaction can be e\u000bectively written\nas\nHRKKY\nij =F1(R;\u0016)~Si\u0001~Sj+F2(R;\u0016)(~Si\u0002~Sj)x\n+F3(R;\u0016)Sx\niSx\nj;(12)\nwhere\nF\u000b(R;\u0016) =\u0000J2v2\n32\u00192X\nmm0Zkc\n0dkdk0D\u000b\nkm;k0m0(R)\n\u0002kk0(EkmEk0m0\u0000\u0018km\u0018k0m0\u0000\u00012\n0)\nEkmEk0m0(Ekm+Ek0m0);(13)\nwith\u000b=1, 2, or 3, kcis a large momentum cuto\u000b,\nEkm=p\n\u00182\nkm+ \u00012\n0is the excitation spectrum of the\nBCS quasi-particles, which can be obtained by diago-\nnalizing Hamiltonian (9). In Eq. (13), we also have\nD1\nkm;k0m0(R) =J0(kR)J0(k0R)\u0000mm0J1(kR)J1(k0R),\nD2\nkm;k0m0(R) =m0J0(kR)J1(k0R) +mJ1(kR)J0(k0R),\nD3\nkm;k0m0(R) = 2mm0J1(kR)J1(k0R), andJ0;1(x) are the\nBessel functions of the \frst kind. From Eq. (12), we\nnote that the RKKY interaction at the interface con-\ntains three di\u000berent kinds: the Heisenberg-like term, the\nDzyaloshinskii-Moriya (DM)-like term, and the Ising-like\nterm. On a face level, the overall behavior is qualitatively\nsimilar to that on a TI surface [21] due to the SOC e\u000bects\nin both systems. However, the presence of the supercon-\nducting part introduces crucial di\u000berences, as re\rected in\nEq. (13) and discussed in more detail below.\nIn general, the oscillation period of the RKKY interac-\ntion is determined by the Fermi wavelength \u0015F= 1=kF.\nAs shown in Fig. 2(a), an increase in the superconduct-\ning gap \u0001 0suppresses both the oscillation magnitude\nand period of the RKKY interaction, exhibiting a fast4\nTABLE I. Robust Curie temperatures for systems of di\u000berent\nsuperconducting gaps, obtained with x=0.03.\n\u00010 0meV 5meV 10meV 15meV\nTMF\nc 3.282K 3.228K 3.234K 3.243K\ndecay of the long-rang part of the interaction to be close\nto zero. These behaviors stem from two physical as-\npects. First, the proximity-induced superconductivity\nwill introduce a gap of 2\u0001 0at the Fermi level by form-\ning Copper pairs; because every excitation of the quasi-\nparticles has to overcome the superconducting gap, the\ncorresponding RKKY interaction mediated by the quasi-\nparticles will be suppressed in magnitude, especially the\nlong-range part. Secondly, since the occupied states close\nto the superconducting gap dominate the contribution to\nthe RKKY interaction, the corresponding wave vector is\nsmaller than kF, leading to a modi\fcation in the oscilla-\ntion period. In Fig. 2(b), the Fermi energy dependence\nof the RKKY interaction is also presented.\nFrom the RKKY interaction described above, we can\nobtain the collective behavior of the magnetic impurities\nunder the realistic assumption that their spatial distri-\nbution is random. The positional randomness combined\nwith Eq. (11) makes the in-plane interaction frustrated,\nwhile the ferromagnetic interactions between the zcom-\nponents of the local spins can be optimized. Accordingly,\naz-direction aligned ferromagnetic ground state is ex-\npected for the multiple magnetic impurity system, even\nthough the atomic s-dhybridization is isotropic. The\nmean-\feld virtual crystal approximation (MF-VCA) can\nbe employed to estimate the Curie temperature TMF\nc,\ngiven as [24, 25]\nkBTMF\nc=2x\n3X\ni(i6=0)J0i; (14)\nwhere the sum extends over the virtual sites, and xis the\nconcentration of the magnetic impurities on those virtual\nsites. The continuum limit is reached with kBTMF\nc =\n4\u0019ni\n3R1\n0rJ(r)dr, whereniis the density of the magnetic\nimpurities. For Bi 2Se3, the virtual sites are the locations\nof the Bi atoms. By setting x= 3%,a=4.14 \u0017A;J= 0:5\neV, andEf= 0:1 eV, the estimated Tcfor di\u000berent su-\nperconducting gaps are listed in Table I. As shown in\nFig. 2(a), the behaviors of the RKKY interaction are dra-\nmatically in\ruenced by the superconducting gap, while\nthe MFTcshows nearly constant values. These intrigu-\ning phenomena stem from a subtle compensation e\u000bect\nbetween ferromagnetism and anti-ferromagnetism: For\n\u00010= 0, the magnitude of the long-range RKKY inter-\naction shows a spatial dependence as 1 =R2[21], favor-\ning anti-ferromagnetism, while the short-range correla-\ntion always favors ferromagnetism. For \u0001 06= 0, both the\nmagnitude and long-range oscillation of the RKKY in-\nteraction will be suppressed, which again mutually com-\nFIG. 3. (Color online) (a) The e\u000bective exchange \feld Vex\n(black) and superconducting gap \u0001 0(red) as a function of\nthe temperature. Tsindicates the topological phase tran-\nsition temperature, which is below the ferromagnetic Curie\ntemperature Tc. (b) The \frst Chern number as a function\nof the e\u000bective exchange \feld Vex. The insets illustrate the\nbulk band spectra (black solid) and edge states (red dashed)\nin the helical and chiral states, calculated with Vex= 5meV\nandVex= 25meV, respectively. Other parameters include\n\u00010= 15meV, J= 0:5eV, Ef= 0meV, and x= 0:03.\npensate each other, leading to robust Curie temperatures\nas listed in Table I.\nChiral TSC and topological phase transition .|Now we\ndiscuss the topological state of the superconductor-TI in-\nterface in the presence of random magnetic impurities.\nBased on the MF approximation, we \frst estimate the\ne\u000bective exchange \feld induced by the randomly dis-\ntributed magnetic impurities, given by Vex= 3JxhSzi.\nWithin the picture that a given magnetic impurity inter-\nacts with an e\u000bective Zeeman \feld Beff=xP\niJ0ihSzi\nde\fned by all the other magnetic impurities, its mag-\nnetic polarization is given by hSzi=SB(BeffS\nkBT), where\nB(x) =2S+1\n2Scoth (2S+1\n2Sx)\u00001\n2Scoth (1\n2Sx) is the Bril-\nlouin function. Therefore, the self-consistent solution of\nhSziandBeffcan give rise to the temperature depen-\ndence ofVex, as shown in Fig. 3(a). Importantly, the\nvery existence of the Vexbreaks the TRS by producing a\ngap at the Dirac point of the TI surface state, which in\nturn characterizes the chiral nature of the superconduct-\ning system, as further elaborated below.\nIn analogy with Ref. [11], by de\fning the\nBogliubov quasi-particle operators as \r(r) =P\n\u001bu\u001b(r) y\n\u001b(r) +\u0017\u001b(r) \u001b(r), and solving the\nBdG equationHBdG\t(r) =E\t(r) with \t( r) =\n[\u0017\"(r);\u0017#(r);u#(r);u\"(r)]Tat geometrical edges, we\ncan \fnd two types of Majorana edge states by\nvaryingVex. First, forp\n\u00162+ \u00012\n0> Vex> \u0016 ,\nthere are two helical edge states, given by \t \u0006(x) =\n1\nN\u0006(p\nV\u0000;\u0007ip\nV+;\u0007ei\u001ep\nV+;iei\u001ep\nV\u0000)Te\u0000\u0011x, where\nV\u0006=Vex\u0006\u0016,\u0011= (\u0001 0\u0006p\nV+V\u0000)=\u0017FandN\u0006\nare the normalization parameters. It is easy to\nverify\ry(ky) =\r(\u0000ky), which implies that the so-\nlutions are Majorana edge modes. In order to give5\nan intuitional picture of the helical edge states,\nwe evaluate the low-energy \\ k\u0001p\" Hamiltonian as\nHh=p\n1\u0000(\u0016=Vz)2\u0017Fky\u001cz, where\u001czis the Pauli\nmatrix. Secondly, for Vex>p\n\u00162+ \u00012\n0, there are\ntwo degenerate chiral Majorana edge states \t \u0006(x) =\n1\nN\u0006(p\nV\u0000;\u0000ip\nV+;\u0007ei\u001ep\nV+;\u0006iei\u001ep\nV\u0000)Te\u0007\u0011x, where\nN\u0006are the normalization parameters. The chiral nature\ncan be illustrated by the low-energy \\ k\u0001p\" Hamilto-\nnian, given byHc=p\n1\u0000(\u0016=Vz)2\u0017Fky. Therefore,\nby varying the exchange \feld Vex, we can expect a\ntopological phase transition from a helical to chiral TSC\natVex=p\n\u00162+ \u00012\n0, and the corresponding transition\ntemperature is marked by Tsin Fig. 3(a).\nAs a quantitative measure for the occurrence of the\ntopological phase transition, we calculate the \frst Chern\nnumber for systems before and after the transition.\nThe \frst Chern number can be de\fned as the inte-\ngral of the Berry curvature over the \frst Brillouin Zone\n[17]:C1=1\n2\u0019R\nBZ(@kxAky\u0000@kyAkx)dk, whereAk\u000b=\n\u0000iP\nnhun(k)j@k\u000bjun(k)iis the Berry connection, \u000b=\nx;y, and the index nruns over all the occupied states.\nHamiltonian (1) can be regularized on a square lattice\nwith the substitution px;y!a\u00001sin (px;ya), whereais\nthe lattice constant. The results for \u0016= 0 are shown\nin Fig. 3(b). There are two sets of subbands due to spin\ndegrees of freedom. When Vex<\u00010, the resulting Chern\nnumbers from the two sets are equal in magnitude but\nopposite in sign, and the total Chern number C1= 0 sig-\nni\fes a non-chiral TSC state. When Vex>\u00010, one set of\nthe subbands will be inverted by the exchange \feld, and\nthe corresponding Chern number will also reverse sign,\nresulting inC1= 1, indicating a chiral TSC state.\nSo far, we have focused on realizing chiral TSC at\nthe superconductor-TI interface. As a natural extension,\nhere we also brie\ry discuss the proposed scheme in con-\nnection with recent experiments [26, 27]. In particular,\nwhen Bi 2Se3was grown on the d-wave superconductor\nof Bi 2Sr2CaCu 2O8+\u000e, ans-wave superconducting gap as\nlarge as 15 meV was observed on the top surface of the\nTI [27]. Based on these experiments, we expect that the\nproposed mechanism can also be exploited to realize chi-\nral TSCs on tops of TI/superconductor heterostructures.\nIn summary, we have proposed an alternative and\nconceptually simpler scheme to realize a chiral TSC,\nachieved by doping magnetic impurities directly at a\nsuperconductor-TI interface. We have found that, for\nrandomly distributed magnetic impurities, the RKKY\ninteraction gives rise to a persistent ferromagnetic state\nindependent of the superconducting gap. The ferromag-\nnetic state can naturally provide a uniform and strong\nexchange \feld, which in turn breaks the time reversal\nsymmetry, driving the system from a helical TSC phaseinto a chiral TSC phase at su\u000eciently low temperatures.\nThe proposed scheme is in principle also applicable on\ntop of a TI/superconductor heterostructure, or when the\nTI is replaced by a normal semiconductor with strong\nRashba SOC. These \fndings therefore provide new plat-\nforms for realizing chiral TSC, observing Majorana zero\nmodes, and executing TQC.\nThis work was supported by the NSFC Grant\nNo.11034006 and National Key Basic Research Program\nof China (2014CB921103).\n[1] A. Y. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003).\n[2] C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das\nSarma, Rev. Mod. Phys. 80, 1083 (2008).\n[3] G. Moore and N. Read, Nucl. Phys. B360 , 362 (1991).\n[4] X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n[5] L. J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788\n(1981).\n[6] M. Matsumoto and M. Sigrist, J. Phys. Soc. Jpn. 68, 994\n(1999).\n[7] X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B\n82, 184516 (2010).\n[8] D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001).\n[9] N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).\n[10] S. Das Sarma, C. Nayak, and S. Tewari, Phys. Rev. B\n73, 220502(R) (2006).\n[11] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407\n(2008); Phys. Rev. B 79, 161408 (2009).\n[12] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,\nPhys. Rev. Lett. 104, 040502 (2010).\n[13] J. Alicea, Phys. Rev. B 81, 125318 (2010).\n[14] V. Mourik et al. , Science 336, 1003 (2012).\n[15] A. Das et al. , Nature Phys. 8, 887 (2012).\n[16] M. T. Deng et al. , Nano Lett. 12, 6414 (2012).\n[17] D. Xiao, M. C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010).\n[18] H. J. Zhang et al. , Nature Phys. 5, 438 (2009).\n[19] Q. Liu, C. X. Liu, C. K. Xu, X. L. Qi, and S. C. Zhang,\nPhys. Rev. Lett. 102, 156603 (2009).\n[20] J. S. Dyck, P. Hajek, P. Lostak, and C. Uher, Phys. Rev.\nB65, 115212 (2002).\n[21] J. J. Zhu, D. X. Yao, S. C. Zhang, and K. Chang, Phys.\nRev. Lett. 106, 097201 (2011).\n[22] A. V. Balatsky, I. Vekhter, and J. X. Zhu, Rev. Mod.\nPhys. 78, 373 (2006).\n[23] R. R. Biswas and A. V. Balatsky, Phys. Rev. B 81,\n233405 (2010).\n[24] G. Bouzerar, J. Kudrnovsky, L. Bergqvist, and P. Bruno,\nPhys. Rev. B 68, 081203(R) (2003).\n[25] H. Chen, W. G. Zhu, E. Kaxiras, and Z. Y. Zhang, Phys.\nRev. B 79, 235202 (2009).\n[26] M. X. Wang et al. , Science 336, 52 (2012).\n[27] E. Wang et al. , Nature Phys. 9, 621 (2013)."    },    {        "title": "0806.1744v1.Geometrical_control_of_the_magnetization_direction_in_high_aspect_ratio_PdNi_ferromagnetic_nano_electrodes.pdf",        "content": " 1 Geometrical control of the magnetization direction in high aspect-ratio \nPdNi ferromagnetic nano-electrodes \nJ. J. Gonzalez-Pons, J. J. Henderson, and E. del Barco* \nDepartment of Physics, University of Central Florida, 4000 C entral Florida Blvd., Orlando, \nFlorida 32816-2385 \nB. Ozyilmaz \nDepartment of Physics, National University of Singapore, 2 S cience Drive 3, Singapore 117542 \n*E-mail: delbarco@physics.ucf.edu \n \nAbstract:  \nWe present a study of electron-beam evaporated Pd 0.4 Ni 0.6 alloy thin films by means of \nferromagnetic resonance measurements on extended films  of varying thickness and anisotropic \nmagnetoresistance measurements of lithographically patte rned high aspect-ratio ferromagnetic \nelectrodes, respectively. The results reveal that the direction of the magnetization strongly \ndepends on the electrode lateral dimensions, transitioning from in-plane magnetization for \nextended films to out-of-the-plane magnetization for electrode widths below 2-3 microns, \nreaching ∼58 degrees off-plane for 100 nm-wide nanoelectrodes.  2  Recently Pd 1-xNi x alloy has attracted considerable attention as ferroma gnetic electrodes in \ncarbon based lateral spin valves. Its excellent wetti ng properties on carbon nanotubes \n(CNT) [1-3], leads to low ohmic contacts (transparent), while its room temperature \nferromagnetic behavior [ 4] provides a means for spin injection. Surprisingly in the  case of CNTs \na tunneling barrier between the PdNi and the CNT itself seems not to be necessary for spin \ninjection, making PdNi alloys the ideal material for ma gnetic electrodes in low dimensional \ncarbon-based electronic devices. Key to these experiment s is anisotropy, since the successful \ndemonstration of spin injection is based on the observ ation of a giant magnetoresistance (GMR) \neffect  [5], which is determined by the relative orientation betwee n the magnetization of the two \nferromagnetic electrodes in a spin-valve device. In parti cular, it has been shown that in all-\nmetallic spin-valves in which the magnetization vectors  of both ferromagnets are not collinear \n(0 < θ < 180), the current-induced switching of the magnetization state of the device can be \nobtained in much shorter times, making them highly effici ent systems for information processing \n[6]. In carbon-based spin-valve devices the planar arrangem ent of the system complicates the \nrealization of non-collinear magnetizations, and the a bility to control the magnetization direction \nwith respect to the plane of the electrode becomes esse ntial. However little is known about the \nmagnetic properties of such ferromagnetic alloys when lithographically patterned into narrow \nelectrodes with large aspect ratios. A detailed understan ding of the magnetic characteristics of \nthis material is therefore crucial to both understand the  switching characteristics of such a device \nand optimize the electrode dimension and configuration when used as ferromagnetic spin \ninjectors. \n In this letter we present a detailed study of the effec t of the geometry of electron-beam \npatterned Pd 0.4Ni 0.6  ferromagnetic thin film structures on the equilibrium  direction of the  3 magnetization with respect to the film plane. For ext ended films, room temperature \nferromagnetic resonance (FMR) measurements show that t he magnetization remains in the film \nplane preferentially for 4-80 nm-thick films. A substantia l out-of-the-plane uniaxial anisotropy \nwhich tends to pull the magnetization off-plane competes w ith the demagnetizing fields which \nset the magnetization in-plane. For laterally constr ained large aspect ratio Pd 0.4 Ni 0.6  thin film \nelectrodes, anisotropic magneto-resistance (AMR) transpo rt measurements show that the \nmagnetization cants out of the film plane for elect rode widths below ~2-3 µm, reaching an angle \nwith respect to the film plane of ~58 o for electrode widths down to ~100 nm.  \n Pd 0.4 Ni 0.6  alloy extended films were fabricated by electron beam  evaporation of the bulk \nmaterial on Si/SiO 2 wafer in a UHV system with a base pressure of 7 ×10 -7 Torr. In addition, 25 \nnm-thick Pd 0.4 Ni 0.6  films were patterned in the shape of high aspect ratio e lectrodes of length \n20 µm and various widths (100 nm – 10 µm). In a second lithographic step the PdNi electrodes \nwere contacted with standard Ti/Au electrodes (5/50nm) nece ssary for transport measurements. \n FMR measurements were carried out at room temperature with a high-frequency \nbroadband (1-50 GHz) micro-coplanar-waveguide ( µ-CPW)  [7] using the flip-chip method [ 8-\n10 ]. A 1.5 Tesla rotateable electromagnet was employed t o vary the applied field direction from \nthe in-plane ( φ = 0o) to normal to the film plane ( φ = 90 o) directions. The change of the resonance \nfield value of the 15 GHz FMR absorption peak as a functio n of the relative angle between the \nexternal magnetic field and a 20 nm-thick NiPd film is show n in Fig. 1. Inset in the figure there \nis a geometrical representation of the applied magnetic field vector and the angle, φ, that was \nrotated through. Also inset in Fig. 1 there are two abso rption curves corresponding to two \norientations of the field, φ = 0 (in-plane) and 90 o (out-of-the-plane). Similar behavior was \nobserved for all the studied films. The data are typica l of polycrystalline ferromagnetic films  4 with in-plane magnetization [ 9,11]. The resonant field value is lowest at 0 or 180 degrees,  when \nthe applied field is in the same plane as the magnetiza tion vector.  \n The magnetic energy of a ferromagnetic film in the pr esence of a magnetic field, H, \napplied at an angle, φ, with respect to the plane of the film is given by [ 9,11] \nm m m s m m s K KK M HME φ φ φ πφφφφ4\n22\n2 122sin sin )2( sin 2)sin sin cos (cos + +− + + −=     (1) \nwhere Ms is the saturation magnetization, φm is the angle between the magnetization and the film \nplane and K1 and K2 are the first and second order out of-the-plane uniaxial anisotropy constants, \nrespectively. The first term represents the Zeeman ene rgy associated to the coupling of the \nmagnetization and the external field, which tries to al ign both along the same direction. The \nsecond term is the magnetostatic energy, which forces  the magnetization into the plane. The last \nterms represent the out-of-the-plane uniaxial anisotropy, which minimizes the energy in a \ndirection normal to the plane. The final orientation of the magnetization of the ferromagnetic \nfilm results from a competition between these three e nergies. The data in Fig. 1 can be fitted \nusing the equation of condition of resonance given by the  Smit and Beljers formula [ 12], \n21HHγω= , where h/Bgµγ= is the gyromagnetic ratio, and H1 and H2 depend on H, φm, φ, \nMs, K1 and K2 (see Refs. [ 9] and [11] for the exact expression). A good fitting of the data in \nFig. 1 (continuous line) is obtained for Ms = 290 emu/cm 3 (according to a 60%-Ni composition \nwith magnetization density  MsNi = 485 emu/cm 3), g = 2.22, K1 = 2.26 ×10 -5 erg/cm 3 and \nK2 = 0.22 ×10 -5 erg/cm 3.  \n When the field is applied at φ = 0 or 90 o, the resonance condition reduces to:  5 ( )\n⊥\n⊥−=\n\n\n+ =\n\n\n\n]4 []4 [// 2\n// \neff res eff res res \nM HM HH\nπγωπγω\n                                              (2) \nwhere s s eff MKMKMs M /4/24]4 [2 1 // − −=π π , and s eff MKMs M /24]4 [1−=⊥π π . According \nto this, the difference between the frequency depen dences of the FMR peak in the parallel and \nperpendicular configurations (see Eq. 2) enables an  independent determination of the anisotropy \nparameters. The longitudinal and perpendicular freq uency behaviors of the FMR peak of a 20 nm \nthick film are shown in Fig. 2. The slope of the cu rves allows the determination of the \ngyromagnetic ratio, hence the Landé g-factor [13], while the intersept with the x-axis gives the \neffective in-plane and out of plane magnetizations,  [(4πMeff ])//  and [(4πMeff ])⊥. The difference \nobserved between the two effective demagnetization fields is indicative of the presence of a \nsecond order anisotropy term, which can be deduced from ([4 πMeff ]//  - [4 πMeff ]⊥ = 4K2/Ms). The \nthickness dependence of the first and second order anisotropies, K1 and K2, extracted by this \nmethod is shown in inset to Fig. 2. The results ind icate that the out-of plane ( K1, K2 > 0) easy-\naxis anisotropy first decreases with increasing fil m thickness, for thickness values below 10nm, \nand reaches a minimum ( K1 = 1.3 ×10 -5 erg/cm 3) at t = 10 nm. K1 then increases up to \n2.9 ×10 -5 erg/cm 3 for a film thickness of 25 nm, above which it beco mes practically thickness \nindependent (results obtained up to 80 nm, not show n here, support this evidence). The magnetic \nquality factor of the thin film is defined as the r atio between the anisotropy energy and the \ndemagnetizing (magnetostatic) energy, Q = (K1+K2)/2 πMs2. In the limit of Q < 1, the \ndemagnetization field dominates over the out-of-the -plane anisotropy. For the thicknesses \nstudied here we obtain Q < 0.7, in agreement with the in-plane alignment of  the magnetization \nextracted from the angular dependence of the FMR fi eld. In general, the demagnetizing field  6 vector is determined by the shape of the sample and  characterized by the demagnetizing factors \nNx, Ny and Nz, with Nx + Ny + Nz = 4π. In the case of an extended thin film, the only co mponent \nof the demagnetizing field is normal to the plane ( i.e. Nz = 4π and Nz = Ny = 0), and will strongly \noppose to the magnetization to tilt away from the f ilm plane. \n Anisotropic magneto-resistance (AMR) measurements were performed in laterally \nconstrained Pd 0.4 Ni 0.6  large aspect ratio electrodes (i.e. nanowires) of thickness t = 25 nm, length \nl = 20 µm and widths varying from 100 nm to 10 µm. An AFM image of one of the electrodes is \nshown in the inset to Fig. 3. The magneto-resistanc e varies as a function of the relative angle \nbetween the applied electrical current and the magn etization vector, Mr\n. The largest (lowest) \nresistance is obtained when Mr\n and Ir\n are collinear (perpendicular) [ 14]. The magneto-resistance \nis proportional to cos 2(θ) [ 15,16], where θ is the angle between the magnetization and the \ncurrent, which in our case flows along the length o f the nanowire. Yet, the direction of the \nmagnetization is determined by the direction and st rength of the applied magnetic field, the \nintrinsic uniaxial anisotropy and the demagnetizing  field of the film, which will in turn depend \non the geometry (width) of the nanowire. \n The AMR curves obtained with the external dc field  applied along three different \ndirections relative to the electrode geometry are s hown in Fig. 3 for a 25 nm-thick Pd 0.4 Ni 0.6  \nelectrode 250 nm wide and 20 µm long. The largest change in resistance, a 0.68 % increase of \nthe AMR at saturation, is observed when the magneti c field is aligned along the axis of the \nelectrode (L). On the contrary, when the magnetic f ield is oriented perpendicular to the film, only \na 0.352 % decrease of the AMR is observed. In both cases, the saturation of the AMR response is \nachieved for fields over ~ 2 kG, in agreement with the value of the effective demagnetizing fields \nobserved by FMR in extended films (Figs. 2a-b). How ever, when the field is oriented across the  7 wire there is no appreciable change in AMR with fie ld for this particular electrode width, for \nfield values as high as 8 Tesla. This is indicative  of a modified demagnetizing field vector \nimposed by the constrained lateral dimension of the  structure, which prevents the magnetization \nto orient across the wire. For widths larger than 2 -3 µm, the transverse AMR saturates at the \nsame value than the perpendicular AMR, as will be d iscussed below. \n One can extract information about the direction of  the magnetization by looking at the \nabsolute change in AMR for different field orientat ions. In the case of the electrode shown in \nFig. 3, the fact of having the largest AMR change w hen the field is applied along the nanowire \naxis, ΔL = 0.68 %, is indicative of an initial magnetic con figuration in which most of the spins \nwere aligned away from this direction, contributing  to a large increase of the device resistance \nwhen aligning with the current by the action of the  longitudinal field (L). In addition, the smaller \nAMR change (decrease) with the field applied perpen dicular to the film plane (P) is an indication \nof the fact that most of the spins were already clo se to this orientation before the field \napplication. Since for this particular nanowire the re is no change in magneto-resistance, ΔT = 0, \nwhen the field is applied transversally to the wire  (T), we can assume that all the spins are \nlocated in the L-P plane. In this situation and ass uming a rigid magnetization vector, the \nnormalized AMR in the longitudinal direction, ΔLN = ΔL/( ΔL+ΔP), can be understood as the \nprojection of the magnetization along the P-axis. C onsequently, one can estimate the angle \nbetween the magnetization and the film plane using φm = sin -1(ΔLN), which for the 250 nm wire \nin Fig. 3 corresponds to φm = sin -1(0.787) ~ 52 o. \n Fig. 4a shows the normalized longitudinal AMR chan ge as a function of the electrode \nwidth, w. It can be observed how the longitudinal AMR chang e reaches its maximum value \n(ΔLN ~ 0.85) for a 100 nm nanowire. Following the previ ous arguments, this corresponds to an  8 angle of 58 degrees between the magnetization and t he film plane. ΔLN gradually decreases with \nincreasing widths until arriving to a final value o f 0.5 for widths over 2-3 µm. Before going into \na detailed discussion about the magnetic configurat ion of the widest electrodes it is important to \nnote that for electrode widths over ~1 µm there starts to appear a change in the transverse  AMR, \nΔT, which eventually arrives to the same saturation v alue than the perpendicular AMR change, \nΔP, for widths over 3 µm. This indicates that the transverse direction (T)  ceases to be a hard \nanisotropy axis for large electrode widths and an a pplied magnetic field has the same effect on \nthe magnetization for different orientations within  the plane, as in the case of an extended film. \nNaturally, different orientations of the magnetizat ion within the plane correspond to different \nAMR values, since the current is applied along the L-axis. Note that for electrode widths over \n3 µm, the absolute value of the normalized AMR change at saturation is the same for the three \norientations of the field with respect to the elect rode axes ( ΔL = ΔP = ΔT = 0.5). This indicates \nthat the magnetization lies in the plane of the ele ctrode without a preferential orientation.  \n The results indicate that the magnetization of the  electrodes gradually depart from the \nfilm plane for widths below 3 µm, reaching a maximum angle of 58 o for the lowest electrode \nwidth studied here ( w = 100 nm). Note that an out-of-the-plane equilibri um configuration of the \nmagnetization has already been observed in thin 100  nm-wide Pd 0.6 Ni 0.4  at low temperature [ 3].  \nFor large electrode widths, the situation becomes e quivalent to the one found by FMR \nmeasurements on extended thin films, indicating tha t the lateral constriction of the electrodes \nforces the magnetization to lie out of the plane, a llowing a geometrical control of the \nmagnetization direction of PdNi ferromagnetic elect rodes. We deduce that the constrained lateral \ndimension of the electrode modifies the demagnetizi ng fields of the device allowing the uniaxial \nanisotropy of the PdNi film to overcome the magneto static energy, pulling the magnetization off  9 plane. Note that the small width of the electrodes generates another non-zero component of the \ndemagnetizing vector, Nx > 0. Since, as discussed above, the sum of the thr ee demagnetization \nfactors must remain constant ( Nx + Ny + Nz = 4π), this implies a reduction of the demagnetizing \nfactor perpendicular to the film, Nz < 4π. Consequently, a weaker perpendicular demagnetizin g \nfield will not be sufficient to sustain the in-plan e magnetization of the extended film in its \ncompetition with the out-of-the-plane uniaxial anis otropy — an effect that becomes more intense \nthe thinner the electrode. The effect of the device  geometry on the reconfiguration of the \nequilibrium magnetization direction has been studie d in other single-ferromagnet nanoscale \nfilms. For example, in Co/Au nanodots a transition from in-plane (at room temperature) to out-\nof-the-plane (at low temperature) magnetization is observed to be modulated by the thickness of \nthe ferromagnetic layer [ 17]. \n The possibility to control the orientation of the magnetization in ferromagnetic electrodes \nis of crucial importance for producing efficient sp in-transfer devices. Our results show that in \nPdNi alloy ferromagnetic thin films this can be ach ieved by tuning the geometry of the electrode. \nSpecifically, the magnetization of Pd 0.4 Ni 0.6  electrodes can be engineered to transit from in-pl ane \nto out-of-the-plane (up to 58 degrees) by varying t he electrode width from 3 µm down to 100 nm \nrespectively. The demonstrated tunability of this m aterial could be used to fabricate planar \ndevices in which different electrodes show differen t magnetization directions, allowing to probe \nthe physics of spin-transfer seen in all-metallic s pin-valves now in carbon-based devices, which \ncould eventually lead to novel applications in emer ging nanotechnologies.  \n \n \n  10  ACKNOWLEDGEMENTS: \n We want to acknowledge fruitful discussions with J ean-Marc L. Beaujour and Werner \nKeune. J.C.G acknowledges support from UCF Undergra duate Research Initiative program. \nJ.C.G, J.J.H and E.d.B acknowledge support from the  US National Science Foundation \n(DMR0706183 and DMR0747587). \n \n \n  11  REFERENCES: \n[1]  S. Sahoo, T. Kontos, C. Schonenberger and C. Surger s, App. Phys. Letters  86 , 112109 \n(2005). \n[2]  S. Sahoo, T. Kontos, J. Furer, C. Hoffmann, M. Grab er, A. Cottet and C. Schonenberger, \nNature Physics , 1,  99-102 (2005).  \n[3]  H. T. Man, I. J. W. Wever and A. F. Morpurgo, Phys. Rev. B 73 , 241401(R) (2006) \n[4]  W. A. Ferrando et al. , Phys. Rev. B 5, 4657 (1972). \n[5]  M. N. Baibich et al . Phys. Rev. Lett.  61 , 2472 (1988); G. Binasch, P. Grünberg, F. Saurenbach \nand W. Zinn, Phys. Rev. B 39 , 4828 (1989).  \n[6]  A. D. Kent, B. Ozeymalz and E. del Barco, Appl. Phys. Lett.  84 , 3897 (2004). \n[7]  W. Barry, I.E.E.E Trans. Micr. Theor. Techn. MTT 34 , 80 (1996). \n[8]   G. Counil et al. , J. Appl. Phys.  95, 5646 (2004). \n[9] J-M. L. Beaujour, W. Chen, K. Krycka, C. –C. Kao, J . Z. Sun and A. D. Kent,  Eur. Phys. J. \nB 59 , 475-483 (2007).  \n[10]  J.-M. L. Beaujour, W. Chen, A. D. Kent and J. Z. Su n, J. Appl. Phys.  99 , 08N503 (2006). \n[11]  C. Chappert, K. L. Dang, P. Beauvillain, H. Hurdequ int and D. Renard, Phys. Rev. B 34  (5), \n3192 (1986). \n[12]  S. V. Vonsovskii, Ferromagnetic Resonance , Pergamon, Oxford, (1996). \n[13]  The average effective value of the g-factor is 2.3 for the thickness range studied. Thi s value \nis larger than the Landé g-factor of bulk Ni ( gNi  = 2.18), which may be attributed to the \nmoment induced in Pd by the surrounding Ni . \n[14]  J. W. Loram and K. A. Mirza, J. Phys. F: Met. Phys.  15 2213 (1985). \n[15]  T. R. McGuire and R. I. Potter, IEEE Trans. Magn . 11 , 1018 (1975). \n[16]  L. Gridneva et al. , Phys. Rev. B  77 , 104425 (2008)  12  FIGURE CAPTIONS \nFigure 1:  Behavior of the FMR peak position of a 20nm-thick Pd 0.4 Ni 0.6  film as a function of the \nangle of application of the magnetic field from the  plane of the field upon application of 15GHz \nmicrowave excitation. The sketch at the bottom of t he figure represents the experimental \nconfiguration of the sample, which is placed upside -down on top of the µ-CPW transmission \nline. The magnetic field direction is also indicate d. The inset on the left shows the corresponding \nFMR absorption versus field for angles φ = 0 and 90 o.  \n \nFigure 2:  Frequency behavior of the FMR peak of a 20nm-thick  Pd 0.4 Ni 0.6  film in the parallel \n(left axis, solid data) and perpendicular (right ax is, open data) configurations, respectively. The \ninset shows the first ( K1) and second ( K2) order out-of-the-plane uniaxial anisotropies as a  \nfunction of the film thickness. \n \nFigure 3:  (Color online) Room temperature AMR response of a 25nm-thick, 250nm-wide, \n10 µm-long Ni 0.6 Pd 0.4  electrode as a function of an external magnetic fi eld applied in the \ndirections indicated in the figure. The inset on th e left shows an AFM image of the electrode.  \n \nFigure 4:  a) Normalized longitudinal AMR change as a functio n of the electrode width. The \nopen circles were taken on a second sample that was  fabricated independently. b) Ratio between \nthe transverse and perpendicular AMR changes as a f unction of the width. The sketch in the \nmiddle represents the orientation of the magnetizat ion with respect to the film plane for different \nelectrode widths.   13  \n \nFIGURE 1 \n \n \n \n \nFIGURE 2 \n  14  \n \nFIGURE 3 \n \n \nFIGURE 4 "    },    {        "title": "1306.0915v1.Interlayer_coupling_in_spin_valves_studied_by_broadband_ferromagnetic_resonance.pdf",        "content": "arXiv:1306.0915v1  [cond-mat.mes-hall]  4 Jun 2013Interlayer coupling in spin valves studied by broadband fer romagnetic resonance\nD. E. Gonzalez-Chavez, R. Dutra, W. O. Rosa, T. L. Marcondes, A . Mello, and R. L. Sommer\nCentro Brasileiro de Pesquisas F´ ısicas,22290-180 Rio de J aneiro, RJ, Brazil\n(Dated: July 11, 2018)\nMagnetization dynamic response of coupled and uncoupled sp in valves with structure\nNiFe(20nm) /Cu(tCu)/NiFe(20nm) /IrMn(10nm) is probed using broadband ferromagnetic resona nce\nabsorption measurements. The coupling intensity between t he free and pinned layers is tailored by\nvarying the Cu thickness t Cu. Broadband spectra exhibit two resonant modes for each valu e of\napplied field. It is observed that the coupling among NiFe lay ers modifies the amplitude of the\nabsorption peaks and also the shape of the dispersion relati ons for each mode, which becomes\nparticularly distorted at the anti-parallel magnetizatio n state. The observed phenomena is well\ndescribed by applying a semianalytical model that properly takes into account the coupling inter-\nactions and allows an efficient numerical calculation of the a bsorption peak amplitudes, and the\ndispersion relation shapes.\nI. INTRODUCTION\nInterlayer coupling is an important ingredient for\nseveral devices as spin valves1,2and magnetic tunnel\njunctions3–6(MTJ), multilayered materials and any sys-\ntemsbasedontwoormoreferromagneticlayersseparated\nby a nonmagnetic spacer.\nOn spin valves and MTJ, a strong interlayer coupling\nis a key issue for devices using synthetic free or pinned\nlayers7–9, while a weak coupling is usually observed be-\ntween the free and pinned layers10–12. In both cases the\ndynamical behavior is influenced by the strength of in-\nterlayer coupling, both, in saturated and not saturated\nmagnetic states.\nThis coupling was extensively studied in the past by\nseveral experimental techniques as magnetization mea-\nsurements and magneto-resistance13,14, ferromagnetic\nresonance15–18, Brillouin light scattering (BLS)19–21, and\nothers22,23.\nAn interesting and recent approach to study the effect\nof interlayer coupling on the high frequency response is\ntheuseofbroadbandferromagneticresonance. Thistech-\nnique is based on the use of a vector network analyser\n(VNA), andisusuallyknownasVNA-FMR24. Usingthis\ntechnique we are able to measure the dynamic properties\n(permeability or absorption) in a frequency range from a\nfew MHz to dozens of GHz. Moreover, all measurements\ncan be performed in the range −Hmax≤0≤+Hmax,\nwhere H maxcan be adjusted from a few Oe to several\nkOe. Therefore, besides measuring the saturated states\nas in traditional FMR, a broadband measurement can\nbe performed on non saturated states, and eventually at\nzero field.\nIn this work, we study the static and dynamic prop-\nerties of spin-valve systems using VNA-FMR and mag-\nnetometry measurements. Our samples are consisted of\nPy/Cu/Py/IrMnlayersdescribedasfollows. Thebottom\nPy = permalloy (Ni 81Fe19) layer acts as a free magnetic\nlayer (F) while the top Py layer is coupled to an antifer-\nromagnet (Ir 20Mn80) and behaves as a pinned layer (P).\nWe are able to address the behavior of each layer and\nthe effect of the interaction mediated by the Cu spacer.By varying Cu layer thickness we are able to control the\ninteraction between the Py layers, which produces new\nfeaturesonbroadbandspectraatnon-saturatedmagnetic\nstates. In particular, we observe complex dispersion re-\nlations, including frequency jumps and absorbed power\nintensities depending on the oscillation modes.\nA semianalytical model based on the magnetic free en-\nergy for the macro spins, together with the Landau Lif-\nshitz Gilbert equation (LLG) is proposed and applied to\nthese systems. This model allows an efficient numerical\ncalculation of the broadband absorption amplitudes and\ndispersion relations describing remarkably well the ex-\nperimental results. Moreover, the model provide further\ninsights on the magnetization dynamics of spin-valve like\nsystems in both, saturated and non-saturated magnetic\nstates.\nII. EXPERIMENT\nWe produced spin-valves with structure\nPy(20nm)/Cu( tCu)/Py(20nm)/IrMn(15nm), where\ntCu= 0.75nm, 1.0nm and 2.5nm, were produced using\na Magnetron Sputtering system onto a Si(100) sub-\nstrate with both buffer and capping layers of Ta(5nm).\nChamber pressure condition for such depositions was\n5mTorr/50sccm Ar pressure/flow, after a 5 ×10−8Torr\nbase pressure in the whole chamber. A RF power source\nwas used for Py depositions, while DC sources were used\nfor Ta, Cu and IrMn depositions. All deposition rates\nwere calibrated using low angle x-ray reflectometry.\nDuring the growth process, an in-plane magnetic field\nof about 200Oe was applied in order to induce an\nunidirectional anisotropy at the FM/AFM interface,\nleading to the pinning of the top FM layer through\nexchange bias effect.\nWe performed static magnetic measurements ( Mvs.\nH) using a VSM under DC fields of ±300Oe. For\nthe dynamic measurements, we used a broadband ferro-\nmagnetic resonancesetup composed by a Rhode&Shwarz\nZVA24VectorNetworkAnalyser, combined with acopla-\nnar waveguide for frequencies in the range of 0.1 - 7.02\nGHz and DC magnetic fields in range of ±300Oe. For\nthese measurements, each sample was placed on top of a\ntwo port coplanar waveguide, where the external field H\nwas applied along the propagation direction, as shown in\nFig. 1. The transmission S21and reflection S11coeffi-\ncientsweremeasuredinsuchspecifiedfieldandfrequency\nrange. The absorbed power ratio in the waveguide was\ncalculated using25\nPLoss/PIn= 1−|S11|2−|S21|2. (1)\nThe ferromagnetic resonant spectra (magnetic absorp-\ntion) were obtained by measuring this ratio with respect\nto a reference measurement of the dielectric losses, ac-\nquired with the sample saturated along the direction of\nthe rf field.\nG\nS\nGG\nS\nG\nFIG. 1. Schematic diagram of the coplanar waveguide (CPW)\nstructure and a sample placed on top of it. The central con-\nductor of the CPW is about 260 µm wide. High-frequency\nmicro-probes and coaxial cables (not shown) were used to\nconnect the structure through VNA. The sample’s anisotropy\naxis (a.a.) is aligned to the direction of the applied extern al\nfield H.\nIII. EXPERIMENTAL RESULTS\nOur samples were engineered in order to have different\ncoupling intensities between the FM layers. From the\nstatic hysteresis loops, measured with the external field\napplied along the easy axes (as shown in Fig. 2), we can\nclearly note how the thickness of Cu spacer intermedi-\nates the intensity of the coupling among the FM layers.\nThesamplewith tCu=2.5nm(Fig. 2(a)) hasshowedthe\ntypical spin-valve behavior, with well-known parallel and\nanti-parallelmagnetizationstates, displayingashiftedre-\nsponse for P and a centered response for F layers. Such\nfeatures indicate non appreciable coupling between the\nFM layers. On the other hand, for tCu= 1.00nm (Fig.\n2(b)), the coupling now manifests itself as a small shift\nin the response of the F layer. A larger coupling is ob-\ntained for tCu= 0.75nm, where the shift of the F layer\nis larger and the anti-parallel state is no longer observed.\nInstead of that, a gradual rotation of the magnetization\nis actually the main switching process.−101(a)\n−101M/MS(c)\n−150 0 150−101(e)\n135(b)\n135\nFrequency (GHz)(d)\n−150 0 150 300\nH (Oe)135(f)\nFIG. 2. Measured and calculated magnetic hysteresis loop\n(left) and the broadband FMR spectra (right) for tCu=\n2.50nm (top) tCu=1.00nm(middle) and tCu= 0.75nm (bot-\ntom). The symbols correspond to the experimental data and\nthe solid line to the calculated curve.\nThe right side of Fig. 2 shows the measured absorbed\npower spectra for our samples. The color scale denotes\nthe amplitude from blue (minimum) to red (maximum).\nThe maximum amplitude on branches correspond to the\nresonant modes. In these measurements, we are able to\nobserve two clear resonant responses (Fig. 2(d)) for the\nsample without coupling ( tCu= 2.50 nm), one centered\nand the other field shifted, correspondingto F and P lay-\ners, respectively. As already observed26in simple or ex-\nchangebiased magnetic systems, the changein the slopes\nof the branches occurs at the switching fields of the re-\nspective layers. On the other hand, for the samples with\ncoupled FM layers ( tCu= 1.00nm and tCu= 0.75nm), a\npairofresonantbranchesis still observedin parallelmag-\nnetization states, while a completely different and a new\nbehavior is observed in non-saturated states, including\nfrequency jumps in the resonant branches for both layers\nat their switching fields. These features will be addressed\ninthe followingsectionV,afterwepresentourmodeland\nnumericalcalculationsforthesesystems. Inallcases(sat-\nurated and non-saturated samples) we observe different\nabsorption intensities on the resonant branches. In order\nto get afurther insight on the absorptionofthe saturated\nstates, we plot the absorption profile for these samples at\n3.7 GHz in Fig. 3. In this figure, four absortion peaks\nare observed for all samples. In the uncoupled case (Fig.3\n3(a)), the small peaks correspondto the oscillation of the\nP layer, while the larger peaks are associated to the res-\nonance of the F layer. Therefore, the difference in the\nheight of peaks is clearly ascribed to the larger damp-\ning parameter αfor the exchange biased P layer27. For\nthe samples with interactionbetween the FM layers(Fig.\n3(b-c)), the inner peak amplitudes decrease with respect\nto the outer peaks. Such decreasing seems to depend on\nthe coupling intensity and it will be explained onward by\nour model and numerical calculations.\n(a)P(a.u.)(b)\n−300 −150 0 150 300\nH (Oe)(c)\nFIG. 3. Absorbed power profiles at 3.7GHz for the samples\nwith different Cu spacer thicknesses (a) tCu= 2.5nm, (b) tCu\n= 1.0nm and (c) tCu= 0.75nm. The symbols correspond\nto the experimental data and the solid line to the calculated\ncurve. Arrows represent the oscillating vectors, see Fig 5 f or\nfurther details.\nIV. SEMIANALYTICAL MODEL AND\nNUMERICAL CALCULATIONS\nIn order to understand the features observed in our\nMvs. H curves and broadband measurements, we have\nadopted a macro spin model which takes into account\nthe usual free energy density terms for each ferromag-\nnetic layerplus a term describing the effective interaction\nbetween the free (F) and pinned (P) layers, as follows:\nE=EPinned+EFree+EInteraction (2)\nThefreeenergydensityforeachlayer EPinnedandEFree\nincorporate the Zeeman, in-plane uniaxial anisotropy,\nFIG. 4. Schematic diagram of the theoretical system con-\nsidered for the numerical calculations. The magnetization\nvectors (filled arrows) lay on the plane of the samples, their\norientations are defined by the θandϕangles measured from\nthe sample’s normal and the anisotropy axis (a.a.), respec-\ntively. The oscillating vectors (empty arrows) are paralle l to\nthe ˆϕdirections (not show). The radio frequency field hrfis\nalso parallel to the sample and perpendicular to a.a.\nshape anisotropy and out-of-plane anisotropy terms.\nEPinnedincludes also the exchange bias interaction term\nthatkeepsthecorrespondinglayerpinnedandtherelated\nrotatableanisotropy27. Inoursystem, asshowninFig. 4,\nbothlayershavethesamethickness tandsaturationmag-\nnetization MSandweexpresstheenergydensityinterms\nof the polar θand azimuthal ϕangles of the magnetiza-\ntionsandtheanisotropyaxes. Sincetheshapeanisotropy\nenergy is the dominant in our system, the magnetization\nvector always lays on the plane of thin films, therefore\nθ=π/2. Considering also that the anisotropies have\nan in-plane easy axis direction that is parallel to ϕ= 0\nand having the external field H applied at ϕH, then the\nnormalized free energy density ( η=E/MS) for F and\nP layers, keeping only the ϕdependent terms, can be\nwritten as follows:\nηPinned=−Hcos(ϕH−ϕP)−HEBcos(ϕP)\n−1\n2HP\nkcos2(ϕP)\nηFree=−Hcos(ϕH−ϕF)−1\n2HF\nkcos2(ϕF)(3)\nwhere HP\nk, HF\nk,ϕPandϕFare the uniaxial anisotropy\nfields and the in-plane magnetization angles of the P and\nF layers, respectively; H EBis the exchange bias field act-\ning on the pinned layer. The adopted interaction energy\ndensity reads:\nηInt=−HJ\n1cos(ϕP−ϕF)+HJ\n2cos2(ϕP−ϕF) (4)\nwith HJ\n1=J1\ntMSand HJ\n2=J2\ntMS, where J 1and J2are the\nbilinear and biquadratic interaction constants between\nthe two layers, respectively.\nBy minimizing η=ηPinned+ηFree+ηIntfor a given H,\nthe equilibrium angles ϕPandϕFof the magnetization\nvectors can be obtained.\nThemagnetizationdynamicsinoursystemisdescribed\nby the Landau-Lifschitz-Gilbert equation(LLG) adapted4\nto our purpose:\ndMi\ndt=−γ(Mi×Hi)+αi\nMS(Mi×dMi\ndt) (5)\nwithi= F, P. One should expect that each layer fol-\nlows independently this equation, hence in angular coor-\ndinates it can be expressed as:\ndθi\ndt=γ\n(1+α2\ni)(Hϕi+αiHθi)\nsinθidϕi\ndt=γ\n(1+α2\ni)(αiHϕi−Hθi)(6)\nwhereHϕandHθare the azimuthal and polar compo-\nnents of the effective field, αthe dimensionless damping\nparameter and γis the gyromagnetic ratio, which is the\nsame for both layers. The effective field components can\nbe expressed as:\nHθi=−1\nMS∂E\n∂θi+hrf·ˆθi\nHϕi=−1\nMSsinθi∂E\n∂ϕi+hrf·ˆϕi(7)\nwherehrfis the dynamic component ofthe applied exter-\nnal field, ˆθi= cosϕicosθiˆx+sinϕicosθiˆy−sinθiˆzand\nˆϕi=−sinϕiˆx+cosϕiˆy\nIn our specific case sin θP= sinθF= 1, which means\nthat we are able to rewrite Eq.6, for the F and P layers,\nas follows:\n\n˙θP\n˙ϕP\n˙θF\n˙ϕF\n=−γ\nMS[Λ]\n∂E/∂θ P\n∂E/∂ϕ P\n∂E/∂θ F\n∂E/∂ϕ F\n+γ[Λ]\nhrf·ˆθP\nhrf·ˆϕP\nhrf·ˆθF\nhrf·ˆϕF\n(8)\nwhere\n[Λ]=\n1\n1+α2\nP/parenleftbigg\nαP1\n−1αP/parenrightbigg\n0\n01\n1+α2\nF/parenleftbigg\nαF1\n−1αF/parenrightbigg\n(9)\nA. Susceptibility tensor\nThe differential susceptibility tensor [χ]=dM/dH\ncharacterizes the dynamic magnetic response of as sys-\ntem to an external field. In our system, [χ]gives us the\nrelation between the radio frequency field hrfand the\noscillating part ˙Mof the magnetization vectors\n˙MF+˙MP=[χ]˙hrf(10)\nwhere˙Mi=MS[sinθi˙θiˆθi+ ˙ϕiˆϕi] for each layer. In order\ntogiveanequivalentexpressioninfunctionoftheangular\ncoordinates, we define Ω= (θP, ϕP, θF, ϕF), withΩ=\nΩ0+δΩrf, whereδΩrf∝ejωtare the small deviationsaround the equilibrium positions Ω0. The magnetization\ndeviations δΩrfare driven by the radio frequency field\nhrf, thus they oscillate at the same frequency ω. The\nprojections hrf\nΩ=hrf·ˆΩ0arerelatedtothemagnetization\noscillations by a pseudo susceptibility tensor [X]defined\nby:\nδΩrf=[X]hrf\nΩ (11)\nthen, if we expand the energy terms around Ω0as in\n∂E/∂Ωrf\nk=/summationtext\nl∂2E\n∂Ωk∂ΩlδΩrf\nl, Eq. 8 can be expressed as:\n−γ\nMS[Λ][EΩΩ][X]hrf\nΩ+γ[Λ]hrf\nΩ=jω[X]hrf\nΩ(12)\nwhere the matrix [EΩΩ]has elements EΩΩkl=∂2E\n∂Ωk∂Ωl.\nFor our particular system, the non-zero values of [EΩΩ]\nare:\nEθPθP=MS[4πMS−H⊥+Hcos(ϕH−ϕP)\n+HP\nkcos2ϕP+HEBcosϕP+HR\n+HJ\n1cos(ϕP−ϕF)\n−2HJ\n2cos2(ϕP−ϕF)]\nEϕPϕP=MS[Hcos(ϕH−ϕP)\n+HP\nkcos(2ϕP)+HEBcosϕP+HR\n+HJ\n1cos(ϕP−ϕF)\n−2HJ\n2cos(2(ϕP−ϕF))]\nEθFθF=MS[4πMS−H⊥+Hcos(ϕH−ϕF)\n+HF\nkcos2ϕF\n+HJ\n1cos(ϕP−ϕF)\n−2HJ\n2cos2(ϕP−ϕF)]\nEϕFϕF=MS[Hcos(ϕH−ϕF)+HF\nkcos(2ϕF)\n+HJ\n1cos(ϕP−ϕF)\n−2HJ\n2cos(2(ϕP−ϕF))]\nEθPθF=EθFθP\n=−MS[HJ\n1+2HJ\n2cos(ϕP−ϕF)]\nEϕPϕF=EϕFϕP\n=−MS[HJ\n1cos(ϕP−ϕF)\n+2HJ\n2cos(2(ϕP−ϕF))](13)\nwhere H Rand H ⊥are the effective rotatable and per-\npendicular anisotropy fields. For an arbitrary field hrf\noscillating at a frequency ωwe can obtain the pseudo\nsusceptibility tensor using:\n[X]=/parenleftbigg\njω\nγ[Λ]−1+[EΩΩ]/parenrightbigg−1\n(14)\nThis equation can be efficiently solved by standard nu-\nmerical methods, resulting in the susceptibility tensor for\neach applied external field H and excitation frequency ω.5\nB. Resonant Frequencies\nOne of the important features in our systems are the\nresonant frequencies. These can be obtained from:\nγ\nMS[Λ][EΩΩ]δΩ=−jωrδΩ (15)\nThis equation can be solved as an eigensystem using\nnumerical methods. The eigenvalues provide us the\nresonant frequencies ωrand the eigenvector values of\nδΩat that frequency. Two positive values of ωrare\nfound for each external field H. The acquired values of\nδΩ= (δθPδϕPδθFδϕF) show that the amplitude of\nthe out-of-plane oscillations is negligible, i. e., almost\nzero (δθP≈δθF≈0) as expected. The analysis of the\nin-plane oscillations δϕPandδϕPof a given eigenvector\nallows us to determine which is the most oscillating layer\nat the frequency of the corresponding eigenvalue. When\n|δϕP|>|δϕF|, we associate the obtained eigenvalue to\nthe natural resonant frequency ωPof the pinned layer.\nThe opposite case ( |δϕP|<|δϕF|) is associated to the\nnatural frequency ωLof the free layer. One must notice\nthatδΩvalues obtained by this method are multiplied\nby an unknown amplitude and phase, thereafter they are\nnotsuitableforcalculatingtheabsorbedpowerortocom-\npare them at different fields H. However, they provide\nrelevant information on the relative phase and amplitude\nof oscillation of each layer over the dispersion relation.\nC. Absorbed Power\nIn order to compare directly our calculations to the\nexperimental results, it is important to write down the\naveragepowerabsorbedbyoursystematagivenfieldand\nfrequency. To proceed, we start describing the instant\npower, per unit of volume, absorbed by our system:\nP=−hrf·(˙MF+˙MP) (16)\nIt must be noticed that the amplitude of Pdepends on\nthreefactors: (a)therelativeorientationbetween hrfand\nthe oscillating vectors ˙Mi; (b) the temporal phase differ-\nence between ˙MFand˙MP; (c) the relative orientation\nof the oscillating vectors that depends on the direction\nof the magnetization at the equilibrium position for each\nlayer. Agraphicalrepresentationofseveralpossible cases\nare presented in Fig. 5.\nThe average absorbed power, per unit of volume, over\nan oscillatory cycle will also depends on the temporal\nphase difference between hrfand the magnetization re-\nsponse, which can be calculated by:\n< P >=−ωMSIm/bracketleftBigg/summationdisplay\nphrf\nΩpδΩrf\nq/bracketrightBigg\n=−ωMSIm/bracketleftBigg/summationdisplay\np,qhrf\nΩpXpqhrf\nΩq/bracketrightBigg(17)FIG. 5. Geometrical representation of the magnetization ve c-\ntors (filled arrows) and oscillating vectors (empty arrows) .\nMagnetizations are shown in parallel states (a and b), anti-\nparallel state (c) and noncollinear states (d and e). The os-\ncillations are in phase in a, c and d or out of phase in b and\ne. Ifhrfis along the vertical direction, a and e should have\nlarger absorbed power than b c or d\nV. DISCUSSION\nHereweseparatethe discussioninsamplesthat exhibit\ncoupling between the ferromagnetic layers and the sam-\nple with uncoupled layers. The coupling strength were\nobtained by comparing the calculated and experimental\ndata. All the simulation parameters are resumed in table\nI\nTABLE I. Parameters used in simulations.\nCommon parameters to all samples\nMS(emu/cm3) 80028HF\nK(Oe) 5\nγ(MHz/Oe) 17.5928HP\nK(Oe) 15\nαPinned 0.018 H R(Oe) 9\nαFree 0.010\nSample dependent parameters\ntCu 2.5nm 1.0nm 0.75nm\nHEB(Oe) 70 83 81\nHJ\n1(Oe) 0 13 35\nHJ\n2(Oe) 0 1 .0 4 .5\nH⊥(Oe) 0 600 600\nϕH 4◦2◦5◦\nWe choose the damping constants values in such a way\nthattheyreproducethefieldwidthsobservedat3.7 GHz\n(see Fig. 3). No frequency dependence of the damping\nparameters were considered in this work.\nA. No coupling\nWhen there is no coupling between the FM layers, our\nsystem behaves as two independent systems. The hys-\nteresis loop can be treated as the sum of two square\nloops, one (centered) corresponding to F and the other\none field shifted by H EBcorresponding to P. The broad-6\nband response is also the addition of the individual re-\nsponse of each layer. In our model, the matrix [EΩΩ]is\nthen formed by two independent block of matrices along\nthe main diagonal. Thus, an independent solution can be\nfound for each block, correspondingto the F and P layers\nof our samples. The solutions for the resonant frequen-\ncies, when the damping is neglected, are the well-known\nKittel relations:\nωP\nr=γ/radicalBig\n4πMS−H⊥±H±HEB+HP\nk+HR\n×/radicalBig\n±H±HEB+HP\nk+HR\nωL\nr=γ/radicalBig\n4πMS−H⊥±H+HF\nk/radicalBig\n±H+HF\nk(18)\nthe±sign should be chosen accordingly to the direc-\ntion of the respective magnetic layer, + for ϕ= 0 and\n−forϕ=πcorresponding to the right or left reso-\nnant branches experimentally observed. The resonant\nbranches cross each other when the layers are in the anti-\nparallel state and where the external field is\nH = H 0−1\n2(HEB+HP\nk+HR−HF\nk) (19)\nAt this point, the total absorbed power is the sum of the\nindividual absorbed powers of each layer.\nB. Coupled FM layers\nIn this subsection, Fig. 6 resumes both the experi-\nmental and numerical results in a expanded H scale. In\nthis figure, the colors were assigned red and blue for the\npinned and free layers, respectively. Also, the filled ar-\nrows represent the magnetization vectors whilst empty\narrows represent the oscillating vectors.\nHaving Fig. 6 in mind, we realize that the hysteresis\nresponse of the F layer is no longer centered. For a pos-\nitive bilinear interaction (J 1>0), the loops of each F\nlayer are field shifted toward the position of the P layer\nloop. If the coupling is not large, as for the sample with\ntCu= 1.0nm, the square shape the loops is maintained,\nindicatingthatthemagnetizationflipsbetweentheparal-\nlel and anti-paralleldirection with respect to the external\nfield. For the sample with tCu= 0.75nm, whose coupling\nintensity is larger, the hysteresis loop is no longer square,\nbut instead acquires a rounded shape due the simultane-\nous rotation of the FM layers (see Fig. 6(d)).\nWhenexcitedbyanexternalrffield, themagnetization\nof each layer does not oscillate independently. Instead,\nthey oscillate coherently but with correlated amplitude\nand phase difference. In this case, we find that for a\nresonant mode with frequency ωr, the phase difference\nof the oscillations ∆ φ=arg[δϕF]−arg[δϕP] depends on\nthe natural frequency of the companion layer: a natural\nfrequency higher or lower than ωrgives rise to a phase\ndifference φ≈0◦orφ≈180◦for each case, respectively.\nThis behavior has the effect of changing the the resonant\npeak amplitudes as seen in Fig. 3.\nf (GHz)(a)M/MS(c)fr(GHz)(e)∆φ(g)\n−60−30 0f (GHz)(i)\n123\n(b)\n−101\n(d)\n123\n(f)\n0π\n(h)\n−60−30 0\nH (Oe)123\n(j)\nFIG. 6. Experimental and simulation details in the non-\nsaturated states for tCu= 1.0nm (left) and tCu= 0.75nm\n(right). Experimental broadband spectra (a and b), simu-\nlated magnetization curves (single branch) (c and d), sim-\nulated dispersion relation (e and f), relative phase betwee n\nthe oscillations (g and h) and simulated broadband average\nabsorbed power (i and j). Filled arrows represent the mag-\nnetization vectors and empty arrows represent the oscillat ing\nvectors.\nIndependently of the coupling intensity, there is an ex-\nternal field value H 0where both layers oscillate at the\nsame frequency (see the dashed line in Fig. 6). As long\nas the anti-parallel state holds, which is the case of tCu\n= 1.0 nm, this field takes the same value as in the un-\ncoupled case (see Eq.19). In all other cases, as for tCu\n= 0.75 nm, the magnetization angles must be taken into\naccount for calculating H 0, giving a complicate analyti-\ncalexpression. The resultingvalues are, however, usually\nclose to the former cases (see the dotted line on the right\npanel of Fig. 6).\nAt this field H 0, two resonant frequencies, rather than\noneasinthe uncoupledcase, arefound. Thegapbetween7\nthese frequencies is proportional to the intensity of the\ngiven coupling. At this magnetization state, both layers\noscillate with the same amplitude ( |δϕP|=|δϕF|) at any\ngiven frequency. This may indicates that the dominant\ninteraction at this state is the coupling energy.\nFrom Fig. 6(a-b), we notice two arc-shaped (lower and\nupper) branches over the non saturated regime. These\nfeatures can be reproduced by our model, which results\nin the dispersion relations and the simulated broadband\naverage absorbed power shown in Fig. 6. Besides, our\nmodel allowus to identify that these branchesare formed\nby both oscillating modes, corresponding to the F and\nP layers respectively. The frequency gap between the\nbranches correspond to the frequency jump at H 0.\nWe calculated the oscillating vectors by taking into\naccount both the magnetization state and the relative\nphase ∆φ(shown in Fig. 6(g-h)).\nInterestingly, for tCu= 1.0nm, the absorption in the\nlower arc is favoured when the oscillating vectors are in\nthe same direction. In this case, the oscillating vectors\nare also in the same direction of the rf field. On the other\nhand, the sample with tCu= 0.75nm shows an apprecia-\nble absorption only in the upper inverted arc. Here the\noscillating vectors (calculated at H 0) are no longer par-\nallel to the rf field. Instead, their vector sum is nearly\nparallel to the rf field for the upper branch and nearly\nperpendicular to the rf field for the lower arc.\nVI. CONCLUSIONS\nIn summary we reported the broadband reso-\nnance spectra in coupled and uncoupled magnetic\nlayers in a single spin-valve configuration, namelyNiFe(20nm)/Cu( tCu)/NiFe(20nm)/IrMn(15nm) where\ntCu= 0.75nm, 1.0nm and 2.5nm controls the coupling\nintensity. For coupled cases, we observed that, at low\nfield, the experimental broadband spectra is complex,\nwhile at high field the spectra show the typical behavior\nof coupled saturated samples. The coupling between the\nferromagnetic layers was observed to modify the relative\namplitudes of the absorption peaks.\nWe were able to reproduce remarkably well the broad-\nband experimental results, both in saturated and non-\nsaturated states, by using our numerical method based\non the macro spin approximation, obtaining the disper-\nsion relations from Eq.15 and the broadband average ab-\nsorbed power from Eq.17. The method provide further\ninsights on the magnetization dynamics in coupled sys-\ntems, predicting frequency gaps and complex dispersion\nrelations in non-saturated magnetic states. Such states,\nbesides of their importance in applications, are usually\nneglected by the traditional descriptions of both regular\nFMR and broadband FMR experiments.\nAs final comments we would like to point that our ma-\ntrix mathematical approach allows any one to easily de-\nscribe magnetic systems with an arbitrary number of in-\nteracting macro spins. It also allows us an easy and fast\nsoftware implementation of the method by using well es-\ntablished numerical subroutines29.\nACKNOWLEDGMENTS\nThe authors thank to Dr. Marcio Assolin Corrˆ ea\nfor the fruitful discussion and revision. This work has\nbeen supported by the Brazilian agencies CNPq, FINEP,\nFAPERJ and CAPES.\n1B. Dieny, V. S. Speriosu, S. S. P. Parkin,\nB. A. Gurney, D. R. Wilhoit, and D. Mauri,\nPhys. Rev. B 43, 1297 (1991).\n2K.-f. 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Peterson, et al.,\n“SciPy: Open source scientific tools for Python,” (2001–\n)."    },    {        "title": "1107.4493v2.Interaction_of_Josephson_and_magnetic_oscillations_in_Josephson_tunnel_junctions_with_a_ferromagnetic_layer.pdf",        "content": "Interaction of Josephson and magnetic oscillations in Josephson\ntunnel junctions with a ferromagnetic layer\nS. Mai, E. Kandelaki, A.F. Volkov, and K.B. Efetov\nTheoretische Physik III, Ruhr-Universit at Bochum, D-44780 Bochum, Germany\n(Dated: October 27, 2011)\nWe study the dynamics of Josephson junctions with a thin ferromagnetic layer F [superconductor-\nferromagnet-insulator-ferromagnet-superconductor (SFIFS) junctions]. In such junctions, the phase\ndi\u000berence'of the superconductors and magnetization Min the F layer are two dynamic parameters\ncoupled to each other. We derive equations describing the dynamics of these two parameters and\nformulate the conditions of validity. The coupled Josephson plasma waves and oscillations of the\nmagnetization Ma\u000bect the form of the current-voltage ( I-V) characteristics in the presence of a\nweak magnetic \feld (Fiske steps). We calculate the modi\fed Fiske steps and show that the magnetic\ndegree of freedom not only changes the form of the Fiske steps but also the overall view of the I-V\ncurve (new peaks related to the magnetic resonance appear). The I-Vcharacteristics are shown\nfor di\u000berent lengths of the junction including those which correspond to the current experimental\nsituation. We also calculate the power Pabsorbed in the system if a microwave radiation with an\nac in-plane magnetic \feld is applied (magnetic resonance). The derived formula for the power P\nessentially di\u000bers from the one which describes the power absorption in an isolated ferromagnetic\n\flm. In particular, this formula describes the peaks related to the excitation of standing plasma\nwaves as well as the peak associated with the magnetic resonance.\nPACS numbers: 74.20.Rp, 74.50.+r, 03.65.Yz\nI. INTRODUCTION\nA great attention in recent years has been paid to the\nstudy of Josephson junctions (JJ) with a magnetic layer\n(or layers)1{4. Although the exchange \feld in the fer-\nromagnetic layer F essentially suppresses the Josephson\ncurrentIJ, the interaction of the exchange \feld and sin-\nglet Cooper pairs results in new, interesting, and nontriv-\nial e\u000bects. For example, the singlet pair wave function\npenetrating from the superconducting leads into the F\nlayer due to the proximity e\u000bect oscillates in space. In\ncase of a uniform F layer, the pair wave function consists\nof two components: one is the singlet component and\nanother is the triplet component with zero projection of\nthe total spin on the direction of the magnetization vec-\ntorMin the ferromagnet. The condensate wave func-\ntion decays in the ferromagnet on a short distance from\nthe superconductor-ferromagnet (SF) interfaces, which\nin the di\u000busive limit, is of the order \u0018F=p\nD=2Eexc,\nwhereD=vFl=3 is the di\u000busion constant and Eexcis\nthe exchange energy. Here, vFandldenote the Fermi\nvelocity and the electron mean free path, respectively.\nOscillations of the Cooper pair wave function in space\nlead to a change of sign of the critical Josephson current\nIJc. This e\u000bect was predicted long ago5,6but observed\nonly recently7{14.\nIf the magnetization in the F layer is not uniform (for\nexample, this occurs in the case of a domain structure\nor multilayered ferromagnet-superconductor (FS) struc-\ntures with noncollinear magnetization directions in the\nF layers), due to the proximity e\u000bect a so-called odd-\nfrequency triplet component arises3,4,15. In contrast to a\nconventional triplet component that is an odd function\nof momentum and is suppressed by scattering o\u000b ordi-nary impurities16, the odd-frequency triplet component\nis an even function of momentum (in the di\u000busive case)\nand is not destroyed by scattering o\u000b ordinary impurities.\nThis component also is not sensitive to the exchange \feld\nand therefore can penetrate into the ferromagnet over a\nlong distance up to \u0018N=p\nD=2\u0019Tat temperature T.\nConvincing data in favor of existence of this long-range\ntriplet component have been obtained in a number of\nrecent experimental works17{23.\nAnother interesting e\u000bect arises in SFIFS junctions.\nIt turns out that at the antiferromagnetic magnetization\norientation in the F layers, the Josephson critical current\nIJcis increased24. Its value may even exceed the critical\ncurrentIJcin similar JJs without ferromagnetic layers.\nThis prediction was also con\frmed experimentally25.\nAlongside with the study of the dc Josephson cur-\nrent in SFS or SF 1F2F1S JJs, dynamic properties of\nthese junctions and also of tunnel SIFS or SFIFS JJs\nhave been investigated both experimentally26{29and\ntheoretically30{32. Here and throughout the paper, S and\nI, respectively, represent a superconducting and insulat-\ning layers and F 1=2denotes two distinct ferromagnetic\nlayers. Interesting dynamic phenomena in JJs with a fer-\nromagnetic layer or a magnetic particle occur when the\ndynamics of the superconducting phase di\u000berence '(t)\nand the magnetization M(t) come into play.\nThe coupling between these two degrees of freedom\nmay be realized in di\u000berent ways. For example, the\nJosephson current produces a torque acting on magne-\ntization vectors in multilayered SF 1F2S junctions. Since\nthe Josephson current IJ['(t)] is determined by the mu-\ntual orientations of magnetization vectors M1=2, the dy-\nnamic behavior of the Josephson current will depend on\nthe dynamics of M(t)33{35. Another mechanism of thearXiv:1107.4493v2  [cond-mat.supr-con]  27 Oct 20112\nsupercurrent action on magnetization was considered by\nKonschelle and Buzdin36. They studied dynamics of SFS\njunctions with a non-centrosymmetric ferromagnet. In\nthis case, the Josephson current IJacts directly on the\nmagnetization Mleading to its precession. In a nonsta-\ntionary case, the interplay between IJ(t) andM(t) leads\nto a complicated behavior of the phase di\u000berence '(t) in\ntime.\nIn several papers,37{39dynamics of SmS\n(superconductor-magnetic impurity-superconductor)\nJJs have been studied, where m stands for a magnetic\nimpurity. Interaction between tunneling Cooper pairs\nand the magnetic moment of the impurity not only\nchanges the current-phase relation IJ(') but also results\nin interesting dynamics of the magnetic moment.\nThe most interesting dynamic e\u000bects arise in tunnel\nJJs with a ferromagnetic layer (or layers). In this case,\nthe interaction between the magnetization in F and the\nJosephson current is realized in the simplest way. As is\nwell known, even a weak in-plane magnetic \feld strongly\na\u000bects the Josephson current IJ('). In case of JJs of the\nSIFS or SFIFS type, such a magnetic \feld is produced\nby the F layer itself. Therefore any perturbations of the\nmagnetization vector Mchange the current IJ(') and\nin addition the Meissner currents in the superconducting\nleads change the orientation of the Mvector.\nIn absence of the F layer, Josephson plasma waves can\npropagate in SIS junctions and their spectrum is40{42:\n!2= \n2\nJ+k2v2\nJ, where \n Jis the Josephson \\plasma\"\nfrequency and vJis the velocity of Swihart waves. On\nthe other hand, in the F \flm, spin waves can be excited\nwith the spectrum: !2= \n2\nM(1 +k2l2\nM)2, where \n Mis\nthe magnetic resonance frequency and lMis a \\magnetic\"\nlength43. If \nJ<\nM, then these dispersion curves cross\n(usuallyl2\nM\u001cl2\nJ\u0011v2\nJ=\n2\nJ), and the interaction between\nmagnetization and Josephson currents leads to a coupling\nbetween Josephson \\plasma\" and spin waves and to a re-\npulsion of the corresponding dispersion \\terms.\" The\ncoupling between magnetic and superconducting oscilla-\ntions can be observed by studying the I-Vcharacteristics\nof the junction in the presence of a weak external mag-\nnetic \feld. In this case, the so-called Fiske steps arise on\ntheI-Vcurve, but their particular positions and form de-\npend on parameters characterizing the magnetic system.\nNew peaks related to magnetic resonances appear on the\ncurrent-voltage characteristics (CVC). These results have\nbeen obtained in a short paper by two of us30.\nIn the current paper, we study dynamic phenomena in\nthe same systems (SIFS or SFIFS JJs) as in Ref. 30.\nHowever, we present in more detail the derivation of\nequations describing the dynamics of the coupled mag-\nnetic and superconducting systems (see Sec. II). In par-\nticular, we formulate conditions (frequency range) under\nwhich these equations are valid. As in Ref. 30, we ana-\nlyze Fiske steps in SFIFS junctions, but the CVC will be\npresented for a wider range of parameters of these junc-\ntions. The CVC will be displayed not only for junctions\nwithL=lJas it was done in Ref. 30, but for junctions\nxyz\ndF\nSuperconductorsFerromagnetsInsulating barrier\ndFdFIG. 1. (Color online) Schematic construction of a SFIFS\njunction of the \\overlap\"geometry.\nlonger or shorter than the Josephson length ( L < lJ).\nThe latter case corresponds to the current experimental\nsituation.\nThe coupled magneto-plasma modes will also be dis-\ncussed in more detail (see Sec. IV). Finally, in Sec. V, we\npresent a formula for the power absorption Pin SFIFS\njunctions when a weak ac in-plane magnetic \feld is ap-\nplied, that is, we study the ferromagnetic resonance in\nthe system. This formula drastically di\u000bers from the\nknown formula for ferromagnetic resonance in an isolated\nF \flm. In particular, it describes plasma resonances in\ntunnel JJs, which also occur in absence of the F \flm. The\nfrequency dependence of Pwill be presented for various\nsystem parameters. In Sec. VI, we discuss the obtained\nresults and analyze possibilities to observe the predicted\ne\u000bects in experiments.\nII. MODEL AND BASIC EQUATIONS\nWe consider a planar SFIFS junction of the \\overlap\"\ngeometry as shown schematically in Fig. 1 (the results ob-\ntained are also applicable to an SIFS junction). Our aim\nis to generalize the equation for the phase di\u000berence '\nbetween the superconducting layers describing the static\nand dynamic properties of an SIS JJ to the case of SFIFS\nJJs.\nThis equation reads40{42,44,45\n\n\u00002\nJ\u0012@2'\n@t2+\rR@'\n@t\u0013\n\u0000l2\nJr2\n?'+ sin(') =\u0011; (1)\nwhere \nJ= (2ejc=C\u0003~)1=2is the Josephson \\plasma\"\nfrequency,\rR= (R\u0003C\u0003)\u00001,C\u0003=\u000f=4\u0019d, andR\u0003are\nthe capacitance and resistance of the junction per unit\narea, respectively, dis the thickness of the insulating\nlayer,l2\nJ=v2\nJ=\n2\nJ,vJ=cp\nd=2\u000f\u0015Lis the plasma wave\npropagation velocity (Swihart waves), \u0015Lis the London\npenetration depth, and r?represents the tangential or\nin-plane gradient with respect to the interfaces in the x-y\nplane.3\nWe single out the term on the right-hand side of\nEq. (1),\u0011=j=jc, which describes the normalized bias\ncurrent through the junction. Although it may depend\nony, the normalized current \u0011will be considered as con-\nstant along the ydirection. Strictly speaking, this is only\ntrue for \\overlap\" junctions41,42considered here in which\nthe system geometry is arranged in such a way that the\nintersection region of superconducting layers is approx-\nimately one-dimensional. However, the form of Eq. (1)\nis most convenient for analysis of CVC for the system\nunder consideration and, moreover, neglecting the yde-\npendence of normalized current \u0011does not change qual-\nitatively the \fnal results. The critical current density jc\nis considered as a known quantity. It was calculated in\nRefs. 24, 46{48.\nThe resistance R\u0003depends on the voltage Vacross the\njunction. This dependence is especially strong in the case\nof tunnel SIS JJs if the voltage Vis close to the energy\ngap \u0001. We assume that the characteristic frequencies\n(\nJand \nM) are smaller than \u0001 =~. In addition, we are\ninterested in the form of the CVC at voltages Vclose\nto~\nJ=2e;~\nM=2e, whereR\u0003and, therefore, \rRcan\nbe regarded as constant. Of course, the overall form of\nthe CVC will be modi\fed as a direct consequence of the\nvoltage-dependent damping coe\u000ecient \rR(V).\nWe consider planar JJs of the SFIFS, SFIS, or SFS\ntype and assume that the layer separating the two su-\nperconductors is characterized by the magnetic suscep-\ntibility\u001f(!;k). In particular, this layer may be a mag-\nnetic insulator or metallic ferromagnet. The derivation\nof an equation for the phase di\u000berence 'in SFIFS junc-\ntions is quite similar to that in the case of tunnel SIS\njunctions41,42,49,50. We assume that there is no mag-\nnetic \feld normal to the interfaces in the superconductors\nor, in other words, no Abrikosov vortices pierce the su-\nperconducting \flms, and the lateral dimensions Lx;yare\nmuch larger than the thickness dFof the F layers and the\nJosephson penetration depth \u0015L. Since the normal com-\nponent of the magnetic induction Bzis continuous at the\nsuperconductor-ferromagnet (SF) interfaces, it also van-\nishes in the ferromagnetic layers and, hence, according to\nBz=Hz+ 4\u0019Mzone hasHz=\u00004\u0019Mzin the F \flms.\nIn order to \fnd the relation between the magnetic \feld\nHin the superconductor (note that in the S layers H\ncoincides with the magnetic induction B) and the phase\ndi\u000berence', we express the tangential component of the\ncurrent density in the S \flm j?\u0011jxnx+jynyusing the\nvector potential A?(nz\u0002@A=@z=B?) and the tan-\ngential gradient of the phase in the superconductor r?\u001f\nas\nj?=c\n4\u0019\u00152\nL(1 +\rqp)\u0012\n\u0000A?\u0000\b0\n2\u0019r?\u001f\u0013\n; (2)\nwhere\rqp(!) = 4\u0019i!\u001b (!)\u00152\nL=c2is a damping parameter\ndescribing e\u000bects of quasiparticles on the supercurrent\nand \b 0=hc=2e>0 is the magnetic \rux quantum. The\nparameter\rqpis very small for not very high frequenciesbecause the frequency c=\u0015Lis very large. For example,\ntaking\u0015L= 5\u000110\u00006cm we obtain c=\u0015L= 0:6\u00011016s\u00001,\nwhich actually allows us to omit the parameter \rqp.\nWriting Eq. (2) we imply a local relation between the\ntangential current density j?and the gauge invariant\nquantity in brackets, which is legitimate in the limit\nk\u0015L\u001c1, wherekis the modulus of the in-plane wave\nvector of perturbations. Subtracting the expressions for\nthe current density, Eq. (2), written for the right and\nleft superconductors from each other we \fnd the change\nof the tangential current density [ j?] =j?(edF=2)\u0000\nj?(\u0000edF=2) across the junction\n[j?] =c\n4\u0019\u00152\nL(1 +\rqp)\u0012\nedFfnz\u0002B?g\u0000\b0\n2\u0019r?'\u0013\f\f\f\fedF\n2;\n(3)\nwhereedF=dFin the case of an SIFS or SFS junction\nandedF= 2dFin the case of an SFIFS junction. The\nparameterdFis the thickness of the F \flm, which is as-\nsumed to be smaller than the London penetration length\n\u0015L, and for any quantity Q, we denote the di\u000berence\nQ\f\f\nS(R)\u0000Q\f\f\nS(L)by [Q], whereS(R) andS(L) are the\nright and left superconductors, respectively.\nThe assumption dF\u001c\u0015Lallows one to neglect the\nchange of A?along thezdirection caused by Meissner\ncurrents in the F layer and to write the change of the\nvector potential A?in the form [ A?] =edF(nz\u0002B?)\nwithB?= 4\u0019M?+H?. The \feld H?is approximately\nthe same to the right and to the left from the SF inter-\nfaces and does not contribute to the jump of the tangen-\ntial current density [ j?]. The Meissner currents in the F\nlayers and, therefore, the variation of H?there are much\nsmaller than in the superconductors for the following rea-\nson. The total screening Meissner current IScrin the F\nlayer is proportional to \u0015\u00002\nLFedFA, where the inverse Lon-\ndon penetration depth \u0015\u00001\nLFis proportional to the density\nof Cooper pairs, \u0015\u00002\nLF\u0018nSF, and, thus, is much smaller\nthan\u0015\u00002\nL. The phase di\u000berence 'between the two S\nlayers has the (gauge-invariant) de\fnition:\n'= [\u001f] +2e\n~cZS(R)\nS(L)dzAz; (4)\nand completely describes the JJ because we choose a\ngauge with Az= 0 and [\u001f] =\u001f(edF=2)\u0000\u001f(\u0000edF=2).\nEquation (3) determines the boundary conditions of\nthe London equation in the superconductors. Indeed,\nconsidering the Maxwell equation at the points z=\n\u0006zSF\u0019\u0006edF=2,\nr\u0002B?\f\f\f\n\u0006edF=2=4\u0019\ncj?\f\f\f\n\u0006edF=2; (5)\nwherezSFdenotes the coordinate of the right SF inter-\nface, we obtain by successively taking the cross product4\nwithnzin both sides and subtracting the two equations\nfrom each other\n\u0000@B?\n@z\f\f\fedF=2=2\u0019\ncnz\u0002[j?]: (6)\nHere, we used the relation\n@B?\n@z\f\f\f\fedF=2=\u0000@B?\n@z\f\f\f\f\n\u0000edF=2(7)\ntaking into account the symmetry of the SFIFS system.\nRecalling that the magnetic \feld component Bznormal\nto the interfaces is assumed to be zero in the S layers\nand considering only the zdependence of B?, we have\nto solve in the superconductors the equation\n@2B?\n@z2\u0000\u00142B?= 0 (8)\nwith\u00142=\u0015\u00002\nL(1 +\rqp)\u0019\u0015\u00002\nL. The solution reads for\njzj>edF=2,\nB?(z) =B?\u0010\nedF=2\u0011\nexp(\n\u0000jzj\u0000edF=2\n\u0015L)\n:(9)\nInserting this expression for B?into Eq. (6) we obtain\nby use of Eq. (3)\nB?\u0010edF\n2\u0011\n=\u0000\b0\n4\u0019e\u0015L(nz\u0002r?')\u00002\u0019edF\ne\u0015LM?\f\f\fedF=2;(10)\nwhere we have set e\u0015L=\u0015L+edF=2. The magnetic \feld\nBdecays exponentially with increasing zprovided the\nthickness of the S layers exceeds the London penetration\nlength\u0015L.\nIn order to obtain an equation for the phase di\u000berence\n'of the superconductors we use the Maxwell equation\n(r \u0002 H)z\u0000c\u00001@ Dz=@ t = (4\u0019=c)jzand the stan-\ndard expression for the Josephson current according to\nthe Stewart-McCumber model51,52. This simple model\n[also known as the resistively and capacitively shunted\njunction (RCSJ) model] provides a good description of\nthe CVC of a real JJ, although e\u000bects due to \fnite di-\nmensions of the contacts and nonlinearities of the quasi-\nparticle current are neglected. Using the Josephson rela-\ntion\n@'\n@t=\u00002eV\n~(11)\nand the standard expression for the Josephson current\nwe obtain within this modelc\n4\u0019(r\u0002H)z=\u0000~C\u0003\n2e@2'\n@t2\u0000~\n2eR\u0003@'\n@t\u0000jcsin(') +j:\n(12)\nFinally, with the help of Eq. (10) and taking into account\nthat in the S layers B=H,\n\n\u00002\nJ\u0012@2'\n@t2+\rR@'\n@t\u0013\n\u0000l2\nJr2\n?'+ sin(') =\n=\u0011+cedF\n2e\u0015Ljc(r\u0002M?)z(13)\nwhere here, too, \n J= (2ejc=C\u0003~)1=2is the Joseph-\nson \\plasma\" frequency, \rR= (R\u0003C\u0003)\u00001,C\u0003(!) =\n\u000f(!)=4\u0019dandR\u0003(!) are the capacitance and resistance\nof the junction per unit area, respectively, dis the thick-\nness of the insulating layer, l2\nJ=v2\nJ=\n2\nJ,vJ=cq\nd=2\u000fe\u0015L\nis the plasma wave propagation velocity (Swihart waves),\nand\u0011=j=jcis the normalized bias current through the\njunction. The capacitance C\u0003and the resistance R\u0003of\nthe junction may depend on frequency !(in the Fourier\nrepresentation). A simpler equation for the phase di\u000ber-\nence'in the stationary case has been reported previ-\nously in Ref. 53. In a general, non-stationary case, this\nequation was derived in Ref. 30. Note that a slightly\ndi\u000berent approach for the study of dynamic processes in\nSFS junctions was used in a recent paper31. In particu-\nlar, Eq. (13) can be easily derived from Eqs. (A3){(A6)\nof this work.\nIn order to obtain a closed set of equations for the\nphase di\u000berence 'of the superconductors and the mag-\nnetization M?of the ferromagnetic layer, we need to use\na dynamic equation for M?as well.\nThe dynamics of the magnetization Min the F layer\nis described by the well-known Landau-Lifshitz-Gilbert\n(LLG) equation (see, e.g., Refs. 43 and 54), which allows\none to describe the temporal development of Min an\ne\u000bective magnetic \feld He\u000bincluding all internal and\nexternal contributions.\nWe decompose the magnetization vector Maccording\ntoM=M0ne+m, where the unit vector nedenotes\nthe easy axis direction and m?neis the dynamic part\nwhich evolves in time as described by the LLG equa-\ntion. Assuming that in equilibrium the magnetization\ncoincides with the static part along the easy axis, i.e.\nM0\u0019jMj\u001djmj, and using Bz= 0, we obtain\n@m\n@t=\u00004\u0019\u000bM e\u000b\u0010\n1\u0000el2\nMr2\n?\u0011\n(M\u0002m) +\n+Me\u000bM\u0002B?+\rM\njMjM\u0002@m\n@t; (14)\nwhereMe\u000b=gjej=2mc,g<0 is the gyromagnetic factor,\n\u000bis a parameter related to the anisotropy constant43,elM5\nis a characteristic length related to spin waves, and \rM\nis the dimensionless Gilbert damping constant.\nWe further neglect the Gilbert damping term ( \rM=\n0), align the easy axis along the zdirection (e\u0011z),\nand substitute B?F= 4\u0019M?+H?(M?\u0011m) into\nEq. (14), where B?Fis the magnetic induction in the F\nlayer and H?is the magnetic \feld, which is assumed to\nbe independent of the zcoordinate (screening e\u000bects in\nthe F layer are negligible). The \feld H?is continuous\nacross the SF interface, i.e., H?=B?\u0010\nz!edF=2\u0011\n, and\nis given by Eq. (10).\nFinally, we obtain\n@m\n@t= \nM\u0014\u0000\n1 +s\u0000l2\nMr2\n?\u0001M\u0002m\nM0\u0000\n\u0000\b0\n(4\u0019)2(\u000b\u00001)e\u0015Lr?'#\n;(15)\nwhere \nM= 4\u0019(\u000b\u00001)jMe\u000bjM0is the resonance fre-\nquency of magnetic moment precession ( \u000b > 1),s=\nedF=[2(\u000b\u00001)e\u0015L],l2\nM= [\u000b=(\u000b\u00001)]el2\nM.\nEquations (13) and (15) fully describe di\u000berent dynam-\nical processes in the junctions under consideration. Note\nthat the Josephson current is coupled to the magnetiza-\ntion through the spatial derivative of the phase di\u000berence\nr?'[the last term on the right-hand side of Eq. (15)].\nTherefore, in a spatially homogeneous case there is no\ncoupling between the Josephson e\u000bect and dynamics of\nthe magnetization.\nIII. FISKE STEPS\nIn this section, we consider a SFIFS Josephson junc-\ntion in a weak external magnetic \feld Hextassuming that\nit is constant in space and time and is directed parallel\nto the interfaces along the ydirection. As is well known,\nin this case so-called Fiske steps arise on the CVC due\nto excitation of eigenmodes in the junction. The phase\ndi\u000berence'(x;t) depends on the xcoordinate and, there-\nfore, dynamics of the magnetic and super\ruid systems\nare coupled together. We consider the case when the\nmagnetization vector in the stationary state is directed\nperpendicular to the SF interfaces, i.e. M0=M0nzand\nH0=\u00004\u0019M0. As the typical values for the magnitude\nof the stationary magnetization M0are hundreds of Gau\u0019\nand the small external magnetic \feld Hextis of the order\nof a few Gau\u0019, one can neglect the in-plane magnetization\nMy=\u0000Hext=(4\u0019) compared to M0. The resulting pre-\ncessional motion of the magnetization Min presence of a\ncurrent through the JJ implies that the in-plane compo-\nnents m?nzofMare excited. Therefore, we represent\nMasM(x;t) =M0+m(x;t). Components mx;yare\neasily found from Eq. (15):my=\nM(1 +s)\ni!mx\n=1\n(1 +s)L!F\b0\n(4\u0019)2(\u000b\u00001)e\u0015L@'\n@x; (16)\nL!F=!(!\u0000i\rM)\n\n2\nM(1 +s)2\u00001: (17)\nEquations (16), (17) are written under the assumption\nthat all relevant quantities depend on time as exp( i!t)\nand, what is more important, spatial derivatives in the\nequation for m(x;t) are neglected. The latter assump-\ntion is justi\fed provided the magnetic length lMis much\nshorter than the Josephson length lJ:lM\u001clJ. It is\nnot di\u000ecult to analyze a more general case of arbitrary\nrelation between lMandlJ, but the corresponding for-\nmulas become too cumbersome. Substituting Eq. (16)\ninto Eq. (13) we obtain\n\u0000\u0014!(!\u0000i\rR)\n\n2\nJ+el2\nJ(!)@2\n@x2\u0015\n'(x;!)+ (18)\n+Ffsin(')g(x;!) =\u0011;\nwhereFfsin(')g(x;!) is the Fourier transform of\nsin['(x;t)] with respect to time tand\nelJ(!) =lJ\u0014\n1 +s\n(1 +s)L!F\u00151=2\n(19)\nis a renormalized Josephson length containing L!Fand,\ntherefore, depending on frequency !. Equation (18) is\nthe favored generalization of Eq. (1) for SFIFS junctions.\nIn order to \fnd the CVC, we represent the phase\ndi\u000berence'of the superconducting layers in the form\n'='0(x;t) + (x;t) (see Ref. 50). The \frst term is\ngiven by'0(x;t) =\u0014Hx+ \nVtwith\u0014H= 4\u0019e\u0015LHext=\b0\n[see Eq. (10)] and \n V= 2eV=~. The function  (x;t)\nis assumed to be small allowing us to linearize Eq. (13)\nwith respect to  :\n\u0000bPf g(x;t) = sin ['0(x;t)] (20)\n= sin(\nVt) cos(\u0014Hx) + cos(\n Vt) sin(\u0014Hx)\nwhere the operator bPis de\fned as\nbP= \n\u00002\nJ\u0012@2\n@t2+\rR@\n@t\u0013\n\u0000el2\nJ(\nV)@2\n@x2: (21)\nThe current correction \u000e\u0011to the dc current \u00110=\n(2eV=~)=\nJ= \nV=\nJis given by\n\u000e\u0011=\n (x;t) cos ['0(x;t)]\u000b\n; (22)6\nwhere the angular brackets denote the average with re-\nspect to space and time.\nEquation (22) determines the constant normalized\ncurrent through the junction as a function of voltage\nV, which gives a current-voltage (I-V) curve. Equa-\ntion (20) contains parts oscillating in space and time.\nIt should be solved taking into account the boundary\nconditions41,42,49,50\n@ \n@x\f\f\f\f\nx=\u0006L= 0; (23)\nwhereLdenotes the length of the junction along the x\ndirection. The right-hand side of Eq. (20) can be writ-\nten in the form Im fexp(i\nVt)[cos(\u0014Hx) +isin(\u0014Hx)]g\nand, therefore, the solution of Eq. (20) can be written as\n (x;t) = Imfexp(i\nVt) 1(x)g, where the function  1(x)\nobeys the equation\n\u0000bP\nf 1g(x) = cos(\u0014Hx) +isin(\u0014Hx) (24)\nwith the boundary condition Eq. (23).\nThe operator bP\ncoincides with bPafter replacing @=@t\nbyi\nV. The solution can be easily found and equals\n 1=1\nP\n(V;H)n\ncos(\u0014Hx) +Ccos(\u0014Vx) +isin(\u0014Hx) +\n+iSsin(\u0014Vx)o\n(25)\nwhere P\n(V;H) =a2\u0000el2\nJ(\nV)\u00142\nH,a2= \n\u00002\nJ(\n2\nV\u0000\ni\rR\nV) and\nC=\u0000\u0012H\n\u0012Vsin\u0012H\nsin\u0012V; S =\u0000\u0012H\n\u0012Vcos\u0012H\ncos\u0012V(26)\nwith\u0012H=\u0014HL; \u0012V=\u0014VL; \u00142\nV=a2el\u00002\nJ(\nV). Substi-\ntuting the function  (x;t) expressed through  1(x) into\nEq. (22), we \fnd the dependence, \u000e\u0011(V)\u0011\u000ej(V)=jc,\n\u000e\u0011= Im\u001a1\nP\n(V;H)\u0014\n1\u0000\u00122\nH\n\u0012V(\u00122\nH\u0000\u00122\nV)\u0002\n\u0002cos(2\u0012V)\u0000cos(2\u0012H)\nsin(2\u0012V)\u0015\u001b\n:(27)\nSince we assumed that the correction  =\nImfexp (i\nVt) 1(x)gto the phase di\u000berence 'in\nthe superconducting layers is small, Eq. (27) is only\nvalid for normalized voltages \n V=\nJ>(\rR=\nJ)\u00001.\nThis can be seen from Eq. (25) where one should verify\nthat the prefactor P\u00001\n\n(V;H) is small.\nLet us discuss the current results and compare them\nwith those obtained in Ref. 30. The prefactor P\u00001\n\n(V;H)\nin Eq. (27) contains the renormalized Josephson lengthelJde\fned in Eq. (19), which corresponds to the quantity\nlVof Ref. 30. The formulas for Fiske steps in Ref. 30\nwere given for small values of the parameter s. If the\nparameter sis not very small, one can reproduce the\ncorrect result by replacing there \n2\nM!(1 +s)\n2\nM, i.e.,\nEq. (19). [Note that in the de\fnition of \n Ms, Eq. (10)\nof Ref. 30, there is a misprint. The factor of two in the\nexponent at the right-hand side is missing so that the\ncorrect formula reads \n2\nMs= \n2\nM(1 +s)2.] The modi-\n\fed dependence of the normalized Josephson length elJ\non the parameter schanges the form of the I-Vcharac-\nteristics and reveals that the e\u000bect of the ferromagnetic\nlayer is much more pronounced compared to the results\nof Ref. 30 even for small sbecause the denominator in\nEq. (10) of Ref. 30 is very small at voltages correspond-\ning to peaks in the CVC and, therefore, is very sensitive\nto the parameter s. Thus, we update the \fgures showing\nthe dependence \u000e\u0011(Vnorm:) as a function of normalized\nvoltageVnorm:= \nV=\nJ. Finally, we also present I-V\ncharacteristics for di\u000berent values of normalized junction\nlengthsL=lJincluding those which correspond to the ex-\nperimental values of Ref. 28 ( L=lJ<1). As in Ref. 30,\nfor simplicity, we assume that the damping coe\u000ecient \rR\nis constant, i.e., it does not depend on voltage V.\nIn Figs. 2 and 3, we plot the current correction \u000e\u0011\nas a function of normalized voltage Vnorm:for di\u000berent\nvalues of the parameter s=edF=(2(\u000b\u00001)e\u0015L) and nor-\nmalized junction length L=lJ. Taking into account the\nexperimental values of LandlJ(see Ref. 28), we dis-\nplay the current correction \u000e\u0011for short junctions with\nL=lJ= 0:75 [see Figs. 2(a) and 2(b)] and, in addition,\nfor longer junctions with L=lJ= 2 [see Figs. 2(c) and\n2(d)] andL=lJ= 10 [see Figs. 3(a) and 3(b)]. Black\ncurves represent the limit s!0 where we have no F lay-\ners in the system and the CVC correspond to ordinary\nFiske steps. Due to the fact that in experiments, only\nthe strength of the external magnetic \feld can be varied,\nwe display our result for di\u000berent values of the parameter\n\u0014HlJ/Hextkeeping all other system parameters such\nas \nM=\nJ;L=lJ, andsconstant.\nThe strongest in\ruence of the ferromagnetic layers\non the current-voltage characteristics develops for exter-\nnal magnetic \felds such that the parameters \u0014HlJand\n\nM=\nJcoincide. By comparing Figs. 2(a) and 2(b) [or\nFigs. 2(c) and 2(d), respectively] one can observe that\nthe change of the current correction is clearly recog-\nnizable for \u0014HlJ= \nM=\nJand nonzero s, while for\n\u0014HlJ6= \nM=\nJ, it only becomes pronounced for larger\nvalues ofs.\nAs can be seen from Figs. 2(a) and 2(c) that the nor-\nmalized junction length L=lJdetermines the form of the\nCVC even in the case s= 0, i.e., the number of Fiske\nsteps close to the normalized magnetic resonance fre-\nquency \n M=\nJmay vary for di\u000berent values of L=lJ.\nProvided for s= 0 there appears a single peak close to\n\nM=\nJ, increasing the parameter sleads to a double\nsplitting of the dominant peak. For even larger values\nofs, the pair of peaks moves more and more apart from7\n4 6 8 10 12\nΩV/ΩJ00.050.10.15δη = δj/jcs = 0\ns = 0.02\ns = 0.1(a)\n6 8 10\nΩV/ΩJ00.020.040.060.08δη = δj/jcs = 0\ns = 0.02\ns = 0.1(b)\n4 6 8 10 12\nΩV/ΩJ00.060.120.18δη = δj/jcs = 0\ns = 0.02\ns = 0.1(c)\n6 8 10 12 14\nΩV/ΩJ00.030.060.09δη = δj/jcs = 0\ns = 0.02\ns = 0.1(d)\nFIG. 2. (Color online) Correction to the I-Vcharacteristics of an SFIFS JJ in a weak external magnetic \feld due to interaction\nof Josephson oscillations and spin-wave modes. The correction is plotted as a function of normalized voltage Vnorm:= \nV=\nJ\nfor di\u000berent values of the parameter s. The \fgures are presented for the following parameters: (a) \n M=\nJ=\u0014HlJ= 8 and\nL=lJ= 0:75, (b) \n M=\nJ= 8;\u0014HlJ= 12;L=l J= 0:75; (c) \n M=\nJ=\u0014HlJ= 8;L=l J= 2; (d) \n M=\nJ= 8;\u0014HlJ= 12;L=l J= 2.\nThe damping coe\u000ecients are \rR=\nJ= 0:4;\rM=\nJ= 0:3.\neach other [see Fig. 2(a)]. A similar e\u000bect can be seen\nfor a larger number of Fiske steps close to \n M=\nJ, e.g.,\nFig. 2(c) displays essentially two Fiske steps in the vicin-\nity of \nM=\nJ= 8 that both split up into two peaks\nmoving apart from each other with increasing s.\nFor distinct values of the parameters \u0014HlJand \nM=\nJ\n[see Figs. 2(b) and 2(d)], there also emerge additional\npeaks in the I-Vcharacteristics close to the normalized\nmagnetic resonance frequency, but the detailed impact of\nthe F layers on the CVC is not as obvious as is the case\nfor\u0014HlJ= \nM=\nJ. From Fig. 2(d), one can already con-\njecture that for long junctions, the ferromagnetic layers\nsimply induce a single additional peak close to \n M=\nJ.\nIn Fig. 3, where the current correction \u000e\u0011is shown for\nthe limit of large values of L=lJ(L=lJ= 10), this fea-\nture becomes more apparent. For coinciding values of the\nmagnetic resonance frequency \n M=\nJand the parame-\nter\u0014HlJ[see Fig. 3(a)], we \fnd a single peak for s= 0\nand a double peak for s6= 0 in the vicinity of \n M=\nJ.\nFor \nM=\nJ6=\u0014HlJ, there emerges a single peak close to\n\nM=\nJand\u0014HlJ, respectively, where the former is no-\ntably smaller in magnitude [see Fig. 3(b)]. Below we also\nderive analytical expressions for these peak positions.\nThus the presence of the F layers leads not only to ashift of the peaks in the dependence \u000e\u0011(Vnorm:) but also\nto a change of the overall form of this dependence. The\nadditional peaks arising on the I-Vcurves can be at-\ntributed to the ferromagnetic resonance and the nonzero\ncoupling between Josephson and magnetic moment oscil-\nlations. In order to observe these peaks experimentally,\none should perform measurements with di\u000berent samples\nthat contain ferromagnetic layers of varying thickness.\nThen, according to our theoretical result, one would be\nable to di\u000berentiate between ordinary Fiske steps and\npeaks caused by interaction of Josephson and magnetic\noscillations in the F layers. Note that in the limit of a\nvery short junction ( L=lJ\u001c1) there is no coupling be-\ntween Josephson and magnetic moment oscillations. In-\ndeed, in this limit we obtain from Eq. (27)\n\u000e\u0011= Im\u001a\n2\nJ\n\nV(\nV\u0000i\rR)\u001b\n: (28)\nIt is seen that magnetic characteristics such as \n Mof the\nF layers drop out from this expression.\nIn the limit of long junctions, L=lJ\u001d1, the expression\nfor the current correction can be approximated by8\n4 6 8 10 12\nΩV/ΩJ00.070.140.21δη = δj/jcs = 0\ns = 0.02\ns = 0.1(a)\n6 8 10 12 14\nΩV/ΩJ00.050.10.15δη = δj/jcs = 0\ns = 0.02\ns = 0.1(b)\nFIG. 3. (Color online) Current correction \u000e\u0011as a function of\nnormalized voltage Vnorm:= \nV=\nJfor long JJ with normal-\nized junction length L=lJ= 10. The function \u000e\u0011is displayed\nfor (a) \n M=\nJ=\u0014HlJ= 8; (b) \n M=\nJ= 8;\u0014HlJ= 12\nand for di\u000berent values of the parameter s. The damping\ncoe\u000ecients are \rR=\nJ= 0:4;\rM=\nJ= 0:3.\n\u000e\u0011= Im\u001a1\nP\n(V;H)\u001b\n(29)\n= Im\u001a\n2\nV\u0000i\rR\nV\n\n2\nJ\u0000\u00142\nHl2\nJ\u0014\n1 +s\n(1 +s)L\nVF\u0015\u001b\u00001\n:\nIn accordance to Fig. 3 we obtain for s= 0 a single peak\nat normalized voltage Vnorm:=\u0014HlJwhile fors6= 0 and\n\u0014HlJ= \nM=\nJthere exist two peaks at\nVnorm:=\nM\n\nJq\n(1 +s)\u0006p\ns(1 +s): (30)\nFinally, for the general case s6= 0 and\u0014HlJ6= \nM=\nJ\nwe \fnd in leading order in the parameter stwo peaks\nlocated at normalized voltages\nV(1)\nnorm:=\u0014HlJr\n1 +s\u0001x2\n1\u0000x2(31a)\nV(2)\nnorm:=\nM\n\nJr\n1 +s\u00011\u00002x2\n1\u0000x2(31b)wherex= (\nM=\nJ)=(\u0014HlJ). These analytical expres-\nsions perfectly describe the peak locations of the current-\nvoltage characteristics in the limit L=lJ\u001d1 as exemplar-\nily shown in Fig. 3 for junctions with L=lJ= 10.\nIV. COUPLED COLLECTIVE MODES\nIn this section, we analyze the spectrum of coupled\ncollective modes in long Josephson junctions with a fer-\nromagnetic layer. So far we have derived essentially two\n(coupled) equations, Eqs. (13) and (15), that describe re-\nspectively the dynamics of the phase di\u000berence 'of the\nS layers and the magnetization Mof the ferromagnetic\nlayers. Here, we consider again the case when the mag-\nnetization M0is aligned normal to the interface so that\nin equilibrium B0= 0. Small perturbations near the\nequilibrium result in precessional motion of the magnetic\nmoment Mand in a variation of the phase di\u000berence\n'in space and time. In order to \fnd the spectrum of\ncollective modes in the system, we represent the phase\ndi\u000berence'and the magnetic moment Min the form\n'='0+ ;M=M0nz+m?; (32)\nwhere nzis the unit vector normal to the SF interface\nand the functions  andm?are assumed to be small,\nj j\u001cj'0jandjm?j\u001cjM0j. Linearizing Eq. (13) with\nrespect to , we \fnd that the function  (x;t) obeys the\nequation\n\n\u00002\nJ\u0012@2 \n@t2+\rR@ \n@t\u0013\n\u0000l2\nJr2\n? + = (33)\n=cedF\n2e\u0015Ljch\nr\u0002m?i\nz:\nThe perturbation m?of the magnetic moment is parallel\nto the SF interface and is described by the equation\n@m?\n@t= \nM(\n\u0000\n1 +s\u0000l2\nMr2\n?\u0001h\nnz\u0002m?i\n\u0000 (34)\n\u0000\b0\n(4\u0019)2\fe\u0015Lr? )\n+\rM\u0014\nnz\u0002@m?\n@t\u0015\n;\nwhere we included again the Gilbert damping term,\nwhich was neglected in Eq. (15). Fourier transforming\nthe perturbations '(r;t) and m?(r;t) to (k?;!) repre-\nsentation and combining Eqs. (33) and (34) into a single\nequation, we obtain\nM\u0012\n'(k?;!)\nm?(k?;!)\u0013\n= 0; (35)\nM=0\n@\n\u00002\nJ\u0000\n!2\nJ\u0000!2\u0001\nibky\u0000ibkx\niakx\u0000i! !M\niaky\u0000!M\u0000i!1\nA; (36)9\n0 1 2 3q0510Z(q)\ns = 0.01\ns = 0.1\ns = 0.20 1 2 3036 s = 0(a)\n-1 0 1\nδq-2-1012δZ(δq)\ns = 0\ns = 0.01\ns = 0.1(b)\nFIG. 4. (Color online) Spectrum of coupled spin and plasma-\nlike modes. (a) The function Z(q) = (!(q)=\nJ)2determined\nby Eq. (40) is shown for \fnite values of sand in the inset for\nthe cases= 0. (b) The dependence \u000eZ(\u000eq) that represents\nthe spectrum close to the crossing point is plotted according\nto Eq. (39). The following parameters are chosen: ZM=\n(\nM=\nJ)2= 4; lM=lJ= 0:1.\nwhere!2\nJ\u0011!2\nJ(k;!) = \n2\nJ\u0000\n1 +k2l2\nJ\u0001\n\u0000i\rR!; !M\u0011\n!M(k;!) = \nM\u0000\n1 +s+k2l2\nM\u0001\n\u0000i\rM!,b=s\fc=jc,a=\ns\nMl2\nJ=b, andk=jk?j.\nThe homogeneous equation (35) has a non-vanishing\nsolution provided the determinant of Mequals zero. Set-\nting det(M) equal to zero we obtain the dispersion rela-\ntion\n\u0002\n!2\u0000!2\nJ\u0003\u0002\n!2\u0000!2\nM\u0003\n=sv2\nJ\nM!Mk2: (37)\nFrom Eq. (37) we can conclude that the spin and\ncharge excitations decouple only in the limit when the\nright-hand side of this equation can be neglected. In\nthis case the spin waves with spectrum !M(k;!) and the\nplasmalike Josephson waves with spectrum !J(k;!) ex-\nist separately. In the general case, Eq. (37) describes the\nspectrum of coupled spin waves and plasma-like modes in\nthe system. The most interesting behavior corresponds\nto the case \n M>\nJ. In this situation, the two branches\nof the spectrum cross each other in the absence of the cou-\npling, while a \fnite coupling leads to mutual repulsion of\nthese branches.In order to show this explicitly, we consider the case\nwithout damping, \rR=\rM= 0, and assume that s\u001c1\nandlM\u001clJ, which means that we neglect the spatial\ndispersion of spin waves on the Josephson length (these\nconditions are usually ful\flled experimentally). It is con-\nvenient to write Eq. (37) in the dimensionless form\n\u0002\nZ\u00001\u0000q2\u0003\n[Z\u0000ZM] =sq2ZM; (38)\nwhereZ= (!=\nJ)2; ZM= (\nM=\nJ)2andq=lJk. One\ncan see that for s= 0 the two dispersion curves Z= 1+q2\nandZ=ZMcross each other at q2\n0=ZM\u00001. To \fnd\nthe form of the dispersion curve in the vicinity of the\ncrossing point q0, we represent Zandq, respectively, as\nZ=ZM+\u000eZandq=q0+\u000eq. Then, one can easily\nobtain from Eq. (38)\n\u000eZ=q0h\n\u000eq\u0006p\n\u000eq2+sZMi\n: (39)\nIn Fig. 4 we plot the spectrum of coupled spin and\nplasma-like modes and the function \u000eZ(\u000eq) close to the\ncrossing point. Here, we take into account a \fnite value\nof the parameter lM=lJso that Eq. (38) that determines\nthe function Z(q) takes the form\n\u0002\nZ\u00001\u0000q2\u0003h\nZ\u0000eZMi\n=sq2q\neZMZM;(40)\nwitheZM=ZM[1+s+q2(lM=lJ)2]2. The inset of Fig. 4(a)\nindicates that the two branches indeed cross each other\nfors= 0, whereas for s6= 0 we \fnd a \\repulsion\"of the\nspin and Josephson excitations. Figure 4(b) displays the\nfunction\u000eZ(\u000eq) that represents the behavior of the spec-\ntrum in the vicinity of the crossing point and distinctly\nemphasizes the mutual repulsion. Both the dispersion\ncurvesZ(q) and\u000eZ(\u000eq) given by Eqs. (39) and (40), re-\nspectively, are presented for several values of sand the\nparameters \n M=\nJ= 2; lM=lJ= 0:1.\nV. FERROMAGNETIC RESONANCE\nIn this section, we study the response of the sys-\ntem to an external oscillating magnetic \feld Hext(t) =\nH\u0017sin(\u0017t) with a small amplitude H\u0017\u001cM0and fre-\nquency\u0017. The applied \feld is supposed to be directed\nalong theyaxis, i.e., Hext(t) =Hext(t)ny. We as-\nsume again that the equilibrium magnetization M0is\noriented in the zdirection, M=M0nz. The external\nmagnetic \feld Hext(t) causes precessional motion of the\nmagnetization vector Mand a variation of the phase dif-\nference'in space and time. As before (see Sec. III),\nwe, respectively, represent magnetization and phase dif-\nference in the form M(x;t) =M0nz+my(x;t)nyand\n'(x;t) ='0+ (x;t). Here,'0is a constant determined\nby a bias current jb=jcsin('0) and (x;t); my(x;t) are10\n1 2 3\nω/ΩJ00.511.522.53Pns = 0\ns = 0.05\ns = 0.2(a)\n0 1 2 3 4 5\nω/ΩJ00.20.40.60.8Pns = 0\ns = 0.05\ns = 0.2(b)\n0 2 4 6\nω/ΩJ05101520Pns = 0,    γR/ΩJ = 0.1\ns = 0.2, γR/ΩJ = 0.1\ns = 0.2, γR/ΩJ = 3(c)\n0 2 4 6 8\nω/ΩJ00.511.522.53Pns = 0,    γR/ΩJ = 1, γM/ΩJ = 0.1\ns = 0.2, γR/ΩJ = 1, γM/ΩJ = 0.1\ns = 0.2, γR/ΩJ = 3, γM/ΩJ = 1(d)\nFIG. 5. (Color online) Frequency dependence of the normalized absorbed power Pnin the JJ as a function of the normalized\nfrequency!=\nJfor di\u000berent values of the parameter sand for di\u000berent damping coe\u000ecients \rR=\nJand\rM=\nJ. The \fgures (a)\nand (b) are shown for the parameters Ln= 3;\rM=\nJ= 0:1;(\nM=\nJ)2= 5 and, respectively, (a) \rR=\nJ= 0:1, (b)\rR=\nJ= 1.\nWith regard to \fgures (c) and (d) we have chosen Ln= 1;(\nM=\nJ)2= 5 and in (c) \rM=\nJ= 0:1.\nsmall perturbations due to the external ac magnetic \feld\nHext(t) =H\u0017Im [exp(i\u0017t)],j j \u001c j'0j;jmyj \u001c jM0j.\nDue to the coupling of Mand', we expect modi\fcations\nof the ferromagnetic resonance in the system appearing\nas additional features in absorption spectra.\nThus, to study ferromagnetic resonance, we need to\ncalculate the power P(per unit area) absorbed in the\nsystem. The absorbed power Pcan be found as the time-\naveraged di\u000berence between the energy \rux Sin;outcom-\ning in and out of the system. These \ruxes are expressed\nin terms of Poynting vectors S43\nP=Z\ndzdynx\u0001hSin\u0000Souti; (41)\nwhereSin;out= (c=4\u0019)[E\u0002H]x=\u0006Land the angular\nbrackets denote averaging with respect to time t.\nThe electric \feld E=nzEis directed along the z\u0000axis\nand is related to the time derivative of the phase di\u000ber-\nence via the Josephson relation\nE=\u0000(1=d)(~=2e)@ =@t: (42)\nTherefore, in order to \fnd the Poynting vector S, we have\nto calculate the function  (t) which is determined by anapplied weak ac magnetic \feld Hext(t). This vectorSdif-\nfers from zero only in the insulating layer of thickness d.\nThe magnetic \feld consists only of the applied ac \feld,\nH(x=\u0006L) =Hext(t)nyand, therefore, the Poynting\nvectors are directed parallel to the x\u0000axis. We repre-\nsent the phase di\u000berence in form of the Fourier trans-\nform (x;t) =R\nd!=(2\u0019) exp(i!t) (x;!);  (x;\u0000!) =\n \u0003(x;!).\nThe function  (x;!) obeys an equation that is derived\nin a way similar to the derivation of Eqs. (18){(22) and\nhas the form\n@2\n@x2n (xn;!)\u0000\u00142\n! (xn;!) = 0; (43)\nwhere we have introduced the dimensionless variable\nxn=x=lJand have set\n\u00142\n!=L!J\n1 +s=[(1 +s)L!F]\u0011L!J\na!(44)\nwithL!J\u0011cos('0)\u0000!(!\u0000i\rR)=\n2\nJ,a!= 1+s=[(1+\ns)L!F], andL!Fis de\fned in Eq. (17). Equation (44) is\nsupplemented by the boundary conditions11\n0 2 404812\n0 2 4 6 8 10\nω/ΩJ0102030405060Pns = 0\ns = 0.05\ns = 0.2(a)\nx10\n0 2 4 6 8 10\nω/ΩJ0123456Pns = 0\ns = 0.05\ns = 0.2(b)\nFIG. 6. (Color online) Frequency dependence of the normalized absorbed power Pnas a function of the normalized frequency\n!=\nJfor really short junctions. Here, too, the \fgures are displayed for di\u000berent values of the parameter sand for the choice\n\rR=\nJ= 1;\rM=\nJ= 0:1;(\nM=\nJ)2= 5. In \fgure (a) Ln= 0:2 whereas in \fgure (b) Ln= 0:6.\na!@\n@xn (x;!)\f\f\f\f\nx=\u0006L=\u0000Hext(!)\nH0andH0=\b0\n4\u0019e\u0015LlJ\n(45)\nthat can be obtained from Eqs. (9) and (10). As a con-\nsequence, the solution for Eq. (43) has the form\n (\u0006L;!) =\u0007Hext(!)Ln\nH0tanh(\u0012!)\na!\u0012!(46)\nwhere\u0012!=\u0014!Ln\u0011\u00120\n!+i\u001200\n!; Ln=L=lJ. Fourier trans-\nforming Eq. (46) back into the time representation we\nobtain\n (\u0006L;t) =\u0007H\u0017Ln\nH0Re\u0014ei\u0017t\na\u0017\u0012\u0017tanh(\u0012\u0017)\u0015\n: (47)\nTaking into account that all the quantities do not depend\nony, the absorbed power Pcan be represented as\nP= 2Ly~c\n4\u0019e\u001c@ (L;t)\n@tHext(t)\u001d\n; (48)\nwhereLyis the length of the junction along the ydirec-\ntion. Substituting Eq. (47) into Eq. (48) and relabeling\nthe external \feld frequency \u0017!!, we \fnally arrive at\nP=\b0H0\n(2\u0019)2\u0012H!\nH0\u00132LxLy\nlJ!Im\u0014tanh (\u0012!)\na!\u0012!\u0015\n;(49)\nwhereLx\u0011L. This formula di\u000bers drastically from a\nstandard formula for the absorbed power Pin ferromag-\nnetic \flms because it describes the power absorption not\nonly in the F \flm, but also in the Josephson junction. In\nparticular,P6= 0 even in the absence of the ferromag-\nnetic layer. In this case, Eq. (49) describes the power\nneeded to excite standing plasma waves.In Fig. 5, we plot the frequency depen-\ndence of the normalized absorbed power Pn=\n(!=\nJ)Im [tanh(\u0012!)=(a!\u0012!)] as a function of normalized\nfrequency!=\nJat di\u000berent sand normalized junction\nlengthL=lJ. Generally speaking, from Fig. 5(a), we see\nthat ats= 0 (no ferromagnetic layer), there are periodic\nresonances related to excitation of standing waves in\nthe Josephson junction (Josephson plasma resonances).\nInterestingly, in the presence of the ferromagnetic layers,\ns6= 0, additional peaks appear on the curves. These\npeaks are caused by the ferromagnetic resonance in\nthe F layer at frequencies !\u0019\nM. With increasing\nsthe in\ruence of the F layer becomes more and more\npronounced. One can see this from the fact that, for\ninstance, the spectrum close to !\u0019\nMappears to have\na more complicated structure and the peaks increase in\nheight.\nTo indicate the in\ruence of the (normalized) damp-\ning parameters \rRand\rM, we display in Fig. 5(b) the\nnormalized absorbed power Pnfor the case \rR\u001d\rM.\nThis shows that the periodic resonances in the junction\nare strongly suppressed and the absorption spectrum is\ndominated by the e\u000bect of the ferromagnetic layer. In\naddition to that, the normalized length of the junction\nLn=Lx=lJalso determines the absorption spectrum. In\nFigs. 5(c) and 5(d), Pnis shown for the case Ln= 1\nand here, too, for di\u000berent values of the parameters s;\rR;\nand\rM. We \fnd that the distance between periodic res-\nonances is larger for short junctions and compared to\nFig. 5(a), where Ln= 3, the in\ruence of the F layer\non the absorption spectrum is weaker. The blue curve\nin Fig. 5(c) reveals that the periodic resonances can be\nalmost completely suppressed by increasing the damping\nparameter \rR. Eventually, Fig. 5(d) indicates that in\nsystems where both damping parameters \rRand\rMare\nlarge and of the same order of magnitude, the e\u000bect of\nthe ferromagnetic layer becomes negligible.\nIn Fig. 6, we show the frequency dependence of the ab-\nsorbed power Pnfor short junctions of length Ln= 0:212\n[see Fig. 6(a)] and Ln= 0:6 [see Fig. 6(b)]. In Fig. 6(a),\nthe peak at !=\nJ\u00192:3 is related to the ferromagnetic\nresonance in the F \flm. In contrast to this, slightly longer\njunctions feature a much stronger in\ruence of the fer-\nromagnetic layer as becomes apparent from Fig. 6(b).\nMore importantly, we \fnd that the relative magnitudes\nof peaks due to the Josephson plasma resonances and the\nferromagnetic resonance are even in the case of \rR\u001d\rM\nof the same order of magnitude for short junctions con-\ntrary to Fig. 5(b), Ln= 3, where the Josephson plasma\nresonances are considerably smaller for the same choice\nof parameters.\nNote that our analysis is valid for not too high fre-\nquencies as it is assumed that the penetration depth is\nnot frequency dependent. This means that the inequal-\nity!\u001c\nJlJ=\u0015L=vJ=\u0015Lshould be ful\flled. For this\nreason, Figs. 6(a) and 6(b) indicate only a small number\nof Josephson plasma resonances.\nVI. DISCUSSION\nWe studied dynamic properties of Josephson junctions\nwith a magnetically active layer characterized by the\nmagnetic susceptibility \u001f(!;k). These junctions may be\nof the SFIFS or SIFS type with conducting or insulat-\ning ferromagnets. In the former case, we assumed that\nboth vectors M1andM2characterizing the stationary\norientation of magnetization in the F layers were aligned\nalong thezdirection, and our results are applicable only\nin this situation.\nWe calculated the form of the CVC for SFIFS junctions\nin the presence of a weak magnetic \feld and found a\nmodi\fcation of Fiske steps due to the presence of the\nferromagnetic layer. The position of these steps depends\non the relation between di\u000berent parameters, especially\nbetween\u0014HlJand \nM=\nJ.\nWe have also analyzed the spectrum of the collectivecoupled modes in long JJs with a ferromagnetic layer.\nIf the frequency of the ferromagnetic resonance \n Mis\nhigher than the characteristic Josephson frequency \n J,\nthen coupled magneto-plasma modes (spin waves and\nJosephson plasma-like modes) occur in the region of\ncrossing terms.\nThe analysis of the ferromagnetic resonance in the F\nlayer incorporated in JJs of the SFS or SFIFS types shows\nthat the peaks in the frequency dependence of the ab-\nsorbed power P(!) correspond both to the ferromagnetic\nresonance in the F \flm and to the Josephson plasma res-\nonances in the tunnel JJ.\nIt is not easy to compare our results with available\nexperimental data. The dynamic properties of ferromag-\nnetic layers play a crucial role in determining the form of\nthe CVC (Fiske steps). Meanwhile, little is known about\nthese properties in experiments. It would be useful to\nstudy experimentally magnetic resonance in the F lay-\ners at temperatures above the critical temperature of the\nsuperconducting transition Tc. The frequencies of the\nJosephson oscillations \n Jand magnetic resonance \n M\nshould not be very di\u000berent. In addition, we assumed\nthat the easy-axis magnetization is perpendicular to the\nSF interface. There are no data about magnetization\norientation in junctions studied experimentally.\nAs to magnetic resonance, we are only aware of\nRefs. 32, 55, and 56 where ferromagnetic resonance was\nmeasured on SF structures. However, the authors of\nRef. 32 measured the CVC of a SFS junction with a\nstrong damping, but not the absorbed power. In Ref. 55\nand 56, the absorbed power was measured, however not\nin SIFS junctions, but in SF bilayers. 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Lett. 100, 047002 (2008)."    },    {        "title": "1306.3652v2.Zero_Field_Fiske_Resonance_Coupled_with_Spin_waves_in_Ferromagnetic_Josephson_Junctions.pdf",        "content": "arXiv:1306.3652v2  [cond-mat.supr-con]  11 Jun 2014Typeset with jpsj2.cls <ver.1.2.2> Full Paper\nZero-Field Fiske Resonance Coupled with Spin-waves in Ferr omagnetic\nJosephson Junctions\nShin-ichi Hikino1,3, Michiyasu Mori2,3, and Sadamichi Maekawa2,3\n1Computational Condensed Matter Physics Laboratory, RIKEN , Wako, Saitama 351-0198, Japan\n2Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai, Ibaraki 319-1195, Japan\n3CREST, Japan Science and Technology Agency (JST), Kawaguch i, Saitama 332-0012, Japan\nAC Josephsoncurrentdensity in a Josephsonjunction with DC bias is spatially modulated\nby an external magnetic field, and induces an electromagnetic (EM) field inside the junction.\nThe current-voltage ( I-V) curve exhibits peaks due to the resonance between the EM field\nand the spatially modulated AC Josephson current density. This is ca lledFiske resonance .\nSuch a spatially modulated Josephson current density can be also ind uced by a non-uniform\ninsulating barrier and the Fiske resonance appears without extern al magnetic field. This is\ncalled zero-field Fiske resonance (ZFFR). In this paper, we theore tically study the ZFFR\ncoupled with spin-waves in a superconductor/ferromagnetic insula tor/superconductor junc-\ntion (ferromagneticJosephsonjunction) with anon-uniformferr omagneticinsulatingbarrier.\nThe resonant mode coupled with spin-waves can be induced without e xternal magnetic field.\nWe find that the I-Vcurve shows resonant peaks associated with composite excitation s of\nspin-waves and the EM field in the junction. The voltage at the reson ance is obtained as a\nfunction of the normal modes of EM field. The ZFFRs coupled with spin -waves are found as\npeak structures in the DC Josephson current density as a functio n of bias voltage.\n1. Introduction\nTheDCJosephsoneffectischaracterized bytheDCcurrentflowi ngwithoutavoltage-drop\nbetween two superconductors separated by a thin insulating barrier.1When a DC voltage Vis\nappliedtothejunction,theACJosephsoncurrentwithfrequ ency(2e//planckover2pi1)Vflowsinthejunction\ndriven by the difference of phases in two superconducting orde r parameters, i.e., Josephson-\nphaseθ. If both the DC voltage and a magnetic field are applied to the j unction, whose width\nLis smaller than the Josephson penetration depth λJ, the AC Josephson current density is\nspatially modulated and generates the electromagnetic (EM ) field inside the junction. In this\ncase, the current-voltage ( I-V) curve exhibits peaks due to the resonance between the AC\nJosephson current density and the EM field. This is called Fiske resonance .2–6\nJosephson junctions composed of ferromagnetic metal (FM) a nd superconductors (Ss) are\nextensively studied for the last decade. The S/FM/S junctio ns exhibit fascinating phenomena\nwhich are not observed in the conventional Josephson juncti ons.7–11The interaction between\nCooper pairs and spin waves in the FM is of importance in the tr ansport properties in the\n1/10J. Phys. Soc. Jpn. Full Paper\nS/FM/S and S/I/FM/S junctions. The dynamics of θcoupled with spin-waves in the FM\nhas been investigated theoretically12–20and experimentally.21However, the Fiske resonance\ncoupled with spin-waves is not yet observed experimentally .\nAnother type of Josephson junction with ferromagnetic insu lator (FI) instead of the FM\nis also examined. It is reported that the dissipation effect in the S/FI/S junction is smaller\nthan that the S/FM/S junction.22–24The damping of spin-waves is also very small in the FI\ncompared to the case of the FM.25–27Therefore, the coupling between θand spin-waves can\nbeobserved more clearly in the S/FI/S junction. In fact, in t he S/FI/S junction, it is expected\nthat the Fiske resonance has clear multiple structures asso ciated with spin-wave excitation.28\nHere, we note that the Fiske resonance in the conventional Jo sephson junction is also\ninduced by the non-uniform insulating barrier in the juncti on, since AC Josephson current\ndensity driven by a DC voltage is spatially modulated and the n the EM field is generated\ninside the junction. In this case, the Fiske resonance occur s without external magnetic field.\nItiscalled zero-field Fiskeresonance(ZFFR), whichorigin ates fromtheresonancebetweenthe\nEM field and the spatially modulated AC Josephson current den sity due to the non-uniform\ninsulating barrier. This phenomenon has been widely studie d experimentally and theoretically\nin the Josephson junction.6,29–32\nIn this paper, we theoretically study the ZFFR coupled with s pin-waves in an S/FI/S\njunction with a non-uniform FI. The merit of such a non-unifo rm geometry of junction is\nthat the spatially modulated AC Josephson current density c an be induced with no external\nmagnetic field and thus the Fiske resonance occurs without ex ternal magnetic field. By solving\nthe equation of motion of θcoupled with spin-waves, it will be found that the I-Vcurve shows\nresonant peaks. Thevoltage at the resonances is obtained as a function of the normal modes of\nEM field, which indicates composite excitations of the EM fiel d and spin-waves in the S/FI/S\njunction. Dependence of those resonances on distributions of the Josephson critical current\ndensity is presented.\nThe rest of this paper is organized as follows. In Sec. II, we f ormulate the Josephson\ncurrentinaJosephsonjunctionwithanon-uniformferromag neticinsulator.InSec.III,theDC\ncomponentofJosephsoncurrentdensityattheZFFRwithspin -wavesiscalculatedanalytically\nand numerically. Summary is given in Sec. IV.\n2. FORMULATION of JOSEPHSON CURRENT in FERROMAGNETIC\nJOSEPHSON JUNCTION with MAGNETIC INSULATOR\nThe system considered is a Josephson junction with a FI sandw iched by two superconduc-\ntors with s-wave symmetry as shown in Fig. 1. The geometry of the junctio n is assumed to\nbe a non-uniform junction to impose non-uniform Josephson c urrent density without external\nmagnetic field.33The magnetization in the FI is parallel to the z-direction. Here, we assume\n2/10J. Phys. Soc. Jpn. Full Paper\nS\ny\nxz//M\nSLFIM\nFig. 1. (Color online) Schematic of a superconductor/ferromagne tic insulator/superconductor\n(S/FI/S) junction with the magnetization Min the FI. The non-uniform geometry of the junction\nwith the width Lis schematically illustrated.\na simple model of non-uniform Josephson current density giv en by,\nJ(y,t)=Jc(y)sin[ωJt+θ(y,t)], (1)\nJc(y)=J0.P(y)/bracketleftbigg\n(1−ζ)cosh[κ(1−2y/L)]\ncosh(κ)\n+ζsinh[κ(1−2y/L)]\nsinh(κ)/bracketrightbigg\n, (2)\nP(y)=/braceleftBigg\n1 for 0≤y≤L,\n0 fory <0 orL < y,(3)\nwhereJc(y) andωJ= (2e//planckover2pi1)Vare the Josephson critical current density and Josephson\nfrequency with bias voltage V, respectively.34J0is the Josephson critical current density\nfor the uniform geometry of junction, i.e., ζ=κ= 0. The electromagnetic dynamics induces\nθ(y,t)dependingonspaceandtime. Thedistributionof Jc(y)isdeterminedbytwoparameters\nκandζ, where 0 ≤ζ≤1 is imposed. In the S/FI/S junction, spin-waves can be excit ed by\nthe EM field inside the FI due to the AC Josephson current. In th is situation, the equation\nof motion for θ(y,t) coupled with spin waves is described by,28\n∂2θ(y,t)\n∂y2=1\nc2/bracketleftbigg∂2θ(y,t)\n∂t2+1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)∂2θ(y′,t′)\n∂t′2\n+ Γ∂θ(y,t)\n∂t+Γ1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)∂θ(y′,t′)\n∂t′/bracketrightbigg\n+1\nλ2\nJ/angbracketleftJc(y)/angbracketrightJ(y,t)+1\nλ2\nJ/angbracketleftJc(y)/angbracketright1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)J(y′,t′),(4)\n/angbracketleftJc(y)/angbracketright=1\nL/integraldisplayL\n0dyJc(y). (5)\n3/10J. Phys. Soc. Jpn. Full Paper\nThe effective velocity of light in the FI cis given by c=/radicalbig\nd/[(d+2λL)ǫµ0] , Josephson\npenetration depth λJ=/radicalbig\n/planckover2pi1/[2eµ0(d+2λL)J0], dielectric constant ǫand permeability µ0.\nThe London penetration depth is denoted by λLand Γ≡(ǫR)−1means the damping factor\ncaused by quasi-particle resistivity Rin the FI. The magnetic susceptibility of the FI in the\nlinearized Landau-Lifshitz-Gilbert equation is given by,25\nχ(q,ωJ) =γMzΩS+iαωJ\nΩ2\nS−(1+α2)ω2\nJ+i2αΩSωJ, (6)\nwhereMz,α, andγare thez-component of the magnetization, Gilbert damping factor, a nd\nthe gyromagnetic ratio, respectively. Magnetic susceptib ility and spin-wave energy /planckover2pi1ΩSin\na magnetic material are generally modified by geometry and th ickness. On the other hand,\nEq. (6) is obtained by assuming a uniform FI. This is justified , because the magnetic sus-\nceptibility and /planckover2pi1ΩSare insensitive to the thickness of FI, provided that the con formation of\nthe ferromagnetic materials changes on a scale of nanometer s.35Therefore, we adopt Eq. (6)\nand/planckover2pi1ΩSobtained in the uniform FI36for the non-uniform FI as an approximation, since\nwe consider the thickness change of FI to be in a range of a few n anometers. In the FI, the\ndispersion relation of spin-waves with the frequency Ω Sis given by\nΩS=ΩB+η\n/planckover2pi1q2, (7)\nwhere Ω B=γ(HK−Mz/µ0). The anisotropic field and the stiffness of spin-waves in the F I are\ndenoted by HKandη, respectively. The spin-wave having a finite wave number qis neglected\nin the Fiske resonance because of the following reason: In Eq . (7), the first term Ω Bis caused\nby the anisotropic and demagnetizing fields, and the wave num berqis given by nπ/L. In a\nconventional FI, /planckover2pi1ΩBis about tens of µeV.25On the other hand, ηq2is of the order peV due\nto the small stiffness of spin-waves37whenLis a few mm. Below, we only consider q= 0\nmode for spin-waves with the constant frequency Ω B.\n3. DC Josephson current density with ZFFR and numerical results\nIn order to obtain the solution of Eq (4), we expand θ(y,t) in terms of the normal modes\nof the EM field generated by the AC Josephson current as follow s,\nθ(y,t) = Im/bracketleftBigg∞/summationdisplay\nn=0gneiωJtcos(kny)/bracketrightBigg\n, (8)\nwheregnis a complex number and kn=nπ/L. This equation of θ(y,t) satisfies [ ∂θ/∂y]y=0=\n[∂θ/∂y]y=L= 0, which is Kulik’s boundary condition.5,6We consider θ(y,t) to be a small\nperturbation and solve Eq. (8) by taking J(y,t) to beJc(y)sin(ωJt). Substituting Eq. (8) into\nEq. (4),gnis determined as,\ngn=−c2J0\nλ2\nJ/angbracketleftJc(y)/angbracketright\n4/10J. Phys. Soc. Jpn. Full Paper\n×1+χ(−ωJ)/µ0\nω2\nJ[1+χ(−ωJ)/µ0]−ω2n+iΓωJ[1+χ(−ωJ)/µ0]\n×[(1−ζ)Bn+ζCn], (9)\nBn=2cos(nπ/2)\ncosh(κ)/integraldisplay1\n0dycos(nπy/2)cosh(κy),\nCn=2sin(nπ/2)\nsinh(κ)/integraldisplay1\n0dysin(nπy/2)sinh(κy),\nwhereωn= (cπ/L)n.\nNext, we calculate the DC Josephson current density JDCcoupled with spin waves as a\nfunction of V. The function, sin( ωJt+θ(y,t)), is expanded in terms of θ(y,t) andJDCis given\nby\nJDC≈lim\nT→∞1\nT/integraldisplayT\n0dt1\nL/integraldisplayL\n0dyJc(y)cos(ωJt)θ(y,t). (10)\nIntroducing Eqs. (8) and (9) into Eq. (10), the analytic form ula ofJDCwithout external\nmagnetic field is obtained as,\nJDC≈∞/summationdisplay\nn=0c2κJ0\nλ2\nJ(1−ζ)tanh(κ)Ψn(ωJ)/bracketleftBigg\n(1−ζ)κcos2/parenleftbignπ\n2/parenrightbig\ntanh(κ)\nκ2+(nπ/2)2+ζκsin2/parenleftbignπ\n2/parenrightbig\ntanh−1(κ)\nκ2+(nπ/2)2/bracketrightBigg2\n,(11)\nΨn(ωJ)≡ΓωJ[1+2χ1(ωJ)/µ0]+ω2\nnχ2(ωJ)/µ0+ΓωJ[χ2\n1(ωJ)+χ2\n2(ωJ)]/µ2\n0/bracketleftbig\nω2\nJ[1+χ1(ωJ)/µ0]−[ω2n+ΓωJχ2(ωJ)/µ0]/bracketrightbig2+/bracketleftbig\nΓωJ[1+χ1(ωJ)/µ0]+ω2\nJχ2(ωJ)/µ0/bracketrightbig2,(12)\nwhereχ1(ωJ) = Re[χ(ωJ)],χ2(ωJ) = Im[χ(ωJ)]. Equation (11) clearly demonstrates that zero-\nfield resonant modes depend on parameters κandζwhich determine the distribution of the\nJosephsoncritical current density flowing through theFI. H ence, onecan easily findthat three\ncases are possible for the zero-field resonance. When ζ= 0 (ζ= 1), the zero-field resonance\nonly appears at even (odd) numbers of n. On the other hand, when ζ/negationslash= 0,1, the zero-field\nresonance appears at all integers n.\nNext, we derive a condition for the ZFFR in the present system by analyzing Eq. (12).\nWhen the denominator of Ψ n(ωJ) is minimum with respect to ωJ, Ψn(ωJ) takes a maximum,\nso that the DC Josephson current exhibits the resonant behav ior. The DC voltage, at which\nthe resonance occurs, is determined by neglecting the dampi ng term of Eq. (12) as α= Γ = 0.\nSetting the denominator of Ψ n(ωJ) to be zero, the voltage is given by\nV±=/planckover2pi1\n2e/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nω2n+Ω2\nS+γMzΩS\nµ0±/radicalBigg/parenleftbigg\nω2n+Ω2\nS+γMzΩS\nµ0/parenrightbigg2\n−4ω2nΩ2\nS\n.(13)\nWe have two DC voltages, V+andV−, at which the ZFFR occurs for each n. The integer nis\ndetermined by the mode of the EM field in the junction. Eq. (13) clearly shows that there are\ntwo dispersion relations, which result from the coupling be tween the EM field and spin-waves\nin the FI. Note that the amplitude of ZFFR strongly depends on κandζas we will see in the\n5/10J. Phys. Soc. Jpn. Full PaperJc(y)/Jc\ny/L n=1 \n()L/ /2 V e ωℏ00.2 0.4 0.6 0.8 100.2 0.4 0.6 0.8 1\n1 2 3 400.02 0.04 0.06 0.08 0.1 \nFig. 2. (Color online) DC Josephson current density ( JDC) as a function of DC voltage ( V) in the\nS/FI/S junction. The black solid line is the total DC Josephson curre nt density. The dashed line\n(Red) is the DC Josephson current density in n= 1, where nis the mode number of EM field (see\nEq. (11)). The inset is the distribution of Josephson critical curre nt density as a function of y.\nnext section.\nAt last, we numerically evaluate Eq. (11). Parameters are se t to beMz= 0.1 T,α=\n1×10−4,27ΩB/ωL= 3, Γ/ωL= 0.5,38γ= 2.2×105m/A·s,39andωL≡cπ/L= 30 GHz.\nInstead of plotting an I-Vcurve,JDCat the resonances will be shown as a function of the\nvoltageVbelow. The amplitude of JDCis associated with a height of resonant peak or a\njump in the I-Vcurve (for instance, see Ref6). In the following numerical calculations, we\nexclude the contribution of n= 0 in Eq.(11), since we discuss about the resonance between\nthe spatially modulated AC Josephson current and standing w ave of EM field.\nFigure 2 shows JDCinduced by ZFFR as a function of V40forκ= 0 and ζ= 0.4. With\ntheseparameters, Jc(y)islinearlydistributedinthejunction(seetheinsetofFi g.2).Theblack\n(solid) and red (dashed) lines are JDCand the component with n= 1 in Eq. (11), respectively.\nThis result clearly demonstrates that the Fiske resonance o ccurs without external magnetic\nfield, i.e., ZFFR. The additional resonance peak around V/(ωL/planckover2pi1/2e)≈3.3 arises from the\npresence of spin-wave excitation in the FI. This resonance c omes from the inhomogeneity of\nJosephsoncritical currentdensityinducedbythenon-unif ormgeometry ofjunction.Moreover,\nin the present case, Eq. (11) becomes\nJDC=c2J0\nλ2\nJ(1−ζ)∞/summationdisplay\nn=0Ψn(ωJ)ζ2/bracketleftbiggsin(nπ/2)\n(nπ/2)/bracketrightbigg4\n. (14)\nIn Eq. (14), it is found that the ZFFR only occurs at odd number ofn. Since resonant peaks\nof ZFFR with n >1 are much smaller than that with n= 1, main contribution to the ZFFR\nas depicted in Fig. 2 is the mode of n= 1.\n6/10J. Phys. Soc. Jpn. Full Paper\n1 2 3 400.02 0.04 0.06 \nJc(y)/Jc\nn=1 \nn=2 \nn=3 y/L \n()L/ /2 V e ωℏ00.2 0.4 0.6 0.8 100.2 0.4 0.6 0.8 1\nFig. 3. (Color online) DC Josephson current density ( JDC) as a function of DC voltage ( V) in the\nS/FI/Sjunction. The blacksolid line is JDC. Red (solid), blue (dashed), and green(chain) lines are\nthe DC Josephson current densities in each n, wherenis mode number of EM field (see Eq. (11)).\nInset is the distribution of Josephson critical current density as a function of y.\nFigure 3 is the case for κ= 2 and ζ= 0.4. The black (solid) line is JDC. Red (solid),\nblue (dashed), and green (chain) lines are each component wi thnin Eq. (11). It is found\nthat ZFFR peaks of JDCclearly appear at n≥1 in Fig. 3 in contrast to Fig. 2. The reason\nis simply due to the non-linear Josephson critical current d ensity to contain both symmetric\nand antisymmetric components with respect to y. The present non-linear distribution of the\nJosephsoncritical currentdensitywillbemorerealistic. Therefore,wecanexpectthatmultiple\nresonant peaks such as Fig. 3 is practically observed withou t external magnetic field.\n4. Summary and Discussion\nWehavetheoretically studiedthezero-fieldFiskeresonanc e(ZFFR)intheS/FI/Sjunction\nwith several patterns of spatial variation in the Josephson critical current density, which is\ninduced by a non-uniform ferromagnetic insulating barrier . Such a non-uniform AC Josephson\ncurrent density can excite the EM field inside the FI without e xternal magnetic field. It\nis found that the current-voltage ( I-V) curve shows two resonant peaks without external\nmagnetic field in the present system, i.e., the ZFFR coupled w ith spin-waves occurs. Voltage\nat the resonances is obtained as a function of the normal mode s of EM field, which indicates\ncomposite excitations of the EM field and spin-waves in the S/ FI/S junction.\nThe present study will provide a platform to study the dynami cs of Josephson phase and\nthe magnetic excitation. Furthermore, in the non-uniform S /FI/S junction, several applica-\ntions such as spin-current emitter by utilizing spin-wave e xcitation in the FI41may be also\npossible in analogy with the emission of coherent THz radiat ion in the high-T ccuprate,42–44\n7/10J. Phys. Soc. Jpn. Full Paper\nalthough Josephson junctions based on the high-T ccuprate are usually laminated structures\ndifferently from the single Josephson junction discussed her e. In fact, the inhomogeneity of\nthe junction was one of essential factors to realize the emis sion without external magnetic\nfield.42–44However, novel devices using the S/FI/S junction are beyond the scope of the\npresent paper and will be studied elsewhere.\nAcknowledgements\nThis work is supported by Grant-in-Aid for Research Activit y Start-up (No. 25887053)\nfrom the Japan Society for the Promotion of Science and Grant -in-Aid for Scientific Research\nfrom MEXT (Grant No.24540387, No.24360036, No.23340093, a nd No.25287094), Center for\nComputational Science and e-Systems of JAEA, and the inter- university cooperative research\nprogram of IMR, Tohoku University.\n8/10J. Phys. Soc. Jpn. Full Paper\nReferences\n1) B. D. Josephson, Phys. Lett. 1(1962) 251.\n2) M. D. Fiske, Rev. Mod. Phys. 36(1964) 221.\n3) R. E. Eck, D. J. Scalapino, and B. N. Taylor, Phys. Rev. Lett. 13(1964) 15.\n4) D. D. Coon and M.D Fiske, Phys. Rev. 138(1965) A744.\n5) I. O. Kulik, JETP Lett. 2(1965) 84.\n6) A.Barone and G.Patern´ o, Physics and Applications of the Josephson Effect (Wiley, New York,\n1982).\n7) A. A. Golubov, M. Yu. Kupriyanov, and E. Ilfichev, Rev. Mod. Phy s.76(2004) 411.\n8) A. I. Buzdin, Rev. Mod. Phys. 77(2005) 935.\n9) V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V. Vereten nikov, A. A. Golubov, and J. Aarts,\nPhys. Rev. Lett. 86(2001) 2427.\n10) T. Kontos, M. Aprili, J. Lesueur, F. Genet, B. Stephanidis, and R. Boursier, Phys. Rev. Lett. 89\n(2002) 137007.\n11) J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, and M. G. Blamire, P hys. Rev. Lett. 97(2006)\n177003; J.W.A.Robinson, S.Piano, G.Burnell, C.Bell, and M.G.Blam ire, Phys.Rev.B 76(2007)\n094522.\n12) Z. Nussinov, A. Shnirman, D. P. Arovas, A. V. Balatsky, and J. X. Zhu, Phys. Rev. B 71(2005)\n214520.\n13) S.Takahashi, S.Hikino, M.Mori, J.Martinek and S.Maekawa, Ph ys.Rev.Lett. 99(2007) 057003.\n14) C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Phys. Rev. Lett .100(2008) 047002.\n15) M. Houzet, Phys. Rev. Lett. 101(2008) 057009.\n16) S. Hikino, M. Mori, S. Takahashi, and S. Maekawa, J. Phys. Soc. Jpn.77(2008) 053707.\n17) F. Konschelle and A. Buzdin, Phys. Rev. Lett. 102(2009) 017001.\n18) T. Yokoyama and Y. Tserkovnyak, Phys. Rev. B 80(2009) 104416.\n19) A. F. Volkov and K. B. Efetov, Phys. Rev. Lett. 103(2009) 037003.\n20) S. Mai, E. Kandelaki, A. F. Volkov, and K. B. Efetov, Phys. Rev. B84(2011) 144519.\n21) I. Petkovi´ c, M. Aprili, S. E. Barnes, F. Beuneu, and S. Maeka wa, Phys. Rev. B 80(2009) 220502.\n22) S. Kawabata, S. Kashiwaya, Y. Asano, Y. Tanaka, and A. Golub ov: Phys. Rev. B 74(2006)\n180502(R).\n23) S. Kawabata, Y. Asano, Y. Tanaka, A. A. Golubov, and S. Kash iwaya Phys.Rev. Lett. 104(2010)\n117002.\n24) Both the S/FM/S and the S/FI/S junctions are called ferromag netic Josephson junction, that\nmeans the latter one in this paper.\n25) B. Hillebrands and K. Ounadjela, Spin dynamics in confined magnetic structures II (Springer-\nVerlag Berlin Heidelberg, New York, 2003)\n26) D.D.Stancil andA.Prabhakar, Spin Waves Theory and Applications (SpringerScience+Business\nMedia, LLC 2009)\n27) Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizugu chi, H. Umezawa, H. Kawai,\nK. Ando, K. Takanashi, S. Maekawa, and E. Saitoh Nature 464(2010) 262.\n28) S. Hikino, M. Mori, S. Takahashi, and S. Maekawa, J. Phys. Soc. Jpn.80(2011) 074707.\n29) M. Russo and R. Vaglio, Phys. Rev. B 17(1978) 2171.\n9/10J. Phys. Soc. Jpn. Full Paper\n30) T. C. Wang, J. Appl. Phys. 50(1979) 2859.\n31) C. Camerlingo, M. Russo, and R. Vaglio, J. Appl. Phys. 53(1982) 7609.\n32) C. Nappi, E. Sarnelli, M. Adamo, and M. A. Navacerrada, Phys. R ev. B74(2006) 144504.\n33) The non-uniform geometry of the junction is the essential poin t, since such a geometry introduces\nthe non-uniform current distribution. In this paper, the Josephs on critical current density have a\nlinear current distribution with yin this geometry when κ= 0 and ζ/negationslash= 0.\n34) Note that we use the voltage bias model for simplicity, since we on ly focus on the DC component\nof Josephson current.6\n35) For instance, Y. Sun, Y. Song, H. Chang, M. Kabatek, M. Jant z, W. Schneider, M. Wu, H.\nSchultheiss, and A. Hoffmann, Appl. Phys. Lett. 101(2012) 152405.\n36) The distribution of Josephson current density with y(see Eq. (2)) can be obtained by changingthe\nthickness of insulating barrier along ydirection, since the Josephson current density exponentially\ndecreases with increasing the thickness of insulating barrier. For in stance, see Ref.6\n37) M. Pajda, J. Kudrnovsk´ y, I. Turek, V. Drchal, and P. Bruno , Phys. Rev. B 64(2001) 174402.\n38) M.P.Lisitskiy and M.V.Fistul, Phys.Rev.B 81(2010) 184505.We adopted the value of damping\nΓ/ωLin a SIS junction as reported in this paper.\n39) S. Chikazumi, Physics of Magnetism (Oxford University Press, New York, 1997), p.559-560.\n40) ForL= 1 mm and c= 107m/s, the magnitude of DC voltage ( V) is about tens of µV.\n41) S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura, Spin Current (Oxford University Press,\nOxford, 2012).\n42) L.Ozyuzer,A.E.Koshelev,C.Kurter,N.Gopalsami,Q.Li,M.T achiki,K.Kadowaki,T.Yamamoto,\nH. Minami, H. Yamaguchi, T. Tachiki, K. E. Gray, W.-K. Kwok, U. Welp, S cience318(2007) 1291.\n43) K. Kadowaki, H. Yamaguchi, K. Kawamata, T. Yamamoto, H. Mina mi, I. Kakeya, U. Welp, L.\nOzyuzer, A. Koshelev, C. Kurter, K.E. Gray, W.-K. Kwok, Physica C 468(2008) 634.\n44) A. E. Koshelev and L.N. Bulaevskii, Phys. Rev. B 77(2008) 014530.\n10/10"    },    {        "title": "0906.3135v1.The_Kondo_effect_in_ferromagnetic_atomic_contacts.pdf",        "content": "arXiv:0906.3135v1  [cond-mat.str-el]  17 Jun 2009The Kondo Effect in Ferromagnetic Atomic Contacts\nM. Reyes Calvo,1Joaqu´ ın Fern´ andez-Rossier,1Juan Jos´ e\nPalacios,1David Jacob,2Douglas Natelson,3and Carlos Untiedt1,∗\n1Departamento de Fisica Aplicada, Facultad de Ciencias,\nUniversidad de Alicante, San Vicente del Raspeig, E-03790 A licante, Spain\n2Department of Physics and Astronomy, Rutgers University, P iscataway, New Jersey 08854, USA.\n3Department of Physics and Astronomy, Rice University, Hous ton, Texas 77005, USA.\nIron, cobalt and nickel are archetypal ferromagnetic metal s. In bulk, electronic conduction in\nthese materials takes place mainly through the sandpelectrons, whereas the magnetic moments\nare mostly in the narrow d-electron bands, where they tend to align. This general pict ure may\nchange at the nanoscale because electrons at the surfaces of materials experience interactions that\ndiffer from those in the bulk. Here we show direct evidence for such changes: electronic transport\nin atomic-scale contacts of pure ferromagnets (iron, cobal t and nickel), despite their strong bulk\nferromagnetism, unexpectedly reveal Kondo physics, that i s, the screening of local magnetic mo-\nments by the conduction electrons below a characteristic te mperature1. The Kondo effect creates\na sharp resonance at the Fermi energy, affecting the electric al properties of the system;this ap-\npears as a Fano-Kondo resonance2in the conductance characteristics as observed in other art ificial\nnanostructures3,4,5,6,7,8,9,10,11. The study of hundreds of contacts shows material-dependen t lognor-\nmal distributions of theresonance widththat arise natural ly from Kondotheory12. These resonances\nbroaden and disappear with increasing temperature, also as in standard Kondo systems4,5,6,7. Our\nobservations, supported by calculations, imply that coord ination changes can significantly modify\nmagnetism at the nanoscale. Therefore, in addition to stand ard micromagnetic physics, strong elec-\ntronic correlations along with atomic-scale geometry need to be considered when investigating the\nmagnetic properties of magnetic nanostructures.\nAtomic-scale contacts can be fabricated by techniques\nsuch as scanning tunnelling microscopy13or the use of\nelectromigrated break junctions (EBJs)14, where the size\nof a macroscopic contact between two leads is reduced\nuntil they are in contact through only a few atoms and,\neventually, through only one. The conductance of metal-\nlicmonatomiccontactsisknowntobearound2 G0, where\nG0=e2/histhespin-resolvedquantumofconductance13\n(ebeing the elementary charge and hPlanck′s constant).\nTo identify the atomic contacts, histograms are con-\nstructed from the evolution of the conductance recorded\nduring the breaking of different contacts (Fig. 1a, b).\nThe position of the first peak of these histograms is iden-\ntified as the conductance of the monatomic contact. For\niron, cobalt and nickel, the conductance is larger than\n2G0owing to the contribution of the spanddorbitals to\nthe transmission15,16,17.\nWe have studied the low-temperature conductance\ncharacteristics of hundreds of atomic-scale contacts of\nthe three transition-metal ferromagnets iron, cobalt and\nnickel using a home-built STM. More than the 80%\nof the differential conductance (d I/dV) curves at the\nmonatomic contact show peaks or dips around zero bias\nsuch as those shown in Fig. 1c, which are very similar to\nthose observed in STM spectroscopy of single magnetic\nadatoms on non-magnetic surfaces9,10,11. Thus, as in the\ncase of these Kondo systems, we can also fit our d I/dV\ncurvestothe sumofaflat component, g0, andaFano-like\nresonance that typically amounts for 10% of the signal:\ndI\ndV=g0+A\n1+q2(q+ǫ)2\n1+ǫ2(1)Hereǫ= (eV−ǫs)/kBTKis the bias shifted with re-\nspect to the centre of the resonance, ǫs, and normalized\nby the natural width of the resonance, kBTK;TKis the\nKondo temperature; qis the dimensionless Fano param-\neter that determines de simmetry of the curve; Ais the\namplitude of the feature; and kBis Boltzmann’s con-\nstant.\nIt is clear from the d I/dVcharacteristics in Fig. 1c\nthat the width of the Fano feature is different for each\nmaterial. This is confirmed by statistical analysis of the\ndata. Figure 2 shows histograms of TKobtained from\nfitting our conductance curves for hundreds of different\niron, cobalt and nickel monatomic contacts. Notably,\nthese histograms follow a log-normal distribution; that\nis, the natural logarithm of TKis normally distributed.\nBecause many different atomic configurations result in\nmonatomic contacts, their electronic properties, such as\nconductance (Fig. 1b), density of states and the asso-\nciated energy scales, are expected to be normally dis-\ntributed. Instead, a normal distribution of log(TK) can\nonly be understood if TKcan be expressed as the expo-\nnential of normally distributed quantities. The Kondo\nmodel12naturally relates TKto the exponential of the\ntypical energy scales in the problem.\nFitting the histograms to a log-normal distribution\nyields most frequent values for the resonance widths in\nthe different materialscorresponding to TK=90 K, 120 K\nand280Kforiron, cobaltandnickel, respectively, follow-\ning the same trend ( TFe\nK< TCo\nK< TNi\nK) for these chemi-\ncal species when deposited as adatoms on non-magnetic\nsurfaces18. In simple terms, the Kondo temperature de-\ncreases as the size of the screened magnetic moment in-2\nFIG. 1: Conductance of a monatomic contact. a, Example of a trace where we record the conductance while str etching a nickel\nwire using a scanning tunnelling microscope (STM) at 4.2 K. I nset, model of a monatomic contact. b, Conductance histograms\nconstructed for iron, cobalt and nickel from thousands of su ch traces. The position of the first peak of in each histogram\ncorresponds to the conductance of the monatomic contact. c, Differential conductance curves recorded at the monatomic\ncontact as a function of the applied voltage. A characterist ic resonance appears at small bias that fits the Fano line shap e. All\npossible symmetries are found in the spectroscopy of iron, c obalt and nickel contacts, and the width of the resonance is t he\nmain difference between the spectra of the three materials. T his width is proportional to the Kondo temperature.\nFIG. 2: Histograms of inferred Kondo temperatures for iron,\ncobalt and nickel. The histograms are constructed from more\nthan 200 fittings andnormalized to thetotal numberof curves\nfitted. The continuous lines show the fits of the data to log-\nnormal distributions of TKwith a different most probable\nvalue for each material.\ncreases, as we change from nickel to cobalt to iron. The\nKondo temperature of cobalt nanocontacts is very sim-\nilar to the one reported in ref.11for cobalt in Cu(100)\nprobed using a STM in the contact regime.\nIn addition to the statistical analysis described above,\nwemeasuredthe temperatureevolutionoftheKondofea-\ntures for a single contact. To preserve the atomic stabil-\nity of the junction while changing the temperature19, we\nused an EBJ such as the one shown in the inset of Fig.\n3b. It is important to note that in this alternative imple-\nmentation we observe exactly the same Fano resonances,\nwith the same distribution of Kondo temperatures, as weobtained in the STM experiments using nickel and cobalt\nsamples. In all the cases, we see a reduction of the ampli-\ntude of the Fano features as the temperature increases,\nas shown in Fig. 3a for the case of cobalt. In Fig. 3b\nwe plotA(T), as defined in equation (1), as a function\nofTon a logarithmic scale. The curve so obtained has a\nlow-temperature plateau followed by a linear decay, very\nsimilar to that of quantum dots and molecules in the\nKondo regime4,5,6,20.\nIn summary, our atomic contacts fabricated with two\ndifferent methods show the same d I/dVcurves(Fig. 1c),\nthe same chemical trends (Fig. 2) and the same tempera-\nture evolution (Fig. 3a, b) as other standard, chemically\ninhomogeneous Kondo systems. This indicates that the\ncontactatom(s)innanocontactsofiron,cobaltandnickel\nare in the Kondo regime. This is unexpected for two\nreasons. First, the Kondo effect has always been associ-\nated with chemically inhomogeneous systems containing\natleasttwokinds ofatom: thosewherethe localizedlevel\nresides and those providing the itinerant electrons. Here\nthe same chemical species hosts both the itinerant states\nand the local magnetic moments. Second, in most re-\nports on the Kondo effect, the itinerant electrons are not\nspin-polarized, as a large spin polarization is expected\nto destroy the effect. or a carbon nanotube21contacted\nwith ferromagnetic leads has been justified by the op-\npositely directed spin polarizations in the electrodes, in\nagreement with refs.22,23,24. Although the situation re-\ngarding the magnetization of the leads might be similar\nin our system, we argue that the Kondo effect is still\npossible even if there is no domain wall pinned in the\ncontact.\nBeing in the Kondo regime implies that the atoms at\nthecontactmusthost,atleast,alocalized d-electronlevel\nwhose magnetic moment is screenedas a result ofantifer-3\nFIG.3: EvolutionoftheFanoresonances withincreasing tem -\nperature. a, Characteristics of differential conductance ver-\nsus bias voltage for a cobalt atomic contact, showing how the\nFano resonance disappears as the temperature is increased.\nb, The amplitude of the Kondo resonance from a decreases\nlogarithmically with the temperature. Inset, an artificial ly\ncoloured example of a lithographic device for the EBJ exper-\niments (before electromigration); the cobalt junction is b lue.\nThe atomic contacts created by this method are suitable for\nstudying the temperature dependence of the Fano resonance.\nromagnetic coupling to the spconduction electrons. We\ncan show how local moment formation and antiferromag-\nnetic coupling occur in nickel nanocontacts in the follow-\ning way (the cases of iron and cobalt can be understood\non similar grounds). The mean-field solution of the An-\nderson model25,which describes a localized dlevel, with\nenergyǫd, on-site repulsion Uand hybridization with the\nitinerant spelectrons, Vsp−d, defines the conditions for\nthe formation of a local moment in such a dlevel. The\nmodel can also be used to derive the antiferromagnetic\nsdexchange coupling, JAF\nsd(ref.26), which arises from\nthesp−dhybridization term. Here we used the local\nspin-densityapproximation(LSDA) todensityfunctional\ntheory and its generalization LSDA+U as the mean field\nto determine the formation of a local moment and its\nFIG. 4: Electronic structure for a nickel chain and a nickel\nnanocontact. a, LSDA minority-spin energy bands for a\nmonostrand nickel chain, where the wavevector kruns over\nthe first Brillouin zone. The bands are labelled according to\ntheir symmetry group: the E1 and E2 bands ( d-band like,\ndoubly degenerate and decoupled from the sband) and the\nA1 ord3z2−r2band (hybridized with the sband). The lat-\ntice constant is 2.09 ˚A .b, As in a, but for majority-spin\nelectrons. The yellow background indicates occupied state s.\nc, LSDA+U minority-spin results for the total DOS and the\nDOS projected on the dx2−y2orbital for a tip atom of a nickel\nnanocontact, with U= 3 eV.d, As inc, but for majority-spin\nelectrons.\nantiferromagnetic coupling to the spcarriers. We con-\nsider both monostrand chains and nanocontacts. Their\nsmaller atomic coordination, compared with that of the\nbulk, results in a stronger electronic localization and a\nlarger magnetic moment per atom (values of 1.17 Bohr\nmagnetons, compared with 0.6 Bohr magnetons in the\nbulk, were obtained using LSDA). This favours the ap-\npearance of the Kondo effect. In Fig. 4a, b, we show the\nLSDA bands obtained for the nickel chain. Out of the\nsix minority-spin bands crossing the Fermi level, the two\ndegenerate E2 bands are the narrowest, hosting highly\nlocalized electrons. These are the bands that are less\nwell described by the LSDA because of the inherent self-\ninteraction problem of this approximation.\nWe modelled actual nanocontacts with two identical\npyramids facing each other in the [001] direction. As pre-4\nviouslydoneforchains27, andtoavoidtheself-interaction\nproblem, we computed the electronic structure of the\nnanocontact using LSDA+U. In Fig. 4c, d, we show the\ntotal density of states (DOS) projected on a tip atom\nand the DOS projected on the corresponding dx2−y2or-\nbitals for both the minority (Fig. 4c) and the major-\nity (Fig. 4d) spins (here we have set U=3 eV). In gen-\neral, solutions obtained for values of Uranging from 3\nto 5 eV also show that the dx2−y2orbital, forming one\nof the E2 bands in chains, hosts an integer local mag-\nnetic moment. Different geometries may favour the for-\nmation of the local moment in other strongly localized\norbitals. The hybridization of the dx2−y2orbital with\nthe surrounding sporbitals results in antiferromagnetic\nkinetic exchange26. We estimate that JAF\nsp−d≃1 eV,\ntakingǫd≃5 eV,|Vsp−d|2≃2 eV and Ueff≃8 eV\nfrom our LSDA+U calculations, where Ueffis the spin\nsplitting of the dx2−y2orbital (Supplementary Informa-\ntion). We note that, in contrast to the contact atom(s),\nthe intra-atomic sp−dhybridization vanishes for bulk\natoms owing to the very small anisotropy of the crystal\nenvironment. This makes the antiferromagnetic coupling\nJAF\nsp−dlarger in the contact than in the bulk. This cou-\npling competes with the ferromagnetic coupling JFM\nsp−d,\nwhich we estimate from the splitting of the spband at\nthe boundary of the Brillouin zone for chains to be ≃0.2\neV (Fig. 4a, b). Thus, an overall antiferromagnetic cou-\npling between delectrons and spconduction electrons is\npossible in nickel nanocontacts. Additionally, the local\nmoment, m, responsible for the Kondo effect is subject\nto the ferromagnetic coupling Jddto the neighbouring\natoms, and this interaction also competes with the anti-\nferromagnetic sp−dcoupling:\nHexch=m·/bracketleftBigg\n(JAF\nsp−d−JFM\nsp−d)Ss−Jdd/summationdisplay\nimi/bracketrightBigg\n(2)\nHere the miare the local moments of the neighbouring\natoms and Ssis the spin of the selectrons. In ref.28, the\ncoupling Jddwas calculated for iron, cobalt and nickel by\nimplementing the magnetic force theorem with a LSDA\nground state. This method yields values for the spin-\nwave dispersion of the materials that compare well with\nexperiment29, and gives Jdd=19 meV, 15 meV and 2.7\nmeV for iron, cobalt and nickel, respectively. Thus, Jdd\nis significantly smaller than JAF\nsp−d.\nThe Kondo effect in nanocontacts is favoured by three\nfactors. First, a local moment forms in the contact\natoms because of their smaller coordination. Second,\nthe reduced symmetry of the contact, compared with\nthe bulk, enhances the intra-atomic contribution to the\nsp−dhybridizationand, thus, the antiferromagneticcou-\nplingJAF\nsp−d. Third, the smaller coordination also re-\nduces the influence of the direct ferromagnetic ddcou-\npling with neighbouring atoms. As a result, the local\nmoment formed in the contact is antiferromagnetically\ncoupled to the sp itinerant electrons and results in theKondo effect in this system, in contradiction to conven-\ntional wisdom.\nAcknowledgments\nWe thank to E. Tosatti, R. Aguado and J. Ferrer for\ndiscussions, G. Scott and G. Saenz-Arcefor experimental\nsupport and V. Esteve for technical support. This work\nwas partly supported by the European Union through\nMolSpinIQ and Spanish MEC (grants MAT2007-65487,\n31099-E and CONSOLIDER CSD2007-0010). D.J. ac-\nknowledges funding by the US National Science Foun-\ndation (NSF) under grant DMR-0528969. D.N. acknowl-\nedgesthesupportofNSFgrantDMR-0347253,theDavid\nand Lucille PackardFoundation and the W.M. KeckPro-\ngram in Quantum Materials.\nAPPENDIX A: METHODS\nFor the statistical results on the Kondo parameters,\nwe used a home-made STM at 4.2K to fabricate the con-\ntacts by indentation between two pieces of metal13. For\nthe temperature-dependence measurements, the contacts\nwereproduced bythe controlledelectromigrationat4.2K\nof 100-nm-wide junctions fabricated by electron beam\nlithography19. In both cases, the spectroscopic curves\nwere obtained by the addition of an a.c. voltage with a\npeak-power amplitude of 1mV and a frequency of 1 kHz\nto the d.c. bias voltage to allow the lock-in detection of\nthe differential conductance.\na. Fabrication of atomic-scale contacts by scanning\ntunnelling microscopy. Two pieces of metal wire of 0.1-\nmm diameter were cleaned and sonicated in acetone and\nisopropanol. These pieces were mounted in a home-built\nSTM. The set-up was pumped down to high vacuum and\nimmersed in a liquid helium bath until the temperature\nreached 4.2 K. The two pieces of wire were brought into\ncontact and then pulled apart until the contact was of\natomicdimensionand, finally,untilonlyoneatomformed\nthe contact13. Strongindentationbeforethe formationof\nevery contact ensured the cleanliness of the atomic con-\ntacts. The histograms obtained by this technique (Fig.\n1a) show similar results to the ones obtained by the me-\nchanicallycontrolledbreakjunction technique, where the\nsurfacesbroughtinto contactare createdunder cryogenic\nconditions.\nb. Fabrication of atomic-scale contacts using EBJs.\nAs a first step, a small junction about 100 nm wide\nwas fabricated from cobalt using electron beam lithog-\nraphy and electron beam evaporation over a silicon diox-\nide substrate. To make the junction suitable for con-\ntact by macroscopic probes, two gold electrodes are de-\nposited over the edges of the junction in a second lithog-\nraphy step, following the procedure described in ref.19.\nThe samples are then placed in a probe station that\nwas pumped down and immersed in a liquid helium bath5\nuntil the sample reached a temperature close to 4.2 K.\nUnder these conditions, the controlled electromigration\nprocess14,19was performed, decreasing the size of the\njunction to the atomic scale.\nAPPENDIX B: SUPPLEMENTARY\nINFORMATION ON LSDA AND LSDA+U\nRESULTS\nThe electronic structure of a Ni nanocontact model\nas that shown in Fig. 5 has been calculated with den-\nsity functional theory in the local spin-density approx-\nimation (LSDA)30using our established ab-initio trans-\nport methodology16. Fig. 5a shows the DOS and par-\ntialdDOS for one of the tip atoms of the nanocontact.\nThe majority-spin d-levels are completely filled while the\nminority-spin d-levels are partially occupied with an av-\nerage occupation of 0.8. This results in a net magnetic\nmoment of µ≃1 for the tip atom similar to the magnetic\nmoment of the one-dimensional chain.\nIt is, however, well known that LSDA suffers from the\nso-called self-interaction problem which blue-shifts occu-\npied levels. Here the spurious self-interaction brings all\nthe minority d-orbitals up to the Fermi level leading to a\npartial and almost equal occupation of 0.8 of all five d-\norbitals. The self-interaction error of LSDA is corrected,\ne.g., inthe LSDA+Umethod31. In the LSDA+Umethod\nthe effective LSDA Kohn-Sham potential of the strongly\ninteracting d-electrons is corrected by adding a Hartree-\nFock term for an on-site Coulomb repulsion Uand ex-\nchange interaction J:\n/angbracketleftiσ|VLSDA+U|iσ/angbracketright=U(Nd−nσ\ni)−J(Nσ\nd−nσ\ni)−Edc(B1)\nHere,Ndis the total occupation ofall the d-levelsof an\natom while nσ\niis the occupation of an individual d-level\nper spin σ, andNσ\ndis the total occupation of d-levels per\nspinσ.Edcaccountsforthe fact that the Coulombrepul-\nsion and exchange interaction have already been taken\ninto account in some way at the LSDA level, and thus\nhave to be subtracted from the effective LSDA Kohn-\nSham potential in order to avoid double-counting31.\nDue to the low coordination of the two tip atoms\n(compared to the rest of the nanocontact and to bulk\natoms) Coulomb interaction effects are expected to be\nmuch stronger for the tip atoms than for the rest of the\nnanocontact. Hence, we only treat the two tip atoms ofthe nanocontact at the LSDA+U level while the rest of\nthenanocontactistreatedattheLSDAlevel. ForbulkNi\nawidelyacceptedvalueis U=3eVforthedirectCoulomb\nrepulsionand J=1eVfortheexchangeinteraction32. Due\nto the lower coordination of the tip atoms with respect\nto bulk atoms, the screening of the Coulomb repulsion\nshould be lower than in bulk. Therefore the Coulomb\nrepulsion is expected to be bigger than U=3 eV for the\ntip atoms.\nU|tsd||tpd|ǫdǫd+˜UJAF\nsp−d\n3 eV0.3 eV0.6 eV-4.5 eV1.5 eV1.6eV\n5 eV0.3 eV0.6 eV-5 eV3 eV1.0eV\nTABLE I: Table of hoppings andexcitation energies extracte d\nfrom an LSDA+Ucalculation with U=3eVand U=5eV,and\nJ=1 eV for all d-orbitals on the two tip atoms, as well as the\nresulting antiferromagnetic exchange coupling JAF\nsp−d.\nIndeed, as can be seen from Figs. 5a,b, for U≥3 eV\nfour of the minority d-orbitals are now well below the\nFermi energy and completely occupied, while the minor-\nitydx2−y2orbitaliscompletelyempty. Thusthe magnetic\nmoment of the tip atom is now entirely carried by the\ndx2−y2-level. This shows that a local moment can form\non the tip atom which is carried by a single d-orbital.\nThe screening of the magnetic moment of this d-orbital\nby thesp-conduction electrons can thus give rise to a\nKondo resonance at the Fermi level.\nNow we estimate the exchange coupling JAF\nsp−dof the\ndx2−y2orbital of the tip atoms from the hoppings and\nexcitation energies of the effective Hamiltonian of the\nLSDA+Ucalculation. The dx2−y2-orbitalofthe tip atom\nof one of the pyramids is coupled via hoppings tsdand\ntpdto thesandpz-orbitals of the four atoms next to the\ntip atom in the same pyramid as detailed in Tab. 1. The\ntotal antiferromagnetic exchange coupling is given by:\nJAF\nsp−d=|Vsp−d|2/parenleftbigg1\n|ǫd|+1\nǫd+Ueff/parenrightbigg\n(B2)\nwhereVsp−dis the effective total hybridization with\nthesp-channels:\n|Vsp−d|2= 4|tsd|2+4|tpd|2(B3)\n∗Corresponding author: untiedt@ua.es\n1J. Kondo, Prog. Theor. Phys. 32, 37 (1964).\n2U. Fano, Phys. Rev. 124, 1866 (1961).\n3D. 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Rev. B 69, 081401 (2004).\n16D. Jacob, J. Fern´ andez-Rossier, and J. J. Palacios, Physi-\ncal Review B 71, 220403 (2005).\n17M. R. Calvo, D. Jacob, M. J. Caturla, C. Untiedt, and\nJ. J. Palacios, IEEE Transactions in Nanotechnology 7,\n165 (2008).\n18T. Jamneala, V.Madhavan, W. Chen, andM. F. Crommie,\nPhys. Rev. B 61, 9990 (2000).\n19Z. K. Keane, L. H. Yu, and D. Natelson, Applied Physics\nLetters88, 062514 (pages 3) (2006).20W. G. van der Wiel, S. D. Franceschi, T. Fujisawa, J. M.\nElzerman, S. Tarucha, and L. P. Kouwenhoven, Nature\n289, 2105 (2000).\n21J. R. Hauptmann, J. Paaske, and P. Lindelof, Nat. Phys.\n4, 373 (2008).\n22J. Martinek, Y. Utsumi, H. Imamura, J. Barna´ s,\nS. Maekawa, J. K¨ onig, and G. Sch¨ on, Phys. Rev. Lett.\n91, 127203 (2003).\n23J. e. a. Martinek, Phys. Rev. Lett. 91, 247202 (2003).\n24M.-S. Choi, D. S´ anchez, and R. L´ opez, Phys. Rev. Lett.\n92, 056601 (2004).\n25P. W. Anderson, Phys. Rev. 124, 41 (1961).\n26J. Schrieffer and P. A. Wolff, Phys. Rev. 149(1966).\n27M. Wierzbowska, A. Delin, and E. Tosatti, Physical Re-\nview B (Condensed Matter and Materials Physics) 72,\n035439 (2005).\n28M. Pajda, J. Kudrnovsk´ y, I. Turek, V. Drchal, and\nP. Bruno, Phys. Rev. B 64, 174402 (2001).\n29H. A. Mook and D. M. Paul, Phys. Rev. Lett. 54, 227\n(1985).\n30W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n31V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev.\nB44, 943 (1991).\n32A. Grechnev, I. D. Marco, M. I. Katnelson, A. I. Lichten-\nstein, J. Wills, and O. Eriksson, Phys. Rev. B 76, 035107\n(2007)."    },    {        "title": "1611.05798v2.Inductive_detection_of_field_like_and_damping_like_AC_inverse_spin_orbit_torques_in_ferromagnet_normal_metal_bilayers.pdf",        "content": "arXiv:1611.05798v2  [cond-mat.mtrl-sci]  26 Oct 2017Inductive detection of field-like and damping-like AC inver se spin-orbit torques in\nferromagnet/normal metal bilayers\nAndrew J. Berger,1Eric R. J. Edwards,1Hans T. Nembach,1\nAlexy D. Karenowska,2Mathias Weiler,3,4and Thomas J. Silva1,∗\n1Quantum Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305, U.S.A.†\n2Department of Physics, University of Oxford, Oxford, U.K.\n3Walther-Meißner-Institut, Bayerische Akademie der Wisse nschaften, Garching, Germany\n4Physik-Department, Technische Universit¨ at M¨ unchen, Ga rching, Germany\n(Dated: October 9, 2018)\nFunctional spintronic devices rely on spin-charge interco nversion effects, such as the reciprocal\nprocesses of electric field-driven spin torque and magnetiz ation dynamics-driven spin and charge\nflow. Both damping-like and field-like spin-orbit torques ha ve been observed in the forward process\nof current-driven spin torque and damping-like inverse spi n-orbit torque has been well-studied via\nspin pumping into heavy metal layers. Here we demonstrate th at established microwave transmis-\nsion spectroscopy of ferromagnet/normal metal bilayers un der ferromagnetic resonance can be used\nto inductively detect the AC charge currents driven by the in verse spin-charge conversion processes.\nThis technique relies on vector network analyzer ferromagn etic resonance (VNA-FMR) measure-\nments. We show that in addition to the commonly-extracted sp ectroscopic information, VNA-FMR\nmeasurements can be used to quantify the magnitude and phase of all AC charge currents in the\nsample, including those due to spin pumping and spin-charge conversion. Our findings reveal that\nNi80Fe20/Pt bilayers exhibit both damping-like and field-like inver se spin-orbit torques. While the\nmagnitudes of both the damping-like and field-like inverse s pin-orbit torque are of comparable scale\nto prior reported values for similar material systems, we ob served a significant dependence of the\ndamping-like magnitude on the order of deposition. This sug gests interface quality plays an impor-\ntant role in the overall strength of the damping-like spin-t o-charge conversion.\nI. INTRODUCTION\nSpin-charge transduction effects for ferromag-\nnet/nonmagnet (FM/NM) multilayers couple electric\nfields to magnetic torques in the forward process\n(so-called spin-orbit torque (SOT)), and they couple\nmagnetization dynamics to currents in the inverse\nprocess (iSOT). In general, these torques can be\nphenomenologically separated into two components:\ndamping-like and field-like. Both are perpendicular to\nthe FM magnetization, but the damping-like torque\nis odd under time-reversal and dissipative, whereas\nthe field-like torque is even under time-reversal and\nconservative1. A classic example of a field-like torque is\nthe action of an Oersted field on a FM magnetization\ndue to a charge current in an adjacent conducting layer.\nBy Onsager reciprocity, the inverse process is captured\nby Faraday’s law: magnetization dynamics in the FM\ngeneratechargecurrentsin the NM. Recently, it hasbeen\nfound that spin-orbit coupling (SOC) in multilayers can\ngive rise to both field- and damping-like SOTs2,3, but\nwith substantially different scaling than that achieved\nwith Oersted fields. Unlike the Oersted effect, these\nspin-orbitronic effects are short-range, making them\nhighly advantageous for microelectronic applications\nthat require device scaling to high densities such as\n∗thomas.silva@nist.gov\n†Contribution of the National Institute of Standards and Tec h-\nnology; not subject to copyright.nonvolatile memory and alternative state-variable\nlogic4,5.\nDamping-like torquesdue to the spin Hall effect (SHE)\nin heavy NM layers such as Pt and β-Ta are well-studied\nand understood, and have been investigated in both\nforward4and inverse configurations6–8. Substantial field-\nlike torques have also been measured for FM/NM inter-\nfaces in the forward configuration2,9–11. However, an in-\nversemeasurement ofthe field-like torque in Ni 80Fe20/Pt\nhas not yet been unambiguosly reported12. Here, we\npresent simultaneous measurements of inverse field-like\nand damping-like torques in Ni 80Fe20/Pt bilayers via\nwell-established coplanar waveguide (CPW) ferromag-\nnetic resonance (FMR). Time-varying magnetic fields\nproduced by a FM/NM sample under FMR excitation\nwill inductively couple into the CPW, altering the to-\ntal inductance of the microwave circuit. Such fields\nare produced by: (1) the Py precessing magnetization,\n(2) Faraday effect induced AC currents in the Pt layer,\nand (3) spin-orbit AC currents due to damping-like and\n(4) field-like processes. We show that through proper\nbackground normalization, combined with Onsager reci-\nprocity for the specific phenomenology of these measure-\nments, commonly-used vector network analyzer (VNA)\nFMR spectroscopy allows accurate identification of the\nprocesses that contribute to spin-charge conversion.\nThe paper is organized as follows. In Sec. II, by\nappealing to Onsager reciprocity we provide the phe-\nnomenological background relating the forward and in-\nverseprocessesthatproducemagnetictorquesandcharge\nflow in a ferromagnet/normal metal system under elec-2\ntrical bias or with excited magnetization dynamics. Sec.\nIII describes the quantitative VNA-FMR technique, and\nderives the expressions we use to calculate the sample’s\ncomplex inductance. This section also introduces the ef-\nfective conductivity ˜ σNMthat quantifies the magnitude\nand symmetry of magnetic torques due to applied charge\ncurrents,andreciprocally,oftheACchargecurrentsflow-\ning in a sample in response to the driven magnetization\ndynamics. In Sec. IV, we present data acquired from\nNi80Fe20/Pt bilayers and Ni 80Fe20/Cu control samples.\nThe magnitude of the phenomenological parameter ˜ σNM\nextracted from these data is well within the range of re-\nported values, and it obeys the usual symmetry proper-\ntiesassociatedwith thestackingorderoftheNi 80Fe20and\nPt layers. Finally, we discuss the results in Sec. V by\ncomparing our extracted iSOT parameters to the micro-\nscopic spin-chargeconversionparameters of spin Hall an-\ngle and Rashba parameter. In all cases, the magnitudes\nof the extracted spin Hall angle and Rashba parameter\nare in rough agreement with what has been reported in\nthe literature, though this agreementis contingenton the\nassumption of typical values for the interfacial and bulk\nspin transport parameters. However, we find that theextracted spin Hall angle changes by a factor of almost 4\ndepending on the growth order of the multilayer stacks,\nwith a larger spin Hall angle when the Pt is grown on\ntop of the Ni 80Fe20. This suggests that the spin trans-\nport parameters are in actuality highly dependent on the\nstack growth order.\nII. ONSAGER RELATIONS FOR SPIN-ORBIT\nTORQUE\nOnsager reciprocity relations13are well known for cer-\ntain pairs of forces and flows. For example, for thermo-\nelectriceffects, appliedelectricfieldsorthermalgradients\ncan drive both charge and heat flow. In this section,\nwe establish Onsager relations for charge current and\nmagnetic torque as the flows that are driven by applied\nelectric fields and magnetization dynamics in a FM/NM\nmultilayer1.\nBy analogy to Ohm’s Law, J=σE, we can write a\ngeneral matrix equation relating driving forces (magne-\ntization dynamics ∂ˆm/∂tand electric field E) to flows\n(magnetic torque density Tand charge current density\nJ)1:\n\n/parenleftbigg2e\n/planckover2pi1/parenrightbigg\n+dFM/integraldisplay\n0T(z)dz\n\n\n+dFM/integraldisplay\n−dNMJ(z)dz\n\n=\nG\nGmag sgn(ˆz·ˆn)/parenleftbig\n−σF\ne+σSOT\ne−σSOT\no[ˆm×]/parenrightbig\nsgn(ˆz·ˆn)/parenleftbig\n−σF\ne+σSOT\ne−σSOT\no[ˆm×]/parenrightbig\n−1\nZeff\n\n∗\n/parenleftbigg/planckover2pi1\n2e/parenrightbigg∂ˆm\n∂t\nˆz×E\n(1)\nwhere ˆmis the magnetization unit vector, /planckover2pi1is Planck’s\nconstant divided by 2 π,eis the electron charge, dFMand\ndNMare the FM and NM thicknesses. The terms in the\n2×2 conductivity matrix are described below. The sign\nof the off-diagonal terms are determined by sgn(ˆ z·ˆn),\nwhere ˆnis an interface normal pointing into the FM.\nThe coordinate unit vector ˆ zis defined by the sample\nplacement on the CPW, as shown in Fig. 1(a), and z= 0\nis defined by the FM/NM interface. Gis a 2×2 matrix\nimposing geometrical constraints: (1) magnetic torques\nareorthogonalto ˆ mand(2) chargecurrentscanflowonly\nin thex,yplane:\nG=/bracketleftbigg\n[ˆm×] 0\n0 [ˆz×]/bracketrightbigg\n(2)Thediagonalelementsoftheeffectiveconductivityma-\ntrix describe the Gilbert damping of the FM and charge\nflow in the metallic bilayer in response to an applied elec-\ntric field. That is,\n/parenleftbigg2e\n/planckover2pi1/parenrightbigg\n+dFM/integraldisplay\n0T(z)dz\n=/parenleftbigg/planckover2pi1\n2e/parenrightbigg\nGmag/parenleftbigg\nˆm×∂ˆm\n∂t/parenrightbigg\n(3)\n\n+dFM/integraldisplay\n−dNMJ(z)dz\n=−1\nZeffˆz×(ˆz×E) (4)\nwhereGmag≡ −dFM(2e//planckover2pi1)2(αMs/γ),αis the Gilbert\ndamping parameter, and γis the gyromagnetic ratio,3\nsuch that Eq. 3 is the usual Gilbert damping term from\nthe Landau-Lifshitz-Gilbert equation:\n∂ˆm\n∂t=−γµ0ˆm×H−/parenleftbiggγ\nMsdFM/parenrightbigg+dFM/integraldisplay\n0T(z)dz(5)In Eq. 4, Zeffis the effective frequency-dependent\nimpedance of the bilayer. Eq. 4 reduces to Ohm’s Law\nin the DC limit ( Zeff→R/squareasf→0).\nThe off-diagonal terms describe the electromagnetic\nreciprocity between Faraday’s and Ampere’s Law14,15, as\nwell as spin-orbit torques (SOT) and their inverse, using\nthe effective conductivities σF\ne,σSOT\ne, andσSOT\no.\n/parenleftbigg2e\n/planckover2pi1/parenrightbigg\n+dFM/integraldisplay\n0T(z)dz\n= sgn(ˆz·ˆn)ˆm×/parenleftbig\n−σF\ne+σSOT\ne−σSOT\no[ˆm×]/parenrightbig\n(ˆz×E) (6)\n\n+dFM/integraldisplay\n−dNMJ(z)dz\n= sgn(ˆz·ˆn)/parenleftbigg/planckover2pi1\n2e/parenrightbigg\nˆz×/parenleftbig\n−σF\ne+σSOT\ne−σSOT\no[ˆm×]/parenrightbig∂ˆm\n∂t(7)\nHere, the superscripts indicate the source of the torque\nor current as due to the Faraday effect or SOT. The sub-\nscripts indicate “even” or “odd” with respect to time-\nreversal, which determines the torque direction or phase\nof the corresponding current with respect to the driving\nelectric field or magnetization dynamics.\nFirst consider the Faraday conductivity, σF\ne. In the\nforward process an electric field Eproduces a charge cur-\nrent, which by Ampere’s Law produces a magnetic field.\nThis field exerts a torque Ton the magnetization of the\nFM layer. In the reverse process, magnetization dynam-\nics∂tˆmproduce an AC magnetic field, which by Fara-\nday’s Law induces a chargecurrent Jin the NM layer. In\nthis way, σF\nequantifies the reciprocity between Ampere’s\nand Faraday’s Law (see Eq. 31 for an estimate of the σF\ne\nmagnitude based on material properties). Inclusion of\nthe terms in Eq. 1 due to electrodynamic reciprocity is\ncritical for the proper interpretation of inverse spin orbit\ntorque experiments12.\nAlsopresentin the off-diagonaltermsareSOTconduc-\ntivities due to spin-charge conversion. In Eq. 6, these\nmanifest as electric-field driven damping-like torques,\nwhich are proportional to ˆ m×(ˆm×(ˆz×E)), and field-\nlike torques, which are proportional to ˆ m×(ˆz×E). The\nconstantsofproportionalitybetweenappliedelectricfield\nand SOTs are σSOT\noandσSOT\ne. In the reverse direction\n(Eq. 7), these effects are responsible for spin-to-charge\nconversion (e.g., inverse spin Hall effect (iSHE)16or in-\nverse Rashba-Edelstein effect (iREE)17).\nReporting effective conductivities, as opposed to spin-\ncharge conversion parameters like the spin Hall angle,\ndirectly relates the microwave inputs and charge current\noutputs of an iSOT device without the need for separate\ncharacterization of spin-mixing conductance or spin dif-\nfusion length. Reciprocally, in a spin torque experiment\nwith charge current inputs and magnetization dynam-\nics (or switching) as output, the effective conductivities\nprovide the ideal figure of merit for determining magne-\ntization oscillation and switching thresholds of the ap-plied current. To estimate the critical current density Jc\nneeded to switch the magnetization of a ferromagnetic\nlayer18,19, one simply needs to equate the Gilbert damp-\ningtorque(Eq. 3)andodd(anti-damping-like)SOT(Eq.\n6):\nJc=αMsdFMω\nγ2e\n/planckover2pi1/parenleftbiggσ\nσSOTo/parenrightbigg\n(8)\nwhereωis the FMR frequencywith noapplied fields (e.g.\nfor in-plane magnetization, ω=µ0γ/radicalbig\nHk(Ms+Hk),\nwith anisotropy field Hk). Using αas determined for\nthese Ni 80Fe20/Pt films (see SI), Ms= 700kA /m,Hk=\n160kA/m (for thermal stability considerations), bulk Pt\nresistivity20, and the measured σSOT\no(see Table I), we\nestimate a critical current density of 2 ×1012A/m2for a\n2nm Ni 80Fe20film.\nWhile the effective conductivity is the directly mea-\nsured quantity, in Sec. VA we nevertheless derive ex-\npressions relating the effective conductivities to micro-\nscopic spin-charge conversion parameters. Extraction of\nthe microscopic parameters is necessarily contingent on\nthe details of the model employed and parameters as-\nsumed.\nThe effective conductivities can also be related to the\noften-used quantity of effective flux density per unit cur-\nrent density21Beff/J, with units of Tm2A−1via the\nequation Beff/J=σSOT\ne,o/planckover2pi1/(2MsσdFMe) (where σis the\nordinary charge conductivity of the NM). However, our\ndefinitionfortheeffectiveconductivityismoregeneralin-\nsofaras it allowsone to calculate the actual SOT without\nthe need to independently determine the sample magne-\ntization, conductivity, or actual thickness.\nEq. 1 is consistent with the phenomenological formu-\nlation presented by Freimuth, Bluegel, and Mokrousov1,\nalthough it has been expanded to include the purely elec-\ntrodynamic contributions. Our use of the descriptors\n“even” and “odd” are different from that of Freimuth,4\net al., who use the symmetry of the spin orbit torques\nwith respect to magnetization-reversal as the symmetry\nidentifier. We instead use the symmetry of the torque\nwith respect to time-reversal because this is the relevant\nsymmetry with regard to the off-diagonal components in\nthe phenomenological Eq. 1. Any process for which the\ntorque is odd under time-reversal qualifies as microscop-\nically non-reversible in the sense of Onsager reciprocity,\nwhere microscopic reversibility pertains solely to forces\nthat are even functions of velocity, as well as position13.\n(We also note that all axial vectorssuch as magnetic field\nare odd under time reversal.)\nIII. EXPERIMENTAL TECHNIQUE\nThe broadband, phase-sensitive FMR measurements\nutilize a coplanar waveguide (CPW) as both the exci-\ntation and detection transducer (see Fig. 1(a)). Any\nsource of AC magnetic flux generated by the bilayer is\ninductively detected in the CPW. Therefore, the induc-\ntive load that the sample contributes to the CPW circuit\nconsists of four terms: (1) The real-valued L0due to the\noscillating magnetic dipolar fields produced by the res-\nonating FM magnetization, (2) the Faraday-effect cur-\nrents induced in the NM layer by the precessing FM\nmagnetization, (3) currents produced by damping-like\niSOT effects (e.g., spin pumping + iSHE), and (4) cur-\nrents produced by field-like iSOT effects (e.g., iREE).\nThe latter three inductances, which we collectively de-\nfine as complex-valued LNM, are produced by currents in\nthe NM which generate Oersted fields that inductively\ncouple to the CPW. We quantify these currents with the\neffective conductivities σF\ne,σSOT\no, andσSOT\ne, described\nabove. Importantly, as shown below, while L0is inde-\npendent of frequency, LNMis linear in frequency, as the\ncurrents in the NM are driven by ∂tˆm. Hence, frequency-\ndependent measurements allow us to disentangle L0and\nLNM.\nFigure1(b)and(c)showschematicsofthesefoursignal\nsources at two instants in time: when the dipolar and\neven SOT effects are maximal (Fig. 1(b)) and when the\nodd SOT effect is maximal (Fig. 1(c)). Fig. 1(d) shows\nthe time dependence of each of these signal sources, and\ntheir distinct phase relationships to the driving field hy,\nwhich we exploit below to determine their contributions\nseparately.\nFor our measurements, we place samples onto a copla-\nnar waveguide (CPW) with the metallic film side fac-\ning down (see Fig. 1). This setup is positioned be-\ntween the pole pieces of a room-temperature electromag-\nnet capable of producing fields up to ≈2.2T. Using aVNA, we measure the change in microwave transmis-\nsion through the CPW loaded with the bilayer sample\nas an out-of-plane DC magnetic field ( H0∝ba∇dblˆz) is swept\nthrough the FMR condition of the Ni 80Fe20(Permalloy,\nPy) layer. We acquire the microwave transmission S-\nparameter S21≡Vin,2/Vout,1whereVin(out),1(2)is the\nvoltage input (output) at port 1 (2) of the VNA. Field\nsweeps were repeated to average the transmission data\nuntil an appropriate signal-to-noise ratio was obtained.\nTypically, VNA-FMR measurements focus on the res-\nonance field and linewidth. Our method additionally\nmakes use of the signal magnitude and phase in order to\ndirectlyprobethe ACchargecurrentsproducedbyiSOT.\nPrevious studies of AC charge currents in spin pumping\nexperiments have relied on intricate experimental setups\nor techniques that suppress or are insensitive to spurious\nbackground signals12,22,23. Our technique remains sensi-\ntive to currents induced by the Faradayeffect, but is able\nto separate them from spin-charge conversion currents\nthrough the combination of phase-sensitive analysis and\ncomparison to control samples in which the heavy metal\nNM (here, Pt) is substituted with a Cu layer of nomi-\nnally negligible intrinsic spin-orbit effects. Furthermore,\nbecause the CPW is inductively coupled to the sample,\nno electrical connections need to be made directly to the\nFM/NM sample.\nThe sampleaddsa complexinductance Lin serieswith\nthe impedance of the bare CPW, Z0(here, 50Ω). The\nchange in microwave transmission ∆ S21is therefore that\nof a simple voltage divider24:\n∆S21=−1\n2/parenleftbiggiωL\nZ0+iωL/parenrightbigg\n≈−iωL\n2Z0(9)\nforZ0>> ωL, whereωis the microwave frequency. The\nfactor of 1 /2 is needed because the port 2 voltage mea-\nsurement is between the CPW signal and ground (and\nnot between port 2 and port 1).\nA. Inductance Derivations\nIn order to extract values for the SOT effects from the\nmeasured ∆ S21, we derive expressions for each contribu-\ntion toL.\n1. Inductance due to dipole field of dynamic magnetization\nTo derive the inductance due to AC dipolar fields pro-\nduced by the precessing FM magnetization, we follow\nRef. 24.5\n(a) (c)\n(b)(d)\nFigure 1. (a) Sample on CPW, showing out-of-plane field H0and sample length l. The microwave driving field points primarily\nalong ˆyat the sample. (b) Schematic of the bilayer, with precessing magnetization m(t) at time t0whenm=∝angbracketleftmx,0,mz∝angbracketright.\nBilayer is insulated from CPW using photoresist spacer laye r (not shown). At this instant in time, JF\ne(due to the Faraday\neffect in the NM) and JSOT\ne(e.g., due to inverse Rashba-Edelstein effect) are maximal a long±ˆx, andhyis also at its maximum\nstrength. The corresponding Oersted fields from JF\neandJSOT\neare superposed. The spin accumulation (with orientation ˆ s)\nandJSOT\neare produced at the FM/NM interface. Interface normal is giv en by ˆn. (c) Same as (b), except at time t1when\nm=∝angbracketleft0,my,mz∝angbracketright. Here, odd-symmetry SOT current JSOT\no(e.g., due to inverse spin Hall effect), and the dynamic fields HSOT\no\nandHdipoleare at maximum amplitude. Note that the dipolar signal is pro portional to ∂t(Hdipole·ˆy), and not simply to Hdipole.\nSpin flow direction ˆQˆsdue to spin pumping into the NM is also shown. (d) Amplitude of driving field hyand different signal\nsources as a function of time (left), and viewed in the comple x plane at time t0(right). Relative amplitudes not indicated. For\nfurther discussion of signal phases, see SI Sec. III.6\nL0=µ0ℓ\nWwgdFMI2\n+∞/integraldisplay\n−∞dydFM+dwg/integraldisplay\ndwgdz[q(y,z)·χ(ω,H0)·h1(y,z,I)]\n\n∗\n+∞/integraldisplay\n−∞dydFM+dwg/integraldisplay\ndwgdz[q(y,z)·h1(y,z,I)]\n\n∼=µ0ℓ\nWwgdFMI2χyy(ω,H0)h2\ny(I,z)d2\nFMW2\nwg\n∼=µ0ℓ\nWwgdFMI2χyy(ω,H0)/parenleftbiggI\n2Wwgη(z,Wwg)/parenrightbigg2\nd2\nFMW2\nwg\n=µ0ℓdFM\n4Wwgχyy(ω,H0)η2(z,Wwg) (10)\nwhereµ0isthe vacuumpermeability, lthe samplelength,\ndFMthe FM thickness, Wwgthe width of the CPW\nsignal line (here, 100 µm), and χyy(ω) the frequency-\ndependent magnetic susceptibility. η(z,Wwg)≡\n(2/π)arctan(Wwg/2z) is the spacing loss, ranging from 0\nto1, duetoafinitedistance zbetweensampleandwaveg-\nuide. We have assumed the coordinate system described\nin Fig. 1 (ˆ xalong the CPW signal propagation direction,\nˆzalong the CPW and sample normal). The function\nq(y,z) describes the normalized spatial amplitude of the\nFMR mode. For the uniform mode, q(y,z) = 1 over the\nentire sample. The first set of integrals in brackets cap-\ntures the integrated amplitude of the mode as excited by\nthe driving microwave field h1=hyˆy, while the second\ndescribes the inductive pickup sensitivity of the CPW.\nThe firstapproximationassumesuniform microwavefield\nover the sample dimensions. The second approximation\nutilizes the Karlqvist equation25to approximate the mi-\ncrowave field as hy(I,z)∼=I/(2Wwg)η(z,Wwg).\n2. Inductance due to AC current flow in NM\nFollowing Rosa26, we model the sample and CPW as\ntwo thin current-carrying sheets of width w=Wwg, sep-\narationz, and length l, so that the mutual inductance is\ngiven by\nL12=µ0\n4π2l/bracketleftbigg\nln/parenleftbigg2l\nR/parenrightbigg\n−1/bracketrightbigg\n(11)\nwhereRis defined as\nR≡/radicalbig\nw2+z2/parenleftbiggz√\nw2+z2/parenrightbigg(z\nw)2\n∗exp/parenleftbigg2z\nwarctan/parenleftigw\nz/parenrightig\n−3\n2/parenrightbigg\n(12)Viewing the sample-CPW system as a voltage trans-\nformer (two mutually-coupled inductors), the voltage in-\nduced in the CPWdue to current INMin the NM and the\nmutual inductance L12is given by V=−L12(∂INM/∂t).\nIf instead we consider the system to be a single lumped-\nelement inductor, the voltage due to the self-inductance\ncontributed by the sample LNMand applied current\nICPWisV=LNM(∂ICPW/∂t). Therefore, we can cal-\nculateLNMas\nLNM=−L12INM\nICPW(13)\nThe charge current we are interested in is that driven\nby the magnetization dynamics of the FM layer, and\ngiven by the off-diagonal term of Eq. 1:\nINM= ˆx·\n+dFM/integraldisplay\n−dNMJ(z)dz\nWwg\n= ˆx·/bracketleftbig\nˆz×(−σF\ne+σSOT\ne−σSOT\no[ˆm×])∂tˆm/bracketrightbig\n∗sgn(ˆz·ˆn)/parenleftbigg/planckover2pi1\n2e/parenrightbigg\nWwg (14)\nAssuming a linear solution to the Landau-Lifshitz-\nGilbert equation of motion for the magnetization, we\nwrite a simple relation between the dynamic component\nof the magnetization mand microwave field h1.\n∂tˆm=iωχ\nMsh1 (15)\nTo convert the vector cross products in Eq. 14 to the\ncomplex plane, we use χin the frequency domain27:7\nχ=γµ0Ms\nω2res−ω2+iω∆ω/bracketleftbigg/parenleftbig\n1+α2/parenrightbig\nωy−iαω iω\n−iω/parenleftbig\n1+α2/parenrightbig\nωx−iαω/bracketrightbigg\n(16)\nwhereωx,y≡γµ0Hx,y,Hx,yis the stiffness field in the x\norydirection(includingexternal,anisotropy,anddemag-\nnetizing fields), ωres≡√ωxωy, and ∆ω≡α(ωx+ωy).\nFor compactness in the following derivation, we utilize\nthe tensor components of the susceptibility as defined in\nEq. S1.\nEq. 14 has even terms along ˆ z×∂tˆmand odd terms\nalong ˆz×(ˆm×∂tˆm). Using Eq. 15 for ∂tˆm, we can\nwork out these cross products assuming ˆ m∝ba∇dblˆz(small-\nangle FMR excitation). The vector components of the\neven terms are given by:\nˆz×∂tˆm= ˆz×/parenleftbigg/bracketleftbigg\nχxxχxy\nχyxχyy/bracketrightbigg/bracketleftbigg\n0\nhy/bracketrightbigg/parenrightbigg/parenleftbiggiω\nMs/parenrightbigg\n= ˆz×(χxyhyˆx+χyyhyˆy)/parenleftbiggiω\nMs/parenrightbigg\n= (−χyyhyˆx+χxyhyˆy)/parenleftbiggiω\nMs/parenrightbigg\n(17)\nSimilarly, we find for the odd terms:\nˆz×(ˆm×∂tˆm) = ˆz×(−χyyhyˆx+χxyhyˆy)/parenleftbiggiω\nMs/parenrightbigg\n= (−χxyhyˆx−χyyhyˆy)/parenleftbiggiω\nMs/parenrightbigg\n(18)\nNoting from Eq. 16 that χxy=iχyy(ignoring terms\nof order αorα2, and working near resonance such that\nωx=ω), the vector relationships of Eq. 17 and 18 are\nsubstituted into Eq. 14. After evaluating the ˆ xprojec-\ntion as prescribed by Eq. 14 and grouping even and odd\nterms, we find:\nINM=/bracketleftbig\n(σF\ne−σSOT\ne)+iσSOT\no/bracketrightbig\nsgn(ˆz·ˆn)iχyyhy\nMs/parenleftbigg/planckover2pi1ω\n2e/parenrightbigg\nWwg\n(19)\nfrom which we define ˜ σNM= (σF\ne−σSOT\ne)+iσSOT\no. On\nresonance, χyy=−iγµ0Ms/(2αeffωres), such that Eq. 19\nproduces the current phases depicted in Fig. 1.\nFinally, using the Karlqvist equation25, we approxi-\nmate the field of the CPW. With these substitutions into\nEq. 13, we arrive at the final result for the inductance\ndue to all AC currents in the NM:\nLNM= sgn(ˆz·ˆn)L12(z,Wwg,l)η(z,Wwg)\n∗/planckover2pi1ω\n4Mseiχyy(ω,H0)˜σNM(20)\nThe different frequency dependencies of L0andLNMis\ncritical for our analysis. When normalized to χyy(ω,H0),L0is a frequency-independent inductance. By contrast,\nLNMhas an extra factor of ω, reflecting the fact that\nboth Faraday and SOT effects are driven by ∂tˆm, rather\nthan bym(t) itself.\nCareful attention needs to be paid to the signal phase\nin order to properly add the inductive effects of L0and\nLNM. As discussed in detail in the SI Sec. III, it is the\ncurrent phase in the CPW that determines the propa-\ngating signal phase. Using the excitation current in the\nCPW as the phase reference, we work out the phase of\nthe induced currents due to the perturbative inductance\nof the sample-on-CPW, and find that the inductances\nadd according to L=L0−iLNM.\nAfter normalizing by the fitted susceptibility ˜L≡\nL/χyy(ω,H0), the real and imaginary normalized induc-\ntance amplitudes are given by:\nRe(˜L) =µ0l\n4/bracketleftbiggdFM\nWwgη2(z,Wwg)−sgn(ˆz·ˆn)η(z,Wwg)\n∗L12(z,Wwg,l)\nµ0lMs/planckover2pi1ω\ne(σF\ne−σSOT\ne)/bracketrightbigg\n(21)\nIm(˜L) =−µ0l\n4/bracketleftbigg\nsgn(ˆz·ˆn)η(z,Wwg)\n∗L12(z,Wwg,l)\nµ0lMs/planckover2pi1ω\neσSOT\no/bracketrightbigg\n(22)\nNote that when the stacking order of FM and NM is\nreversed, so is the sign of the SOT and Faraday currents\n(and therefore their inductance contributions).\n3. Magnetization dynamics driven by forward SOT\nFrom the transformer analogy developed above and\ndiscussed in SI Sec. III, we see that “image currents”\nare produced in the CPW when currents flow in the con-\nducting sample. Reciprocity requires that the excitation\ncurrents in the CPW drive image currents in the sam-\nple. This current will produce Amperian torque and\nforward SOT effects according to Eq. 6, exciting ad-\nditional magnetization dynamics which are then picked\nup by the CPW. This series of transduction effects is\nfully reciprocal with the Faraday and iSOT sequence de-\nscribed above. In the first case, a drive current in the\nCPW excites magnetization dynamics (via the coupling\nfactor,η(z,Wwg)). Those magnetization dynamics drive\ncharge current in the NM via ˜ σNM. Finally, these charge\ncurrents couple into the CPW via the mutual inductance\nL12(z,Wwg,l). In the second case, the order is simply\nreversed: the CPW currents create image currents in the\nNM (via L12(z,Wwg,l)), which drive magnetization dy-\nnamics (via ˜ σNM), which are picked up by the CPW (via8\nη(z,Wwg)). It can be shown that the induced signal due\nto forward Amperian or SOT-driven magnetization dy-\nnamics add together in-phase with their inverse counter-\nparts, increasing the inductive response from each con-\ntribution by a factor of 2. The inductance in Eq. 20\n(and hence 21 and 22) is therefore too small by a factor\nof 2. Therefore, in the below calculation of ˜ σNMbased\non measured values of ˜LNM, we include this factor.\nB. Background Correction\nTo make use of the phase and amplitude information\nin the VNA-FMR spectra, we first fit the raw spectra to:\nS21(ω,H0) =Aeiφχyy(ω,H0)+C0+C1H0(23)\nwhereAis the signal amplitude, φis the raw phase (in-\nclusive of signal line delay), and C0andC1are complex\noffset and slope corrections to the background. Utiliz-\ning the information in this complex background is key to\nour data processing method. The background-corrected\nsignal can be plotted from the measured values of S21as:\n∆S21(ω,H0) =S21(ω,H0)−(C0+C1H0)\nC0+C1H0(24)\nThis corrects the signal phase for the finite length of the\nsignal line between the VNA source and receiver ports\nand the sample, effectively placing the ports at the sam-\nple position. Additionally, it normalizesthe signal ampli-\ntude by the frequency-dependent losses due to the com-\nplete microwave circuit (cables + CPW + sample). In\nFig. 2(a) and (b), we plot the raw and de-embedded\ndata, respectively. The large complex offset on top of\nwhich the resonance signal is superimposed in (a) repre-\nsentsC0andC1.\nComparison of Eqs. 23 and 24 shows that the change\nin microwave transmission can be written as:\n∆S21(ω,H0) =Aeiφ\nC0+C1H0χyy(ω,H0) (25)\nUsing this form for the background-corrected ∆ S21,\nthe inductance amplitude ˜L(f) is calculated as\n[∆S21/χyy(ω,H0)][i2Z0/(2πf)]. When ˜Lis plotted vs.\nfrequency as in Fig. 4, we note that there can be a small\nphase error that causes Im( ˜L)(f→0)∝negationslash= 0. The correc-\ntion for this phase error is discussed in SI Sec. IV.\nC. Calculation of ˜σNMfrom measured L\nUsing the results for Re( ˜L) and Im( ˜L) (Eqs. 21 and\n22), wecanisolatethe ˜ σNMcontributionasfollows. First,\nthe slope of ˜Lis used to isolate the contribution of ˜LNM:-0.7682-0.7680-0.7678-0.7676-0.7674-0.7672-0.7670Re(S21)\n-0.2032-0.2030-0.2028-0.2026-0.2024-0.2022-0.2020Im(S21)Py/Pt @ 20 GHz\nRaw data\nRe(S21)\nIm(S21)\n-0.0015-0.0010-0.00050.00000.00050.0010\u0001S21\n1.60 1.56 1.52 1.48\u00020 H0 (T)Re(\u0000S21)\nIm(\u0003S21)(a)\n(b)\nDe-embedded data\nFigure 2. Example S21spectrum, acquired at f = 20 .0GHz.\n(a) Raw data, with fits. Note the different background offsets\nof the Re and Im data (left and right axes). (b) De-embedded\n∆S21signal.\nd˜L\ndf=−1\n2sgn(ˆz·ˆn)η(z,Wwg)L12(z,Wwg,l)\nMs\n∗h\ne/bracketleftbig\n(σF\ne−σSOT\ne)+iσSOT\no/bracketrightbig\n(26)\nWe normalize d˜L/dfby˜L0in order to remove any resid-\nual differences in sample-CPW coupling from sample to\nsample. Variation in ˜L0(e.g., as seen in Fig. 4) can be\ncaused by sample-to-sample variations in magnetization,\nincluding dead layer effects at the various FM/NM inter-\nfaces, as well as measurement-to-measurement variations\nin the sample-waveguidespacing, which could be affected\nby small dust particles in the measurement environment.\nFinally, we solve for the effective conductivity.\n/bracketleftbig\n(σF\ne−σSOT\ne)+iσSOT\no/bracketrightbig\n=−sgn(ˆz·ˆn)\nd˜L\ndf\n2˜L0\n\n∗µ0l\nL12(z,Wwg,l)MsdFM\nWwgη(z,Wwg)e\nh(27)\nD. Analysis Protocol\nOur quantitative VNA-FMR analysis protocol is sum-\nmarized below28.9\n1. Complex VNA-FMR data is collected and fit with\nEq. 23.\n2. ∆S21is calculated with Eq. 25 to de-embed the\nsample contribution to the inductance.\n3. ∆S21is converted to sample inductance Lusing\nEq. 9.\n4.Lis normalized by magnetic susceptibility χyy,\nyielding the complex inductance amplitude given\nby Eqs. 21 and 22 (Re( ˜L) and Im( ˜L)).\n5. The phase error of ˜Lis corrected as described in SI\nSec. IV.\n6. Linear fits of ˜L(ω) (using Eqs. 21 and 22) are used\nto extract ˜L0and˜LNM(ω).\n7. The effective conductivities σSOT\noand (σF\ne−σSOT\ne)\nareobtainedfrom( ∂˜L/∂f)/˜L0accordingtoEq. 27.\nIV. DATA AND ANALYSIS\nTo demonstrate the quantitative VNA-FMR tech-\nnique, we measured FMR in metallic stacks consisting\nof substrate/Ta(1.5)/Py(3.5)/NM/Ta(3) and inverted\nstacks of substrate/Ta(1.5)/NM/Py(3.5)/Ta(3) (where\nthenumbersinparenthesesindicatethicknessinnanome-\nters). We focus on a Pt(6) NM layer due to its large in-\ntrinsic SOC, and use Cu(3.3) as a control material with\nnominally negligible SOC16,29,30. We collected room-\ntemperature FMR spectra as a function of out-of-plane\nexternal magnetic field H0with microwave frequencies\nfrom 5GHz to 35GHz and VNA output power of 0 dBm.\nExemplary Re(∆ S21) spectra are shown in Fig. 3. Each\nraw spectrum has been normalized by the complex sig-\nnal background (see Sec. IIIB). In the following discus-\nsion, we use a notation for the bilayers which indicates\nthe sample growth order as read from left-to-right. For\nexample, Py/Pt indicates Py is first deposited onto the\nsubstrate, followed by Pt.\nBoth Py/Cu and Cu/Py samples exhibit a mostly real\nnormalized inductance amplitude (symmetric Lorentzian\ndip for Re(∆ S21) in Fig. 3(a) and (b)) with a magni-\ntudelargelyindependentoffrequency,inaccordancewith\n˜LNM≈0. That is, the signal is dominated by the dipolar\ninductance. In contrast, the lineshape and magnitude of\nthe Py/Pt and Pt/Py data in Fig. 3(c) and (d) exhibit a\nclear frequency dependence as expected for ˜LNM∝negationslash= 0. In\nparticular, the data for Py/Pt indicate that ˜LNMadds\nconstructively with L0, such that Re( ˜L) increases with\nincreasing f. The Pt/Py inductance evolves in an oppo-\nsite sensedue to the stackinversion, leadingtoa decrease\nand eventual compensation of Re( ˜L) at high f. The in-\ncreasingly antisymmetric lineshape for both Py/Pt and\nPt/Pyrevealsthat the magnitude ofIm( ˜L) alsoincreases\nwith frequency, with a sign given by the stacking order.\nBy normalizing the spectra in Fig. 3 to the magnetic\nsusceptibility χ(ω,H0) defined in Eq. S2, we extract the\nRe(ΔS21)\n2.0 1.8 1.6 1.4 1.2\nμ0 H0 (T)-2.0-1.5-1.0-0.50.0x10-3\nPy/Cu(a)\n-2.0-1.5-1.0-0.50.0x10-3\n-2.0-1.5-1.0-0.50.0x10-3\n7.5 GHz 35 GHzPy/Pt\nPt/Py(c)\n(d)-2.0-1.5-1.0-0.50.0x10-3\nCu/Py(b)\nFigure 3. FMR spectra for FM/NM bilayers. Re(∆ S21)\nat several excitation frequencies for different samples: (a )\nPy/Cu, (b) Cu/Py, (c) Py/Pt, and (d) Pt/Py. The change\nin lineshape and amplitude for Py/Pt and Pt/Py clearly\nshows the presence of frequency-dependent inductive terms\nnot present in the Py/Cu and Cu/Py control samples. The\ncolored circles indicate the value of Re(∆ S21)∝Re(L) when\nH0satisfies the out-of-plane FMR condition.\ncomplex inductance amplitude ˜L. Re(˜L) and Im( ˜L) are\nshown in Fig. 4 for all investigated bilayers with a length\nlof 8mm. As shown in Eqs. 21 and 22, Re( ˜L) pro-\nvidesinformationaboutthedipolarinductance( ˜L0, zero-\nfrequency intercept), and −(σF\ne−σSOT\ne) (slope). Simi-\nlarly, the slope of Im( ˜L) reflects −σSOT\no. Immediately10\n504 \u0004\n30\n20\n10\n0Py/Pt\nPt/Py\nPy/Cu\nCu/Py- \u0005 \u0006-30-20-100\n35 30 25 20 15 10 5 0F \u0007 \b \t \n \u000b \f \r \u000ePy/Pt\nPt/Py\nPy/Cu\nCu/Py(a)\n(b)\nR\u000f\u0010\u0011\u0012\u0013\u0014\u0015\u0016\n~I\u0017\u0018\u0019\u001a\u001b\u001c\u001d\u001e\n\u001f\nFigure 4. Frequency dependence of real and imaginary induc-\ntances extracted from S21spectra (symbols) and fits to Eqs.\n21 and 22 (lines). (a) Re( ˜L) for all samples with l= 8mm.\nZero-frequency intercept indicates the dipolar inductive cou-\npling, while the linear slope reflects ( σF\ne−σSOT\ne). (b) Im( ˜L)\nfor all samples, as a function of frequency, where the linear\nslope is governed by σSOT\no.\nevident is the reversalofthe slopefor Py/Ptcomparedto\nPt/Py, which is captured by the sgn function (where ˆ nis\nthe FM/NM interface normal, pointing into the FM, and\nˆzis defined by the coordinate system in Fig. 1). This\nsign-reversal is consistent with the phenomenology ex-\npected for interface-symmetrysensitive effects, e.g., com-\nbined spin pumping and iSHE, as well as iREE. There\nis also a marked difference in the slope magnitude for\nPy/Pt and Pt/Py in panel (b), the implications of which\nare discussed below.\nEach of the inductance terms has some dependence on\nsample length: linear for the dipolar contribution, and\nslightly non-linear for the inductances due to charge flow\nin the NM (see Eqs. 10 and 11). We therefore repeated\nthe measurements shown in Fig. 4 for a variety of sample\nlengths from 4 to 10mm. Fig. 5 shows the measured\ninductance terms ˜L0,∂Re(˜L)/∂f(intercept and slope of\ncurves in Fig. 4(a)), and ∂Im(˜L)/∂f(slope of curves\nin Fig. 4(b)) as a function of sample length. Following\nnormalization by its corresonding ˜L0, each data point in\nFig. 5(b) provides a value of ( σF\ne−σSOT\ne) (see Eq. 27).\nSimilarly, datapoints in panel(c) providevaluesof σSOT\no.(a)\n(b)\n(c)\n !\"\nFigure 5. ˜L(f= 0) and ∂˜L/∂fextracted from data as in Fig.\n4 vs. sample length for all samples. (a) Dipolar inductive\ncoupling ˜L0. (b) From ∂[Re(˜L)]/∂f, we extract ( σF\ne−σSOT\ne).\n(c) From ∂[Im(˜L)]/∂f, we extract σSOT\no. Dashed lines are\nguides based on Eqs. M8 and M9 with values of σSOT\noand\n(σF\ne−σSOT\ne) calculated as described in the Methods. Several\nmeasurements were repeated to demonstrate reproducibilit y.\nThese valuesareaveragedto provideasingle ( σF\ne−σSOT\ne)\nandσSOT\nofor each sample type. Results are summarized\nin Table I. The dashed lines in Fig. 5(b) and (c) are\ncalculated curves based on these average values and the\nlength dependence of ˜L.\nBecause σSOT\neandσF\nehave the same phase and fre-\nquency dependence, we use control samples where we re-\nplace the Pt with Cu, wherein it is generally accepted\nthat both the SHE for Cu and the REE at the Py/Cu in-\nterface are negligible16,29,30. Furthermore, the Cu thick-11\nnessischosensothat itexhibits thesamesheetresistance\nas the Pt layer, so that the two samples have identical σF\ne\n(see Eq. 31). Subtraction of the time-reversal-even con-\nductivity for the Py/Cu control samples from the time-\nreversal-even conductivity for the Py/Pt samples there-\nfore isolates σSOT\nespecifically for the Py/Pt interface.\nLikewise, any damping-like contributions to σSOT\nodue to\nthe Ta seed layer should also be removed by subtraction\nof the Py/Cu inductance data.\nAdditional data collected for varied NM thickness (to\nbe presented in a future publication) indicates that the\ncharge currents produced by iSOT effects experience a\nshunting effect, whereby some fraction of the interfacial\ncharge current flows back through the sample thickness,\nreducing the inductive signal. This can be modeled as a\ncurrent divider with some of the iSOT-generated current\ncoupling to the 50Ω CPW via image currents, and the\nremainder shunted by the sheet conductance of the sam-\nple. Final values of the extracted conductivities reported\nin Table I have been corrected to account for current\nshunting (see SI Sec. V for more details). Comparison\nof the shunt-corrected SOT conductivities makes evident\nthatthefield-likechargecurrentsarecomparabletothose\ndue to damping-like spin-charge conversion processes.\nWe can compare our measured values of σSOT\neand\nσSOT\noto measurements made by other groups using dif-\nferent techniques. Garello, et al.9use the harmonic\nHall technique and Miron, et al.2investigate domain\nwall nucleation to quantify the spin-orbit torque ex-\nerted on Co sandwiched between Pt and AlO x. Con-\nverting their measured values of field-like SOT field\nper unit current density to our metric σSOT\ne, they find\n1.1×106Ω−1m−1and 1.9×107Ω−1m−1. Nguyen, et\nal.21find a similar value of ≈1.3×106Ω−1m−1for a\nPt/CobilayerusingharmonicHallmethods. TheGarello\nand Nguyen results are within an order of magnitude of\nour findings ( −1.48±0.07×105Ω−1m−1for Pt/Py and\n−1.8±0.2×105Ω−1m−1for Py/Pt).\nGarello and Nguyen also report damping-like values\nfor their effective SOT fields. Converted to σSOT\no,\nthey find 5 .8×105Ω−1m−1and≈2.9×105Ω−1m−1, re-\nspectively, which are again within an order of magni-\ntude of our values: 2 .4±0.3×105Ω−1m−1(Py/Pt) and\n0.6±0.2×105Ω−1m−1(Pt/Py).\nV. DISCUSSION\nFor comparison to previous measurements and to the-\nory, we can relate the effective conductivities σSOT\neand\nσSOT\noto microscopic spin-charge conversion parameters\nunderthe assumptionsthat the damping-likeiSOTisdue\nto iSHE only, and the field-like iSOT is from iREE only.\nWe alsorelate the Faradaycontribution to the AC charge\ncurrents in the NM—that is, σF\ne—to sample properties.A. Contributions to effective conductivity, ˜σNM\n1. Effective Faraday conductivity, σF\ne\nTo relate the effective Faraday conductivity, σF\ne, to\nsample parameters, we isolate the Faraday component\nof the induced charge current from Eq. 7:\n\n+dFM/integraldisplay\n−dNMJF(z)dz\n=−sgn(ˆz·ˆn)/parenleftbigg/planckover2pi1\n2e/parenrightbigg\nσF\ne(ˆz×∂tˆm) (28)\nThe charge current is driven by the induced e.m.f., Vx,\naccording to:\nˆx·\n+dFM/integraldisplay\n−dNMJF(z)dz\n=Ix\nw\n=Vx\nZeffl(29)\nThe induced e.m.f. is derived from inductive\nreciprocity31\nVx=−∂φ\n∂t=−µ0Ms/integraldisplay\nVFM[h(r)·∂tˆm]d3r(30)\nwhereh(r) is the magnetic sensitivity function for a cur-\nrent of unit amplitude in the NM layer. We assume this\nfield can be approximated with the Karlqvist equation,\nand use the results for ∂tˆmfrom Sec. IIIA. Subsituting\nEq. 30 into Eq. 29, and equating the result with Eq. 28\nyields the final expression for σF\ne:\nσF\ne=eµ0MsdFM\n/planckover2pi1Zeff(31)\n2. Rashba parameter and σSOT\ne\nWe can relate the even spin-orbit torque conductivity\nσSOT\neto the Rashba parameter αR. We start from the\nfield-like interfacial spin torque per spin tflintroduced by\nKim, et al. (Eq. 12 in Ref. 32):\ntfl= sgn(ˆz·ˆn)kRvs/bracketleftig\nˆm×(ˆj׈z)/bracketrightig/parenleftbigg/planckover2pi1\n2/parenrightbigg\n(32)\nwherekR= 2αRme//planckover2pi12is a wavevector corresponding to\nthe Rashba energy parameter αR,meis the mass of the\nelectron, and vs=PJintgµB/(2eMs) is the spin velocity,\nwith charge current density Jintat the FM/NM interface\nat which the Rashba effect is present, spin polarization\nof the charge current P, Land´ e g-factor g, and Bohr12\nSample ( σF\ne−σSOT\ne)meas (σSOT\no)meas (σSOT\ne)corr (σSOT\no)corr\nPy/Pt −0.45±0.03 1 .0±0.1 −1.48±0.07 2 .4±0.3\nPt/Py −0.69±0.05 0 .31±0.06 −1.8±0.2 0 .6±0.2\nPy/Cu 0 .143±0.006 0 .07±0.03\nCu/Py 0 .04±0.03 0 .06±0.01\nTable I.Effectiveconductivities(inunitsof105Ω−1m−1)andmicroscopic spin-charge conversionparameters (Rash baparameter\nαRand spin Hall angle θSH). Measured values are calculated from measured inductance s (Fig. 5). Corrected values are\ncalculated by subtraction of Cu control to remove the Farada y contribution (in the case of σe) and any contribution from the\nTa interfaces, followed by application of the shunting corr ection (see SI Sec. V).\nmagneton µB. Note that tfl/(/planckover2pi1/2) has units of Hz; that\nis, the same units as ∂tˆm. We can therefore relate Eq.\n32 to the volume-averaged magnetic torque density T\nfrom Eqs. 5 and 6 through the time rate of change of\nthe magnetization: tfldintδ(z)/(/planckover2pi1/2) =∂tˆm, where we\nhave added dintδ(z) to account for the interfacial nature\nof this torque (where dintis an effective thickness of the\ninterface).\n2\n/planckover2pi1dFM/integraldisplay\n0tfldintδ(z)dz=−γ\nMsdFM/integraldisplay\n0T(z)dz (33)\nkRvsˆm×(ˆj׈z)dint=−γ\nMs/planckover2pi1\n2eσSOT\neˆm×(ˆz×E)(34)\nThe final line results from subsituting Eq. 32 and the\neven SOT term from Eq. 6 into Eq. 33. Making the\nsubstitutions for kRandvs, and using E= (Jint/σint)ˆj\nyields:\nαR=/planckover2pi12\n2meσSOT\ne\nσint1\nPdint(35)\nHere,σintis the interfacial conductivity of the FM/NM\ninterface(extractedbymeasuringresistancevs. Pythick-\nness; see SI Sec. VI) and Pis the spin polarization at\nthe FM/NM interface. We use P= 0.6 as determined\nvia spin-wave Doppler measurements in Ref. 33, and as-\nsumedintis one Py lattice constant (0 .354nm)34. We\ntherefore find αR=−5.8±0.3meVnm for the Py/Pt\nsample, and −7.5±0.7meVnm for Pt/Py. These values\naresmallerthan thosemeasuredwith angle-resolvedpho-\ntoelectron spectroscopy (ARPES) for the surface state\nof Au(111) (33meVnm)35, Bi(111) (56meVnm)36, and\nGe(111) (24meVnm)37, and much smaller than the\nBi/Ag(111) interface (305meVnm)38.\nWe can also compare our results for the Rashba\nparameter to a recent theoretical calculation. Kim,\nLee, Lee, and Stiles (KLLS)32have shown that SOT\nand the Dzyaloshinskii-Moriya interaction (DMI) at a\nFM/NM interface are both manifestations of an under-\nlying Rashba Hamiltonian, and predict a straightfoward\nrelationship between the Rashba parameter αR, inter-\nfacial DMI strength Dint\nDMI, and the interfacial field-like\nSOT per spin tfl:αR=/planckover2pi12\n2me/parenleftbiggDint\nDMI\n2A/parenrightbigg\n=/planckover2pi1\nme/parenleftbiggtfl\nvs/parenrightbigg\n(36)\nwhereAis the exchange stiffness.\nFor the Pt/Py stack, the ratio of interfacial DMI,\nDint\nDMI, to bulk exchange Awas previously measured via\na combination of Brillouin light scattering (BLS) and\nsuperconducting quantum interference device (SQUID)\nmagnetometry for samples prepared under nearly identi-\ncal growth conditions, albeit with a stack geometry that\nwas optimized for optical BLS measurements39. The ra-\ntio is a constant value of −0.25±0.01nm−1over a Py\nthickness range of 1.3 to 15nm. As such, this material\nsystem is an ideal candidate to test the quantitative pre-\ndiction of the KLLS theory. Using the experimentally-\ndetermined value for Dint\nDMI/Awith Eq. 36 predicts a\nRashbastrengthof −4.8±0.2meVnm, whichagreeswell\nin sign and magnitude with the result of our iSOT mea-\nsurement for the Pt/Py sample of the same stacking or-\nder, as well as the Py/Pt sample with opposite stacking\norder. Together, the spin wave spectroscopy and iSOT\nmeasurementsclarifytheroleoftheRashbaspin-orbitin-\nteraction as the underlying physical mechanism for both\nDMI and field-like SOT in the Py/Pt system.\n3. Spin Hall angle and σSOT\no\nIn order to develop intuition for Eq. 7 we first derive\nan approximate relationship between σSOT\noand the spin\nHall angle, θSH, applicable when the NM thickness is\nmuch thicker than its spin diffusion length. We assume\nseries resistors 1 /G↑↓+ 1/Gext(interfacial spin-mixing\nconductance + spin conductance of the NM) in a voltage\ndivider model for the spin accumulation at the FM/NM\ninterface due to spin pumping\nµs(z= 0+)ˆs=/planckover2pi1\n2/parenleftbigg\nˆm×∂ˆm\n∂t/parenrightbigg/parenleftbiggG↑↓\nG↑↓+Gext/parenrightbigg\n(37)\nwhereµs(z= 0+) is the spin accumulation at the\nFM/NM interface. Using the result of Eq. 6 from Ref.\n40 for the effective one-dimensional spin conductance of\na NM (where we have set GNM\n2= 0 because we are inter-\nested in only a FM/NM bilayer, not a FM/NM1/NM2\nmultilayer):13\nGext=σ\n2λstanh/parenleftbiggdNM\nλs/parenrightbigg\n(38)\nwhereλsis the spin diffusion length in the NM. The\nintegrated charge current in the NM layer driven by the\nresulting spin chemical potential gradient −∇µs=Qs\nand the inverse spin Hall effect ( Jc∝Qs׈s) is given by\ndNM/integraldisplay\n0Jc(z)dz=dPt/integraldisplay\n0/bracketleftbigg\nσSH−∇µs(z)\ne׈s/bracketrightbigg\ndz(39)\n=σSHµs(z= 0+)\ne(−ˆz׈s) (40)\nassuming dNM>> λ s. The spin Hall conductivity is\nrelated to the spin Hall angle via the Pt charge conduc-\ntivity:σSH=θSHσPt. If we combine Eqs. 37, 38, and\n40 and equate the integrated charge current to that from\nσSOT\noin Eq. 7 we arrive at the final result:\nσSOT\no=σ\n\nθSHRe\nG↑↓\nσ\n2λstanh/parenleftbiggdNM\nλs/parenrightbigg\n+G↑↓\n\n\nǫ(41)\nThe model also accounts for less-than-unity efficiency ǫ\nfor spin transmission into the NM (such that (1 −ǫ) is the\nspin loss fraction, which has been attributed to processes\nsuch as spin memory loss41or promixity magnetism42).\nA more accurate version of Eq. 41 is obtained by re-\nplacing the unitless term in curly brackets with Eq. 11\nfrom Ref. 43:\nσSOT\no=σ/braceleftbigg\nθSH(1−e−dNM/λs)2\n(1+e−2dNM/λs)\n∗|˜G↑↓|2+Re(˜G↑↓)tanh2/parenleftbiggdNM\nλs/parenrightbigg\n|˜G↑↓|2+2Re(˜G↑↓)tanh2/parenleftbiggdNM\nλs/parenrightbigg\n+tanh4/parenleftbiggdNM\nλs/parenrightbigg\n\nǫ\n(42)\nwhere˜G↑↓=G↑↓2λstanh(dNM/λs)/σ. This properly ac-\ncounts for the boundary condition that the spin current\ngoes to zero at the distant surface of the NM.\nEq. 42canbe usedto calculate θSHifweassumevalues\nforλs,G↑↓, andǫ. Iftheseparametersarepresumediden-\ntical for the two stacking orders, we would find spin Hall\nangles that differ by a factor of 4 depending on whether\nPt is deposited on Py, or vice versa. Instead, the large\ndiscrepancy in σSOT\nofor the two stacking orders suggests\ndifferences in the FM/NM interface that affect G↑↓and\nǫ. Given the data presented here, it is possible for us to\nestimate the efficiency with which spins are pumped intothe Pt layer as follows. The total Gilbert damping αtot\nis the sum of intrinsic processes αint, spin pumping into\nthe Pt and Ta layers αPt(Ta), and possible spin memory\nlossαSML.\nαtot=αint+αPt+αTa+αSML (43)\nWe can apply Eq. 43 to each of the stacking or-\nders (Py/Pt and Pt/Py), and use the damping measure-\nments for Py/Cu and Cu/Py control samples as a mea-\nsure ofαint+αTafor Py/Pt and Pt/Py, respectively.\nWe note that that the total Gilbert damping for the\ntwo stacking orders differs by only 8% (see Table S2),\nwhile the odd SOT conductivity differs by a factor of\n4. This suggests that the damping-like procceses con-\ntributing to σSOT\no(i.e. iSHE) add only a small amount\nof enhanced damping, while the majority of spin current\npumped out of the FM is lost and not available for iSHE\nconversion41. If we therefore assume that αSMLis iden-\ntical for the two stacking orders, and that the difference\ninσSOT\nofor the two stacks is due entirely to a differ-\nence in spin-mixing conductance, such that αPt(Py/Pt)\n= 4αPt(Pt/Py), then the resulting system of equations\nis solvable for αPt(Py/Pt) and αPt(Pt/Py), as well as\nαSML. Using the results, we can estimate the spin pump-\ning efficiency factor ǫ≡αsp/(αsp+αSML). We find that\nonly 33% or 13% of the spin current pumped through the\nPt interface is available for iSHE conversion, for Py/Pt\nand Pt/Py samples respectively.\nA more rigorous calculation can be done to estimate\nG↑↓,ǫ, andθSHby simultaneously fitting Eq. 42 and Eq.\n43 for the two stacking orders (using the corrected values\n(σSOT\no)corrfrom Table I and total damping values from\nTableS2). Toperformthisoptimization, weusethefunc-\ntional form for the spin pumping damping contributions\nas presented in Ref. 40, such that αPt(Ta)depends on λs,\nG↑↓, andσin order to implement the spin current back-\nflow correction. We obtained a value for the Pt charge\nconductivity σ= 4.16×106Ω−1m−1from four-probe re-\nsistance measurement on a series of Py/Pt samples with\nvarying Pt thickness, to allow isolation of the Pt contri-\nbution to the total conductivity. Using a value of λs=\n3.4nm from Ref. 41, we obtain a spin Hall angle of θSH\n= 0.28. This falls within the range of published values\nfrom DC spin Hall measurements (0.01–0.33)7,12,44–50.\nIn good agreement with the estimate above, we find\nefficiencies of 34% and 18% for Py/Pt and Pt/Py re-\nspectively. Furthermore, this optimization yields G↑↓=\n8.9×1014Ω−1m−2(for Py/Pt) and 2 .3×1014Ω−1m−2\n(for Pt/Py). Both of these values are below the Sharvin\nconductance51(G↑↓=1×1015Ω−1m−2), which serves as\nthe theoretical upper bound for the spin-mixing conduc-\ntance. This result demonstrates clearly that when Py is\ndeposited on Pt, the FM/NM interface is detrimental to\nspin transport.14\nVI. CONCLUSION\nIn summary, we have quantified both field- and\ndamping-like inverse spin-orbit torques in Ni 80Fe20/Pt\nbilayers using phase-sensitive VNA-FMR measurements\nand an analysis of the sample’s complex inductance that\narises in part from the AC currents due to spin-charge\nconversion. The magnitude of these currents is deter-\nmined by their respective SOT conductivities, a key fig-\nure of merit for characterizating and optimizing oper-\national spintronic devices. Because our technique en-\ntails straightforward post-measurement data processing\nfor an experimental technique that is well-established in\nthe field, it provides a powerful way to unpick a highly\ncomplex experimental system and represents a broadlyapplicable tool for studying strong SOC material sys-\ntems. The technique could evenbe applied to previously-\nacquired VNA-FMR data sets in which only spectro-\nscopic analysis was performed. The measurements pre-\nsented here demonstrate that both Rashba-Edelsteinand\nspin Hall effects must be considered in FM/NM metal-\nlic bilayers. Together with the observation of significant\nvariation in σSOT\nowith respect to FM/NM stacking or-\nder, these results point to interfacial engineering as an\nopportunity forenhancing current-controlledmagnetism.\nACKNOWLEDGMENTS\nThe authors would like to thank Mark Stiles and Mark\nKeller for many helpful discussions and illuminating in-\nsights.\n1F. Freimuth, S. Bl¨ ugel, and Y. 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Buhrman,\nPhysical Review Letters 106, 036601 (2011).\n48M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pern-\npeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao,\nY.-T. Chen, H. Jiao, G. E. W. Bauer, and S. T. B. Goen-\nnenwein, Physical Review Letters 111, 176601 (2013).\n49M. Obstbaum, M. H¨ artinger, H. G. Bauer, T. Meier,\nF. Swientek, C. H. Back, and G. Woltersdorf,\nPhysical Review B 89, 060407 (2014).\n50C.-F. Pai, Y. Ou, L. H. Vilela-Le˜ ao, D. C. Ralph, and\nR. A. Buhrman, Physical Review B 92, 064426 (2015).\n51Y. Liu, Z. Yuan, R. Wesselink, A. A. Starikov, and P. J.\nKelly, Physical Review Letters 113, 207202 (2014).1\nSupplementary Information\nI. SAMPLE FABRICATION\nSampleDeposition Order\nPy/PtSubstrate/Ta(1.5)/Py(3.5)/Pt(6)/Ta(3)\nPt/PySubstrate/Ta(1.5)/Pt(6)/Py(3.5)/Ta(3)\nPy/CuSubstrate/Ta(1.5)/Py(3.5)/Cu(3.3)/Ta(3)\nCu/PySubstrate/Ta(1.5)/Cu(3.3)/Py(3.5)/Ta(3)\nTable S1. Sample deposition orders and metallization thick -\nnesses (in nanometers).\nAll samples were prepared by DC magnetron sputter-\ning in an Ar base pressure of ≈0.07Pa (≈0.5mTorr) and\na chamber base pressure of 3 ×10−6Pa (2×10−8Torr)\non 3-inch wafers of thermally oxidized (100) Si (nominal\nresistivity = 3Ωcm). The wafers were rotated at 1Hz\nto 2Hz during deposition to eliminate growth-induced\nanisotropy, and the sample holder was held at room tem-\nperature. All samples were grown on a 1 .5nm Ta seed\nlayer to promote (111) textured growth, which was then\nfollowed by the FM/NM (or NM/FM) bilayer. X-ray\ndiffraction shows that the Ta seed layer is unordered. A\n3nm Ta cap layer prevents oxidation of the FM and NM\nlayers. It is expected that 1nm to 2nm of the cap layer\nforms the insulator TaO when exposed to air. Depositionorder and film thicknesses are shown in Table S1. The Pt\nand Cu thicknesses were chosen so that the DC conduc-\ntivities (as characterized by a four-probe measurement)\nofthesampleandcontrolwereequal,toensureequalityof\nFaraday induced currents. The wafers were subsequently\ncoatedwith 8 µm ofphotoresistto provideelectricalinsu-\nlation from the CPW and reduce the capacitive coupling\nof the CPWto the metallic layers. The waferswere diced\nto precise sizes using an automatic dicing saw.\nII. MAGNETIC CHARACTERIZATION\nA. Magnetic Susceptibility\nForourgeometry,thedrivingmicrowavemagneticfield\nlies primarily along ˆ y, and we are concerned with the\nAC component of magnetization along ˆ y(see Fig. 1 in\nthe main text for coordinate system). Therefore, the S21\nspectra are fit to the χyycomponent of the complex mag-\nnetic Polder susceptibility tensor in order to extract res-\nonance field, linewidth, amplitude, and phase.\n/bracketleftbigg\nMx\nMy/bracketrightbigg\n=/bracketleftbigg\nχxxχxy\nχyxχyy/bracketrightbigg/bracketleftbigg\nhx\nhy/bracketrightbigg\n(S1)\nχ(ω,H0) =Ms/parenleftigg\n(H0−Meff)2−/parenleftbiggω\nγµ0/parenrightbigg2\n+i2αeffω(H0−Meff)\nγµ0/parenrightigg\n(H0−Meff)iω\nγµ0\n−iω\nγµ0(H0−Meff)\n (S2)\nwhereH0is the externally applied DC field, Meff=Ms−\nH⊥\nkis the effective magnetization, Msis the saturation\nmagnetization, H⊥\nkis the perpendicular anisotropy field,\nωis the drivingfrequency, γis the gyromagneticratio, µ0\nthe vacuum permeability, and αeff=α+γµ0∆H0/(2ω) is\nthe effective damping parameter, with Gilbert damping\nconstant αand inhomogeneous broadening ∆ H0.\nThe frequency dependence of the resonant field Hres\nand linewidth ∆ Hallow extraction of the effective mag-\nnetization Meff=Ms−H⊥\nk, spectroscopic g-factor g,\ninhomogeneous broadening ∆ H0, and Gilbert damping\nparameter α. We used SQUIDmagnetometrytomeasure\nthe magnetization per unit area for all samples. Magne-\ntization, g-factor, and damping values are summarized in\nTable S2.B. Resonance Field Dispersion\nFrom the susceptibility fits to the S21spectra, we ex-\ntract the resonance field as a function of microwave fre-\nquency. This is expected to followthe Kittel dispersionS1\nfor out-of-plane field H0.\nω=µ0γ(Hres−Meff) (S3)\nA plot of µ0Hresvs.f=ω/2πis shown in Fig. S1,\nwith slope set by the gyromagnetic ratio γ=gµB//planckover2pi1, and\ny-intercept set by µ0Meff.2\n2.0\n1.5\n1.0\n0.5\n0.0\n#0Hres(T)\n35302520151050\nFreq(GHz)Py/Pt\nPt/Py\nPy/Cu\nCu/Py\nFigure S1. Resonance field vs. frequency dispersion, to ex-\ntract spectroscopic g-factor, and Meff.\n/s32/s33\n/s34/s33\n/s35/s33\n/s36/s33\n/s37/s33\n/s38/s33\n/s33/s39/s33/s40/s41/s42/s40/s43/s44/s45/s46\n/s36/s34/s40/s47/s38/s33/s48 /s36/s33/s37/s34/s37/s33/s38/s34/s38/s33/s34/s33\n/s49/s50/s51/s52/s40/s43/s53/s42/s54/s46/s40/s55/s56/s57/s55/s58/s40\n/s40/s55/s58/s57/s55/s56/s40\n/s40\n/s40/s55/s56/s57/s59/s60/s40\n/s40/s59/s60/s57/s55/s56/s40\n/s40\nFigure S2. Resonance linewidth vs. frequency, to extract\nGilbert damping constant αand inhomogeneous broadening.\nC. Linewidth and Damping\nThe resonance linewidth is determined by the Gilbert\ndamping constant αand inhomogeneous broadening\n∆H0according to\nµ0∆H=µ0∆H0+2ωα\nγ(S4)\nData and fits of Eq. S4 for the 6mm long samples for\neach deposition order are shown in Fig. S2.\nD. SQUID Measurement\nWe measured in-plane hysteresis curves at room tem-\nperaturetodetermine the saturationmoment ofoursam-ples. This total moment was normalized by the sample\narea to obtain MsdFM(see Table S2).\nIII. DETERMINATION OF SIGNAL PHASE\nWe consider the sample and CPW in a lumped ele-\nment circuit model, in which the sample contributes an\nimpedance iωLto the circuit, in series with the char-\nacteristic impedance Z0of the CPW. Therefore, at the\nsample (ordevice-under-test), the currentis simply given\nby:\nIDUT=V1\nZ0+iωL\n≈V1\nZ0/parenleftbigg\n1−iωL\nZ0/parenrightbigg\n(S5)\n=ICPW+∆I\nforωL << Z 0, and where ICPWis the current in the\nunloaded CPW (with a positive Real current flowing in\nthe +ˆxdirection). Therefore:\n∆I=−/parenleftbiggiωL\nZ0/parenrightbigg\nICPW (S6)\nUsing the dipolar inductance of Eq. 10, and considering\nthe current response at the FMR condition, such that\nχyy=−iγµ0Ms/(2αeffωres) (for CCW precession), we\nfind:\n∆Idip=−γµ2\n0lMsdFMη(z,Wwg)\n8Z0αeffWwgICPW (S7)\nFrom Eq. S7 we see that the change in current is in-\nphase with, but opposite in sign to the current responsi-\nble forhy(asdepicted in Fig. S3(a)). This changein cur-\nrent could be viewed as a change in the CPW resistance.\nThat is, the sample inductance creates a purely dissipa-\ntive response at the FMR condition, which is clearly seen\nin Fig. 3(a) and (b), and is expected for a spin system\non resonance.\nLet us now consider the phase of the currents in the\nCPW due to currents in the NM (from the Faraday and\niSOT processes). These effects are captured by Fig. 1(b-\nd) and the derivation of Sec. IIIA2. For simplicity, we\nfirst focus on the Faraday-type currents in the NM. At\ntimet0, this current is maximum along the ˆ xdirection.\nViathe mutualinductance betweensampleandCPW,an\n“image current” flows in the CPW opposite to the Fara-\ndaycurrentin the NM. Extending this logicto all current\nsources in the NM layer, we produce the phasor diagram\nof Fig. S3(b). This demonstrates clearly that at the\nFMR condition, currents with even time-reversal sym-\nmetry create a dissipative response in the CPW, while\nodd-symmetry currents create a reactive response. The3\nSample Meff(kA/m) g µ0∆H0(mT) α M sdFM(µA)\nPy/Pt 663 .5±0.7 2 .079±0.001 1 .2±0.8 0 .0261±0.0003 2069 ±1\nPt/Py 647 ±1 2 .079±0.003 2 ±2 0 .0241±0.0008 2121 ±1\nPy/Cu 674 ±1 2 .075±0.001 1 .1±0.5 0 .0115±0.0001 2341 ±2\nCu/Py 642 ±1 2 .077±0.001 1 .7±0.9 0 .0129±0.0002 2077 .0±0.4\nTable S2. FMR and SQUID parameters for Py/Pt and Py/Cu bilaye rs.\n(a) (b)\nFigure S3. (a). Phasor diagram describing phase of current\ndue to dipolar coupling to precessing magnetization m, rela-\ntive tohyat the FMR condition. The current ∆ Idipcreates\na dissipative response. (b) Same as (a), but for currents in\nthe CPW due to currents INMcaused by Faraday and iSOT\neffects. Even currents appear dissipative or resistive, odd cur-\nrents appear reactive. Note that all currents are defined suc h\nthat a positive Real current in the CPW flows in the +ˆ xdi-\nrection, and relative magnitudes are not indicated.\nSample φcorr(deg)\nPy/Pt 12 ±1\nPt/Py 11 .6±0.4\nPy/Cu 1 .8±0.8\nCu/Py 7 .2±0.3\nTable S3.\ncontribution of even and odd currents to dissipative or\nreactive response changes as field is swept through the\nresonance condition, resulting in the evolving lineshapes\nobserved in Fig. 3(c) and (d).\nIn order to coherently add the perturbative currents\ndue toL0andLNMto satisfy the above discussion (i.e.\nto combine the effects of Fig. S3(a) and (b) with the\nproper phase assignment), we find:\n∆Itot= ∆IL0+∆ILNM\n=/parenleftbigg\n−iωL0\nZ0−ωLNM\nZ0/parenrightbigg\nICPW (S8)\n=−/parenleftbiggiωLtot\nZ0/parenrightbigg\nICPW (S9)\nwhereLtot≡L0−iLNM. Using this result, we recover\nthe complex inductance relationships given by Eqs. 21\nand 22.\nIV. PHASE ERROR OF ∆S21\nThe background correction procedure of Sec. IIIB re-\nquires one further phase correction in order to enforce\nRe(L) (fH)40\n30\n20\n10\n0Im(L) (fH)\n-30-20-100\n35 30 25 20 15 105 0\nFreq (GHz)Py/Pt\nPt/Py\nPy/Cu\nCu/Py\n35 30 25 20 15 105 0\nFreq (GHz)Raw Phase-Corrected~ ~\nFigure S4. Correction of phase to enforce Im( ˜L)(f= 0) = 0\nforl= 6mm sample. Raw data (left panels) show a small,\nnon-zero component of Im( ˜L) atf= 0, which is unphysical.\nWe therefore apply a small correction to eliminate this non-\nzeroy-intercept, resulting in the phase-corrected data (right\npanels).\nthat Im( ˜L)(f= 0) = 0, as any finite Im( ˜L) at zero fre-\nquency would be unphysical. However, as can be seen\nin the raw data of Fig. S4, the intercept of Im( ˜L) at\nf= 0 is indeed a small, finite number (left panels). In\naddition to the background phase correction described in\nEq. 25, we therefore force an additional phase correction\nφcorr= arctan[Im( ˜L)(f= 0)/Re(˜L)(f= 0)]. The φcorr\nnecessary for each sample is shown in Table S3.\nV. SHUNTING CORRECTION\nOursamplesexhibitashuntingeffectwhenthemetallic\nthicknesses are such that the sheet resistance of the sam-\nple drops below 50Ω ( Z0, the characteristic impedance\nof our CPW). This is similar to the shunting effect de-\nscribed in Ref. S2. However, in that case, the atten-\nuation of voltage signals as sample thickness increases\nfollows immediately from Ohm’s law and the decreasing\nresistance across which the iSHE voltage is measured.\nIn our inductive measurements, the AC currents driven\nbyiSOTgeneratesignalvoltagesacrossthecharacteristic\nimpedanceoftheCPW, Z0. However,whenthesampleis\nthick enough, there is also a current return path through\nthe thickness of the sample. For very thick samples, the4\nintegrated current through the sample thickness is zero\n(equal forward and return currents), and the inductive\nsignal drops to zero.\nWethereforemodeltheiSOTeffectsasacurrentsource\nwhich drives current through parallel resistances Z0and\nRs, whereRsis the measuredsheet resistanceofoursam-\nple. For all samples in this study Rswas found to be\n≈34Ω. In this model, only the fraction of the total cur-\nrent generatedby iSOT that flowsthroughthe Z0branch\ncan generatean inductive signal, correspondingto a frac-\ntionRs/(Z0+Rs)≈0.4ofthe totalcurrent. Wetherefore\nscaleσSOT\neandσSOT\noby≈2.5. Note that the Faraday\neffect acts as a source of emf, such that the currents due\nto the Faraday effect are observed to increase linearly\nwith sample thickness, in accordance with Ohm’s Law.\nTherefore, we do not correct σF\neby the same shunting\nfactor.\nVI. MEASUREMENT OF PERMALLOY\nRESISTIVITY\nIn order to determine the interface conductivity σint\nused for determination of αRin main text Eq. 35, wemeasuredthe resistivity ofTa(1.5)/Py( dPy)/Pt(6)/Ta(3)\nand Ta(1.5)/Pt(6)/Py( dPy)/Ta(3) films (thicknesses in\nnanometers) as a function of Py film thickness, dPy(Fig.\nS5). Ineachcase, wefindthatthedataarewell-described\nby a simple model in which the Py resistivity is indepen-\ndent of thickness, and adds as a parallel resistance with\nthe Pt and Ta conducting layers. That is, the total sheet\nresistance Rsis given by: 1 /Rs=dPy/ρ0+ 1/Rother,\nwhereρ0is the Py bulk resistivity, and Rotheris the com-\nbined sheet resistance of the Pt and Ta layers. We mul-\ntiply the measured sheet resistance by the Py thickness,\nsuch that\nRsdPy=dPy\ndPy\nρ0+1\nRother(S10)\nFrom the fits shown in Fig. S5, we find ρ0=\n21.9±0.2×10−8Ωm and Rother= 49.5±0.4Ω for the\nPy/Pt sample, and , ρ0= 22.78±0.04×10−8Ωm and\nRother= 60.7±0.1Ω for the Pt/Py sample. To calulate\nσintfor Eq. 35, we simply use the inverse of these bulk\nresistivity values.\n[S1] C. Kittel, Introduction to Solid State Physics , 8th ed.\n(Wiley, 2004).\n[S2] H. Jiao and G. E. W. Bauer,\nPhysical Review Letters 110, 217602 (2013)./s32/s33/s34/s35/s36/s32/s34/s37/s38\n/s32/s34/s34\n/s33/s34\n/s34/s39/s40/s35/s41/s42/s43/s35/s44/s45/s35/s46/s47\n/s48/s34/s32/s33/s32/s34/s33/s34\n/s41/s42/s43/s35/s44/s49/s46/s47/s35/s42/s50/s44/s51/s47/s52/s42/s43/s44/s41/s42/s43/s47\n/s35/s42/s43/s44/s41/s42/s43/s47/s52/s42/s50/s44/s51/s47\n/s35\n/s35\nFigure S5. Measured sheet resistance vs. Py\nthickness dPyfor both stacking orders of Py\nand Pt: Ta(1.5)/Py( dPy)/Pt(6)/Ta(1.5) and\nTa(1.5)/Pt(6)/Py( dPy)/Ta(1.5). Eq. S10 is used as the\nfit function."    },    {        "title": "2109.06840v1.Quantum_magnetism_of_ferromagnetic_spin_dimers_in__α__KVOPO__4_.pdf",        "content": "Quantum magnetism of ferromagnetic spin dimers in \u000b-KVOPO 4\nPrashanta K. Mukharjee,1,\u0003K. Somesh,1,\u0003K. M. Ranjith,2M. Baenitz,2Y.\nSkourski,3D. T. Adroja,4, 5D. Khalyavin,4A. A. Tsirlin,6,yand R. Nath1,z\n1School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695551, India\n2Max Planck Institute for Chemical Physics of Solids, N othnitzer Str. 40, 01187 Dresden, Germany\n3Dresden High Magnetic Field Laboratory (HLD-EMFL),\nHelmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany\n4ISIS facility, Rutherford Appleton Laboratory, Chilton Oxon OX11 0QX, United Kingdom\n5Highly Correlated Matter Research Group, Physics Department,\nUniversity of Johannesburg, Auckland Park 2006, South Africa\n6Experimental Physics VI, Center for Electronic Correlations and Magnetism,\nInstitute of Physics, University of Augsburg, 86135 Augsburg, Germany\n(Dated: September 15, 2021)\nMagnetism of the spin-1\n2\u000b-KVOPO 4is studied by thermodynamic measurements,31P nuclear\nmagnetic resonance (NMR), neutron di\u000braction, and density-functional band-structure calculations.\nFerromagnetic Curie-Weiss temperature of \u0012CW'15:9 K and the saturation \feld of \u00160Hs'11:3 T\nsuggest the predominant ferromagnetic coupling augmented by a weaker antiferromagnetic exchange\nthat leads to a short-range order below 5 K and the long-range antiferromagnetic order below TN'\n2:7 K in zero \feld. Magnetic structure with the propagation vector k= (0;1\n2;0) and the ordered\nmagnetic moment of 0.58 \u0016Bat 1.5 K exposes a non-trivial spin lattice where strong ferromagnetic\ndimers are coupled antiferromagnetically. The reduction in the ordered magnetic moment with\nrespect to the classical value (1 \u0016B) indicates sizable quantum \ructuations in this setting, despite\nthe predominance of ferromagnetic exchange. We interpret this tendency toward ferromagnetism as\narising from the e\u000bective orbital order in the folded chains of the VO 6octahedra.\nI. INTRODUCTION\nAnderson's superexchange theory [1] requires that in-\nteractions in 3 dmagnetic insulators are predominantly\nantiferromagnetic (AFM) in nature. Ferromagnetic\n(FM) interactions are only possible when magnetic ions\nare close apart, but in this case equally strong AFM inter-\nactions will typically occur between more distant atoms,\nunless special conditions such as charge order [2, 3] or\norbital order [4, 5] are met. In contrast to dozens of\nwell documented material candidates for quantum AFM\nspin-1\n2chains, only a handful of FM spin-chain com-\npounds have been reported. This disparity is in fact\nnot undesirable, because spin-1\n2antiferromagnets show\nintriguing behavior caused by underlying quantum \ruc-\ntuations [6, 7], which do not occur in the ferromagnetic\ncase.\nHere, we introduce a material that remarkably departs\nfrom this general paradigm. Featuring V4+as the mag-\nnetic ion,\u000b-KVOPO 4is dominated by short-range FM\ninteractions. It further reveals an antiferromagnetically\nordered ground state caused by residual AFM long-range\ninteractions, wherein the reduction in the respective or-\ndered magnetic moment signals quantum \ructuations\nthat appear despite the strong proclivity for ferromag-\nnetism.\n\u0003These authors have equal contribution.\nyaltsirlin@gmail.com\nzrnath@iisertvm.ac.in\u000b-KVOPO 4belongs to the group of vanadyl com-\npounds with the general formulas AVOXO4and\nAVXO4F, whereX= P or As, and Ais a monova-\nlent cation. The common crystallographic feature of this\nfamily is the presence of structural chains of the VO 6oc-\ntahedra linked by corner-sharing. The tetrahedral XO4\ngroups connect the chains into a three-dimensional net-\nwork, yet leaving large channels for the A+ions that\nremain su\u000eciently mobile, especially at elevated tem-\nperatures. This feature triggered interest in AVOXO4\nas potential battery materials [8{10], whereas concurrent\nmagnetism studies have also led to very encouraging re-\nsults.\nFrom the magnetism perspective, the crucial feature of\nAVOXO4is that their structural chains are not the direc-\ntion of predominant magnetic interactions, even though\nthe intrachain V{V distances are much shorter than the\ninterchain ones [11, 12]. It has been shown that mono-\nclinic NaVO XO4[13, 14] and AgVOAsO 4[11, 15], as well\nas triclinic \"-LiVOPO 4[16], all adopt an intriguing pat-\ntern of crossed bond-alternating spin chains, which are\nperpendicular to the structural chains. These compounds\nreveal a \feld-induced quantum phase transition and an\nunusual double-dome regime of Bose-Einstein condensa-\ntion (BEC) of magnons in high magnetic \felds [12].\nIn the following, we report a comprehensive study of \u000b-\nKVOPO 4[17, 18], which was so far not on radar of mag-\nnetism studies. It is compositionally similar but struc-\nturally di\u000berent (structure: orthorhombic, space group:\nPna21) from the AVOXO4compounds with A= Li,\nNa, and Ag. The larger K+ion stabilizes the non-\ncentrosymmetric KTiOPO 4-type variety of the crystalarXiv:2109.06840v1  [cond-mat.mtrl-sci]  14 Sep 20212\nc\nb(a)\n(b)\nJ2zV1zV2J1\n[110]\nV1K\nV2\nFIG. 1. (a) Crystal structure of \u000b-KVOPO 4features chains of\nVO6octahedra arranged along the [110] direction. (b) Each\nchain is folded and shows a nearly orthogonal con\fguration of\nthe magnetic V dxyorbitals because of the di\u000berent directions\nof the local z-axes (short V{O bonds), which are denoted with\nthe gray arrows. VESTA software [19] was used for crystal\nstructure visualization.\nstructure, where chains of the VO 6octahedra are folded\nand comprise two nonequivalent vanadium sites, V1 and\nV2 (Fig. 1). This structural modi\fcation has drastic\nrepercussions for the magnetism. Whereas other com-\npounds of the family are AFM in nature, \u000b-KVOPO 4is\ndominated by a FM interaction, yet it shows prominent\nquantum e\u000bects revealed by the reduced ordered mag-\nnetic moment.\nII. METHODS\nPolycrystalline sample of \u000b-KVOPO 4was synthesized\nby a solid-state reaction from the stoichiometric mix-\nture of KPO 3and V 2O4(Aldrich, 99.995%). The KPO 3\nprecursor was obtained by heating KH 2PO4(Aldrich,\n99.995%) for 5 hrs at 300\u000eC in air. The reactants were\nground thoroughly, pelletized, and \fred at 550\u000eC for 48\nhrs in \rowing argon atmosphere with intermediate grind-\nings. The phase purity of the sample was con\frmed by\npowder x-ray di\u000braction (XRD) measurement at room\ntemperature using a PANalytical powder di\u000bractometer\n(CuK\u000bradiation,\u0015avg'1:5418 \u0017A).\nFigure 2 shows the powder XRD pattern at room\ntemperature. Rietveld re\fnement of the acquired XRD\ndata was performed using the FULLPROF software package\n[20]. The initial structure parameters were taken from\nRef. [18]. All the peaks could be re\fned using Pseudo-\nVoigt function. The best \ft was obtained with a good-\nness of \ft\u001f2'3:4. The obtained lattice parameters\n[a= 12:7612(2) \u0017A,b= 6:3658(2) \u0017A,c= 10:5052(1) \u0017A,\nand V cell= 853.40(4) \u0017A3] are in close agreement with the\nprevious report [18].\nMagnetization ( M) was measured as a function of\ntemperature (2.1 K \u0014T\u0014380 K) and magnetic \feld\n(H) using the vibrating sample magnetometer (VSM) at-\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48\n/s32/s32 /s73\n/s111/s98/s115\n/s32 /s73\n/s99/s97/s108/s99\n/s32/s66/s114/s97/s103/s103/s32/s112/s111/s115/s105/s116/s105/s111/s110/s115\n/s32 /s73\n/s111/s98/s115/s32/s45/s32 /s73\n/s99/s97/s108/s99/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/s114/s101/s101/s115/s41/s50\n/s32 /s32/s51/s46/s52/s32FIG. 2. Powder x-ray di\u000braction of \u000b-KVOPO 4collected at\nT= 300 K. The solid black line denotes the Rietveld re\fne-\nment \ft of the data. The Bragg peak positions are indicated\nby green vertical bars and bottom blue line indicates the dif-\nference between experimental and calculated intensities.\ntachment to the Physical Property Measurement System\n[PPMS, Quantum Design]. Speci\fc heat ( Cp) as a func-\ntion of temperature was measured down to 0 :35 K using\nthe thermal relaxation technique in PPMS by varying\nmagnetic \feld from 0 to 14 T. For T\u00142 K, measure-\nments were performed using an additional3He attach-\nment to PPMS. High-\feld magnetization was measured\nin pulsed magnetic \feld at the Dresden high magnetic\n\feld laboratory [21, 22].\nThe NMR measurements were carried out using pulsed\nNMR techniques on31P (nuclear spin I= 1=2 and gy-\nromagnetic ratio \rN=2\u0019= 17:235 MHz/T) nuclei in the\ntemperature range 1.8 K \u0014T\u0014250 K. The NMR mea-\nsurements were done at a radio frequency of 24 :76 MHz.\nSpectra were obtained by sweeping the magnetic \feld\nat a \fxed frequency. The NMR shift K(T) = [Href\u0000\nH(T)]=H(T) was determined by measuring the resonance\n\feld of the sample [ H(T)] with respect to the nonmag-\nnetic reference H 3PO4(resonance \feld Href). The31P\nspin-lattice relaxation rate 1 =T1was measured as a func-\ntion of temperature using the inversion recovery method.\nNeutron powder di\u000braction (NPD) measurements [23]\nwere carried out at various temperatures down to 1.5 K\nusing the time-of-\right di\u000bractometer WISH at the ISIS\nFacility, UK [24]. The Rietveld re\fnements of the NPD\ndata were executed using the FullProf software pack-\nage [20].\nDensity-functional-theory (DFT) band-structure cal-\nculations were performed in the FPLO code [25] using\nPerdew-Burke-Ernzerhof (PBE) \ravor of the exchange-\ncorrelation potential [26]. Correlation e\u000bects in the V\n3dshell were taken into account on the mean-\feld level\n(DFT+U) with the on-site Coulomb repulsion Ud= 3 eV,3\n1\n0\n10TN\nExperiment (1 T)\nExperiment (1 T)\nCurie-Weiss fitcoupled FM dimers\nJ J J( model)1 3/c45 /c45b\n100\n100 200 300 400300\n0\n0600900T(K)\nT(K)234\nχ(10 cm /mol)/c452 3\nχ/c45 /c45 /c451 2 3 1(10 cm /mol)4TN\n6 8 2 0d dTχ/ (arb. units)\nFIG. 3. Upper panel: magnetic susceptibility \u001f(T) measured\nin the applied \feld of \u00160H= 1 T. The solid line is the \ft\nwith the model of coupled FM dimers with J1=\u0000150 K,\nJ3= 12 K, and Jb= 1:5 K (see also Fig. 12c and Sec. III E).\nLower panel: inverse susceptibility (1 =\u001f) as a function of tem-\nperature and the CW \ft using Eq. (2).\nHund's coupling Jd= 1 eV, and double-counting correc-\ntion in the atomic limit [27]. Exchange couplings entering\nthe spin Hamiltonian,\nH=X\nhijiJijSiSj (1)\nwhereS=1\n2and the summation is over bonds hiji, were\nobtained by a mapping procedure from total energies of\ncollinear spin con\fgurations [28, 29]. Alternatively, we\nused superexchange theory of Refs. [30, 31], as further\nexplained in Sec. III E.\nQuantum Monte-Carlo (QMC) simulations were per-\nformed using the loop [32] and dirloop_sse [33] algo-\nrithms of the ALPS package [34] on the L\u0002L=2 \fnite\nlattices with periodic boundary conditions and L\u001436.\nThe spin lattice used in the simulations is detailed in\nSec. III E.\nIII. RESULTS\nA. Magnetization\nMagnetic susceptibility \u001f(T) [\u0011M(T)=H] measured\nin an applied \feld of \u00160H= 1 T is displayed in the\nExperiment (1.5 K)\n0.6\n0.4\n0.2\n0.0\n5 0\nμ0H(T)10 15 20 250.81.0\nM M/sat20 15 10 5 0 25\ndM dH/ (arb. units)coupled\nFM dimersFIG. 4. Magnetization vs \feld measured at T= 1:5 K and\nthe \ft with the model of coupled FM dimers ( J1\u0000J3\u0000Jb)\nusing the same parameters as in Fig. 3. The inset shows\ndM=dH vsH. The colored arrows indicate the \feld range\nabove\u00160Hs'11:3 T where the residual curvature of M(H)\nis observed (see text for details).\nupper panel of Fig. 3. With decreasing temperature,\n\u001f(T) increases in a Curie-Weiss (CW) manner, as ex-\npected in the high-temperature regime, and then passes\nthrough a broad maximum at Tmax\n\u001f'5 K, indica-\ntive of an AFM short-range order. A small kink at\nTN'2:7 K indicates the magnetic ordering transition,\nwhich is more pronounced in the ( d\u001f=dT ) vsTplot\nshown in the inset. Measurements in the weak applied\n\feld of 20 mT under zero-\feld-cooled (ZFC) and \feld-\ncooled (FC) conditions showed no bifurcation, thus ruling\nout spin freezing around or below TN. In contract to the\notherAVOXO4compounds with the prominent Curie\ntail below 5\u000010 K [11, 16], the low-temperature suscep-\ntibility of\u000b-KVOPO 4saturates at a constant value and\nprovides testimony to the high quality of the polycrys-\ntalline sample with a negligible amount of paramagnetic\nimpurities and/or defects.\nThe\u001f(T) data in the paramagnetic (PM) region were\n\ftted with the Curie-Weiss law,\n\u001f(T) =C\nT\u0000\u0012CW(2)\nwhereCis the Curie constant, and \u0012CWis the Curie-\nWeiss temperature. The \ft above 150 K returns C'\n0:386 cm3K/mol and \u0012CW'15:9 K. TheCvalue corre-\nsponds to the paramagnetic e\u000bective moment of 1.756 \u0016B,\nwhich is similar to 1.73 \u0016Bexpected for spin-1\n2. More\ninterestingly, the positive Curie-Weiss temperature sig-\nnals predominant FM couplings despite the susceptibility\nmaximum above TNand the associated short-range AFM\norder, which is typical for low-dimensional antiferromag-\nnets.\nMagnetization curve measured in pulsed magnetic\n\felds up to 60 T at the base temperature of 1.5 K (Fig. 4)\napproaches saturation around 12 T. This \feld de\fnes the\nenergy required to overcome AFM interactions and po-\nlarize the spins. It serves as a witness for AFM couplings\nthat should be weaker than the FM ones, though. Inter-4\nestingly, the magnetization curve is nearly linear. Its cur-\nvature is smaller than in the other low-dimensional V4+\nmagnets [21, 35], including the AVOXO4compounds\nwithA= Li, Na, and Ag [12{14]. Along with the\nFM Curie-Weiss temperature, this reduced curvature of\nM(H) sets\u000b-KVOPO 4apart from other V4+materials\nwith the similar composition.\nAnother peculiarity of \u000b-KVOPO 4is its behavior near\nsaturation, where a residual curvature is observed above\nthe kink at \u00160Hs'11:3 T beforeM(H) becomes com-\npletely \rat around 17 T (Fig. 4, inset). Similar features\nhave been reported in BaCdVO(PO 4)2[36{38] and as-\nsigned to a spin-nematic state expected in a strongly frus-\ntrated square-lattice antiferromagnet in the vicinity of\nsaturation [39]. However, in the \u000b-KVOPO 4case a more\ntrivial explanation, the distribution of saturation \felds\ndepending on the \feld direction, could be an equally\nplausible reason for the residual curvature above Hs. Ex-\nperiments on single crystals would be interesting in order\nto pinpoint the exact origin of this feature.\nB. Speci\fc Heat\nSpeci\fc heat ( Cp) measured in zero magnetic \feld is\nshown in the upper panel of Fig. 5. In the high tem-\nperature region, Cp(T) is entirely dominated by phonon\nexcitations, whereas at low temperatures magnetic con-\ntribution becomes prominent. The \u0015-type anomaly at\nTN'2:7 K con\frms the magnetic transition.\nMagnetic contribution Cmagwas separated by sub-\ntracting the estimated phonon contribution ( Cph) from\nthe total measured Cp(T). To this end, the data above\n40 K were \ftted by a linear combination of four Debye\nfunctions [40, 41]\nCph(T) = 9R4X\nn=1cn\u0012T\n\u0012Dn\u00133Z\u0012Dn\nT\n0x4ex\n(ex\u00001)2dx: (3)\nHere,Ris the universal gas constant, the coe\u000ecients\ncnrepresent the number of distinct atoms in the for-\nmula unit, and \u0012Dnare the corresponding Debye tem-\nperatures. The resulting Cmagand the respective mag-\nnetic entropy Smagobtained by integrating Cmag(T)=T\n(Fig. 5) suggest that about 75% of the total magnetic\nentropyRln 2 = 5:76 J/mol K is released below 10 K.\nAdditional magnetic entropy should then be released at\nhigher temperatures, but it is di\u000ecult to extract because\nthe phonon term becomes predominant. This observation\ncorroborates the scenario of weak AFM couplings that\ncoexist with a much stronger FM exchange. The AFM\ncouplings are responsible for the e\u000bects below 10 K (both\nshort-range and long-range magnetic order revealed by\nthe susceptibility data) and account for the correspond-\ning 75% of the entropy, whereas the FM coupling is re-\nsponsible for the remaining 25% released at higher tem-\nperatures.\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s49/s48/s50/s48/s51/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s49/s50\n/s32/s32\n/s32/s67\n/s112\n/s32/s67\n/s112/s104\n/s32/s67\n/s109/s97/s103/s67\n/s112/s32/s40/s74/s47/s109/s111/s108/s45/s75/s41\n/s84 /s32/s40/s75/s41/s32/s67\n/s109/s97/s103/s47/s84 /s32/s40/s74/s47/s109/s111/s108/s45/s75/s50\n/s41\n/s84 /s32/s40/s75/s41/s48/s50/s52/s54\n/s83\n/s109/s97/s103/s32/s40/s74/s47/s109/s111/s108/s45/s75/s41/s82 /s108/s110/s50FIG. 5. Upper panel: Speci\fc heat ( Cp) vsTfor\u000b-KVOPO 4\nin zero applied \feld. The dashed red line is the phonon con-\ntribution to the speci\fc heat ( Cph) using Debye \ft [Eq. (3)].\nThe solid blue line indicates the magnetic contribution to the\nspeci\fc heat Cmag. Lower panel: Cmag/TandSmagvsTin\nthe left and right y-axes, respectively.\n/s32/s32\n/s32/s48/s32/s84\n/s32/s52/s32/s84\n/s32/s54/s32/s84\n/s32/s55/s32/s84\n/s32/s56/s32/s84\n/s32/s57/s32/s84\n/s32/s49/s48/s32/s84\n/s32/s49/s49/s32/s84\n/s32/s49/s50/s32/s84\n/s32/s49/s52/s32/s84/s67\n/s112/s47/s84 /s32/s40/s74/s47/s109/s111/s108/s45/s75/s50\n/s41\n/s84 /s32/s40/s75/s41/s80/s77\n/s32 /s32 /s72 /s32/s40/s84/s41\n/s84\n/s78/s32/s40/s75/s41/s65/s70/s77\nFIG. 6. Speci\fc heat divided by temperature ( Cp=T) vsTfor\n\u000b-KVOPO 4measured in di\u000berent \felds in the low- Tregime.\nInset:HvsTNphase diagram.5\n/s49/s46/s51/s56 /s49/s46/s52/s48 /s49/s46/s52/s50 /s49/s46/s52/s52/s50/s48/s32/s75/s50/s50/s48/s32/s75\n/s49/s48/s48/s32/s75\n/s56/s48/s32/s75\n/s53/s48/s32/s75\n/s51/s48/s32/s75\n/s49/s54/s32/s75\n/s49/s48/s32/s75\n/s55/s32/s75\n/s53/s32/s75/s51/s49\n/s80/s32/s78/s77/s82\n/s50/s52/s46/s55/s54/s32/s77/s72/s122\n/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s48/s72 /s32/s40/s84/s41/s51/s46/s53/s32/s75\n/s72\n/s114/s101/s102\nFIG. 7. Field-sweep31P NMR spectra at di\u000berent temper-\natures (for T > T N) measured at 24.76 MHz. The vertical\ndashed line corresponds to the31P resonance frequency of\nthe reference sample H 3PO4. The solid line is the \ft to the\n30 K spectrum with the \ftting parameters Kiso'1:03 %,\nKx'1:28 %,Ky'1:16 %, andKz'0:65 %.\nWe also followed the evolution of TNin the applied\n\feld. The transition temperature is nearly unchanged up\nto 4 T and decreases gradually in higher \felds. Such a\nbehavior is intermediate between classical antiferromag-\nnets, where TNis systematically reduced by the \feld [42],\nand low-dimensional quantum antiferromagnets, where\nTNinitially increases and then becomes suppressed, thus\nleading to a non-monotonic phase boundary [43, 44]. The\nabsence of such a non-monotonic phase boundary, despite\nthe presence of the susceptibility maximum due to short-\nrange order, also distinguishes \u000b-KVOPO 4from a typical\nlow-dimensional antiferromagnet.\nC.31P NMR\n1. NMR Spectra\nThe\u000b-KVOPO 4structure contains two nonequivalent\nP sites and both of them are coupled to the V4+ions in\n/s49/s48 /s49/s48/s48/s48/s49/s50/s51/s52\n/s32/s75\n/s120 /s32\n/s32/s75\n/s121 \n/s32/s75\n/s122 \n/s32/s75\n/s105/s115/s111\n/s32/s32/s75/s32 /s40/s37/s41\n/s84 /s32/s40/s75/s41/s51/s49\n/s80/s32/s78/s77 /s82\n/s50/s52/s46/s55/s54/s32/s77 /s72/s122\n/s48/s46/s48/s48 /s48/s46/s48/s49 /s48/s46/s48/s50 /s48/s46/s48/s51 /s48/s46/s48/s52/s48/s49/s50/s51/s52\n/s32/s75\n/s120 \n/s32/s75\n/s121 \n/s32/s75\n/s122 \n/s32/s75\n/s105/s115/s111\n/s32/s32/s75 /s32/s40/s37/s41\n/s49/s46/s52/s32/s84/s32/s40/s99/s109/s51\n/s47/s109/s111/s108/s41FIG. 8. Upper panel: NMR shift components ( Kx,Ky, and\nKz) and the isotropic shift ( Kiso) as a function of T. Lower\npanel:Kvs\u001fmeasured at 1.4 T is plotted with temperature\nas an implicit parameter. The solid lines are the linear \fts as\ndescribed in the text.\neach chain. Both the P sites reside in an almost sym-\nmetric position between the two V4+ions.31P being a\nI= 1=2 nucleus, one expects a single and narrow spectral\nline.\nFigure 7 presents the31P NMR spectra measured at\ndi\u000berent temperatures. We observed a single spectral line\nat high temperatures but the line shape is found to be\nasymmetric, similar to that observed for Zn 2VO(PO 4)2\nand Sr(TiO)Cu 4(PO 4)4[45, 46]. The single spectral\nline implies that both the P-sites are almost equivalent.\nFurther, the asymmetric line shape is likely due to the\nanisotropy in \u001f(T) and/or asymmetric hyper\fne coupling\nbetween the31P nuclei and V4+spins. With decrease\nin temperature, the line broadens and the asymmetric\nline shape becomes more pronounced. At very low tem-\nperatures, when the system approaches TN, the spectra\nbroaden abruptly.\n2. NMR Shift\nAnisotropic components of the NMR shift ( K) as a\nfunction of Twere estimated by \ftting each spectrum.\nThe \ft of the spectrum for T= 30 K is shown in Fig. 7.6\nThe extracted K(T)'s corresponding to three di\u000berent\ncrystallographic directions ( Kx,Ky, andKz) and the\nisotropic NMR shift [ Kiso= (Kx+Ky+Kz)=3] are pre-\nsented in Fig. 8. With decrease in temperature, all the\ncomponents increase in a CW manner and then show a\nbroad maximum around 5 K, similar to \u001f(T). Unlike\nbulk\u001f(T),K(T) is insensitive to free spins and defects\nand allows a more reliable estimate of the intrinsic sus-\nceptibility.\nThe relation between K(T) and\u001f(T) is typically writ-\nten as\nK(T) =K0+Ahf\nNA\u001fspin(T); (4)\nwhereK0is the temperature-independent NMR shift,\nAhfis the hyper\fne coupling constant between the31P\nnuclei and V4+electronic spins, and \u001fspin(T) is the in-\ntrinsic susceptibility. From the linear relation between\nKand bulk\u001f, we determine ( K0'0:006 %,Ax\nhf'\n4927 Oe/\u0016B), (K0'0:01 %,Ay\nhf'4487 Oe/\u0016B), and\n(K0'0:04 %,Az\nhf'2398 Oe/\u0016B) forKx,Ky, andKz,\nrespectively. The average value of Ahf[= (Ax\nhf+Ay\nhf+\nAz\nhf)=3] is calculated to be \u00183937 Oe/\u0016Bwhich matches\nnicely with the Aiso\nhfvalue (\u00183909 Oe/\u0016B) obtained di-\nrectly from the Kisovs\u001fanalysis. This value of Aiso\nhfis\ncomparable with the values reported for other transition-\nmetal phosphate compounds [14, 47, 48].\n3. Spin-lattice relaxation rate 1=T1\nThe local spin-spin correlations can be understood by\nmeasuring the temperature-dependent spin-lattice relax-\nation rate 1 =T1, which yields information on the imag-\ninary part of the dynamic susceptibility \u001f(q;!). The\n31P 1=T1was measured by exciting the sample at the\n\feld corresponding to the central peak position down to\nT= 2 K. As expected, the recovery of the longitudi-\nnal magnetization follows the single-exponential behav-\nior, which is typical for a I= 1=2 nucleus. The longitu-\ndinal recovery curves at various temperatures were \ftted\nwell by the single exponential function,\n1\n2\u0014\n1\u0000M(t)\nM(0)\u0015\n=Ae\u0000t=T1+C: (5)\nHere,M(t) is the nuclear magnetization at a time tafter\nthe inversion pulse, M(0) is the equilibrium magnetiza-\ntion, andCis a constant. Recovery curves at four di\u000ber-\nent temperatures along with the \ft are shown in the inset\nof the upper panel of Fig. 9. The temperature-dependent\n1=T1estimated from the above \ft is plotted in the upper\npanel of Fig. 9.\nAs shown in Fig. 7, the NMR spectra are asymmetric\nwith two shoulders. In view of this, we measured 1 =T1(T)\nat both shoulder positions and found equal values. The\ndata shown in Fig. 9 correspond to the most intense\nshoulder position on the left-hand side. With decreasing\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s48/s46/s48/s48/s48 /s48/s46/s48/s48/s49 /s48/s46/s48/s48/s50 /s48/s46/s48/s48/s51/s48/s46/s49/s49\n/s51 /s49/s48 /s51/s53/s52/s49/s48/s50/s48/s53/s48/s49 /s49/s48 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s49/s47 /s84\n/s49/s32/s40/s109/s115/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s51/s49\n/s80/s32/s78/s77/s82/s32\n/s50/s52/s46/s55/s54/s32/s77/s72/s122\n/s32/s32/s49/s47/s40 /s84\n/s49/s84 /s41/s32/s40/s99/s109/s45/s51\n/s32/s109/s111/s108/s32/s40/s109/s115/s41/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s32/s50/s46/s55/s53/s32/s75\n/s32/s51/s46/s55/s53/s32/s75\n/s32/s57/s32/s75\n/s32/s54/s48/s32/s75\n/s32 /s32 /s49/s47/s50/s91/s49/s45/s32 /s77 /s40/s116/s41/s47 /s77 /s40/s48/s41/s93\n/s116/s32/s40/s115/s101/s99/s41/s32\n/s49/s47/s40 /s84\n/s49/s84 /s41/s32/s40/s99/s109/s45/s51\n/s32/s109/s111/s108/s32/s40/s109/s115/s41/s45/s49\n/s32/s75/s45/s49\n/s41\n/s32/s32\n/s84 /s32/s40/s75/s41FIG. 9. Upper panel: Spin-lattice relaxation rate 1 =T1vs\nTmeasured at 24.96 MHz. Solid line depicts the linear be-\nhaviour for T\u001420 K. Inset: Recovery of the longitudinal\nmagnetization as a function of tfor four di\u000berent tempera-\ntures. Solid lines are \fts using Eq. (5). Lower panel: Plot of\n1=(\u001fT1T) vsT. Inset: The low- T1=(\u001fT1T) data are magni-\n\fed.\ntemperature, 1 =T1remains constant down to 20 K due\nto \ructuating paramagnetic moments [49]. With further\ndecrease in temperature, 1 =T1decreases in a linear man-\nner and then exhibits a sharp peak at TN'2:72 K. This\nsharp peak at TNre\rects the slowing down of the \ruc-\ntuating moments upon approaching TN. BelowTN, 1=T1\ngradually decreases toward zero.\nIn the lower panel of Fig. 9, 1 =(\u001fT1T) is plotted against\nT. At high temperatures, it is almost temperature-\nindependent and then increases slowly below about 25 K.\nIn the inset of the lower panel of Fig. 9, the data near\nTNare magni\fed in order to highlight this slow increase.\nThe general expression for1\nT1Tin terms of the dynamic\nsusceptibility \u001fM(~ q;!0) can be written as[49, 50]\n1\nT1T=2\r2\nNkB\nN2\nAX\n~ qjA(~ q)j2\u001f00(~ q;!0)\n!0; (6)7\nFIG. 10. Temperature evolution of the neutron powder\ndi\u000braction patterns of \u000b-KVOPO 4in the low-Qregime and\nfor temperatures below 3.4 K. An emergence of magnetic re-\n\rections with k= (0;1\n2;0) is evident below TN.\nwhere the sum is over the wave vector ~ qwithin the \frst\nBrillouin zone, A(~ q) is the form factor of the hyper-\n\fne interaction, and \u001f00(~ q;!0) is the imaginary part of\nthe dynamic susceptibility at the nuclear Larmor fre-\nquency!0. Forq= 0 and!0= 0, the real compo-\nnent of\u001f(~ q;!0) represents the uniform static suscepti-\nbility (\u001f). Thus, the temperature-independent 1 =(\u001fT1T)\nin the high-temperature region ( \u001530 K) indicates the\ndominant contribution of \u001fto 1=T1T. The slow increase\nbelow 25 K can be attributed to the growth of AFM cor-\nrelations.\nD. Neutron Di\u000braction\nWe now probe the magnetic order using neutron\ndi\u000braction. Figure 10 shows that three additional mag-\nnetic re\rections develop below TNatQ= 0:6973 \u0017A\u00001,\n0:7757 \u0017A\u00001, and 0:9189 \u0017A\u00001. They can be indexed with\nthe propagation vector k= (0;1\n2;0) suggesting a com-\nmensurate magnetic order.\nThe Rietveld re\fnement was done using the total\ndi\u000bracted intensities at di\u000berent temperatures. All the\n\fve banks of data have been re\fned simultaneously to\nobtain the \fnal parameters. Figure 11(a) represents the\nRietveld re\fnement of the nuclear pattern at T= 10 K\nusing the orthorhombic crystal structure with the space\ngroupPna21[51]. The obtained lattice parameters are\na= 12:7521(3) \u0017A,b= 6:3603(2) \u0017A,c= 10:4985(2) \u0017A,\nandVcell= 851:52(3) \u0017A3. These values are in close agree-\n/s49 /s49/s48/s48/s46/s55 /s48/s46/s56 /s48/s46/s57/s49/s46/s53 /s49/s46/s56 /s50/s46/s49 /s50/s46/s52 /s50/s46/s55/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s32/s32 /s32/s32/s73\n/s69/s120/s112\n/s32/s73\n/s67/s97/s108\n/s32/s78/s117/s99/s108/s101/s97/s114\n/s32/s73\n/s69/s120/s112/s32/s45/s32 /s73\n/s67/s97/s108/s40/s97/s41\n/s84 /s32/s61/s32/s49/s48/s32/s75\n/s84 /s32/s61/s32/s49/s46/s53/s32/s75\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/s81 /s32/s40/s197/s45/s49\n/s41/s32/s77/s97/s103/s110/s101/s116/s105/s99/s40/s98/s41\n/s51\n/s32 /s32 /s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s81 /s32/s40/s197/s45/s49\n/s41/s84 /s32/s61/s32/s49/s46/s53/s32/s75/s32/s45/s32/s49/s48/s32/s32/s75\n/s49\n/s50\n/s32 /s32 /s77/s111/s109/s101/s110/s116/s32/s40\n/s66/s47/s86/s52/s43\n/s41\n/s84 /s32/s40/s75/s41FIG. 11. The neutron powder di\u000braction patterns along with\nthe Rietveld re\fnement for (a) T= 10 K and (b) T= 1:5 K.\nOpen black circles represent the experimental data, red solid\nline represents the calculated curve, and di\u000berence between\nthem is shown as a blue solid line at the bottom. Vertical\nmarks correspond to the position of all allowed Bragg peaks\nfor the nuclear (top row) and magnetic (bottom row) re\rec-\ntions. The inset in (a) shows the temperature variation of\nordered magnetic moment. Inset in (b) presents the Rietveld\nre\fnement of the di\u000berence data (\u0001 T= 1:5\u000010 K) using\nonly the magnetic model.\nment with the re\fned values from the powder XRD data\nat room temperature [18].\nMagnetic structure re\fnement was performed for the\ndi\u000braction data from the paired 2 and 9 detector banks\nof the di\u000bractometer because of the best resolution (at an\naverage 2\u0012value of 58:308\u000e) [52]. Figure 11(b) shows the\ncombined Rietveld re\fnement of the nuclear and mag-\nnetic re\rections at T= 1:5 K. A solution was found in the\nmagnetic space group P\u00161. It corresponds to a collinear\nstructure with equal magnetic moments on the V1 and\nV2 sites. The magnetic moments lie in the acplane with\n\u0016a= 0:53(5)\u0016Band\u0016c= 0:23(8)\u0016Bat 1.5 K. The spin\narrangement is shown in Fig. 12 and will be discussed in\ndetail in the next section along with the relevant mag-\nnetic couplings.\nE. Microscopic magnetic model\nTo analyze magnetic couplings in \u000b-KVOPO 4, we \frst\nconsider band dispersions and associated hopping param-\neters in the band structure obtained on the PBE level\nwithout taking strong correlations into account. Band\nstructures of the V4+compounds usually show a well-8\nJ1\nJ2J3c\nca\naV1\nV2\n(a) (b)\ncb(c)\nJb\nFIG. 12. (a,b) Crystal and magnetic structures of \u000b-KVOPO 4shown in the same projection, with the K atoms omitted\nfor clarity. The shadings in panel (b) highlight the J2bonds with the parallel and antiparallel spin arrangement, thus illus-\ntrating that J2does not stabilize the magnetic order. (c) The coupling Jbconnects the J1\u0000J3chains into layers and causes\nantiferromagnetic order along bwithk= (0;1\n2;0).\n1\n0\n10dx y2 2/c45\ndxyd dxz yz/d3z r2 2/c45\n20 0 X S Y Z U R T Г Г\nDOS (eV /f.u.)/c451234\nEnergy (eV)\nFIG. 13. Energy bands and the corresponding density\nof states (DOS) obtained from the PBE calculation for \u000b-\nKVOPO 4. The Fermi level is at zero energy. Note the crystal-\n\feld splitting of the V 3 dstates with the signi\fcant mixing\nofdxyanddyz=dxzaround the Fermi level.\nde\fned crystal-\feld splitting imposed by the local envi-\nronment of vanadium with the short vanadyl bond to-\nward one of the oxygen atoms. This short bond de-\n\fnes the local z-direction and renders dxythe lowest-\nenergy, half-\flled magnetic orbital. The dxybands\naround the Fermi level are typically well separated from\nthe higher-lying bands formed by the four remaining d-\norbitals [53, 54].\nAn inspection of the \u000b-KVOPO 4band structure\n(Fig. 13) suggests a small but important deviation from\nthis conventional scenario. The band complex between\n\u00000:2 and 0:2 eV includes 12 bands per 8 V atoms, so it\ncan't be formed by the dxystates only. Orbital-resolved\ndensity of states reveals that these bands include the ma-\njority of the dxycontribution, but also a signi\fcant por-\ntion of thedyzanddxzstates. Their admixture is caused\nby the remarkably large xy\u0000yzandxy\u0000xzhoppings ofTABLE I. Interatomic V{V distances (in \u0017A) and the corre-\nsponding exchange couplings (in K) calculated using superex-\nchange theory [ JAFMandJFMfrom Eq. (7)] and mapping\nanalysis (JDFT+ U). The V{V distances are given using the\nstructural parameters from Ref. [18]. The couplings not listed\nin this table are well below 1 K.\ndV\u0000V JAFMJFMJDFT+ U\nJ1 3.415 V1{V2 0 \u0000154\u0000149\nJ2 3.520 V1{V2 16 \u00009\u00003\nJ3 5.622 V1{V2 34 \u00004 17\nJ4 5.731 V1{V2 1 0 \u00003\nJ5 6.138 V2{V2 4 \u00002 1\nJb 6.360 V1{V1 5 \u00004 1\nJ6 6.376 V1{V1 2 \u00003\u00002\n0:25\u00000:30 eV for the V{V pairs with the 3.412 \u0017A sepa-\nration (J1bond).\nWe now use Wannier projections to construct a \fve-\norbital tight-binding model that reproduces all V 3 d\nbands, and introduce the extracted hoppings into the\nsuperexchange theory [30, 31] that yields magnetic cou-\nplings as\nJ=4t2\nxy\nUe\u000b\u0000X\n\u000b4t2\nxy!\u000bJe\u000b\n(Ue\u000b+ \u0001\u000b)(Ue\u000b+ \u0001\u000b\u0000Je\u000b)(7)\nwheretxyare the hoppings between the half-\flled ( dxy)\norbitals,txy!\u000bare the hoppings between the half-\flled\nand empty orbitals, the index \u000bgoes through these empty\nd-orbitals, \u0001 \u000bare the corresponding crystal-\feld split-\ntings,Ue\u000b= 3 eV is the e\u000bective Coulomb repulsion, and\nJe\u000b= 1 eV is the e\u000bective Hund's coupling. The \frst and\nsecond terms of Eq. (7) stand, respectively, for the AFM\n(JAFM) and FM (JFM) contributions to the exchange, as9\n0.80.9\n0.7\n0.6\n0.02 0\n1/L0.04 0.06 0.08 0.10 0.12 0.14ms B( )μ1 0\n0.6\n0.3\n0.0μ μ( )BT(K)\n2 3\nFIG. 14. Finite-size scaling of the staggered magnetization\n(ms) obtained from the static structure factor calculated by\nQMC at\f= 1=(kBT) = 16L. The dashed line is the \ft with\nEq. (8). The inset shows the experimental ordered magnetic\nmoment (\u0016) as a function of temperature and its empirical\nextrapolation, as described in the text.\nlisted in Table I.\nThe largexy\u0000yzandxy\u0000xzhoppings on the J1\nbond render the respective magnetic coupling strongly\nFM. This coupling is augmented by a much weaker AFM\nJ3, whereas all other couplings are below 5 \u00007 K in mag-\nnitude, either FM or AFM. Our DFT+ Umapping anal-\nysis (Table I) leads to essentially similar results, with the\nleading FM coupling J1and the secondary AFM coupling\nJ3. One immediate and rather unexpected consequence\nof this analysis is that the structural chains with the al-\nternating V{V separations of 3.415 \u0017A (J1) and 3.520 \u0017A\n(J2), as shown in Fig. 1b, break into FM dimers formed\nby the former bond. On the other hand, J2features weak\nFM and AFM contributions that nearly compensate each\nother. Indeed, the FM dimers of J1can be clearly dis-\ntinguished in the experimental magnetic structure. In\ncontrast, the spins on the J2bonds are both parallel and\nantiparallel [Fig. 12(b)], thus indicating that J2does not\ncontribute to the stabilization of the long-range order.\nThe two leading couplings, J1andJ3, form alternating\nspin chains with the strong FM and weak AFM couplings\n[Fig. 12(b)]. Experimental magnetic structure suggests\nthat neither J2norJ4\u0000J6stabilize the order between\nthese chains. The most likely candidate for the inter-\nchain coupling is then Jb, which runs along the crystal-\nlographicbdirection and is also responsible for the dou-\nbling of the magnetic unit cell [Fig. 12(c)]. Our DFT\nresults indeed show a small Jb, although between the\nV1 sites only, whereas the respective interaction between\nthe V2 sites should be below 0.1 K and thus negligible.\nThe interactions Jbcouple the aforementioned alternat-\ning spin chains into layers. Long-range magnetic order\nbetween these layers may be caused by residual interac-\ntions, which are too weak to be resolved in DFT, or by\nminute anisotropic terms in the spin Hamiltonian.\nWhile these weak interlayer interactions and/or\nanisotropy require further dedicated analysis, we ar-\ngue that the J1\u0000J3\u0000Jbmodel of coupled alternatingspin chains reproduces main features of the experimental\ndata. To this end, we simulate thermodynamic proper-\nties by QMC and \fnd an excellent agreement with both\n\u001f(T) [Fig. 3] and M(H) [Fig. 4] using the same set of\nparameters: J1=\u0000150 K,J3= 12 K,Jb= 1:5 K, and\ng= 1:98. Further on, we estimate zero-temperature or-\ndered magnetic moment \u00160by calculating the spin struc-\nture factor S(k) for the same set of exchange parameters\nand using the \fnite-size scaling for the staggered magne-\ntization [55],\nms(L)2=\u00162\n0+m1\nL+m2\nL2(8)\nwhereLis the size of the \fnite lattice, and m1andm2\nare empirical parameters. The resulting \u00160'0:68\u0016B\n(Fig. 14) reveals the 32% reduction compared to the\nclassical value of 1 \u0016B. Experimental values of the or-\ndered moment are even lower and a\u000bected by thermal\n\ructuations, because the base temperature of our neu-\ntron di\u000braction measurement is more than half of TN.\nFrom the empirical scaling, \u0016(T) =\u00160[1\u0000(T=T N)\u000b]\f, we\nestimate\u0016exp\n0= 0:66\u0016Bin an excellent agreement with\nQMC (Fig. 14, inset).\nIV. DISCUSSION AND SUMMARY\nWeak FM couplings are not uncommon among the V4+\ncompounds. They usually take place between those ions\nwhere magnetic dxyorbitals lie in parallel, well-separated\nplanes and lack a suitable superexchange pathway [11].\nHowever, more e\u000ecient superexchange pathways are al-\nways found for other pairs of the V4+ions and give rise\nto AFM couplings of a similar or even larger strength. \u000b-\nKVOPO 4stands as an exception in this row, because its\nFM coupling J1'\u0000150 K is by far the strongest among\nthe known V4+compounds and, therefore, predominant.\nIt arises between the dxyorbitals lying in non-parallel,\nnearly orthogonal planes and can be traced back to the\ne\u000bective orbital order between the two distinct vanadium\nsites in the crystal structure, V1 and V2 [Fig. 1(b)]. One\ninteresting question in this respect is why the strong FM\ncoupling is observed for J1and not for J2, despite the\nsimilar V{V distances. This di\u000berence can be traced back\nto the dihedral angles  between the planes of the dxy\norbitals ( = 30:4\u000eforJ1and 35:5\u000eforJ2) and to the\nadditional superexchange pathways that arise from the\nPO4bridges.\nAs a system dominated by FM couplings, \u000b-KVOPO 4\nis not expected to show any signi\fcant quantum ef-\nfects. Nevertheless, several experimental observations {\nthe susceptibility maximum preceding TN, and the reduc-\ntion in the ordered magnetic moment { challenge these\nexpectations. Our model of weakly coupled FM dimers\nreproduces signi\fcant features of the experimental data\nand suggests weak AFM couplings J3andJbas the ori-\ngin of the quantum e\u000bects in \u000b-KVOPO 4. The sepa-\nration of energy scales into strong FM J1=\u0000150 K10\nand weak AFM J3= 12 K further implies that at low\ntemperatures an e\u000bective description in terms of S= 1\nmoments located on the J1dimers may be appropriate.\nThis description is supported by the magnetic entropy\nof 4.3 J/mol K released below 10 K and corresponding to\n1\n2Rln 3 = 4:56 J mol K for 0.5 S= 1 dimers per formula\nunit. The e\u000bective S= 1 description entails S= 1 Hal-\ndane chains formed by J3and coupled into layers by Jb.\nSuch coupled S= 1 chains show a quantum phase transi-\ntion between the magnetically ordered and gapped Hal-\ndane phases [56, 57]. The proximity to this transition\nmay give a further clue to the reduced ordered moment\nand quantum e\u000bects in \u000b-KVOPO 4.\nIn summary, \u000b-KVOPO 4is an unusual spin-1\n2magnet\nfeaturing strong FM and weak AFM couplings. While the\nformer cause the formation of ferromagnetic spin dimers,\nthe latter connect these dimers into layers and trigger\nquantum e\u000bects. Quantum \ructuations manifest them-\nselves by the short-range order that appears below 5 K\nand precedes the long-range order formed at TN= 2:7 K.\nBelowTN, the commensurate and collinear ground state\nfeatures a strongly reduced ordered magnetic moment of\n0.58\u0016Bat 1.5 K. The unusually strong FM coupling J1is caused by the e\u000bective orbital order on the two crys-\ntallographically nonequivalent vanadium sites. Overall,\nfrom the magnetism perspective \u000b-KVOPO 4is entirely\ndi\u000berent from its siblings, such as NaVOPO 4with its\n\feld-induced quantum critical point [14] and AgVOAsO 4\nwith the double-dome regime of magnon BEC [12]. 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Mater. 2, 561\n(2000)."    },    {        "title": "1103.0304v2.Two_orbital_Kondo_effect_in_quantum_dot_coupled_to_ferromagnetic_leads.pdf",        "content": "arXiv:1103.0304v2  [cond-mat.str-el]  16 Apr 2011Two-orbital Kondo effect in quantum dot coupled to ferromagn etic leads\nHitoshi Yoshizumi,1Kensuke Inaba,2,3Tomoko Kita,1,∗and Sei-ichiro Suga1,4\n1Department of Applied Physics, Osaka University, Suita, Os aka 565-0871, Japan.\n2NTT Basic Research Laboratories, NTT Corporation, Atsugi 2 43-0198, Japan\n3JST, CREST, Chiyoda-ku, Tokyo 102-0075, Japan\n4Department of Materials Science and Chemistry,\nUniversity of Hyogo, Himeji, Hyogo 671-2280, Japan\n(Dated: December 5, 2018)\nWe study the Kondo effect of a two-orbital vertical quantum do t (QD) coupled to two ferromag-\nnetic leads by employing an equation of motion method. When t he ferromagnetic leads are coupled\nwith parallel spin polarization, we find three peaks in the si ngle-particle excitation spectra. The\nmiddle one is the Kondo resonance caused by the orbital degre es of freedom. In magnetic fields,\nthe Kondo effect vanishes. However, at a certain magnetic fiel d new two-fold degenerate states\narise and the Kondo effect emerges there. In contrast, when th e ferromagnetic leads are coupled\nwith antiparallel spin polarization, the Kondo effect cause d by the spin (orbital) degrees of freedom\nsurvives (is suppressed) in magnetic fields. We investigate the field dependence of the conductance\nin the parallel and antiparallel spin polarizations of the l eads and find that the conductance changes\nnoticeably in magnetic fields.\nPACS numbers: 72.15.Qm, 73.63.Kv, 72.25.-b, 85.75.-d\nI. INTRODUCTION\nThe Kondo effect, which was originally studied in di-\nlute magnetic alloys, is a typical phenomenon caused\nby local electron correlation. Recent progress on nano-\nprocessing techniques has enabled us to observe the\nKondo effect in quantum dot (QD) systems1,2. Since the\nQDsystemhasmanytunableparameters,variousaspects\nof the Kondo effect can be controlled. For instance, the\nmultiorbital Kondo effect realized in vertical QDs3,4and\ncarbon-nanotube QDs5,6has attracted much interest. In\nparticular, in a vertical QD, the confinement potential\nyields multiply degenerate single-particle QD energy lev-\nels called the Fock-Darwin states7–9, which are regarded\nas effective orbital degrees of freedom. The degeneracy\nof the QD states in the confinement potential can be con-\ntrolled with external magnetic fields. The introduction\nof the orbital splitting of the QD energy levels realizes\na crossover from the SU(4) to the SU(2) Kondo effect,\nwhich results in characteristic transport properties10–14.\nIn the SU(4) Kondo effect, a higher Kondo temperature\nthan that in the SU(2) Kondo effect was observed in\nexperiments4–6.\nSpin-dependent current in metallic leads is another\ninteresting topic in relation to QD systems. Recently,\nthere have been a number of experimental studies of QD\nsystems coupled to ferromagnetic leads in the context\nof spintronics15–18. The splitting and restoration of the\nKondo peak has been observed in experiments on a C 60\nQD system15and a semiconductor QD system, which are\ncoupled to ferromagnetic leads16,17. There have been ex-\ntensive theoretical studies on a single-orbital QD coupled\nto two ferromagnetic leads19–30. When the spin polariza-\ntion of the two ferromagnetic leads is parallel, the Kondo\neffect caused by the spin degrees offreedom is suppressed\neven in the absence of magnetic fields. On the otherhand, the Kondo effect remains when the spin polariza-\ntion of the two ferromagnetic leads is antiparallel25–27.\nThese results motivated us to investigate a multiorbital\nQD coupled to ferromagnetic leads. One can expect that\ntheinterplaybetweenthespinpolarizationandthemulti-\norbital effects causes characteristic features of the Kondo\neffect and the related transport properties. For exam-\nple, in a two-orbital QD coupled to ferromagnetic leads,\nthe appearance of the underscreening Kondo effect was\ndiscussed31. In carbon nanotube QD, the influence of\nferromagnetic leads on the shot noise was analyzed32.\nIn this paper, we investigate the Kondo effect of a\ntwo-orbital vertical QD coupled to two ferromagnetic\nleads. By employing an equation of motion (EOM)\nmethod20,25,28,30,33,34, we calculate the single-particleex-\ncitation spectra (SPES) of the localized electron in the\nQD and the conductance. The SPES for several spin\npolarization values are discussed in connection with the\nsplitting of the QD energy levels obtained by the poor\nman’s scaling approach26,27,35. We also investigate the\neffects of the orbital splitting caused by magnetic fields\non the SPES. The conductance is shown as a function of\nthe orbital splitting.\nThis paper is organized as follows. In Sec. II, we out-\nline the model and the methods. We consider a QD with\ntwo orbitals coupled to two ferromagnetic leads. The\nsystem is modeled as a two-orbital Anderson impurity\nHamiltonian where the conduction electrons have a spin\ndependent density of states (DOS). We outline an EOM\nmethod and a poor man’s scaling approach. In Sec. III,\nwe show the results. In the SPES for the parallel spin\npolarization of the leads, we find that three peaks ap-\npear and the middle one is located close to the Fermi\nenergy. It is considered that this is a Kondo resonance\ncaused by the orbital degrees of freedom. When a mag-\nneticfield isapplied, theenergylevelsofthe QDsplitand2\nthe Kondo effect disappears. However, in a certain mag-\nnetic field two energy levels cross and the Kondo effect\nemerges again. We discuss the conductance as a func-\ntion of the energy-level splitting of the QD. A summary\nis provided in Sec. IV.\nII. MODEL AND METHODS\nA. Two-orbital quantum dot coupled to\nferromagnetic leads\nWe consider a single QD with two-orbital degrees of\nfreedom coupled to two ferromagnetic leads. The system\nis described by the following Hamiltonian,\nH=/summationdisplay\ni=L,R/summationdisplay\nk,l,σεikσc†\niklσciklσ+/summationdisplay\nl,σεld†\nlσdlσ\n+/summationdisplay\ni=L,R/summationdisplay\nk,l,σ(Viklσc†\niklσdlσ+h.c.)\n+U/summationdisplay\nlnl↑nl↓+U′/summationdisplay\nσσ′nlσnlσ′, (1)\nwherec(†)\niklσannihilates (creates) a conduction electron\nwith wave number k, spinσ, and orbital lin the lead\ni(=L,R),εlis the QD energy level for orbital l,d(†)\nlσ\nannihilates (creates) a localized electron with spin σand\norbitallin the QD, and nlσ=d†\nlσdlσ. Intraorbital and\ninterorbitalCoulombinteractionsareexpressedby Uand\nU′, respectively, and Viklσis the tunneling amplitude be-\ntween the leads and the QD. We set UandU′≫ |εl|as\nrealized in many QD systems, so that the double occu-\npancy of the QD electron is forbidden and the electron\nnumber is at most unity. Therefore, we can neglect the\neffects of the exchange interaction. We assume that the\norbitalstatesin the QDhybridizewith the corresponding\nconduction channels in the leads. Although the assump-\ntion of multiple conduction channels is nontrivial, it is\nknown that this is relevant for certain systems, e.g. ver-\ntical QD systems and carbon nanotube QD systems3–5.\nThe spin polarization of the ferromagnetic leads is de-\nfined by p= (ρci↑−ρci↓)/(ρci↑+ρci↓), (0≤p≤1),\nwhereρciσis the spin dependent DOS of the conduction\nelectron in the lead i(=L,R), which is assumed to be\nindependent of orbital l. We use the energy-independent\nconstant ρciσ, which is reasonablein the wide-band limit.\nFor a parallel magnetic (P) configuration, the relative\norientation of the magnetic moments in the two ferro-\nmagnetic leads is parallel as shown in Fig. 1(a). Both\nDOSs in the leads LandRare defined by ρcLσ=ρcRσ=\n[1+(δσ,↑−δσ,↓)p]ρ0withρ0being the DOS of the leads\natp= 0. For an antiparallel magnetic (AP) configura-\ntion, as shown in Fig. 1(b), the DOSs in the leads L\nandRare defined by ρcLσ= [1+(δσ,↑−δσ,↓)p]ρ0and\nρcRσ= [1−(δσ,↑−δσ,↓)p]ρ0, respectively.\nInaverticalQDsystem, thesingle-particleenergylevel\nof the QD in a magnetic field can be described as thelead lead dot (a) Parallel \nlead lead dot (b) Antiparallel \nFIG. 1: (Color on-line) Schematic diagram of the magnetic\nconfigurations in our QD system. The relative orientations o f\nthe magnetic moments in the two ferromagnetic leads are (a)\nparallel and (b) antiparallel.\nFock-Darwin state. The energy levels of two orbitals are\nsplit into two levels, ε1andε2, in magnetic fields. We\nmodel this splitting as\nεl=ε0+(δl,1−δl,2)∆orb,(l= 1,2),(2)\nwhereε0is the center of the QD energy level and ∆ orbis\nthe splitting width. We call ∆ orbthe orbital splitting in\nthe following. Because of the small gfactor in semicon-\nductors, the Zeeman splitting of the semiconductor QD\nsystem is much smaller than ∆ orbin a magnetic field, so\nthat we can ignore the effects of the Zeeman splitting10.\nWe thus obtain the field dependence via the ∆ orbdepen-\ndence using the Fock-Darwin energy diagram9.\nB. Equation of motion method\nWe calculate the retarded Green’s function Gr\nlσ(t) =\n−iθ(t)/an}bracketle{t{dlσ(t),d†\nlσ(0)}/an}bracketri}htforalocalizedelectronoftheQD.\nFor this purpose, we employ an EOM method. We per-\nform the straightforwardEOM calculations until the sec-\nond iterative procedure. In the third iterative EOM\nprocedure, we use the approximation which successfully\nyields the closed equations of the higher-order Green’s\nfunctions in a single-orbital QD system33,34. In addition\nto this approximation, we include the higher-order terms\nexpressing the scattering processes of the electrons in the\nsame orbital with different spins, and in the different or-\nbitals with the same and different spins. We neglect the\nhigher-order terms expressing other higher-order scatter-\ning processes. Details of the calculations are described in\nAppendix.\nIn the condition of the strong interactions UandU′≫\n|εl|, we obtain the equations for the Green’s function as\nGr\nlσ(ω) =1−/an}bracketle{tnlσ/an}bracketri}ht−/an}bracketle{tnlσ/an}bracketri}ht−/an}bracketle{tnlσ/an}bracketri}ht\nω−εl−Σ0\nlσ(ω)−Σ1\nlσ(ω)−Σ2\nlσ(ω)−Σ3\nlσ(ω),\n(3)\nwhere\nΣ0\nlσ(ω) =/summationdisplay\ni=L,R/summationdisplay\nk|Viklσ|21\nω−εikσ, (4)3\nΣ1\nlσ(ω) =/summationdisplay\ni=L,R/summationdisplay\nk|Viklσ|2f(εikσ)1\nω−εikσ+∆˜ε1\nlσ,(5)\nΣ2\nlσ(ω) =/summationdisplay\ni=L,R/summationdisplay\nk|Viklσ|2f(εikσ)1\nω−εikσ+∆˜ε2\nlσ,(6)\nΣ3\nlσ(ω) =/summationdisplay\ni=L,R/summationdisplay\nk|Viklσ|2f(εikσ)1\nω−εikσ+∆˜ε3\nlσ,(7)\nwithσbeing the opposite spin state to σandlbeing the\ndifferent orbital state from l. The Fermi energy is set at\nω= 0. The self-energy Σ1\nlσ(ω) expresses the scattering\nprocess of the electrons in the same orbital with differ-\nent spins. The self-energies Σ2\nlσ(ω) and Σ3\nlσ(ω) express\nthe scattering processes of the electrons in the different\norbitals with the same and different spins, respectively.\nAccordingly, spin fluctuations are introduced via Σ1\nlσ(ω)\nand Σ3\nlσ(ω), while orbital fluctuations are introduced via\nΣ2\nlσ(ω) and Σ3\nlσ(ω). These scattering processes play es-\nsential roles in the appearanceofthe Kondo effect caused\nby spin and orbital fluctuations. We thus argue that the\npresent EOM approximation captures the essentials of\nthe Kondo effect in the two-orbital QD.\nThe electron number /an}bracketle{tnlσ/an}bracketri}htis determined using the\nequation,\n/an}bracketle{tnlσ/an}bracketri}ht=−1\nπ/integraldisplay\nIm[Gr\nlσ(ω)]f(ω)dω. (8)\nIn this way, we complete the self-consistent loop which\nconsists of Eqs. (3)-(8).\nTo improve the numerical results quantitatively, we\nhave further introduced the second self-consistent loop.\nBy analogy with the self-consistent procedure used for\na single-orbital QD coupled to ferromagnetic leads20, we\nhave replaced the bare QD energy splittings in Σ1\nlσ(ω),\nΣ2\nlσ(ω) and Σ3\nlσ(ω) by the renormalized ones ∆˜ ε1\nlσ, ∆˜ε2\nlσ\nand ∆˜ε3\nlσ, respectively, as\n∆˜ε1\nlσ= ˜εlσ−˜εlσ, (9)\n∆˜ε2\nlσ= ˜εlσ−˜εlσ, (10)\n∆˜ε3\nlσ= ˜εlσ−˜εlσ, (11)\n˜εlσ=εl+Re[Σ0\nlσ(˜εlσ)+Σ1\nlσ(˜εlσ)+Σ2\nlσ(˜εlσ)+Σ3\nlσ(˜εlσ)],\n(12)\nwhere ∆˜ ε1\nlσdenotes the difference between the QD lev-\nels of the same orbital with different spins, and ∆˜ ε2\nlσ\nand ∆˜ε3\nlσdenote the difference between the QD levels\nof a different orbital with the same and different spins,\nrespectively. At ω= ˜εlσdetermined self-consistently\nin Eq. (12), the real part of the denominator of theGreen’s function in Eq. (3) becomes zero. This fact sug-\ngests that the additional self-consistent procedure pro-\nvides us with the important information about the renor-\nmalized energy level of the electrons in the QD. For the\nsingle-orbital case in the P configuration, the splitting\nof the Kondo resonance was obtained using this EOM\nmethod20. This feature was confirmed by using the nu-\nmerical renormalization-group calculation21. The split-\nting of the Kondo resonance caused by the ferromag-\nnetic leads is often characterized by the so-called local\nexchange field24.\nSubstituting the self-consistently obtained Green’s\nfunction into the relation\nρlσ(ω) =−1\nπIm[Gr\nlσ(ω)], (13)\nwe obtain the SPES for the localized electron of\nthe QD. Note that the spectral weights obtained\nby the EOM method are rather underestimated\nand are quantitatively not accurate, although the\nspectral profiles are qualitatively correct. The\nbias-linear conductance is expressed as34,36,37G=\n−(2πe2/h)/integraltext\ndωf′(ω)/summationtext\nlσρlσ(ω)ΓL\nlσΓR\nlσ//parenleftbig\nΓL\nlσ+ΓR\nlσ/parenrightbig\n,\nwheref′(ω) =∂f(ω)/∂ωwithf(ω) being the Fermi\ndistribution function and ΓL(R)\nlσis the hybridization\nbetween the left (right) lead and the QD. Note that this\nexpression for the conductance is valid, if the Green’s\nfunction Gr\nlσ(ω) is diagonalized with respect to land\nσ. When we consider the QD system with off-diagonal\ncontributions as well as plural electrons, we have to also\nimprove the present EOM formulation. Such a situation\nis beyond the current interest.\nIn the calculation, we set a flat band and as-\nsume that Viklσis independent of i,k,l, andσ\nasViklσ=V. By these reasonable simplifications,\nwe obtain ΓL\nlσ= [1+( δσ↑−δσ↓)p]Γ0/2 and ΓR\nlσ=\n[1±(δσ↑−δσ↓)p]Γ0/2, where ±denotes the P(+) and\nAP(−) configurations, and Γ 0= 8πρ0V2is the total hy-\nbridization of the left and right leads at p= 0. The\nconductance takes the form\nG=−2πe2\nh/integraldisplay\ndωf′(ω)Γ0\n4/summationdisplay\nlσρlσ(ω)\n×/braceleftBigg\n[1+(δσ↑−δσ↓)p],Pconfiguration ,/bracketleftbig\n1−p2/bracketrightbig\n, APconfiguration .(14)\nC. Poor man’s scaling approach\nWe evaluate the splitting of the renormalized QD en-\nergy levels for the P configuration using a poor man’s\nscaling approach26,27which is known as a powerful tool\nto study the renormalization effect in the Anderson\nmodel35. We reduce the energy scale of the half width\nof the flat band DtoD1, where charge fluctuations are\nquenched. By integrating out the excitations in the en-\nergy range between D1andD, we obtain the renormal-\nized QD energy levels ˜ εscal\nl↑and ˜εscal\nl↓. Because of this4\nrenormalization, the degenerate two energy levels yield\nthe spin splitting, which is given by\n∆˜εscal≡˜εscal\nl↓−˜εscal\nl↑=1\nπpΓ0lnD\nD1,(15)\nwhereD1is determinedfromthe conditionthat the lower\nband edge −D1coincides with the lower renormalized\nQD level ˜ εscal\nl↑according to the procedure in Refs.26,27:\n−D1=ε0+1\n2π(1−p)Γ0lnD\nD1. (16)\nWe obtain D1for given pandDfrom Eq. (16). Substi-\ntuting the obtained D1into Eq. (15), we determine the\npdependence of ∆˜ εscal.\nIII. RESULTS AND DISCUSSIONS\nOnthebasisofthe formulationsobtainedin Sec. II, we\ncarry out the numerical calculations, keeping a vertical\nQD and Ni ferromagnetic leads in mind. We use Γ 0in\nunits of energy. We adopt D/Γ0= 100,ε0/Γ0=−2, and\ntemperature T/Γ0= 0.005. Since the spin polarization\nof the Ni ferromagnetic lead is p∼0.315, we direct our\nattention to the p= 0.3 system and the pdependence up\nto around p= 0.3.\nA. Spin polarization dependence of Kondo effect\nWe discuss the effects of the ferromagnetic leads in the\nP configuration on the Kondo effect for ∆ orb/Γ0= 0. In\nFig. 2, we show ρ(ω) =/summationtext\nσρσ(ω),ρ↑(ω), andρ↓(ω) for\nseveral values of the spin polarization p. Note that the\nFermi energy is ω/Γ0= 0. Since we find that ρlσ(ω) are\nindependent of l, we setρσ(ω) =/summationtext\nlρlσ(ω). In Fig. 2(a),\nforp= 0 one conspicuous peak appears at ω/Γ0∼0.02,\nwhich results from the spin-orbital symmetric SU(4)\nKondo effect12–14forUandU′≫ |εl|. In the SU(4)\nKondosystem, theFriedelsumruleensuresthatthe peak\nof the Kondo resonance appears at ω/Γ0∼TSU(4)\nK/Γ0,\nwhereTSU(4)\nKis the Kondo temperature14. From the nu-\nmerical results, we evaluate TSU(4)\nK/Γ0∼0.02 forp= 0\nand ∆ orb/Γ0= 0. This value is largerthan the used tem-\nperature in the calculation ( T/Γ0= 0.005) and thus the\nSU(4) Kondo effect appears for T/Γ0= 0.005.\nForp/ne}ationslash= 0, the Kondo peak splits into three peaks.\nThe splittings can be clearly seen for p≥0.1 as shown in\nFig. 2(a). The energydifference δω/Γ0between the right\nand left peaks becomes large with increasing p, while the\nmiddle peak remains significantly at ω/Γ0∼0 for finite\np. In Fig. 2(b) and (c), we find that the left and right\npeaks in ρ(ω) originate from ρ↑(ω) andρ↓(ω), respec-\ntively. Accordingly, it is considered that for finite pin\nthe P configuration the Kondo effect caused by the spin\ndegrees of freedom (the spin Kondo effect) is suppressed 0 0.5 1 1.5\n-0.6-0.4-0.2 0 0.2 0.4 0.6ρ  (ω)\nω/Γ0(c) 0.5 1 1.5\nρ  (ω)(b) p=0.0\n0.05\n0.1\n0.2\n0.3 0.5 1 1.5\nρ(ω)(a)Parallel\nΔorb/Γ0 = 0\n 0 1\n-3-2-1 0 1\nFig. 2: (Color on-line) Single-particle excitation spectr a (a)\nρ(ω) =ρ↑(ω)+ρ↓(ω), (b)ρ↑(ω), and (c) ρ↓(ω) for several p\nvalues in the P configuration. The parameters are ∆ orb/Γ0=\n0,T/Γ0= 0.005, and ε0/Γ0=−2. Inset: Expanded scale of\nρ(ω).\n 0 0.2 0.4\n 0 0.1 0.2 0.3pParallel\nΔorb/Γ0 = 0(a) (b)∼\n∼Δεeom/Γ0\nΔεscal/Γ0\n 0 0.2 0.4\n 0 0.1 0.2 0.3δω / Γ0\npParallel\nΔorb/Γ0 = 0(a) (b)∼\n∼\nFig. 3: (Color on-line) (a) Energy splittings between up- an d\ndown-spin QD levels as functions of pfor the P configuration.\n∆˜εeomand ∆˜εscalare obtained by the EOM method and the\nscaling approach, respectively. (b) Energy difference δω/Γ0\nbetween the peaks of ρ(ω) in the positively and negatively\nhigh-energy region as a function of pfor the P configuration.\nFor both (a) and (b), we set ∆ orb/Γ0= 0,T/Γ0= 0.005, and\nε0/Γ0=−2.\nasa result ofthe localexchangefield, while the Kondoef-\nfect caused by the orbital degeneracy (the orbital Kondo\neffect) survives.\nWe investigate the origin of the splitting in terms of\nthe poor man’s scaling approach. In Fig. 3(a), we\nshow the spin splitting of the renormalized QD energies\n(∆˜εscal/Γ0)obtainedbythepoorman’sscalingasafunc-\ntion ofp. We compare it with the spin splitting of the\nrenormalized QD energies (∆˜ εeom/Γ0) obtained by the\nEOM. According to Eqs. (5) and (9), ∆˜ εeom/Γ0can be5\nParallel \n∆orb /Γ0= 0 〈n1↑〉\n0.9 1\n-10 -2〈ntot 〉\nε0/Γ 0p=0.2 〈n2↑〉\n〈n1↓〉\n〈n2↓〉\n〈ntot 〉\n00.2 0.4 0.6 0.8 1\n0 0.1 0.2 0.3 \np\nFig. 4: (Color on-line) QD electron numbers, /angbracketleftn1↑/angbracketright,/angbracketleftn2↑/angbracketright,\n/angbracketleftn1↓/angbracketright,/angbracketleftn2↓/angbracketright, and/angbracketleftntot/angbracketrightas functions of pin the P configura-\ntion./angbracketleftntot/angbracketright=/summationtext\nlσ/angbracketleftnlσ/angbracketright. The parameters are ∆ orb/Γ0= 0,\nT/Γ0= 0.005, and ε0/Γ0=−2. Inset: /angbracketleftntot/angbracketrightas a function of\nε0/Γ0withp= 0.2.\ndefined as\n∆˜εeom≡∆˜ε1\nl↓−∆˜ε1\nl↑. (17)\nWe find that both calculations are in good quantitative\nagreement, although∆˜ εscal/Γ0deviates slightlyfrom the\nlinearpdependenceof∆˜ εeom/Γ0inp/greaterorsimilar0.2. InFig. 3(b),\nwe show the energy difference δω/Γ0between the right\nandleftpeaksinFig. 2(a)asafunctionof p. Wefindthat\nδω/Γ0is proportional to pand agrees with ∆˜ εeom/Γ0\nquantitatively. Therefore, in the P configuration spin-\ndependent chargefluctuations induce the spinsplitting of\nthe renormalized QD energy, which results in the linear\npdependence of δω/Γ0shown in Fig. 3(b).\nWe next investigate the electron number of the QD\nin the P configuration. In Fig. 4, we show /an}bracketle{tnlσ/an}bracketri}htand\n/an}bracketle{tntot/an}bracketri}ht=/summationtext\nlσ/an}bracketle{tnlσ/an}bracketri}htas functions of p. Forp= 0, the\n/an}bracketle{tnlσ/an}bracketri}htvalues are the same irrespective of landσ. As\npincreases, /an}bracketle{tnl↑/an}bracketri}htincreases, while /an}bracketle{tnl↓/an}bracketri}htdecreases. For\nthe up-spin (down-spin) electrons, the hybridization Γ1\n↑\n(Γ1\n↓) is enhanced (suppressed) with increasing p, which\nresults in an increase (decrease) in /an}bracketle{tnl↑/an}bracketri}ht(/an}bracketle{tnl↓/an}bracketri}ht). On the\nother hand, we find /an}bracketle{tn1σ/an}bracketri}ht=/an}bracketle{tn2σ/an}bracketri}htforp/ne}ationslash= 0. This result\nindicates that the orbitaldegeneracyexists evenfor finite\npin the P configuration, which yields the orbital Kondo\neffect as shown in Fig. 2(a). Details will be discussed\nin the next subsection. The total QD electron number\n/an}bracketle{tntot/an}bracketri}htis almost constant irrespective of pand approaches\nunity with decreasing ε0/Γ0as shown in the inset of Fig.\n4. This property originates from the strong Coulomb\nrepulsions UandU′≫ |εl|.\nAs shown in the inset of Fig. 2(a), the hump structure\naroundω/Γ0=−1 caused by charge fluctuations is al-\nmost independent of p. The energy at which the hump\nstructure shows its maximum is shifted from the bare\nQD energy level ǫ0/Γ0=−2 toward the Fermi energy\nbecause of the correlation effects.00.5 1\n-0.6 -0.4 -0.2 0 0.2 0.4 0.6 \nω/Γ 0ρ(ω )\nParallel \np = 0.3 ∆orb /Γ0= 0.16 0.5 1∆orb /Γ0= 0.115 0.5 1∆orb /Γ0= 0.04 0.5 1∆orb /Γ0= 0 \nFig. 5: (Color on-line) Single-particle excitation spectr aρ(ω)\nfor the several values of ∆ orb/Γ0in the P configuration. The\nparameters are p= 0.3,T/Γ0= 0.005, and ε0/Γ0=−2.\nB. Orbital splitting effects\nWe investigate the effects of the orbital splitting in the\nP and AP configurations. In Fig. 5, we show the SPES\nin the P configuration for several ∆ orb/Γ0values at p=\n0.3. For ∆ orb/Γ0= 0, the Kondo peak caused by orbital\nfluctuations appears close to ω/Γ0= 0. As ∆ orb/Γ0is\nincreased, the peak close to ω/Γ0= 0 disappears, which\nindicates that the Kondo effect has vanished. Instead,\nmultipeaks appear. At ∆ orb/Γ0= 0.115, we find a small\npeakatω/Γ0∼0asindicatedbythearrow,showingthat\nthe Kondo effect is restored. When ∆ orb/Γ0is further\nincreased, the peak close to ω/Γ0= 0 disappears again\nand the multipeaks appear.\nThe peakpositionsin ρ(ω) aredetermined bythe ωde-\npendence of the self-energiesΣ1\nlσ(ω), Σ2\nlσ(ω), and Σ3\nlσ(ω),\nwhich reflect ∆˜ ε1\nlσ, ∆˜ε2\nlσ, and ∆˜ε3\nlσ. We show ˜ εlσ/Γ0as\nfunctions of ∆ orb/Γ0in Fig. 6. The vertical dashed lines\nare ∆orb/Γ0= 0.04, 0.115, and 0 .16, which we have used\nin Fig. 5.\nFor ∆ orb/Γ0= 0, we find ˜ ε1σ/Γ0= ˜ε2σ/Γ0and\n˜εl↑/Γ0/ne}ationslash= ˜εl↓/Γ0. According to Eqs. (9), (10), and (11),\n∆˜ε1\nlσand ∆˜ε3\nlσhave two different values, and ∆˜ ε2\nlσ= 0.\nBecause of these properties, for ∆ orb/Γ0= 0, the Kondo\npeakcausedbyorbitalfluctuationsappearsandthe split-\nting two peaks appear in ρ(ω). As ∆ orb/Γ0is increased,\nthe orbital degeneracy is lifted and ˜ ε1σ/Γ0(˜ε2σ/Γ0) in-\ncrease (decrease) with the same ∆ orb/Γ0dependence as6\n-1.2-1-0.8-0.6-0.4\n 0 0.05  0.1  0.15  0.2\nΔorb/Γ0∼\n∼\n∼\n∼Parallel\np = 0.3ε1↑/Γ0ε1↓/Γ0ε2↑/Γ0ε2↓/Γ0\nFig. 6: (Color on-line) Renormalized QD energy levels in the\nP configuration as functions of ∆ orb/Γ0forp= 0.3. The\nvertical dashed lines are ∆ orb/Γ0= 0.04, 0.115, and 0 .16,\nwhich correspond to the values used in Fig. 5. ˜ εl↑/Γ0and\n˜ε2↓/Γ0cross at ∆ orb/Γ0= 0.115.\nshown in Fig. 6. Accordingly, for ∆ orb/Γ0/ne}ationslash= 0, ∆˜ε1\nlσand\n∆˜ε2\nlσexhibit two different values depending on σandl,\nrespectively, and ∆˜ ε3\nlσexhibits four different values. Be-\ncause of these features, the Kondo effect disappears for\n∆orb/Γ0= 0.04 and 0.16, and eight peaks appear in ρ(ω)\nas shown in Fig. 5.\nFor ∆ orb/Γ0= 0.115, we find a noticeable feature:\n˜ε1↑/Γ0and ˜ε2↓/Γ0cross as shown in Fig. 6 and new\ntwo-fold degenerate states arise there. The scattering\nbetween these two degenerate states can be expressed\nusing the pseudospin-flip process. This process is intro-\nduced via the self-energy Σ3\nlσ(ω), resulting in an uncon-\nventionalKondoeffect. Thesmallpeakcloseto ω/Γ0= 0\nindicated by the arrow in Fig. 5 is a Kondo resonance\ncaused by this scattering process. Since the respective\ngradients of ˜ ǫ1σ/Γ0and ˜ǫ2σ/Γ0are the same, we find\nonly one crossing point as a function of ∆ orb/Γ0. Al-\nthough the ∆ orb/Γ0value at the crossing point depends\non the parameters, the result does not change qualita-\ntively: The new degeneracy arises at a certain value of\n∆orb/Γ0, which can be evaluated from the magnitude of\nthe energy splitting ˜ εl↓−˜ǫl↑at ∆orb= 0, and the gradi-\nentsof ˜εlσ/Γ0asafunction of∆ orb. Thesimilarsituation\nhas been observed as a singlet-triplet Kondo effect and a\ndoublet-doublet Kondo effect10,11. It is considered that\nthis Kondo effect can be observedin the field dependence\nof the conductance, which will be described in the next\nsubsection.\nThe results for the AP configuration are shown in Fig.\n7. For ∆ orb/Γ0= 0, the spectral profile is the same\nas that of the SU(4) Kondo effect shown in Fig. 2(a).\nTherefore, the effects of the ferromagnetic leads vanish\nin the AP configuration for a zero magnetic field. For\n∆orb/Γ0= 0.01, we find a single peak and the SU(4)\nKondo effect remains there. As ∆ orb/Γ0is further in-\ncreased, the SU(4) Kondo peak splits into three peaks.\nThe peaks at either end move further apart with increas-\ning ∆orb/Γ0, while the middle peak remains at ω/Γ0∼0\nirrespective of ∆ orb/Γ0. In the inset of Fig. 7, we\nshow the renormalized QD energy levels as a function 0 0.5 1 1.5\n-0.6-0.4-0.2 0 0.2 0.4 0.6ρ(ω)\nω/Γ0Antiparallel\np = 0.3Δorb/Γ0Δorb / Γ0 = 0\n0.01\n0.04\n0.115\n0.16-0.8-0.4\n 0  0.1  0.2∼\n∼\n∼\n∼ε1↑/Γ0\nε1↓/Γ0\nε2↑/Γ0\nε2↓/Γ0\nFig. 7: (Color on-line) Single-particle excitation spectr aρ(ω)\nfor the several values of ∆ orb/Γ0in the AP configuration.\nThe parameters are ∆ orb/Γ0= 0,p= 0.3,T/Γ0= 0.005, and\nε0/Γ0=−2. Inset: Renormalized QD energy levels in the\nAP configuration as functions of ∆ orb/Γ0forp= 0.3.\n 0.3 0.6 0.9\n 0  0.1  0.2G[e2/h]\nΔorb/Γ0p=0.3Antiparallel\nParallel\nFig. 8: (Color on-line) Conductance as functions of ∆ orb/Γ0\nforp= 0.3 in the P and AP configurations. We set T/Γ0=\n0.005. The local maximum appears at ∆ orb/Γ0∼0.115 in\nthe P configuration.\nof ∆orb/Γ0forp= 0.3. We find that ˜ ε1↑/Γ0= ˜ε1↓/Γ0\nand ˜ε2↑/Γ0= ˜ε2↓/Γ0, and that the two levels ˜ ε1σ/Γ0\nand ˜ε2σ/Γ0deviate with increasing ∆ orb/Γ0. The re-\nsults indicate that in the AP configuration the orbital\nKondo effect is suppressed for finite ∆ orb/Γ0, while the\nspin Kondo effect remains.\nC. Conductance\nSubstituting ρ(ω) for various ∆ orb/Γ0values into Eq.\n(14), we obtain the ∆ orb/Γ0dependence of the conduc-\ntanceG. The results for p= 0.3 in the P and AP config-\nurations are shown in Fig. 8. In the P configuration, the\nconductance first decreases with increasing ∆ orb/Γ0be-\ncause of the suppression of the Kondo effect. As ∆ orb/Γ0\nincreases further, the conductance increases nontrivially\nand takes a local maximum at ∆ orb/Γ0∼0.115. This\nnontrivial local maximum is caused by the Kondo effect\nindicated by the arrow in Fig. 5. By contrast, in the7\nAP configuration, the conductance has its maximum at\n∆orb/Γ0∼0.01. We discuss the origin of this maximum.\nFor ∆ orb/Γ0/lessorsimilar0.01, the SU(4) Kondo effect remains as\nshown in Fig. 7. However, the spectral weight close to\nthe Fermi energy at ∆ orb/Γ0= 0.01 becomes slightly\nlarger than that at ∆ orb/Γ0= 0 as shown in Fig. 7. For\nfurther increase in ∆ orb/Γ0, the orbital Kondo effect is\nsuppressed. The contributions of two spectral peaks at\nboth ends to the spectral weight at ω/Γ0∼0 are reduced\nwith increasing∆ orb/Γ0, leadingto amonotonicdecrease\ninG. Accordingly, the conductance has a maximum at\n∆orb/Γ0∼0.01.\nWe now discuss the effects of the polarization pon the\nKondo temperature. In the AP configuration, the SU(4)\nKondo effects at ∆ orb/Γ0= 0 is considered to appear\neven for finite pandTSU(4)\nKis independent of p. On\nthe other hand, when pis introduced for ∆ orb= 0 in\nthe P configuration, the SU(4) Kondo effect is destroyed\nand the SU(2) orbital Kondo effect survives as shown in\nFig. 2(a). We evaluate the Kondo temperature from the\ninflection point of the temperature dependence of G in\nthe P configuration (not shown in Figure). For p= 0,\nwe obtain TSU(4)\nK/Γ0∼0.02 for the SU(4) Kondo ef-\nfect. This value is consistent with that obtained from\nthe peak energy of the Kondo resonance shown in Fig.\n2(a). For p= 0.3, we obtain TSU(2)\nK/Γ0∼0.005 for the\nSU(2) Kondo effect. These results are reasonable, be-\ncauseTSU(4)\nKis expected to be higher than TSU(2)\nK. Ac-\ncordingly, the effective temperature scaled by TSU(4)\nKfor\np= 0 is lower than that scaled by TSU(2)\nKforp= 0.3\nin the P configuration. This result qualitatively explains\nthe conductance at ∆ orb/Γ0= 0 shown in Fig. 8, be-\ncause in the AP configuration the SU(4) Kondo effect\nappears at ∆ orb= 0: At ∆ orb/Γ0= 0, the effective tem-\nperature in the P configuration is higher than that in the\nAP configuration, which yields the smaller G in the P\nconfiguration. It is known that the EOM tends to un-\nderestimate the spectral weights and thus the magnitude\nof the conductance20,25. Actually, the conductance for\np= 0 and ∆ orb= 0 does not approach the value in the\nunitarylimit2 e2/hfortheSU(4)Kondoeffect. Toobtain\nmore quantitatively accurate results, another numerical\ncalculation by, for example, a numerical renormalization-\ngroup may be effective. Such a research is our future\nstudy.\nFinally, we discuss the conductance of a vertical QD\ncoupled to two Ni ferromagnetic leads. To develop a\nquantitative discussion, we use Γ 0= 30 meV, which\nwas estimated in a C 60experiment15. The range of\n∆orb/Γ0= 0.2 in Fig. 8 thus corresponds to ∆ orb= 6\nmeV. According to the Fock-Darwin state with an elec-\ntron confinement energy of 3 meV9, we evaluate that\n∆orb= 6 meV corresponds to the magnetic field B∼\n3.5T. On the basis of these results, we discuss the\nchange in the conductance in external magnetic fields38.\nWe first set the QD coupled to the Ni ferromagnetic\nleads in the AP configuration. The conductance in azero magnetic field is G|B=0∼0.952 e2/h as shown\nin Fig. 8. When the magnetic field is increased, the\nAP configuration changes into the P configuration at\nB∼O(10−1)T15. When the magnetic field is further\nincreased, we observe a nontrivial local maximum at\nB∼1.9T and then obtain G|B=3.5∼0.367 e2/h at\nB∼3.5T. We evaluate the ratio of the change in Gas\n∆G≡(G|B=0−G|B=3.5T)/G|B=3.5T∼1.59. This value\nis nearly twice that in the single-orbital QD coupled to\nthe Ni ferromagnetic leads, where the change ratio in G\nbetween the AP and P configurations is estimated to be\n∆G∼0.81. Note that ∆ Ghas the same meaning as tun-\nnel magnetoresistance. We have demonstrated that the\nlarge gain in ∆ Gcan be controlled by an external mag-\nnetic field in a two-orbital vertical QD coupled to two\nferromagnetic leads.\nIV. SUMMARY\nWe have investigated the Kondo effect of a two-orbital\nvertical QD coupled to two ferromagnetic leads using the\nEOM method. We have shown that in the P configu-\nration the orbital Kondo effect remains, while the spin\nKondo effect is suppressed. In magnetic fields, the or-\nbital Kondo effect also disappears, because the energy\nlevels of the QD are completely split. However, at a cer-\ntain magnetic field, two of the four energy levels cross\nand the Kondo effect newly emerges there. This Kondo\neffect can be possibly observed in experiments. In the\nAP configuration the spin Kondo effect remains, while\nthe orbital Kondo effect is suppressed in magnetic fields.\nWe have demonstrated that the change ratio in the con-\nductance between the AP and P configurations can be\ncontrolled by an external magnetic field. In a typical\ntwo-orbital QD coupled to Ni ferromagnetic leads, the\nchange ratio of the conductance is larger than that in a\nsingle-orbital QD system.\nAcknowledgments\nSomeofthenumericalcomputationswereperformedat\nthe Supercomputer Center at ISSP, University of Tokyo.\nT.K. was supported by the Japan Society for the Pro-\nmotion of Science. This work was supported by Grants-\nin-Aid for Scientific Research (C) (No. 20540390) from\nthe Japan Society for the Promotion of Science and on\nInnovative Areas (No. 21104514) from the Ministry of\nEducation, Culture, Sports, Science and Technology.\nAppendix A: Derivation of Eqs. (3)-(7)\nWe begin our discussion with the retarded Green’s\nfunction Gr\nlσ(t) =−iθ(t)/an}bracketle{t{dlσ(t),d†\nlσ(0)}/an}bracketri}ht. In the first8\niteration of the EOM procedure, we obtain\nid\ndtGr\nlσ(t) =δ(t)+εlGr\nlσ(t)+/summationdisplay\ni=L,R/summationdisplay\nkV∗\niklσGr\nik,lσ(t)\n+UΠlσ(t)+U′Λlσ(t), (A1)\nwhere Gr\nik,lσ(t)≡ − iθ(t)/an}bracketle{t{ciklσ(t),d†\nlσ(0)}/an}bracketri}ht,\nΠlσ(t)≡ −iθ(t)/an}bracketle{t{nlσ(t)dlσ(t),d†\nlσ(0)}/an}bracketri}ht, and Λ lσ(t)≡−iθ(t)/summationtext\nm(/negationslash=l)α/an}bracketle{t{nmα(t)dlσ(t),d†\nlσ(0)}/an}bracketri}ht. We perform the\nsecond iterations of the EOM procedure for Gr\nik,lσ(t),\nΠlσ(t), and Λ lσ(t). ForGr\nik,lσ(t), we obtain the closed\nform asidGr\nik,lσ(t)/dt=εikσGr\nik,lσ(t)+ViklσGr\nlσ(t). For\nthe latter two ones, the higher-order Green’s functions\nappear as\nid\ndtΠlσ(t) =/an}bracketle{tnlσ/an}bracketri}htδ(t)+εlΠlσ(t)−/summationdisplay\ni=L,R/summationdisplay\nkViklσXA\nik,lσ(t)−/summationdisplay\ni=L,R/summationdisplay\nkV∗\niklσXB\nik,lσ(t)+/summationdisplay\ni=L,R/summationdisplay\nkV∗\niklσXC\nik,lσ(t)\n+UΠlσ(t)+U′/summationdisplay\nm(/negationslash=l)αWmα,lσ(t), (A2)\nid\ndtΛlσ(t) =/summationdisplay\nm(/negationslash=l)α/an}bracketle{tnmα/an}bracketri}htδ(t)+εlΛlσ(t)−/summationdisplay\ni=L,R/summationdisplay\nk/summationdisplay\nm(/negationslash=l)αViklσYA\nikmα,lσ(t)−/summationdisplay\ni=L,R/summationdisplay\nk/summationdisplay\nm(/negationslash=l)αV∗\niklσYB\nikmα,lσ(t)\n+/summationdisplay\ni=L,R/summationdisplay\nk/summationdisplay\nm(/negationslash=l)αV∗\niklσYC\nikmα,lσ(t)+U/summationdisplay\nm(/negationslash=l)αWmα,lσ(t)+U′/summationdisplay\nm(/negationslash=l)α/summationdisplay\nn(/negationslash=l)βZmαnβ,lσ(t), (A3)\nwhere /an}bracketle{tnlσ/an}bracketri}htis the electron number of\nthe QD with spin σand orbital l, and\nXA\nik,lσ(t)≡ −iθ(t)/an}bracketle{t{c†\niklσ(t)dlσ(t)dlσ(t),d†\nlσ(0)}/an}bracketri}ht,\nXB\nik,lσ(t)≡ −iθ(t)/an}bracketle{t{d†\nlσ(t)dlσ(t)ciklσ(t),d†\nlσ(0)}/an}bracketri}ht,\nXC\nik,lσ(t)≡ − iθ(t)/an}bracketle{t{nlσ(t)ciklσ(t),d†\nlσ(0)}/an}bracketri}ht,\nWmα,lσ(t)≡ −iθ(t)/an}bracketle{t{nmα(t)nlσ(t)dlσ(t),d†\nlσ(0)}/an}bracketri}ht,\nYA\nikmα,lσ(t)≡ −iθ(t)/an}bracketle{t{c†\nikmα(t)dmα(t)dlσ(t),d†\nlσ(0)}/an}bracketri}ht,\nYB\nikmα,lσ(t)≡ −iθ(t)/an}bracketle{t{d†\nmα(t)dlσ(t)cikmα(t),d†\nlσ(0)}/an}bracketri}ht,\nYC\nikmα,lσ(t)≡ − iθ(t)/an}bracketle{t{nmα(t)ciklσ(t),d†\nlσ(0)}/an}bracketri}ht,\nZmαnβ,lσ(t)≡ −iθ(t)/an}bracketle{t{nmα(t)nnβ(t)dlσ(t),d†\nlσ(0)}/an}bracketri}ht.\nWe perform the third iterations of the EOM procedure\nfor these higher-order Green’s functions which appear\nin the right-hand side of Eqs. (A2) and (A3). In\nthis calculation, we adopt the approximation which\nsuccessfully yields the closed equation of the Green’s\nfunctions in a single-orbital QD system33,34. Accordingto this approximation, we neglect the following types of\nhigher-order terms /an}bracketle{t{c†\niklσ(t)dlσ(t)ci′k′lσ(t),d†\nlσ(0)}/an}bracketri}ht\nand/an}bracketle{t{d†\nlσ(t)ciklσ(t)ci′k′lσ(t),dlσ(0)†}/an}bracketri}htas well as\nthe expectation values /an}bracketle{tc†\niklσdlσ/an}bracketri}htand/an}bracketle{td†\nlσciklσ/an}bracketri}ht,\nand decouple /an}bracketle{t{ciklσ(t)†ci′k′lσ(t)dlσ(t),dlσ(0)†}/an}bracketri}ht ∼\nδii′δkk′f(εikσ)/an}bracketle{t{dlσ(t),dlσ(0)†}/an}bracketri}ht. In addition to this\napproximation, we decouple the number operator of the\nlocalized electron and replace it by its expectation value.\nFurthermore, we turn our attention to the scattering\nprocesses of the electrons in the same orbital with\ndifferent spins, and in the different orbitals with the\nsame and different spins. We include the higher-order\nterms which express these scattering processes and\nneglect those expressing other higher-order scattering\nprocesses. Using these approximations, we obtain the\nfollowing equations of the higher-order Green’s functions\nid\ndtXA\nik,lσ(t) =−\nεikσ−2εl−U−2U′/summationdisplay\nm(/negationslash=l)α/an}bracketle{tnmα/an}bracketri}ht\nXA\nik,lσ(t)−V∗\niklσΠlσ(t)+V∗\niklσf(εikσ)Gr\nlσ(t),(A4)\nid\ndtXB\nik,lσ(t) =εikσXB\nik,lσ(t)+V∗\niklσf(εikσ)Gr\nlσ(t)−ViklσΠlσ(t), (A5)\nid\ndtXC\nik,lσ(t) =εikσXC\nik,lσ(t)+ViklσΠlσ(t), (A6)9\nid\ndtWmα,lσ(t) =/an}bracketle{tnmα/an}bracketri}ht/an}bracketle{tnlσ/an}bracketri}htδ(t)+\nεl+U+U′/summationdisplay\nj(/negationslash=l)s/an}bracketle{tnjs/an}bracketri}ht\nWmα,lσ(t)−/summationdisplay\nkViklσ/an}bracketle{tnmα/an}bracketri}htXA\nik,lσ(t)\n−/summationdisplay\nkV∗\niklσ/an}bracketle{tnmα/an}bracketri}htXB\nik,lσ(t), (A7)\nid\ndtYA\nik,lσ(t) =−\nεikσ−εl−εm+U′+U′/summationdisplay\nj(/negationslash=l)s/an}bracketle{tnjs/an}bracketri}ht+U/an}bracketle{tnlσ/an}bracketri}ht\nYA\nik,lσ(t)+[f(εikσ)−/an}bracketle{tnmα/an}bracketri}ht]V∗\niklσGr\nlσ(t),(A8)\nid\ndtYB\nik,lσ(t) =\nεikσ+εl−εm−U′+U′/summationdisplay\nj(/negationslash=l)s/an}bracketle{tnjs/an}bracketri}ht+U/an}bracketle{tnlσ/an}bracketri}ht\nYB\nik,lσ(t)+[f(εikσ)−/an}bracketle{tnmα/an}bracketri}ht]ViklσGr\nlσ(t),(A9)\nid\ndtYC\nik,lσ(t) =εikσYC\nik,lσ(t)+/an}bracketle{tnmα/an}bracketri}htViklσGr\nlσ(t), (A10)\nid\ndtZmαnβ,lσ(t) =/an}bracketle{tnmα/an}bracketri}htδm,nδα,βδ(t)+(εl+U/an}bracketle{tnlσ/an}bracketri}ht+U′/an}bracketle{tnlσ/an}bracketri}ht)Zmαnβ,lσ(t). 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B\n50, 5528 (1994).\n38Using these parameters, we estimate the Zeeman energy as\nEZ/Γ0∼O(10−4). This value is smaller than the Kondo\ntemperature TK/Γ0∼O(10−2)−O(10−3) and the orbital\nsplitting ∆ orb/Γ0∼O(10−1). Therefore, we can ignore the\nZeeman effect. 0.3 0.6 0.9\n 0  0.1  0.2G[e2/h]\nΔorb/Γ0p=0.3, Parallel\nAntiparallel 0.3 0.6 0.9\n 0.01  0.1\nT/Γ0Δorb/Γ0 = 0, Parallel\np=0\n0.3"    },    {        "title": "1212.5474v1.Shot_noise_of_spin_current_and_spin_transfer_torque.pdf",        "content": "arXiv:1212.5474v1  [cond-mat.mes-hall]  21 Dec 2012Shot noise of spin current and spin transfer torque\nYunjin Yu1, Hongxin Zhan1, Langhui Wan1, Bin Wang1,2,\nYadong Wei1,Qingfeng Sun,2and Jian Wang3\n1College of Physics Science and Technology and Institute of Computa tional\nCondensed Matter Physics, Shenzhen University, Shenzhen 5180 60, China\n2Institute of Physics, Chinese Academy of Sciences, Beijing, China\n3Department of Physics and The Center of Theoretical and Comput ational Physics,\nThe University of Hong Kong, China\nAbstract. We report the theoretical investigation of noise spectrum of spin c urrent\n(Sσ) and spin transfer torque ( Sτ) for non-colinear spin polarized transport in a spin-\nvalve device which consists of normal scattering region connected by two ferromagnetic\nelectrodes (MNM system). Our theory was developed using non-eq uilibrium Green’s\nfunction method and general non-linear Sσ−VandSτ−Vrelations were derived as a\nfunction of angle θbetween magnetization of two leads. We have applied our theory to\na quantum dot system with a resonantlevel coupled with two ferrom agneticelectrodes.\nIt was found that for the MNM system, the auto-correlation of sp in current is enough\nto characterize the fluctuation of spin current. For a system with three ferromagnetic\nlayers, however, both auto-correlation and cross-correlation o f spin current are needed\nto characterize the noise spectrum of spin current. Furthermor e, the spin transfer\ntorque and the torque noise were studied for the MNM system. For a quantum dot\nwith a resonant level, the derivative of spin torque with respect to b ias voltage is\nproportional to sin θwhen the system is far away from the resonance. When the\nsystem is near the resonance, the spin transfer torque becomes non-sinusoidal function\nofθ. The derivative of noise spectrum of spin transfer torque with res pect to the bias\nvoltageNτbehaves differently when the system is near or far away from the re sonance.\nSpecifically, the differential shot noise of spin transfer torque Nτis a concave function\nofθnear the resonance while it becomes convex function of θfar away from resonance.\nFor certain bias voltages, the period Nτ(θ) becomes πinstead of 2 π. For small θ, it\nwas found that the differential shot noise of spin transfer torque is very sensitive to\nthe bias voltage and the other system parameters.\nPACS numbers: 72.25.-b,72.70.+m,74.40.-n\nSubmitted to: Nanotechnology\n1. Introduction\nElectronic shot noise describes the fluctuation of current and is an intrinsic property of\nquantum devices due to the quantization of electron charge. In th e past decade, the\nstudy of shot noise has attracted increasing attention[1] becaus e it can give additionalShot noise of spin current and spin transfer torque 2\ninformation that is not contained in the conductance or charge cur rent. It can be\nused to probe the kinetics of electron[2] and investigate correlatio ns of electronic wave\nfunctions[3]. In the study of shot noise <(∆ˆI)2>, the Fano Factor F=<(∆ˆI)2>\n/2q <ˆI >is often used where <ˆI >is the current. When F >1 it is referred as super-\nPoissonian noise, while F <1 corresponds to sub-Poissonian behavior. In general,\nfor a quantum device, Pauli exclusion suppresses the shot noise an d hence reduces the\nFano factor[4, 5, 6] but Coulomb interaction can either suppress o r enhance shot noise\ndepending on system details[7, 8, 9, 10, 11]. The suppression of sho t noise has been\nconfirmed experimentally in quantum point contact[12, 13], single elec tron tunneling\nregime[14, 15], graphene nano-ribbon[16, 17], and atom-size metallic contacts[18, 19].\nThe enhancement of shot noise was also observed in GaAs based qua ntum contacts\nwhen the system is in the negative differential conductance region[ 20]. Recently, with\nthe development of spintronics, polarized spin current especially pu re spin current\nreceived much more attention. Less attention has been paid on the polarized spin\ncurrent correlation compared with the charge current correlatio n[21, 22, 23, 24, 25, 26].\nShot noise of polarized spin current has been studied in several qua ntum devices\nincluding the MNM(ferromagnet-normal-ferromagnet)[27] and NM N(normal-magnetic-\nnormal)[28]. In these devices, shot noise is expected to provide add itional information\nabout the spin-dependent scattering process and spin accumulat ion. It was shown that\nshotnoisecanbeusedtoprobeattractiveorrepulsive interaction sinmesoscopic systems\nand to measure the spin relaxation time [29]. For a two-probe normal system (NNN\nsystem), it is well known that the charge current correlation betw een different probes\n(cross correlation noise) is negative definitely[30], but for a magnet ic junction, the spin\ncross correlation noise between different probes is not necessarily negative due to spin\nflip mechanism. For example, Ref.[31] showed that the cross correla tion can be positive\nat special Fermi energy due to Rashba interaction.\nRecently, spin transfer torque (STT), predicted by Slonczewski[3 2, 33] and\nBerger[34], has been the subject of intensive investigations[35, 36 , 37, 38]. Spin current\ncan transfer spin angular momentum and be used to switch the magn etic orientation\nof ferromagnetic layers in GMR and TMR devices. Therefore, STT ha s potential\napplications[39] such as hard-disk read head[40], magnetic detectio n sensor[41], and\nrandom access memory (MRAM)[42], etc.. It comes from the absorp tion of the itinerant\nflow of angular momentum components normal to the magnetization direction and relies\non the system spin polarized current. The noise spectrum of STT dr astically affects the\nmagneto-resistance behavior[43]. Many studies have focused on t he spin transfer torque\nin various materials and under dc or ac condition. The correlation effe ct or quantum\nnoise of spin transfer torque has not been studied so far. It is the purpose of this paper\nto fill this gap. In this paper, we have calculated the shot noise of pa rticle current,\nspin current as well as spin transfer torque in the nonlinear regime f or a magnetic\nquantum dot connected with two non-colinear magnetic electrodes . We found that\nfor a MNM spin-valve system, the spin auto-correlation is enough to characterize the\nfluctuation of spin current. For a system with three ferromagnet ic layers (MNMNM),Shot noise of spin current and spin transfer torque 3\n/s32\nFigure 1. The schematic plot of the spin-degenerated quantum dot connect ed by two\nferromagnetic leads. The magnetic moment of the left lead is always a tzdirection and\nthe magnetic moment of the right lead is at an angle θRwith respect to the z-axis in\nthex−zplane.\nhowever, both auto-correlation and cross-correlation are need ed to characterize the\nfluctuation of spin current. For the quantum dot with a resonant le vel, the behavior of\ndifferentialSTTdependsonwhetherthesystemonresonanceoro ffresonance. Whenthe\nsystem is off resonance, the differential STT reduces to the familiar result of tunneling\nbarrier (1 /2)(Is(π)−Is(0))sinθwhereθis the angle between magnetic moments of\nferromagnetic leads. If it is on resonance, the dependence of diffe rential STT on θ\nbecomes non-sinusoidal. The resonance also has influence on the no ise spectrum of\nSTT. If the system is near the resonance, noise spectrum of STT is a concave function\nofθwhile it becomes a convex function far away from the resonance.\nThis paper is organized as follows. Firstly, we derive the general for mulae of spin\nauto-correlation shot noise, spin cross-correlation shot noise an d spin transfer torque\nshot noise from the non-equilibrium Green’s function method. Then w e analyze the\nspin transport properties for the MNM system. Finally, we give the c onclusions.\n2. Theory formalism\nWe start from the Hamiltonian of the quantum dot which is connected by two magnetic\nelectrodes. We assume that the current flows in the y(y′)-direction and the left lead\nmagnetic moment MLalways points at the z-direction, while the right lead magnetic\nmoment MRpoints at an angle θRto thez-direction in x−zplane (see figure 1).\nIn the second quantized form, Hamiltonian is\nˆH=ˆHlead+ˆHdot+ˆHT, (1)\nwhereˆHleadis the Hamiltonian of the leads,\nˆHlead=/summationdisplay\nkασ(ǫkα−σMαcosθα)ˆC†\nkασˆCkασ−/summationdisplay\nkασMαsinθαˆC†\nkασˆCkα¯σ,(2)\nwhereˆC†\nkασcreates an electron in lead αwith energy level kand spin σ,σ=±1 and\n¯σ=−σ. The second term ˆHdotis the Hamiltonian of the isolated quantum dot,\nˆHdot=/summationdisplay\nnσǫnˆd†\nnσˆdnσ, (3)Shot noise of spin current and spin transfer torque 4\nThe third term ˆHTis the Hamiltonian describing the coupling between quantum dot\nand the leads with the coupling constant tkαnσ,\nˆHT=/summationdisplay\nkαnσ[tkαnσˆC†\nkασˆdnσ+c.c.], (4)\nwherec.c.denotes the complex conjugate. By applying the Bogoliubov transf orma-\ntion [44],\nˆckασ=cos(θα/2)ˆCkασ−σsin(θα/2)ˆCkα¯σ, (5)\nwe can diagonalize the Hamiltonian of the electrodes to give the followin g effective\nHamiltonian,\nˆHα=/summationdisplay\nkσ(ǫkα−σMα)ˆc†\nkασˆckασ. (6)\nSoforaferromagneticleadcoupledwithscattering quantumdot, t heline-widthfunction\nΓαcan be written as [45]\nΓα=Rα/parenleftBigg\nΓα↑0\n0 Γα↓/parenrightBigg\nR†\nα, (7)\nwhere\nRα=/parenleftBigg\ncosθα\n2−sinθα\n2\nsinθα\n2cosθα\n2/parenrightBigg\n(8)\nis the rotational matrix.\nThe current operator of the lead αwith spin σis defined as\nˆIασ(t) =qdˆNασ\ndt. (9)\nwhereˆNασ=/summationtext\nkˆC†\nkασˆCkασis the number operator for the electron in the lead α. By\nusing the Heisenberg equation of motion, we have\nˆIασ(t) =−iq\n¯h/summationdisplay\nkm[tkαmσˆC†\nkασ(t)ˆdmσ(t)]+c.c. (10)\nThe average current can be expressed by in terms of Green’s func tion,\n<ˆIασ(t)>=−q\n¯h/summationdisplay\nkm[tkαmσG<\nmσkασ(t,t)+c.c]. (11)\nIf we consider the total charge current flowing through the lead α, the charge current\noperator can be expressed as\nˆIα=ˆIα↑+ˆIα↓ (12)\nand the spin current operator in z-direction is\nˆIs\nα=¯h\n21\nq(ˆIα↑−ˆIα↓). (13)\nSince the local spin current is not conserved, the loss of the spin an gular momentum\nis transferred to the magnetization of the free layer. For spin tra nsfer torque, we are\ninterested in the MNM system and we assume that electron coming fr om the left leadShot noise of spin current and spin transfer torque 5\nwhich is pinned and the right lead is the free layer. The spin transfer t orque can be\ncalculated as follows. The total spin of the right ferromagnetic elec trode is[46]\nˆSθ=¯h\n2/summationdisplay\nkRµνC†\nkRµCkν(R−1χµ)†ˆσ(R−1χν). (14)\nHere, ˆσis Pauli matrices and the spinup state χµ(ν)= (1\n0) forµ(ν) = 1 or the\nspindown state χµ(ν)= (0\n1) forµ(ν) =−1. Note that the equation above is written\ninxyzcoordinate frame while ˆSθare quantized in the x′y′z′frame. Because ˆSθ(t) is\nalong the direction of z′, the total spin torque ˆ τ=∂ˆSθ\n∂t=i\n¯h[ˆHT,ˆSθ] should be along the\ndirection of x′(see figure 1). So we need the expression of the spin operator of t he right\nlead along x′direction, ˆSθ\nx′, which can be obtained from equation (14),\nˆSθ\nx′=¯h\n2/summationdisplay\nkRσ(ˆC†\nkRσˆCkR¯σcosθ−σˆC†\nkRσˆCkRσsinθ). (15)\nAccording to the Heisenberg equation of motion, the spin transfer torque operator is\nˆτR= ˆτx′=i\n¯h[ˆHT,ˆSθ\nx′]\n=−i\n2[/summationdisplay\nkRnσσ′(ˆC†\nkRσRσσ′ˆdnσ′tkRnσ′−t∗\nkRnσ′ˆd†\nnσ′Rσσ′ˆCkRσ)] (16)\nwhere,\nR=/parenleftBigg\nR↑↑R↑↓\nR↓↑R↓↓/parenrightBigg\n=/parenleftBigg\n−sinθRcosθR\ncosθRsinθR/parenrightBigg\n,tkRn=/parenleftBigg\ntkRn↑0\n0tkRn↓/parenrightBigg\n.(17)\nThe average spin transfer torque is[46]\n<ˆτR>=Re{/summationdisplay\nkRnTrσ[tkRnRG<\nn,kR]}\n=/integraldisplaydE\n2π(fL−fR)Tr[GrΓLGa(iΣa\nRR−iRΣr\nR)] (18)\nwhereTrσis over spin space.\nThe correlation of the charge current is given by\n<∆ˆIα(t1)∆ˆIβ(t2)>=/summationdisplay\nσσ′(<∆ˆIασ(t1)∆ˆIβσ′(t2)>) (19)\nand the shot noise of spin current is\n<∆ˆIs\nα(t1)∆ˆIs\nβ(t2)>=1\n4¯h2\nq2/summationdisplay\nσσ′σσ′(<∆ˆIασ(t1)∆ˆIβσ′(t2)>) (20)\nwhere\n∆ˆIασ=ˆIασ−<ˆIασ>, (21)\nandσ=↑↓or±1. Finally, the correlation of spin transfer torque is\nS(t1,t2) =<∆ˆτR(t1)∆ˆτR(t2)>, (22)Shot noise of spin current and spin transfer torque 6\nwhere ∆ˆτR= ˆτR−<ˆτR>.\nWe now derive the correlation of charge current, spin current, an d spin transfer\ntorque. Clearly, all correlation functions contain the following term ,\n<ˆIασ(t1)ˆIβσ′(t2)>=−q2\n¯h2/summationdisplay\nkk′mn\n[tkαmσtk′\nβnσ′<ˆC†\nkασ(t1)ˆdmσ(t1)ˆC†\nk′\nβσ′(t2)ˆdnσ′(t2)>\n+t∗\nkαmσt∗\nk′\nβnσ′<ˆd†\nmσ(t1)ˆCkασ(t1)ˆd†\nnσ′(t2)ˆCk′\nβσ′(t2)>\n−tkαmσt∗\nk′\nβnσ′<ˆC†\nkασ(t1)ˆdmσ(t1)ˆd†\nnσ′(t2)ˆCk′\nβσ′(t2)>\n−t∗\nkαmσtk′\nβnσ′<ˆd†\nmσ(t1)ˆCkασ(t1)ˆC†\nk′\nβσ′(t2)ˆdnσ′(t2)>],\n(23)\nand\n<ˆIασ(t1)><ˆIβσ′(t2)>=−q2\n¯h2/summationdisplay\nkk′mn\n[tkαmσtk′\nβnσ′<ˆC†\nkασ(t1)ˆdmσ(t1)><ˆC†\nk′\nβσ′(t2)ˆdnσ′(t2)>\n+t∗\nkαmσt∗\nk′\nβnσ′<ˆd†\nmσ(t1)ˆCkασ(t1)><ˆd†\nnσ′(t2)ˆCk′\nβσ′(t2)>\n−tkαmσt∗\nk′\nβnσ′<ˆC†\nkασ(t1)ˆdmσ(t1)><ˆd†\nnσ′(t2)ˆCk′\nβσ′(t2)>\n−t∗\nkαmσtk′\nβnσ′<ˆd†\nmσ(t1)ˆCkασ(t1)><ˆC†\nk′\nβσ′(t2)ˆdnσ′(t2)>].\n(24)\nUsing the Wick’s theorem [47], we have,\n<ˆC†\nkασ(t1)ˆdmσ(t1)ˆC†\nk′\nβσ′(t2)ˆdnσ′(t2)>\n=<ˆC†\nkασ(t1)ˆdmσ(t1)><ˆC†\nk′\nβσ′(t2)ˆdnσ′(t2)>\n+<ˆC†\nkασ(t1)ˆdnσ′(t2)><ˆdmσ(t1)ˆC†\nk′\nβσ′(t2)> . (25)\nThe shot noise can be expressed in terms of Green’s function\n<∆ˆIασ(t1)∆ˆIβσ′(t2)>=\n−q2\n¯h2/summationdisplay\nkk′mn[tkαmσtk′\nβnσ′G>\nmσk′\nβσ′G<\nnσ′kασ\n+t∗\nkαmσt∗\nk′\nβnσ′G>\nkασnσ′G<\nk′\nβσmσ\n−tkαmσt∗\nk′\nβnσ′G>\nmσnσ′G<\nk′\nβσ′kασ\n−t∗\nkαmσtk′\nβnσ′G>\nkασk′\nβσ′G<\nnσ′mσ]. (26)\nFrom the Langreth theorem of analytic continuation, we have\nG<,>\nmσk′\nβσ′(t1,t2)\n=/summationdisplay\npσp/integraldisplay\ndt[Gr\nmσpσp(t1,t)t∗\nk′\nβpσpg<,>\nk′\nβσpσ′(t,t2)\n+G<,>\nmσpσp(t1,t)t∗\nk′\nβpσpga\nk′\nβσpσ′(t,t2)], (27)Shot noise of spin current and spin transfer torque 7\nand\nG<,>\nk′\nβσ′mσ(t1,t2)\n=/summationdisplay\npσp/integraldisplay\ndt[g<,>\nk′\nβσ′σp(t1,t)tk′\nβpσpGa\npσpmσ(t1,t)\n+gr\nk′\nβσ′σp(t1,t)tk′\nβpσpG<,>\npσpmσ(t,t2)], (28)\nas well as\nG<,>\nkασk′\nβσ′(t1,t2) =g<,>\nk′\nβσσ′(t1,t2)δkk′δαβ\n+/summationdisplay\npσp/integraldisplay\ndt[Gr\nkασpσp(t1,t)t∗\nk′\nβpσpg<,>\nk′\nβσpσ′(t,t2)\n+G<,>\nkασpσp(t1,t)t∗\nk′\nβpσpga\nk′\nβσpσ′(t,t2)], (29)\nThe self-energy is given by\nΣγ\nαmσnσ′(t1,t2) =/summationdisplay\nkt∗\nkαmσ(t1)gγ\nkασσ′(t1,t2)tkαnσ′(t2). (30)\nFrom the above equations, we can calculate the noise spectrum Sdefined as\nfollows [6]:\nπδ(0)Sσσ′\nαβ=/integraldisplay\ndt1dt2<∆ˆIασ(t1)∆ˆIβσ′(t2)> . (31)\nUsing Eqs.(27)-(30), it is straightforward to write the noise spect rum as\nSσσ′\nαβ=4/summationdisplay\ni=0Sσσ′\ni,αβ. (32)\nHere\nπδ(0)Sσσ′\n0,αβ=q2\n¯h2δαβTrt[G>\nσσ′Σ<\nα,σ′σ+Σ>\nα,σσ′G<\nσ′σ], (33)\nπδ(0)Sσσ′\n1,αβ=−q2\n¯h2Trt[(GrΣ>\nβ+G>Σa\nβ)σσ′\n×(GrΣ<\nα+G<Σa\nα)σ′σ], (34)\nπδ(0)Sσσ′\n2,αβ=−q2\n¯h2Trt[(Σ>\nαGa+Σr\nαG>)σσ′\n×(Σ<\nβGa+Σr\nβG<)σ′σ], (35)\nπδ(0)Sσσ′\n3,αβ=q2\n¯h2Trt[G>\nσσ′(Σr\nβGrΣ<\nα+Σ<\nβGaΣa\nα\n+Σr\nβG<Σa\nα)σ′σ)], (36)\nπδ(0)Sσσ′\n4,αβ=q2\n¯h2Trt[(Σr\nαGrΣ>\nβ+Σ>\nαGaΣa\nβ\n+Σr\nαG>Σa\nβ)σσ′G<\nσ′σ]. (37)\nSince all Green’s functions depend on double time indices, the trace Trtis taken in the\ntime domain.Shot noise of spin current and spin transfer torque 8\nTo get the charge current noise spectrum, we combine equation (1 9) with equation\n(31) and perform Fourier transformation, the well known auto ch arge current noise can\nbe obtained\nSLL=q2\nπ/integraldisplay\ndE(fL−fR)2Tr[(1−T)T], (38)\nhere,T= ΓLGrΓRGais the transmission matrix. We can also get the cross charge\ncurrent shot noise (along z direction):\nSLR=−q2\nπ/integraldisplay\ndE(fL−fR)2Tr[(1−T)T], (39)\nSLL+SLR= 0 confirms the current conservation.\nFor spin current noise spectrum, the situation is very different. We combine\nequation (20) with equation (31), taking Fourier transformation, and using the relation\n/summationdisplay\nσσ′σσ′Aσσ′Bσ′σ=Tr[AσzBσz], (40)\n(σzis pauli matrix), we can obtain the auto spin current shot noise (zer o-temperature\nlimit),\nSσ\nLL=¯h2\n4π/integraldisplay\ndE(fL−fR)2Tr[σzTσz(1−T)]. (41)\nSimilarly, the cross spin current shot noise can be obtained:\nSσ\nLR=−¯h2\n4π/integraldisplay\ndE(fR−fL)2Tr{[GrΓRGa(Σa\nRσz\n−σzΣr\nR)+GrΓRσz](GrΓLσz+GrΓLGa(Σa\nLσz\n−σzΣr\nL)}. (42)\nNow, we derive the shot noise of spin transfer torque S(t1,t2),\nS(t1,t2) =−1\n4</summationdisplay\nkRAn/summationdisplay\nσAσ′\nA(ˆC†\nkRAσARσAσ′\nAˆdnσ′\nAtkRAnσ′\nA−t∗\nkRAnσ′\nAˆd†\nnσ′\nARσAσ′\nAˆCkRAσA)\n/summationdisplay\nkRBm/summationdisplay\nσBσ′\nB(ˆC†\nkRBσBRσBσ′\nBˆdnσ′\nBtkRBnσ′\nB−t∗\nkRBnσ′\nBˆd†\nnσ′\nBRσBσ′\nBˆCkRBσB)>(43)\nand\n<ˆτR(t)>=i\n2</summationdisplay\nkRAn/summationdisplay\nσAσ′\nA(ˆC†\nkRAσARσAσ′\nAˆdnσ′\nAtkRAnσ′\nA−t∗\nkRAnσ′\nAˆd†\nnσ′\nARσAσ′\nAˆCkRAσA)> .(44)\nSimilarly, we define the shot noise spectrum of spin transfer torque like the shot noise\nof spin current, it can be written as\nπδ(0)Sτ=/integraldisplay\ndt1dt2S(t1,t2). (45)\nwhereS(t1,t2) is defined in equation (22).\nBy using the Wick’s theorem, and after Fourier transformation, we can obtain\nexpression of Sτ,\nSτ=¯h2\n4π/integraldisplay\ndE(fL−fR){Tr[GrΓLGaRΓRRShot noise of spin current and spin transfer torque 9\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48/s50/s46/s52/s32\n/s82/s61/s48/s46/s49\n/s32\n/s82/s61/s48/s46/s50\n/s32\n/s82/s61/s48/s46/s52\n/s32/s32\n/s32/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110/s40/s97/s41\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s32\n/s82/s61/s48/s46/s49\n/s32\n/s82/s61/s48/s46/s50\n/s32\n/s82/s61/s48/s46/s52\n/s32/s32\n/s32/s83/s112/s105/s110/s32/s116/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s105/s110\n/s70/s101/s114/s109/s105/s32/s69/s110/s101/s114/s103/s121/s40/s98/s41\nFigure 2. (a)Thechargetransmissioncoefficient( T↑+T↓)and(b)thespintransmission\ncoefficient ( T↑−T↓) versus fermi energy when Γ R↓= 0.1 (red dashed line), Γ R↓= 0.2\n(black solid line), Γ R↓= 0.4 (blue dotted line). The other parameters are θR= 0,\nǫ= 0 ΓL↓= 0.2, ΓL↑= ΓR↑= 0.8. The energy unit is eV.\n+GrΓLGa(Σa\nRR−RΣr\nR)GrΓLGa(Σa\nRR−RΣr\nR)\n+Gr(iRΣr\nR−iΣa\nRR−ΓRR)GrΓLGa(Σa\nRR−RΣr\nR)\n+ (Σa\nRR−RΣr\nR)GrΓLGa(RΓR−iRΣr\nR+iΣa\nRR)Ga]}. (46)\n3. Shot noise of spin current and spin torque for MNM system\nIn this paper, we consider a normal quantum dot connected by two ferromagnetic leads\n(see figure 1) (MNM system). The magnetic moment of left lead is poin ting to the\nz-direction, while the moment of right lead is at an angle θRto thez-axis in the x−z\nplane. Hence the Hamiltonian of quantum dot can be written as\nHdot=/parenleftBigg\nǫ00\n0ǫ0/parenrightBigg\n. (47)\nFirstly, we set the direction of magnetization of the right lead be alon g thez\ndirection, i.e., let θR= 0 and calculate the charge and spin current according to\nLandauer-B¨ uttiker formula\nIc=−q\n¯h/integraldisplaydE\n2πTr[ˆT(E)](fL−fR), (48)\nand the expression of spin current is\nIs=1\n2/integraldisplaydE\n2πTr[σzˆT(E)](fL−fR). (49)\nThe charge andspin transmission coefficients aredepicted in figure2 . In thecalculation,\nwe have chosen Γ L↑= ΓR↑= 0.8eV and fix the energy unit is to be eV. Let\nΓL↓= ΓR↓= 0.2eV (Here, we let Γ α↑/ne}ationslash= Γα↓due to the presence of ferromagnetic\nleads), we found that the charge transmission coefficient reaches two at the resonant\nenergy level E=ǫ0of the quantum dot (solid line in the left panel of figure 2), whileShot noise of spin current and spin transfer torque 10\nthe spin transmission coefficient is zero at resonant energy point (s olid line in the right\npanel of figure 2). For parallel situation ( θR= 0) there is no spin flip so that different\nspin channel can be treated separately. For a symmetric coupling f rom the lead, both\nspin up and spin down electrons have complete transmission at the re sonance. For total\ncharge current they add up together while for total spin current they cancel to each\nother. When we break this symmetry and change Γ R↓while keeping Γ L↓constant, the\nspin down transport will be partially blocked, so the charge transmis sion coefficient will\ndecrease and the spin transmission coefficient will increase (see the dashed lines and\ndotted lines in figure 2).\nFigure 3 gives a comparison between the charge current and spin cu rrent versus\nθRunder the small bias voltage 0.05V. From the figure, we find that for the symmetric\ncoupling with Γ L↑= ΓR↑and ΓL↓= ΓR↓, both charge current and spin current decrease\nasθRincreasing from zero to π(see the solid lines in figure 3(a) and 3(b)). But if we fix\nΓL↓and change Γ R↓, although the charge current still decreases when θRchanges from\nzero toπ, the spin current increases when Γ R↓>ΓL↓and decreases when Γ R↓<ΓL↓and\nchanges sign at θR=π. To understand the behavior, we plot the spin-up current in the\npanel (c) and spin-down current in the panel (d), respectively. O ne can clearly see that\nspin up current always decreases with θRfrom zero to π, but spin down current always\nincreases though it is negative. So the competition between spin up a nd down channels\ndetermines how the total spin current varies with Γ R��. Another interesting result is that\natθR=π, i.e., when the magnetic moments of the two leads are antiparallel, the spin\ndown current does not change when we change the Γ R↓(see figure 4(d)) while keeping\nother parameters the same. In fact, when we change the Γ R↓atθR=π, we actually\nchange the right coupling line-width constant of spin up but not spin d own due to\nΓR(π) =Rα(π)/parenleftBigg\nΓR↑0\n0 ΓR↓/parenrightBigg\nR†\nα(π) =/parenleftBigg\nΓR↓0\n0 ΓR↑/parenrightBigg\n. So we can find that at θR=π,\nthe spin up current is different with different Γ R↓but spin down keeps unchanged.\nTo study the shot noise of spin current, we first examine the differe ntial shot noise\nspectrum versus bias voltage VL=VandVR= 0. At zero temperature, they can be\ncalculated from equations (41) and (42) (AC means auto-correlat ion and CC means\ncross-correlation)\nNAC=4π\nq¯h2∂Sσz\nLL\n∂V=Tr[σzT(E)σz(1−T(E))]|E=qV (50)\nand\nNCC=4π\nq¯h2∂Sσz\nLR\n∂V=−Tr{[GrΓRGa(Σa\nRσz−σzΣr\nR)+\nGrΓRσz](GrΓLσz+GrΓLGa(Σa\nLσz−σzΣr\nL))}|E=qV. (51)\nFor equation (51), we see that if the direction of magnetic moments of both leads\nare parallel the off-diagonal matrix elements of all the physical qua ntity including the\nlinewidth function Γ ασare zero, so σzcommutes with other matrices in equation (51).\nUsing this property and Ga−Gr=iGrΓLGa+iGrΓRGa, we find\nNCC=−Tr[σzTσz(I−T)] =−NAC. (52)Shot noise of spin current and spin transfer torque 11\n/s50/s46/s52/s51/s46/s50/s52/s46/s48/s40/s100/s41\n/s40/s99/s41\n/s82/s32\n/s82/s61/s48/s46/s49\n/s32\n/s82/s61/s48/s46/s50\n/s32\n/s82/s61/s48/s46/s52\n/s32/s32/s67/s104/s97/s114/s103/s101/s32/s99/s117/s114/s114/s101/s110/s116/s40 /s41\n/s82/s48/s101/s108/s101/s99/s116/s114/s111/s110/s32/s99/s117/s114/s114/s101/s110/s116/s40/s97/s41\n/s48/s46/s48/s49\n/s48\n/s32/s82/s82/s48/s48\n/s32\n/s82/s61/s48/s46/s49\n/s32\n/s82/s61/s48/s46/s50\n/s32\n/s82/s61/s48/s46/s52/s115/s112/s105/s110/s32/s99/s117/s114/s114/s101/s110/s116\n/s32/s32/s83/s112/s105/s110/s32/s99/s117/s114/s114/s101/s110/s116/s40/s101/s86/s41\n/s48/s40/s98/s41\n/s45/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s32\n/s32/s115/s112/s105/s110/s117/s112\n/s99/s117/s114/s114/s101/s110/s116\n/s45/s48/s46/s48/s52/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50\n/s32/s115/s112/s105/s110/s100/s111/s119/s110\n/s32/s32/s99/s117/s114/s114/s101/s110/s116\n/s32\nFigure 3. The charge current (panel (a)), total spin current (panel (b) ), spin up\ncurrent (panel (c)) and spin down current (panel (d)) versus θRfor MNM system at\nΓR↓= 0.1 (red dashed line), Γ R↓= 0.2 (black solid line), Γ R↓= 0.4 (blue dotted\nline),respectively. The other parameters are Vbias= 0.05V,ǫ0= 0, Γ L↑= ΓR↑= 0.8,\nΓL↓= 0.2.\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s45/s50 /s45/s49 /s48 /s49 /s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s32\n/s32/s78\n/s65/s67\n/s82/s48/s40/s97/s41\n/s32\n/s32\n/s32/s32\n/s40/s98/s41/s32 /s32/s32\n/s86\n/s98/s105/s97/s115/s32/s40/s86/s41/s40/s99/s41\nFigure 4. MNM system. (a) NACof spin current noise versus the angle θR\nwithVbias= 0.05V; (b) NACversus the bias voltage with two electrode magnetic\nmomentsparallel; (c) NACversusthebiasvoltagewithtwoelectrodemagneticmoments\nantiparallel. The other parameters are ǫ0= 0, and Γ L↑= ΓR↑= 0.8,ΓL↓= ΓR↓= 0.2.\nNow we calculate NACfrom equations (50). The figure 4(a) gives NACversusθR.\nOnecanfindthatthedifferential spinshot NACissmall forparallelsituationandreaches\nmaximumwhenthemagnetizationofleadsareantiparallel. Wealsoplotd ifferential spin\nshot noise versus bias voltage at parallel and antiparallel configura tions in figure 4(b)\nand 4(c). When the magnetization of two lead are parallel, NACincreases abruptly\nwith the bias voltage and reaches a flat plateau between about Vbias= (0.3,0.7)V, then\ndecreases gradually upon further increasing bias voltage. Howeve r, for antiparallel case,Shot noise of spin current and spin transfer torque 12\nNACstarts at a large value compared with that of parallel case and incre ases a bit to\na maximum value at Vbias=±0.26V. For large bias voltage VbiasNACdecreases and\ngradually approaches to zero. Wehave shown that NAC+NCC= 0 in the case of parallel\nand anti-parallel situations. It is found that this relation is still valid w henθRis not\nequal to 0 or π. In general, the relation NAC+NCCis not satisfied. For instance, If we\nstudy a system MNMNM with three ferromagnetic layers or the MM int erface where\ncoupling matrix elements Γ σ¯σ/ne}ationslash= 0, one can get NAC+NCC/ne}ationslash= 0.\nNow we analyze the spin transfer torque and its auto-correlation f unction. From\nequations (18) and (46), we calculate the derivative of spin transf er torque and its\ncorrelation function with respect to the bias voltage as follows:\nTτ=2π\nq∂ <ˆτR>\n∂V=Tr[GrΓLGa(iΣa\nRR−iRΣr\nR)]|E=qV, (53)\nand\nNτ=4π\nq¯h2∂Sτ\n∂V=Tr[GrΓLGa(Σa\nRR−RΣr\nR)GrΓLGa(Σa\nRR��RΣr\nR)\n+GrΓLGaRΓRR\n+Gr(iRΣr\nR−iΣa\nRR−ΓRR)GrΓLGa(Σa\nRR−RΣr\nR)\n+ (Σa\nRR−RΣr\nR)GrΓLGa(RΓR−iRΣr\nR+iΣa\nRR)Ga]}|E=qV.(54)\nSince most of calculations for the spin transfer torque were obtain ed using the\nformula[48, 49, 50] τ0=Is(π)−Is(0)\n2sinθ, we also calculate T′\nτ=2π\nq∂τ0\n∂Vfor comparison. In\nfigure 5, we plot TτandT′\nτversusθR. When the bias voltage is tuned far away from\nthe resonant point ǫ0(figure 5(a)), the profile of TτversusθRobeys sin θRfunction.\nThis gives very good agreement with T′\nτwhich is expected since T′\nτwas derived for\na non-resonant tunneling system. When the system is near resona nce, however, Tτ\ndeviates away from the sinusoidal dependence[46, 51]. This behavio r can be understood\nas follows. When we set Γ L↑= ΓR↑= Γ↑and Γ L↓= ΓR↓= Γ↓, equation (53) can be\nsimplified as\nTτ=1\n2(qV−ǫ0)2(Γ2\n↑−Γ2\n↓)sinθ\n(qV−ǫ0)2(Γ↑+Γ↓)2+[(qV−ǫ0)2−Γ↑Γ↓−1\n4(Γ↑−Γ↓)2sin2θ\n2]2.(55)\nWe examine the denominator of this equation. Since Γ ↑/ne}ationslash= Γ↓, it is clear that near the\nresonance qV∼ǫ0, the term sin2(θ/2) in the denominator cannot be neglected so that\nTτin the upper panel of figure (5) is not the sin θRdependence. But when |qV−ǫ0|is\nlarge enough so that the term sin2(θ/2) is small compared with the term ( qV−ǫ0)2,we\nobtainTτ≈T′\nτ. Actually, we can derive T′\nτby differentiating Is(π) andIs(0) according\nto equation (49) and obtain\nT′\nτ=1\n2(qV−ǫ0)2(Γ2\n↑−Γ2\n↓)sinθ\n[(qV−ǫ0)2+Γ2\n↑][(qV−ǫ0)2+Γ2\n↓]. (56)\nOne can easily find that if we neglect the term sin2(θ/2) in the denominator of Tτ,Tτ\nwill equal T′\nτ.\nFinally, we calculated derivative of the noise spectrum of spin transf er torque with\nrespect to the bias voltage by equation (54). From figure (6), we s ee thatNτas aShot noise of spin current and spin transfer torque 13\n/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s32/s84\n/s32/s84/s39\n/s32/s84 /s32/s32/s97/s110/s100/s32/s32/s84/s39/s40/s98/s41\n/s48\n/s47/s50/s50/s51 /s47/s50\n/s82/s32/s84\n/s32/s84/s39/s32 /s32/s40/s97/s41\nFigure 5. TτandT′\nτversusθRwith different bias voltages V= 0.01V(panel (a)) and\nV= 0.97V(panel (b)). The other parameters are ǫ0= 1.0, ΓL↑= ΓR↑= 0.8,ΓL↓=\nΓR↓= 0.2.\n/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s32/s32/s78\n/s86\n/s98/s105/s97/s115/s61/s48/s46/s49/s55/s86/s40/s97/s41\n/s48/s47/s50/s48/s46/s53/s49/s48/s46/s53/s52/s48/s46/s53/s55\n/s32/s78\n/s86\n/s98/s105/s97/s115/s61/s48/s46/s54/s55/s86/s40/s98/s41\n/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s50\n/s51 /s47/s50\n/s32/s78\n/s82/s86\n/s98/s105/s97/s115/s61/s48/s46/s57/s55/s101/s86/s40/s99/s41\nFigure 6. Nτversus the angle θRwith the bias voltage Vbias= 0.17V(panel (a)),\nVbias= 0.67V(panel (b)) and Vbias= 0.97V(panel (c)). The other parameters are\nǫ0= 1.0, and Γ L↑= ΓR↑= 0.8,ΓL↓= ΓR↓= 0.2.\nfunction of θRgives very different behaviors depending on whether it is near reson ance\nor far away from that. When the bias voltage is close to ǫ0/q, i.e., when the system is\nnear resonance (figure 6(c)), Nτis a concave function of θRwhich is very large at θR= 0\nbut close to zero at θR=π. However, when the system is far away from the resonance,\nNτis is convex function of θRthat is small at θR= 0 but large at θR=π(see figure\n(6)(a)). In the intermediate range of bias voltage, the differentia l noise spectrum of spin\ntransfer torque behaves like sin(2 θR) (see figure (6)(b)). When we change Γ R↓and keep\nthe other parameters the same, we found that the noise spectru m of spin transfer torque\nis very sensitive to Γ R↓whenθRis near zero.Shot noise of spin current and spin transfer torque 14\n4. Conclusions\nIn conclusion, based on the Green’s function approach, the spin cu rrent and spin noise\nof quantum dot coupled by two ferromagnetic leads were investigat ed. The spin auto-\ncorrelation function is always positive while the spin cross-correlatio n noise is negative\ndefinite. Due to the existence of the spin flip, the sum of them can be non-zero for\nsystems with three ferromagnetic layers, i.e, Sσz\nLL+Sσz\nLR/ne}ationslash= 0. As a result, both the spin\nauto-correlation noise and spin cross-correlation noise are neede d to characterize the\nshot noise of spin current. The spin transfer torque and its noise s pectrum were also\ninvestigated. For a system with a resonant level, the differential sp in transfer torque\nwas found to be proportional to sin θfar away from the resonance where θis the angle\nbetween magnetization of two ferromagnetic leads. Near the reso nance, however, a non-\nsinusoidal θdependence was found. The noise spectrum of spin transfer torq ue is found\nto be a concave function of θnear the resonance and becomes a convex function far\naway from the resonance. The noise spectrum of spin transfer to rque was found to be\nvery sensitive to the system parameters and might be used to char acterize the electron\nspin transport properties.\nAcknowledgments\nWe gratefully acknowledge support by the grant from the National Natural Science\nFoundation of China with Grant No.10947018(Y.J. Yu) and No.110741 71(Y.D. Wei),\nNo. 11274364 (Q.F. Sun), and a GRF grant from HKSAR (HKU 705611 P) (J. Wang).\nReferences\n[1] Blanter Y.M. and B¨ uttiker M. 2000 Phys. Rep. 3361.\n[2] Landauer R. 1998 Nature392658.\n[3] Gramespacher T. and B¨ uttiker M. 1998 Phys. Rev. Lett. 812763.\n[4] Khlus V.A. 1987 Sov. Phys. JETP. 661243.\n[5] B¨ uttiker M. 1990 Phys. Rev. Lett. 652901.\n[6] B¨ uttiker M. 1992 Phys. Rev. B 4612485.\n[7] Gonz´ alez T., Gonz´ alez C., Mateos J. and Pardo D. 1998 Phys. Rev. Lett. 802901.\n[8] Beenakker C.W. 1999 Phys. Rev. Lett. 822761.\n[9] Iannaccone G., Lombardi G., Macucci A. and Pwellegrini B. 1998 Phys. Rev. Lett. 801054.\n[10] Kuznetsov V.V., Mendez E.E., Bruno J.D. and Pham J.T. 1998 Phys. Rev. B 58R10159.\n[11] Chen Q. and Zhao H.K. 2008 Europhys. Lett. 8268004.\n[12] Reznikov M., Heiblum M., Shtrikman H. and Mahalu D. 1995 Phys. Rev. Lett. 753340.\n[13] DiCarlo L., Zhang Y., McClure D. T., Reilly D. J., Marcus C. M., Pfeiffer L . 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Lett. 97237205.\n[49] Jia X.T., Xia K., Ke Y.Q. and Guo H. 2011 Phys. Rev. B 84014401.\n[50] Liu D.P., Han X.F. and Guo H. 2012 Phys. Rev. B 85245436.\n[51] Chen X., Zheng Q. R. and Su G. 2008 Phys. Rev. B 78104410."    },    {        "title": "1509.03344v1.Ferromagnetic_resonance_and_magnetoresistive_measurements_evidencing_magnetic_vortex_crystal_in_nickel_thin_film_with_patterned_antidot_array.pdf",        "content": "arXiv:1509.03344v1  [cond-mat.mes-hall]  10 Sep 2015Ferromagnetic resonance and magnetoresistive measuremen ts evidencing\nmagnetic vortex crystal in nickel thin film with patterned an tidot array\nI. R. B. Ribeiro,1,2J. F. Felix,1,3L. C. Figueiredo,3P. C. de Morais,3,4S. O. Ferreira,1W. A. Moura-Melo,1A.\nR. Pereira,1A. Quindeau,5and C. I. L. de Araujo1,a)\n1)Departamento de F´ ısica, Universidade Federal de Vi¸ cosa, Vi¸ cosa, 36570-900, Minas Gerais,\nBrazil\n2)Instituto Federal do Esp´ ırito Santo, Alegre, 36570-900, E sp´ ırito Santo, 29520-000,\nBrazil\n3)Instituto de F´ ısica, N´ ucleo de F´ ısica Aplicada, Univers idade de Bras´ ılia-UnB, Bras´ ılia, 70910-900, Distrito Fe deral,\nBrazil\n4)Huazhong University of Science and Technology, School of Au tomation, 430074, Wuhan,\nChina\n5)Department of Physics and Francis Bitter Magnet Lab, Massac husetts Institute of Technology, Cambridge,\nMA 02139, USA\nFerromagnetic vortices deliver robust out-of-plane magnetizatio n at extremely small scales. Their handling\nand creation therefore has high potential to become a necessary ingredient for future data storagetechnologies\nin order to keep up with the pace of growing information density dema nds. In this study we show that by\nusing onestep nanolithographymethod, we areable to createferr omagneticvortexlattices in thin nickelfilms.\nThe necessary control of the magnetic stray field at the domain ed ges was achieved by actively modifying\nthe ferromagnetic thin film anisotropic properties at nanometer sc ale. We present experimental evidence\nusing ferromagnetic resonance and magnetoresistance measure ments supporting simulations based on the\ntheoretical prediction of the proclaimed vortex structures.\nAnisotropic magnetoresistance (AMR) occurs as a\nweak phenomenon in any soft ferromagnetic material.\nThe exploitation of this effect enabled the develop-\nment of the first industrial magnetic sensors. However,\ndue to fundamental limitations owing to the spin-orbit\ninteraction1, junctions based on AMR did not exceed the\n1%-mark. Later advancements in microstructure design\nallowed the creation of heterostructures involving multi-\nplelayersofmagneticandnonmagneticconductingmate-\nrials. As a result, the ”giant” magnetoresistance (GMR)\nwas discovered, leading to a huge increase in the mag-\nnetic sensor sensitivity2,3. Innovative material composi-\ntions that introduced antiferromagnetic pinning layers to\n(exchange-) bias the coercive fields of the ferromagnetic\nfilms opened the door for the design of very efficient and\nreliable magnetic switches4.\nThe necessity to produce hard disc drives with in-\ncreasing storage density to cope with the advancements\nof the computer industry and the internet demanded\neven larger magnetoresistive effects. Following the trend\nof scaling down computer architecture to the nanome-\nter range, the nonmagnetic conducting films in GMR\njunctions were replaced by ultra thin insulating oxide\nfilms to allow electronic transport only via quantum\ntunneling. The spin filtering properties of those tun-\nnel magnetoresistance (TMR) junctions hence increased\ndramatically5,6. Benefiting from this very high magnetic\nsensitivity, devices such as the magnetoresistive random-\naccess memory (MRAM) became feasible. Today, new\nchallenges arise as a product of recent industrial develop-\nments. Improving the speed and durability of magnetic\na)Electronic mail: dearaujo@ufv.br\nFigure 1. (a) Atomic Force Microscopy (AFM) image of the\nsample, (b) AFM three-dimensional visualization (c) Micro -\nmagnetic simulation of the ground state after magnetizatio n\nrelaxation with randon quirality and polarization. (d) Mag -\nnification of the spin configuration inside a single vortex.\nrecording as well as reducing the power consumption be-\ncomes important to a growing number of applications.\nApproachestopotentialsolutionsinclude the magnetic\ndomain-wall racetrack memory7and magnetic dots8–10.\nAmong the latter, nanodisks appear to be very promis-\ning candidates once they allow magnetic vortices as sta-\nblestates11–13, whichwouldenablecontrollableswitching\nbetween a couple of holes and thus yield tiny logic binary\nelements14. Large arrays of nanodisks have been pro-\nposed to complement thermally assisted MRAM devices\nin order to improve stability and to avoid magnetostatic2\nFigure 2. (a) Ferromagnetic resonance measurements data ob tained with applied magnetic field in unpatterned Nithin film\nin-plane sample (black lines) and out-of-plane (red dots), with rotating angles from (90o) to (0o), (b) in-plane measurements\nperformed in patterned sample and (c) out-of-plane measure ments performed in patterned sample.\ntraps, asthey occurin ordinaryrectangularelements15,16\nand thus demand relatively strong readout fields.\nIn this work, we corroborate experimentally the theo-\nretical prediction that suitably patterned array of plat-\nters, supports a magnetic crystal vortex, due to the\nanisotropy generated by the stray field on its antidot\nborders17. This system brings advantages in relation to\narrays of nanodisks (conventional elements that support\na vortex state), once the vortexare electrically connected\nin the same material, resultingin low powerconsumption\nin vortex manipulated by alternate currents18. We char-\nacterize the as-fabricated samples using ferromagnetic\nresonance (FMR) and magnetoresistance (MR) measure-\nments and bring the observed behavior in context with\ntheoretical predictions of the vortex configuration and\nspin dynamics19.\nFor patterning, a silicon substrate previously covered\nwith a 250 nmpolymethylmethacrylate (PMMA) film\nwas brought into a RAITH e-LINE Plus system, where\nthe exposure of the antidot design was performed. This\nprocedurewasrepeatedseveraltimestoformshapeswith\narea of 100 µm2(see Fig. 1 a and b). After the develop-\nment of the PMMA film, the samples were placed into a\nThermionics e-Beam evaporation system where a 30 nm\nnickel (Ni) film was deposited on top of a 5 nmtitanium\nseed layer. A gold capping layer of 3 nmwas deposited\non top of the sample to prevent Nioxidation. In this\nstep, we also performed a sample with same thickness\nbut without patterning for comparison. The structures\nwere eventually finalized by a lift-off process in an ace-\ntone ultrasonic bath. The magnetization dynamics ofthe\nas-fabricated samples have been investigated via FMR\ncarried out in a X-band spectrometer (Bruker EMX Pre-\nmiumX, equipped with an ER 4102ST resonator). For\nthe MR measurements electric contacts were developed\nby photolithographydefining regionswhere a 50 nmthick\ngold was deposited. Subsequently, a lift-off process pro-\nvided two electrodes in the borders of the antidots sam-\nples with channels of 80 µm length, on which a direct\ncurrentof10 mAwasapplied. Micromagneticsimulationsperformed with the software Object Oriented MicroMag-\nnetic Framework ( OOMMF )20, utilizing square mesh of\n5nmedge and parameters for Nias saturation magne-\ntization 4 .3x105A/m, exchange constant 9x10−12J/m3,\ncubic anisotropy constant −5.7x103J/m3and damping\ncoefficient 0 .01, provided ground state magnetization of\nvortex crystal configuration in which chirality and po-\nlarization appear to be randomly distributed throughout\nthe sample (see Fig. 1 c and d). Upon application of\nexternal magnetic field in parallel to the sample plane\n(in-plane configuration), a relatively small magnetic field\nis sufficient to annihilate the vortex patterns, as demon-\nstrated in previous work17,21. Furthermore, the spins\nseem to relax and align with the external applied mag-\nnetic field except at the borders, where the spin-stray\nfield tends to pin the spins parallel to each border22.\nOnce two adjacent borders are perpendicular to each\nother,thespin-strayfieldcouplingmayrenderthesample\nto be magnetically anisotropic. Indeed, in the compar-\nison between experimental FMR performed in uniform\n30nmNithin film andin patterned sample in function of\nrotating angle in-plane and out-of-plane, depicted in Fig.\n2, is possible to note that in the thin film the peak of res-\nonancefield ( HR) is isotropicin-plane, see Fig. 2 a, while\npresents a slight change in out-of-plane configurationdue\nto the shape anisotropy. However, in the measurements\nperformed in patterned sample in-plane configuration,\npresented in Fig. 2 b, the value for HRisotropic peak\nis lower, around 1 .1kOe (at 0o), which shifts towards\nhigher values as the applied magnetic field in-plane angle\nprogresses towards 90o, thus accounting for the pinning\nof spins at the borders of the platters. This behavior\nis similar to the commonly observed one in anisotropic\nmagnetic systems23. When the FMR measurements were\nperformed starting with the external magnetic field ap-\nplied out-of-plane, besides the FMR peaks at 1 .1kOe,\na huge peak observed at 2 .4kOe (Fig. 2 c) can be at-\ntributed to the resonance field of vortex core polariza-\ntions. This, therefore, tend to align and remain very\nstable along the applied field axis, similarly to what hap-3\npens in nanodisk arrays24. At magnetic field strengths of\naround 11 .6kOe, in the out-of-plane saturation magneti-\nzation regime, more FMR peaks are observed, suggest-\ning formation of magnon excitations, as shown in Fig.\n2 c and in more detail in Fig. 4 c. A similar behavior\nwas observed in perpendicular FMR spectrum analysis\non micrometer-sized disks, recently reported by Castel\net al.19. It was shown that in the saturated regime, one\nmain resonance line along with several peaks decreas-\ning in amplitude on the low-field side is observed in the\ndisk array. These multiple resonance peaks have been\nattributed to standing in-plane spin-wave modes.\nIn the present context, the externally applied mag-\nnetic field is divided into two components, namely in-\nplane and out-of-plane contributions. As the effective\nout-of-plane component decreases with tilting angle, HR\nincreases from 2 .4kOe to 6 .5kOe, as a result of fixed\ncore orientation along out-of-plane axis. On the other\nhand, the in-plane component is responsible for increase\nin magnetization of chiral spins aligned along the exter-\nnal field axis, which causes vortex quenching bellow 15o.\nFor better observation, the behavior of HR(black) and\npeak amplitudes (red dash) in function of rotating angle\nin both in-plane and out-of-plane configuration, for pat-\nterned sample and Nithin film are summarized in Fig.\n3, where squares and circles represent the first and sec-\nond peak, respectively, observed in the patterned sam-\nple, while triangles represent the peak in Nithin film.\nFrom this data is possible to follow the similar behav-\nior of thin film HRpeak and first peak in the patterned\nsample that is suppressed by the vortex peak after 60\no. In our experiment, the ACmicrowave magnetic field\nis perpendicular to the DCsweeping field. As the ab-\nsorption peak intensity is related to the configuration of\nmicrowave incidence in the thin film samples, due to the\nmagnetic material interaction, the values will be always\nlower in-plane than in out-of-plane configuration, as can\nbe notice in a comparison of values in figures 3 a and 3\nb. The intensity increase in thin film peak with satura-\ntion after 60o, represented by red triangles in Fig. 3 a,\ncan be related to the alignment of sample plane with the\nACmicrowave magnetic field and the second peak inten-\nsity represented by red circles behaves like the first peak\n(red squares), after vortex quenching. In Fig. 3 b the\nisotropy expected for peak intensity and HRin the thin\nfilm in-plane configuration was confirmed, while the spin\npinning in the borders of antidots gives rise to anisotropy\nin the patterned sample.\nOur observations were corroborated by theoretical dy-\nnamics investigations performed by micromagnetic sim-\nulation in Mumax3GPU based code25. With same Ni\nparameters described before and following procedure uti-\nlized in references26,27, we have applied an alternating\nfield, which is bases upon the following equation:\n/vectorBac\ny= (1−eλt)/vectorBac,0cos(ωt) (1)\n, with same intensity /vectorBac,0= 10Oeandλ∼f=\nω/(2π) = 9.5GHz, utilized in experiment. For each step/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s50/s46/s48/s52/s46/s48/s54/s46/s48/s56/s46/s48\n/s50/s46/s48/s107/s52/s46/s48/s107/s54/s46/s48/s107/s56/s46/s48/s107\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s49/s46/s48/s107/s50/s46/s48/s107/s51/s46/s48/s107/s52/s46/s48/s107\n/s48/s46/s48/s48/s46/s51/s48/s46/s54/s49/s46/s56/s50/s46/s49\n/s40/s98/s41/s32/s49/s111\n/s32/s112/s101/s97/s107\n/s32/s50/s111\n/s32/s112/s101/s97/s107\n/s32/s102/s105/s108/s109/s32/s112/s101/s97/s107/s40/s97/s41\n/s32\n/s32\n/s32\n/s65/s110/s103/s108/s101/s32 /s40/s111\n/s41\n/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s100/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s40/s97/s46/s32/s117/s46/s41/s32/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s70/s105/s101/s108/s100/s32/s40/s79/s101/s41\nFigure3. (a)Resonancefieldandpeaksintensityperformedi n\nout-of-plane configuration as a function of angle in pattern ed\nsample and Nithin film and (b) Resonance field and peaks\nintensity performed in-plane configuration.\nofDC Bz, 1000 interactions during 5 nswere performed.\nAfter discard the 600 first interactions and apply Fourier\ntransform in the stable mysignal, we have obtained the\nabsorption data presented in Fig. 4 a. The derivative of\nthe data lorentzian fit gives the FMR spectra, which is\ncompared with non-saturated regime experimental data\nin Fig. 4 b and saturated regime in Fig. 4 c. The main\npeaks observed experimentally were also present in the\nsimulations, despite of the difference between the exper-\nimental and theoretical HRand peak intensity, which\ncan be attributed to the zero temperature in the simu-\nlation and defects in sample borders and antidots that\ncan act as magnetostatic traps for spin pinning. Addi-\ntional experimental evidence of the topological crystal\nvortex based on MR measurements is presented in Fig.\n5. The MR measurements were carried out with the ap-\nplied magnetic field in the longitudinal and transverse\nconfiguration (in-plane) and in the perpendicular con-\nfiguration (out-of-plane) at 300 K. As it can be seen on\nthe magnified areasin Fig. 5, an anisotropicbehavior oc-\ncurs, presumablyresultingfromacontributionof Nithin\nfilm spin orbit coupling. The main magnetoresistive sig-\nnal, however, is attributed to the higher resistance given\nby the random orientation of crystal vortex polarizations\nand chiralities at zero field, which enhance the resistance\ndue to the higher density of scattering and spin mixing\nevents. Thelowerresistanceobservednearthesaturation\nmagnetization in the longitudinal and transverse config-4\n/s55/s107 /s56/s107 /s57/s107\n/s49/s48/s46/s52/s107 /s49/s48/s46/s56/s107 /s49/s49/s46/s50/s107 /s49/s49/s46/s54/s107/s48/s46/s48 /s50/s46/s48/s107 /s52/s46/s48/s107 /s54/s46/s48/s107 /s56/s46/s48/s107 /s49/s48/s46/s48/s107 /s49/s50/s46/s48/s107\n/s48/s46/s48 /s50/s46/s48/s107 /s52/s46/s48/s107\n/s49/s107 /s50/s107 /s51/s107 /s52/s107/s32/s83/s105/s109/s117/s108/s97/s116/s105/s111/s110/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s97/s108/s40/s99/s41/s40/s98/s41/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32 /s40/s97/s46/s32/s117/s46/s41/s32/s83/s105/s109 /s117/s108/s97/s116/s105/s111/s110\n/s32/s76/s111/s114/s101/s110/s116/s122/s105/s97/s110/s32/s102/s105/s116/s40/s97/s41\n/s83/s97/s116/s117/s114/s97/s116/s101/s100/s32/s32/s32/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32\n/s100/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s40/s97/s46/s32/s117/s46/s41/s78/s111/s110/s45/s115/s97/s116/s117/s114/s97/s116/s101/s100\n/s65/s112/s112/s108/s105/s101/s100/s32/s68/s67/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s79/s101/s41\nFigure 4. (a) Data obtained from dynamics micromagnetic\nsimulation and lorentzian fit, (b) derivative absorption ob -\ntained from the fit in (a) and comparison in non-saturated\nregime experimental FMR data and (c) derivative absorption\nobtained from simulation and comparison with the saturated\nregime.\nurations (in-plane) as well as the vortex core orientation\nwith the external field (out-of-plane) is provided by the\nonset of a low resistive path caused by vortex core orien-\ntation or in-plane saturation by the out-of-plane external\nmagnetic field. Similar behavior was recently observed\nfor systems with vortex array28. The magnetoresitive\nmeasurements evidently show long-range spin polarized\ntransport throughout the system. In Fig. 6 we present\nresultsforspinpolarizationcalculatedfromMRmeasure-\nments performed in same configurations as in Fig. 5 and\nin different temperatures. The highest polarization was\nachieved upon out-of-plane magnetic field sweeping, cor-\nroborating our observations of crystal vortice with ran-\ndom chirality and core polarization at low magnetic field,\nas highest spin mixing resistive path, and successive low\nresistive path obtained by alignment of vortex polariza-\ntion and chirality with the field. The observed linear\ndecrease of polarization in function of temperature, is in\ngood agreement with the spin diffusion length behavior\nin metallic non-magnetic spacer in multilayermagnetore-\nsitive devices29, corroborating our model.\nIn summary, ferromagnetic resonance measurements\nwere used to assess the anisotropy behavior of as-\nfabricated antidot systems, which is responsible for crys-\ntal vortex formation, as it was recently predicted in the\nliterature. The crystal vortex configuration was fur-\nthermoreconfirmedbyferromagneticresonancemeasure-\nmentswiththeonsetofmagnonstatessignatureathigher\nmagnetic fields. Additional experimental evidence of the\nvortex crystal created by the insertion of the antidots in\nthe nickel thin film was assessed by local magnetoresis-/s50/s51/s46/s56/s49/s54/s50/s51/s46/s56/s50/s52/s50/s51/s46/s56/s51/s50/s50/s51/s46/s56/s52/s48\n/s50/s51/s46/s48/s52/s50/s51/s46/s48/s54/s50/s51/s46/s48/s56/s50/s51/s46/s49/s48\n/s45/s52/s46/s48/s107 /s45/s50/s46/s48/s107 /s48/s46/s48 /s50/s46/s48/s107 /s52/s46/s48/s107/s50/s51/s46/s55/s57/s48/s50/s51/s46/s56/s48/s53/s50/s51/s46/s56/s50/s48/s50/s51/s46/s56/s51/s53/s50/s51/s46/s56/s53/s48\n/s40/s99/s41/s40/s98/s41/s82/s101/s115/s105/s115/s116/s97/s110/s99/s101/s32 /s40 /s41\n/s32\n/s32\n/s65/s112/s112/s108/s105/s101/s100/s32/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s40/s79/s101/s41/s40/s97/s41/s32\n/s32/s45/s49/s46/s48/s107 /s45/s53/s48/s48/s46/s48 /s48/s46/s48 /s53/s48/s48/s46/s48 /s49/s46/s48/s107/s50/s51/s46/s56/s52/s50/s51/s46/s56/s52/s50/s51/s46/s56/s53\n/s32/s32\n/s45/s49/s46/s48/s107 /s45/s53/s48/s48/s46/s48 /s48/s46/s48 /s53/s48/s48/s46/s48 /s49/s46/s48/s107/s50/s51/s46/s48/s55/s50/s51/s46/s48/s56/s50/s51/s46/s48/s57/s32\n/s32/s32\n/s32\nFigure 5. (a) Magnetoresistance measurements performed in\nthe antidot sample in configuration of applied DC electric\ncurrent (a) Longitudinal, (b) Transversal and (c) perpendi c-\nular to the applied magnetic field. In the insets are presente d\nzoom of the peaks with the anisotropic magnetoresistance si g-\nnal expected for ferromagnetic thin films.\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s48/s46/s57\n/s32/s76/s111/s110/s103/s105/s116/s117/s100/s105/s110/s97/s108\n/s32/s84/s114/s97/s110/s115/s118/s101/s114/s115/s97/s108\n/s32/s80/s101/s114/s112/s101/s110/s100/s105/s99/s117/s108/s97/s114/s80/s111/s108/s97/s114/s105/s122/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\nFigure 6. Long range spin polarization measured by local\nmagnetoresistance as a function of temperature.\ntance measurements. The anisotropic magnetoresistance\nexpected for thin magnetic films is non-zero around zero\nmagnetic field and the main isotropic peaks are related\nto the generation of the vortex crystal. The long-range\nspin polarization is highest at low temperatures and de-\ncreases linearly while with increasing temperature, as it\nis expected for metallic ferromagnetic thin films.5\nACKNOWLEDGMENTS\nThe authors thank CNPq, CAPES and FAPEMIG\n(Brazilian agencies) for financial support. They also\nwould like to thank the LABNANO/CBPF for techni-\ncal support during electron microscopy/nanolithography\nwork.\nREFERENCES\n1T.R.McGuire and R.I. Potter, IEEE Transactions on Magnetic s\n11, 1018-1038 (1975).\n2M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F.\nPetroff, P. Etienne, G. Creuzet, A. Friederich and J. Chazela s,\nPhys. Rev. Lett. 61, 2472 - 2475 (1988).\n3G. Binasch, P. Gr¨ unberg, F. Saurenbach and W. Zinn Phys. Rev .\nB39, 48284830 (1989)\n4B. Dieny, V. S. Speriosu, S. S. P. 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This paper \ndiscusses the impact of molecule induced strong coupling on the spin transport. To s tudy \nmolecular coupling effect  organometallic molecular complex (OMC)  was bridged between two \nferromagnetic electrodes of a magnetic tunnel junction  (Ta/Co/NiFe/AlOx/NiFe/Ta)  along the \nexposed side edges. OMCs induced strong iter -ferromagnetic electrode coupling to yield drastic \nchange s in transport properties of the magnetic tunnel junction  testbed  at the room temperature . \nThese OMCs  also transformed t he magnetic properties of magnetic tunnel junctions. SQUID and \nferromagnetic resonance studies provide d insightful data to explain transport studies on the \nmagnetic tunnel junction based molecular spintronics devices.       \nKey Words:  Molecular spintronics ; magnetic tunnel junctions ; magnetic molecules;   \nIntroduction:  Connecting magnetic molecul es between two ferromagnetic electrodes opens \nflood gate of opportunities for observing new phenomenon and making novel devices  [1, 2]. \nInitial studies focused on sandwiching molecules between two ferromagnetic leads  [3]. In a more \npopular approach molecules  were placed in a break -junction  on a nanowire on a planar insulating \nsubstrate  [4]. However, these two approaches ha ve been extremely difficult to mass produce \nrobust molecular spintronics devices  [5]. With conventional approaches i t is also almost \nimpossible to conduct extensive magnetic studies to explore the true effect of molecules on the \nmagnetic properties of the molecular spintronics devices  [6]. To date most of the experimental \nstudies have only focused on the transport studies - no dire ct magnetic measurements were \nperformed  [3, 4, 7]. To overcome the limitations of the conventional molecular spintronics \ndevices magnetic tunnel junctions  (MTJ) , produced by sandwiching an insulator  between two \nferromagnetic electrodes, were utilized as the test bed  [5, 6, 8]. Under this approach a n MTJ with \nthe exposed side edges can enable s the covalent bonding of desired molecular channels onto the \ntwo ferromagnetic electrodes  along the junction perimeter  [5]. These molecules can be single \nmolecular magn ets [9], porphyrin  [10], single ion molec ules, and DNA  [11]. This approach is \nequally capable of utilizing a lkane like simple molecules  with lo w spin orbit coupling and \nZeeman splitting . The MTJ based molecular spintronics device (MTJMSD) approach enabled us \nto study the impact of paramagnetic molecules on the spin transport and magnetic properties of \nMTJ s. This paper discusses experimental studies conducted before and after transforming a n \nMTJ into MTJMSD . We report the observation of paramagnetic molecule induced dramatic \nchange s in spin transport  of an MTJ . We also report complementary SQUID and magnetic \nresonance studies exploring the underlying mechanism behind the impact of covalently bonded \nmolecular channels  between two ferromagnets of an MTJ testbed .    \nExperimental details : To produce MTJ for MTJMSD , a bilayer ferromagnetic electrode  was \ndeposited by sequentially depositing ~ 3 nm tantalum (Ta), 5 -7 nm cobalt (Co) and 5 -3 nm NiFe 2 \n with 79% Ni by weight. In the next step  photolithography was perfo rmed to create a cavity with \nvertical  side edges (Fig. 1b ). This cavity helped defining the lateral dimensions of the deposition \nof a 2 nm thick alumina ( AlOx ) (Fig.  1c), and top electrode comprising a ~7 nm thick NiFe and ~ \n3 nm tantalum ( Ta) (Fig. 1d). The deposition of AlOx and top electrode  via the same phot oresist \n(PR) cavity ensured that along the MTJ edges th e minimum gap between the two ferromagnetic  \nelectrodes is equal to the AlOx insulator thickness (Fig. 1a and g). The liftoff of PR produced \nTa/Co/NiFe/AlOx/NiFe /Ta MTJ (Fig. 1d). This MTJ possessed  exposed side edges (Fig. 1f ). \nAlong the exposed side edges organometallic molecular complex es (OMCs ) [12] were bridg ed \nacross the AlOx to complete the M TJMSD fabrication. These OMCs exhibited S=6 spin state in \nthe bulk powder form at <10 K  [12]. An OMC possessed cyanide -bridged octametallic molecular \ncluster, [(pzTp)FeIII(CN) 3]4[NiII(L)] 4[O3SCF 3]4  [(pzTp) = tet ra(pyrazol -1-yl)borate; L = 1-\nS(acetyl)tris(pyrazolyl)decane] [12] chemical structure. With the help of thiol functional groups, \nan array of OMCs  were  covalently -linked onto the NiFe layer of the top and bottom electrodes  \n(Fig.1 f). The process details for \nattaching OMCs and the \nfabrication of tunnel junction with \nthe exposed side edges are \npublished elsewhere [13, 14]. \nNiFe  film was extensively \nemployed in MTJ MSDs. NiFe \nelectrode possessed many  useful \nattribute  to enable the fabrication \nof a n MTJ MSD  [15, 16]. As \nshown in the reflectance study on \nthe NiFe surface, N iFe is stable  in \nthe ambient  conditions  and start  \noxidizing upon heating around 9 0 \nºC (Fig. 1 g). We also tested the \nefficacy of liftoff based molecular \ndevice fabrication by attaching \nand detaching the OMC between \ngold (Au) and nickel (Ni) \nelectrode  (Fig. 1h) . For the \nmagnetic studies cylindrical MTJs \nwere produced by the utilization \nof photoresist cavities (Fig. 1i). \nAfter the thin film deposition and \nliftoff of photoresist we produced \nan array of ~7000 MTJ cylindrical \npillars  (Fig.  1j). Subsequently , \nMTJ s were transformed into \nMTJMSD s. SQUID and \nferro magnetic resonance  were \nperformed as per the details \nfurnished elsewhere [8].  \nFig.1: MTJMSD fabrication steps: (a) deposit the bottom \nferromagnetic (bottom -FM) electrode on the insulating \nsubstrate, (b) creates photoresist (PR) cavity pattern for the \ndeposition of (c) ~2 nm AlOx and (d) top ferromagnetic \nelectrode (top -FM). (e) Lift off step produces an MTJ with the \nexposed sides where (f) OMC molecules are chemically \nbridged across the AlOx insulator to produce an MTJMSD. (g) \nNiFe films is determined by reflectivity study. (h) OMC were \nconnected and disconnected on a tunnel junction to retain bare \ntunnel junction transport attributes. Fabrication steps for \nmagnetic studies of MTJMSD. (i) In ~7000 photoresist cavities \nthin films were deposited to mass produce (j)an array of \ncylindrical MTJs with exposed edges. (k) Individual MTJ \ncylind ers were transformed into MTJMSD by attaching OMCs.   \n3 \n Results and discussion:  It must be noted that we formed covalent bonding between the  OMCs \nand the NiFe ferromagnet using thiol functional groups. Thiol anchoring group  are known for \nestablishing strong coupling with the metal electrodes in molecular devices [7]. Simple thiol \nassembly on non-magnetic gold surface was effective in developing permanent magnetism [17]. \nAlso,  self-assembly of molecules wit h thiols end group has enhanced environmental stability and \ncorrosion resistance of ferromagnets [18]. We noted that OMCs interfacing with two \nferromagnets  of an MTJ influenced the magnetic properties of the ferromagnetic electrodes  [8]. \nRecent, theories developed for interface  based devices  confirmed that molecule can significantly \nimpact density of states of the ferromagnets [19, 20]. Our prior experimental and theoretical  \nstudies [8] also indicate d that MTJMSD discussed here  are inclined to develop strong coupling \nbetween thiol -terminated OMC and NiFe ferromagnetic electrodes.  \nTo study the impact of OMC we conducted current -voltage (I -V) studies on MTJ testbed before \nand after transforming them into MTJMSDs.  Generally , OMCs lowered the current below the \nbare MTJ’s leakage current level via the ~ 2 nm AlOx tunneling barrier  (Fig. 2a) . In the specific \ncase discussed here, OMCs lowered the current by one order for the Ta/Co/NiFe/AlOx/NiFe/Ta \nMTJ (Fig. 2 b). However, after m ultiple current –voltage studies overall MTJMSD current settled \nin the high current state. Higher current state also appeared after magneto -resistance (MR) \nstudies. The magnitude of high current state could be one to two orders of magnitude higher than \nthe leakage current via the ~ 2  nm AlOx tunneling barrier. We performed MR studies to \ninvestigate the effect of magnetic field on the MTJMSD transport. We observed that application \nof in plane magnetic field was effective in bringing MTJMSD into suppressed cu rrent state by \nseveral orders (Fig. 2c). However, it appears that application of 50 mV voltage  in MR studies  \npromoted the higher current state MTJMSD  (Fig. 2c) . Hence , during MR studies a competition \noccurred between voltage and magnetic field to control the MTJMSD current state. In the first \nMR study  (MR1) , the application of magnetic field switched MTJMSD ’s current state by more \nthan three order s (Fig. 2c) . In the second MR study (MR 2) hi gher current remained stable for up \nto ~170 Oe and the n switched to lower current state (Fig.  2c). In 3rd MR study (MR 3) current \nreduction was not sudden or sharp. MTJMSD transitioned through several current values with \nincreasing magnetic field (Fig. 2c).  The fourth  MR study (MR4)  did not show complete \nFig. 2 : Current -voltage study of the MTJ (a) before and after attaching (b) OMCs across the tunnel \nbarrier along the exposed edges. (c) Magneto -resistance (MR) studies on MTJMSD showing several \norders of change current at 50 mV bias and under in -plane magnetic field. (d) Effect of changing \nmagnetic field direction from inplane to out of the plane on MTJMSD.   \n4 \n transition from sub µA to fully suppressed current state (Fig. 2c). It appears that prolong voltage \napplication promot ed higher current state. In the sixth  MR study (MR 6) MTJMSD lowered to \nsuppress  current s tate and then sprung back to higher current state. Finally, during the seventh  \nMR study (MR 7) MTJMSD settled in the higher current state. This behavior was observed on \nmultiple devices . Another device exhibiting this behavior is shown in the supplementary material  \n(Fig.  1S). Even though similar pattern  repeated on multiple samples , but switching fields were \nnot consistent over multiple MR studies .  \nWe also observed that the application of alternating inplane and out of the plane magnetic field \nwas effective in switching MTJMSD between high and low current state (Fig. 2d). The \napplic ation of 200 Oe magnetic fields  was applied inplane and out of the plane. MTJMSD was \nsubjected to 50 mV bias during this study. Switching  off magnetic field appeared to stabilize the \nhigher current state (Fig. 2d).     \nWe further investigated the suppresse d current state. It is also noteworthy that due to the \ninstrument limitation we were not able to capture exact transport properties of the MTJMSD in \nthe suppressed current state during MR studies discussed in figure 2c -d. The observed sub nA \nlevel suppress ed current state during MR study was more due to the instrument measurement \nlimit. For the study of suppressed current state a Kithley 6430 source meter with fA level \nmeasurement capability was utilized. For this study another sample with exact same \nconfig uration was prepared in the different batch. This sample also showed consistent behavior \nand MTJ testbed current reduced below the leakage level after bridging of the OMC channels \n(Fig.  3a inset). This behavior matched with most of the samples we studied a nd two  other \nsamples (Fig.  2 and Fig 1 S in the supplementary material) discussed in this paper. Knowing that \nmagnetic field produce d suppressed current state we applied magnetic field by using a permanent \nmagnet producing ~1000 Oe inplane magnetic field in  MTJMSD vicinity. As expected \nMTJMSD current reduced and settled in sub pA range at room temperature (Fig. 3a). This \nsuppressed current state was persistent over multiple I -V studies. Flipping the biasing direction \nfrom top electrode to bottom electrode pr oduced minor impact on transport (Fig. 3b).  Results \nfrom three consecutive I -Vs are shown in Fig. 3a and Fig. 3b. However, after leaving the sample \nintact for some time brought the MTJMSD back in the higher current state. To ensure that \nelectrodes to the j unction were intact the transport through the top and bottom electrodes were \nstudied before and after setting MTJMSD into suppressed current state. Current through top \nFig. 3 (a) Stabilized pA level current state on MTJMSD. Inset of this panel shows bare tunnel \njunction current before and after hosting OMC molecular channels. (b) Effect of changing biasing \nlead on pA level current state. (c) Current -voltage s tudy through bottom electrode before and after \nthe stabilization of pA level current state.   \n5 \n electrode was almost the same before and after the setting of suppressed current state. However, \nthe bottom electrode appeared to show reduced current state, but it was still almost six orders \nhigher than the magnitude of the obs erved suppressed current state  (Fig. 3c) .        \nIt is highly intriguing that ~10,000 OMCs  along the MTJ edges produc ed a suppressed current \nstate below the leakage current level through the planar area  (Fig. 2 -3). This observation of OMC \ninduced current suppression was only observed with the magnetic electrodes. On the other hand , \nnonmagnetic tunnel junctions , prepared with gold, copper,  and tantalum electrodes,  exhibited an \nincrease in current due to the addition  of OMC s channels  along the side edges.  The key \ndifference in the magnetic and nonmagnetic electrode is the unequal densi ty of states for the spin \nup and spin down electrons. Spin transport through a MTJ  depends on the spin density of states \nof the ferromagnetic electrodes. A plausible reason behind the current suppression may be due to \nOMC induced changes in magnetic proper ties or spin density of state s of the ferromagnetic \nelectrodes.  \nIn the prior study , we have provided insight regarding OMC induced impact on inter -\nferromagnetic electrode coupling and magnetic ordering of the ferromagnetic electrode itself [8]. \nThis prior work  presented experimental SQUID, Ferromagnetic resonance (FMR), and magnetic \nforce microscopy studies and it also presented theoretical insight from Monte Carlo simulations  \n[8]. However, the top electr ode in the current study differed  with previous study. In the current \nstudy top electrode is made up of ~ 7  nm NiFe with ~ 3 nm Ta on top. In the previous study top \nelectrode was ~10 nm thick NiFe. To i nvestigate direct correlation between magnetic properties \nand the transport studies discussed in Fig.  2 and 3 separate SQUID magnetometer and FMR \nstudies were conducted (Fig. 4). These studies were performed on an array of ~ 7000 cylindrical \nMTJs before an d after treating them with OMCs. The fabrication process for the cylindrical  MTJ \nis shown in Fig.  1i-k. The e xperimental details for SQUID magnetometer and FMR studies are \ndiscussed elsewhere [8]. SQUID study showed that the magnetic moment of the MTJ \nFig. 4 Magnetic studies of 7000 MTJMSD by SQUID and FMR. (a) SQUID magnetometer study of \nMTJ before and after hosting OMC channels across AlOx insulator. Top left inset shows  that OMC \ncan bridge across AlOx and also bo nd with the exposed magnetic layers,  mainly NiFe. Bottom right \ninset show magnetic study of NiFe before and after interacting with OMCs. (b) FMR study of MTJ \nbefore and after hosting OMC channels. Inset graph shows the impact of OMC on NiFe film only.  \n6 \n (TaCoNiFe/AlOx/NiFe/Ta) almost doubled after hosting OMC mole cular channels (Fig. 4a). \nThis is an average effect from 7000 cylindrical MTJs and clearly suggests that OMCs \ndramatically affected the magnetic moment of MTJ. This study also confirmed that OMC impact \nis strong and can spread all over the microscopic junc tion. The mechanism behind doubling  of \nmagnetic moment is not clear to us. However, it is quite well known that a molecule with net \nspin state can enable spin filtering and that may significantly enhance the degree of spin \npolarization. For instance if spi n polarization for ferromagnet changes from ~0.5 to 1 then \nmagnetic moment for each electrode may double to produce observed enhancement (Fig. 4a).  \nIt is noteworthy that only a small fraction of OMCs formed bridge across insulator. Most of the \nOMC s stick to the NiFe ferromagnetic electrode regions away from the tunnel barrier (top left \ninset of the Fig.  4a). SQUID magnetometer study was conducted  before and after treating \nunpatterned NiFe film with OMCs. SQUID magnetometer study did not produce any noticea ble \nchange due to OMC (bottom right inset Fig.4a). This study also confirmed that magnetometer \nperformed accurate measurement . This study also suggested that OMCs that matters in an \nMTJMSD are those which bridged  across insulating AlOx. We hypothesized tha t other magnetic \nmeasurements must be able to record such a strong OMC induced change in MTJ properties. We \nperformed ferromagnetic resonance ( FMR ) at room temperature and found that OMC treated \nsample with ~7000 MTJs was starkly different from the untreat ed MTJ. FMR of the OMC \ntreated sample showed one pronounced resonance mode around ~1200 Oe. However, bare MTJ \nproduced an optical mode (lower intensity mode) around ~650 Oe and acoustic mode (higher \nintensity mode) around 1100 Oe. According to FMR theory  of coupled ferromagnetic films the \nbare MTJ seems  to possess  ferromagnetic coupling  [21]. Luckily, prior theoretical studies have \nexplored the FMR  spectra from  analogous system . The development of very strong coupling \nbetween the f erromagnetic electrodes produced one resonance mode  [22]. Under strong coupling \nlimit two ferromagnetic electrodes behave like one  ferromagnetic electrode  [22]. Similar to \nSQUID magnetometer study we also investigated the FMR response from the NiFe. OMC \ntreatment did n ot produce significant change (i nset Fig.4b). Similar to SQUID mag netometer \nstudy, FMR study also confirmed that OMC bridges strongly influenced the magnetic coupling \nbetween two ferromagnetic electrodes of the MTJ and dominated the spin transport.  \nAt this point we are unsure about why the application of magnetic field produced the suppressed \ncurrent state on MTJMSD (Fig. 2 -3). We hypothesize that the magnetic field play a role in \nconfiguring the large mass of the ferromagnetic elec trodes in accordance with the molecule \ninfluenced area. It is noteworthy that overall magn etic leads are nearly one cm long and hence \nwhole length of ferromagnetic leads cannot be influenced by the molecular junction. The \napplication of magnetic field may optimally aligned magnetic moment of the ferromagnetic \nelectrode. Disciplining the large m ass of ferromagnet may promote or restrict the transport of \nspins through the highly ordered OMC affected junction. We also observed that repeating I -V \nstudies or prolong application of external voltage promoted the higher current state  on an \nMTJMSD . We hypothesize that the  application of voltage assist in transporting unpolarized \nelectrons from the bulk of the leads into the OMC affected regions and hence , promoting \ndisorder to yield settlement of higher current state.  Pinpointing exact  mechanism is yet to be \ndone  and will require first principle theoretical studies and extensive magnetic studies . \nBased on the I -V study we estimated the effective barrier heights and barrier thicknesses using \nBrinkman tunneling models  [23]. A bare MTJ exhibited ~2 .2 nm barrier thickness and ~0.7 eV 7 \n barrier height. After hosting OMC channels  along the exposed side edges an MTJ become \nMTJMSD and showed very different barrier properties in the ~micro amp range high current \nstate and pA range suppressed current state. According to modelling results i n the µA r ange high \ncurrent state  an MTJMSD ex hibited ~ 1.2 nm barrier thicknesses and ~0.4 eV barrier height.  The \nsame MTJMSD in the pA level  suppressed current state (Fig. 3) exhibited ~1.4 nm barrier \nthickness and ~ 2 eV barrier heights . Importantly the barrier thickness after hosting OMCs is \nequiv alent to the length of a deca ne molecule chain that connected the core of OMC molecules to \nthe NiFe electrode in an MTJMSD (Fig.  6a).  \nWe investigated the effective density of states before and after transforming MTJ into an \nMTJMSD. According to prior literature density of state in the molecule affected regions depends \non barrier height (ϕ) and the molecular coupling strength  (ω) [19]. One can calculate the \neffective density of states ( D(E)) the following equation:      \n𝐷(𝐸)=𝜔/2𝜋\n(𝜙)2+(𝜔\n2)2               (1) \nWe estimated the relative increase in the strength of exchange coupling by utilizing the following \nequation [24].  \n). exp(0 d J  −=\n                                (2)  \nAccording to the analysis of charge transport data, barrier thickness ( d) decreased  from ~2 .2 nm to ~1.2 \nnm range. This effective barrier thickness of MTJMSD corresponded to alkane tether length of ~1.2 nm.  \nThe MTJMSD barrie r thickness was comparable in the high and low current state . The barrier height  (ϕ) \ncan be utilized to obtain tunneling decay factor β according to the relation β =10.2 nm-1√(ϕ/eV) [25]. In \nthe equation (2) J0 correspond the coupling energy bet ween two magnetic atoms that decreased \nexponentially with barrier thickness and β. According to literature Ni -Ni atoms coupling energy is ~12 \nµeV [26]. It is noteworthy that in this simplified analysis we have not incorporated spin fluctuation that \nmay dra matically enhance the exchange coupling [2, 27]. The density of the state of bare MTJ was \ncalculated to be ~10-9 states/eV for barrier thickness 2.2 nm and 0.7 eV barrier heights. The MTJMSD’s \ndensity of states in the high current state was ~10-7 states/eV. However, the density of states in suppressed \ncurrent state was ~10-13 states/eV. It is noteworthy that spin polarization has been routinely calculated by \nthe high and low current states for the magnetic electrodes in magnetic tunnel junctions [28-30].  Nearly \nsix orders of magnitude change in the high and low current states suggests that OMC have prod uced  \nnearly 100% spin polarized magnetic electrodes. We believe that OMC’s  spin state played a crucial \nrole in spin filtering [31]. The OMC molecule is capable of a ttaining the  S=6 spin state in the \nisolated state [12, 32] . We presumed t hat this S=6 state is still applicable when an OMC is \nbonded to the ferromagnetic electrod es (Fig. 6a) . In this state , it seems that iron  (Fe) and Ni \natoms of the OMC clusters are open for accommodating only one specific type of spin  due to \nselection rule  and leading to spin filtering  [7].We also hypothesize that method of chemically \nbonding OMC to the magnetic electrode also played crucial role. Several thousands of OMCs are \nconnected to two ferromagnetic electrodes via alkane barrier and thiol chemical bonding  (Fig.  \n6a). We utilized thiol ( -S) bonds to form covalent bonding between OMC and NiFe to produce \nvery strong molecule induced exchange coupling. Prior studies illustrated the strong impact of \nthiolated molecules on the magnetic and transport properties of metallic electrodes [7, 17, 33]. \nOur magnetic studies provide clear evidences that OMC  produced transformative exchange \ncoupling on an MTJ [8] with similar thin film configuration as discussed here. Our expe rimental 8 \n FMR study also confirmed that OMC have dramatically enhanced the exchange coupling \nbetween top and bottom ferromagnetic electrodes (Fig. 5b). We noted that adding Ta on top of \nthe previously studied MTJ has been helpful in observing current switch ing in the high and low \ncurrent state of the MTJMSD. However, we also noted that FMR response from the MTJ with  Ta \n(Fig. 5b)  and without Ta on top [8] were very different. The further exploration about Ta role \nwill be part of future study and beyond the scope of this paper.    \nWe conjectured that strong OMC based  coupling impact ed the whole microscopic junctions and \nchange d the spin polarization of the ferromagnetic electrodes. Recent studies shine light on \nmolecule impact on the spin density of states on ferromagnets near interface regions [20]. \nAccording to our SQUID magnetometer study  on an array of ~7000 MTJ OMC showed almost \n100% increase in the magnetic moment  (Fig. 5a). OMCs seem to  change the spin density of \nstates of the ferromagnetic  electrodes. The spin density of state before OMC interaction (Fig.  6b) \nis presumably adjusted to new configuration (Fig.  6c) leading to high spin polarization. The \noverall tunneling barrier energy band diagram has changed from AlOx dominant (Fig. 6d) to \nOMC dominant (Fig. 6e -f). We hypothesize that to establish high current state majority spin \ndensity of states of the two ferromagnetic electrodes is  parallel  to each other (Fig.  6e). The \neffective barrier height in this state is ~0.4 eV, as calculated from the experimental result (Fig.  \n6e). However, in the suppressed current state we hypothesize that spin transport depends on the \nnet energy barrier with respect to the minority spin density of states on an MTJMSD (Fig.  6f). In \nthis state the majority and minority spins are antiparallel to each other and hence do not allow \ntransport. In this state transport between minority and majority spin is possible (Fig. 6g). Due to \nselection rule spin from majority band cannot tunne l to antiparallel majority band of the other \nmagnetic electrode (Fig. 6g). Similarly, the spin from minority  band of one magnetic electrode \ncannot tunnel to antiparallel minority band of the other magnetic electrode (Fig.  6g).   \nConclusion s: We discussed the observation of molecule  induced dramatic changes in the \nmagnetic and transport properties of th e magnetic tunnel junctions. OMCs were chemically \nFig. 6: (a) Schematic of an OMC chemically bonded with two NiFe layers along the MTJ edge. \nSimplified molecular structure shows energy level s of the Fe and Ni atoms. Density of states of a \nferromagnetic  electrode  (b) before, and (c) after affected by the molecular channels. (d) Tunneling \nbarrier between two ferromagnetic layers for a bare magnetic tunnel junction.  OMC provide \nintermediate qua ntum well like energy levels for the spin transport. The conceptual depiction of \ntunneling barrier between ferromagnet and OMC energy level in the (e) high current state and (f) \nsuppressed current state. (g) Possible spin pathways in the suppressed current  state.    \n \n9 \n bonded to  ferromagnetic electrodes to  bridge them  across the insulating spacer along the exposed \nedges.  These molecules dominated the MTJMSD’s spin transport and produced several orders \ngap in the high and low current state at room temperature . The estimation of transport data  in the \nlow and high current  states indicated that molecules change d the spin density of states . Potential \nreason behind modification in the density of states is the molecule induced spin filtering effect.  \nTransport studies are strongly supported by the experimental magnetic studies.  It is also \nnoteworthy that our experimental studies provide a platform  to connect a vast variety of \nferromagnetic leads to the broader  array  of high potential molecules such as single molecular \nmagnets  [1], porphyrin  [10], and single ion m olecules  [34] etc. The strength of exchange \ncoupling between ferromagnetic electrodes and molecul es can be tailored by utilizing different \ntethers and terminal functional groups  [7]. Th e MTJMSD  can provide advanced computer \ndevices by serving as a testbed for the molecule based quantum computation  devices [34-37].    \nAcknowledgments : Pawan Tyagi thanks Dr. Bruce Hinds and Department of Chemical and \nMaterials engineering at University of Kentucky for facilitating experimental work on MTJMSD \nduring his PhD. OMC were produced Dr. Stephen Holmes’s group. The preparation of  this paper \nand supporting studies were  in part supported by National Science Foundation -Award (Contract \n# HRD -1238802), Department of Energy/National Nuclear Security Agency (Subawa rd No. \n0007701 -1000043016), and  Air For ce Office of Sponsored Research (Awa rd #FA9550 -13-1-\n0152).  Any opinions, findings, and conclusions expressed in this paper  are those of the author(s) \nand do not necessarily reflect the views of any funding agency and author s’ affiliations.    \nReferences  \n[1] Bogani L and Wernsdorfer W 2008 Molecular spintronics using single -molecule magnets Nat. \nMater.  7 179-86 \n[2] Pasupathy A N, Bialczak R C, Martinek J, Grose J E, Donev L A K, McEuen P L and Ralph D C 2004 \nThe Kondo effect in the presence of ferromagnetism Science  306 86-9 \n[3] Petta J R, Slater S K and Ralph D C 2004 Spin -dependent transport in molecular tunnel junctions \nPhys. Rev. 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Phys.  40 2999 -3004  \n \n "    },    {        "title": "1207.4919v1.Measurement_of_the_dynamical_dipolar_coupling_in_a_pair_of_magnetic_nano_disks_using_a_Ferromagnetic_Resonance_Force_Microscope.pdf",        "content": "arXiv:1207.4919v1  [cond-mat.mtrl-sci]  20 Jul 2012Measurement of the dynamical dipolar coupling in a pair of ma gnetic nano-disks\nusing a Ferromagnetic Resonance Force Microscope\nB. Pigeau,∗C. Hahn, G. de Loubens, V. V. Naletov,†and O. Klein\nService de Physique de l’ ´Etat Condens´ e (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Y vette, France\nK. Mitsuzuka, D. Lacour, M. Hehn, S. Andrieu, and F. Montaigne\nInstitut Jean Lamour, UMR CNRS 7198, Universit´ e H.Poincar ´ e, 54506 Nancy, France\n(Dated: October 18, 2021)\nWe perform an extensive experimental spectroscopic study o f the collective spin-wave dynamics\noccurring in a pair of magnetic nano-disks coupled by the mag neto-dipolar interaction. For this, we\ntake advantage of the stray field gradient produced by the mag netic tip of a ferromagnetic resonance\nforce microscope (f-MRFM) to continuously tune and detune t he relative resonance frequencies\nbetweentwoadjacent nano-objects. This reveals theanti-c rossing andhybridization ofthe spin-wave\nmodesinthepairofdisks. Attheexacttuning, themeasuredf requencysplittingbetweenthebinding\nand anti-binding modes precisely corresponds to the streng th of the dynamical dipolar coupling Ω.\nThis accurate f-MRFM determination of Ω is measured as a func tion of the separation between the\nnano-disks. It agrees quantitatively with calculations of the expected dynamical magneto-dipolar\ninteraction in our sample.\nStudies of the collective dynamics in magnetic nano-\nobjects coupled by the dipolar interaction has recently\nattracted a lot of attention [1–8] due to its potential for\ncreatingnovelpropertiesandfunctionalities for the infor-\nmation technology. It affects the writing time of closely\npacked storage media [9], the synchronization of spin\ntransfer nano-oscillators [10], and more broadly the field\nof magnonics [11], which aims at using spin-waves (SW)\nfor information process [12]. Despite the generic na-\nture of the dynamic magneto-dipolar interaction, which\nis present in all ferromagnetic resonance phenomena, its\ndirect measurement has been elusive because it is diffi-\ncult to reach a regime where this coupling is dominant.\nIt requires that the strength of the dynamical coupling\nΩ exceeds both the deviation range of eigen-frequencies\nbetween coupled objects and the resonance linewidth.\nLarge Ω are usually obtained by fabricating nano-objects\nhaving large magnetization and placed nearby. But the\nconstraint of fabricating two nano-objects, whose SW\nmodes both resonate within Ω, is difficult to meet. For\nlong wavelengths, the SW eigen-frequency is indeed very\nsensitive to imperfections in the confinement geometry,\ninherent to uncertainties of the nano-fabrication process.\nMoreover,adirectdeterminationofthecouplingstrength\nbetween anytwosystems, asforinstanceasuperconduct-\ning qubit and electronic spins [13], requires the ability to\ntune anddetune them atleastonthe Ω-range. Sofar, the\nabsence of a knob to do so with the individual frequen-\ncies of nearby magnetic objects has prevented a reliable\nmeasurement of the dynamical dipolar coupling.\nIn this paper we shall demonstrate that ferromagnetic\nresonance force microscopy (f-MRFM) allows this quan-\ntitative measurement of Ω. We shall rely on the field\ngradient of the magnetic tip as a mean to fully tune\nand detune the resonance frequencies of two nano-disks\nby continuously moving the tip laterally above the pairof disks. One can find a position where the stray field\nof the tip exactly compensates the deviation of inter-\nnal field due to the patterning process. At this position,\nthe splitting between the eigen-frequencies of the binding\nand anti-binding modes is exactly equal to the dynamical\ndipolar coupling Ω. By studying Ω as a function of the\nseparation between the nano-disks, we shall demonstrate\nthat f-MRFM provides a reliable mean to measure the\nstrength of the dynamical coupling. It provides also a\nreliable mean to measure mode hybridization and mode\nlinewidth.\nThe magnetic material used for this study is a t=\n26.7 nm thick Fe-V (10% V) film grown by molecular\nbeam epitaxy on MgO(001) [14, 15]. This is a ferro-\nmagnetic alloy with a very high magnetization, 4 πMs=\n1.7×104G, and a very low magnetic Gilbert damping,\nα= 2×10−3. The film is patterned into disks by e-\nbeam lithography and ion milling techniques. The ge-\nometrical pattern (image in FIG.1a) consists in three\npairs of nearby disks having the same nominal diame-\nter 2R= 600 nm but different edge to edge separation:\ns= 200 nm, 400 nm and 800 nm. Each set is separated\nby 3µm in order to avoid cross coupling. An isolated\ndisk of identical diameter is also patterned for reference\npurpose. The sample is then placed in the room tem-\nperature bore of an axial superconducting magnet. The\ndisks are perpendicularly magnetized ( z-axis) by an ex-\nternal field of 1.72 Tesla [16]. This field is sufficient to\nsaturate all the disks. A linearly polarized ( y-axis) mi-\ncrowavefield hrfisproducedbyabroadbandAustrip-line\nantenna of width 5 µm deposited on top of a 50 nm thick\nSi3O2isolating layer, above the magnetic disks. The f-\nMRFM experiment consists in detecting the mechanical\nmotion produced by the magnetization dynamics in the\nFe-V nano-disks of a Biolever cantileverwith an Fe nano-\nsphere of diameter 700 nm glued at its apex (see FIG.1a)2\nFIG. 1. a) Schematic of the f-MRFM setup: an Fe sphere\nglued attheapexof asoft cantileveris scanned laterally ab ove\ndifferent pairs of Fe-V disks excited by a microwave field. b)\nDensity plot of the f-MRFM signal as a function of the dis-\nplacement xof the sphere above an isolated disk. The inset\nis a SEM image of the 2 R= 600 nm Fe-V disk (green) placed\nbelow the microwave antenna (gold).\n[17]. We will consider in the following that the stray\nfield of the tip Hsphreduces to the dipolar field created\nby a punctual magnetic moment msph= 3×10−10emu\nplaced at the center of the sphere. The role of the mag-\nnetic tip in f-MRFM is to create a field gradient tensor\n/hatwideG=∇Hsphon the sample in order to spatially code the\nresonancefrequency and to providea local detection [18].\nThese two features are illustrated by FIG.1b, which\nshows the dependence of the f-MRFM signal measured\nabove the isolated disk as a function of the position of\nthe tip on the x-axis. It displays the behavior of the\nlowest energy SW mode, where all spins are precessing\nin phase at the Larmor frequency around the unit vector\nˆz. The cantilever is scanned at constant height habove\nthe sample surface. The position x= 0 corresponds to\nplacing the probe on the axis of the disk.\nWe first concentrate on the variation of the FMR res-\nonance frequency as a function of the x-position of the\nsphere. It displays a bell curve, whose shape is due to\nthe additional bias field produced by the tip\nω(x) =ωFMR+γ{Hsph,z(x)}, (1)\nwhere the first term is the resonance frequency in the ab-\nsence of the sphere and the second term is the gyromag-\nneticratio γtimesthespatialaverageofthe z-component\nof the stray field of the sphere over the disk volume.\nThe curly bracket indicates that this average should beweighted by the spatial profile of the lowest energy SW\nmode [19]. The maximum shift of frequency occurs close\ntox= 0, where the additional field from the f-MRFM\nsphere is maximal [20]. The slope of the wings is propor-\ntional to the lateral field gradient Gzx. Forh≫2R, it is\nmaximum at x≃0.39h, wherehis the height between\nthe sample surface and the sphere center. At this loca-\ntion, the gradient is about Gzx≈2.7msph/h4. Since it\nis important to keep has large as possible for stability\npurpose, the optimal his reached when γGzxR >Ω. For\nour settings, this occurs at h= 1.8µm, leading to slope\nof about 0 .3 GHz/µm. At this distance, the maximum\nstray field of the sphere is about 140 G, a small variation\ncompared to the static perpendicular field of 1.72 T, en-\nsuring that no significant deformation of the SW modes\nprofile is induced [17].\nWe then turn to the variationofthe amplitude of the f-\nMRFM signal as a function of the position of the sphere\nin FIG.1b. The force acting on the cantilever can be\ncalculated as the vertical force exerted by the tip on the\nsample,Fz=Gzz∆Mz, where ∆ Mzis the variation of\nthesamplemagnetizationinducedbytheFMRresonance\n[17]. The gradient Gzzdecays as the power 1 /x5, for\nlarge lateral displacement x. This decay ensures a local\ndetection. Experimentally, the signal decreases by one\norderofmagnitudewhentheprobeisdisplacedby1.2 µm\nlaterally.\nWe now discuss the same experiment above the pair\nof two 600 nm disks separated by s= 200 nm. The\nresult is displayed on FIG.2a. Here x= 0 corresponds\nto the middle of the pair. At each position xof the f-\nMRFM probe, we can see two modes. The upper branch\nhas two frequency maxima at x1,2=∓400 nm, whose\nseparation corresponds to the center to center distance\nbetween disk 1 and disk 2. The two maxima occur at\nslightly different frequencies, presumably due to a small\ndifference in diameter between the two disks. When the\nprobe is placed in between, x1< x < x 2, the two levels\nanti-cross, which is a characteristic behavior of a coupled\ndynamics. Defining ω1,2as the frequencies of the two\nuncoupled disks, the collective frequencies follow:\nωA,B=ω1+ω2\n2±/radicalBigg/parenleftbiggω1−ω2\n2/parenrightbigg2\n+/parenleftbiggΩ\n2/parenrightbigg2\n(2)\nwith Ω being the dynamical coupling strength. The two\ncoupled eigen-frequencies ωA,Bcorrespond respectively\nto the anti-binding mode (A), where spins are precessing\nout-of-phase between the two disks, and to the binding\nmode(B),wherespinsareprecessingin-phase[21]. Inour\nf-MRFM experiment, ω1,2both depend on x, see Eq.1:\nω1,2(x) =ωFMR+γ{Hsph,z(x−x1,2)}. Usingthesedepen-\ndencies in Eq.2, one can obtain an analytical expression\nfor the frequency difference ωA(x)−ωB(x) observed in\nFIG.2. At x= 0, when ω1=ω2, the splitting ωA−ωB\nexactly measures Ω. Using this analytical expression for3\nFIG. 2. a) Density plot of the experimental f-MRFM spectra\nas a function of the displacement xof the sphere above a\npair of 600 nm Fe-V disks separated by s= 200 nm. b)\nPredicted behavior by micromagnetic simulations. The uppe r\nmode (blue) corresponds to the anti-binding mode (A), while\nthe lower (red) shows the bindingmode (B). Insets: simulate d\nprecession profiles in each disk for modes (A) and (B) at the\nanti-crossing. Thedashedlineswould betheindividualmod es\nof each disks in the absence of dynamical coupling.\nthe spatial dependence of the splitting, we have fitted\nΩ/2π= 50±5 MHz. We emphasize that this splitting is\n2.5 times larger than the linewidth, found to be 20 MHz.\nTheoretically, the coupling Ω for the magneto-dipolar in-\nteraction is defined by [21]\nΩ2= 4γ2h1,2h2,1. (3)\nhi,jrepresents the cross depolarization field produced by\nthe SW in the j-th disk on the i-th disk ( i,j= 1,2)\n[22, 23]. It can be expressed as a function of the cross\ndepolarization tensor elements, which have an analytical\nexpression in the approximation of a uniform precession\n[24]:hi,j= 2πMs({Ni,j\nxx}+{Ni,j\nyy}). This formula reflects\nthat the magneto-dipolar interaction is anisotropic and\nthus, it induces an elliptical precession in the two disks.\nFor the separation s= 200 nm, a numerical application\nyields{N1,2\nxx} ≈ −2{N1,2\nyy} ≈0.0012, which corresponds\nto a coupling field of about 10 G between the two disks,\nor a coupling frequency Ω /2π= 56 MHz, in very good\nagreement with the measured value.\nAnother striking feature in FIG.2a is the strong varia-\ntion of the signal amplitude near the optimum coupling.\nWe haveexplicitly plotted in FIG.3athe amplitude ofthe\nf-MRFM signal as a function of the lateral displacement\nxof the probe, showing both the near extinction of theanti-bindingmode(A)andthestrongenhancementofthe\nbinding mode (B) near x= 0. The ratio of hybridization\nin the two coupled disks follows the expression:\nc1\nc2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nA,B=/parenleftBigg\n(ω1−ω2)∓/radicalbig\n(ω1−ω2)2+Ω2\nΩ/parenrightBigg∓1\n(4)\nIntroducing the spatial dependence of ω1,2described by\nEq.1 in Eq.4 we can calculate the total force Fz∝\nP/bracketleftbig\nc2\n1Gzz(x−x1)+c2\n2Gzz(x−x2)/bracketrightbig\nacting on the can-\ntilever. The power efficiency P=|c1+c2|2h2\nrfis propor-\ntional to the overlap integral between the uniform rf field\nand the collective SW mode (the vector sum of the trans-\nverse magnetization in the two disks) [21]. Using Eq.4,\nthe dependence on xofthe force producedby the binding\nand anti-binding mode gives the continuous lines shown\nin FIG.3a. The difference between the two curves comes\nmainly from the selection rules defined in P. We find\nthat at the optimum coupling (when ω1andω2cross),\nthe anti-binding mode (A) in Eq.4 has c1=−c2,i.e., a\nprecession with equal hybridization weight between the\ntwo disks and out-of-phase. The overlap with the uni-\nform rf field excitation is thus zero at x= 0, leading to\na vanishing amplitude. In contrast, the binding mode\n(B) hasc1= +c2at the anti-crossing, i.e., a precession\nwith equal hybridization weight too, but now in-phase\nbetween the two disks. It represents and enhancement\nof the absorbed power by a factor of 22compared to the\namplitude in one disk.\nWe then study the effect the magneto-dipolar coupling\non the linewidth of the collective mode. We observe that\nthe linewidth does not change much with tuning and\nthe observed variation with xis below the 5% range.\nAt the optimal tuning x= 0, the linewidth measured\nis ∆f= 22.3±0.5 MHz (see FIG.3b) and it becomes\nslightly larger ∆ f= 23.1±0.5 MHz at the maximum\ndetuning x=x1,2. For comparison we have displayed\nin FIG.3c the linewidth observed above the single disk,\nwhose value ∆ f= 21.4±0.5 MHz. A small increase of\nthe ratio ∆ f/fis indeed expected for the dynamically\ncoupled modes. This comes from the fact that this ratio\nis equal to ∆ f/f=α(Hx+Hy)//radicalbigHxHy, whereαis the\nGilbert damping, and HxandHyrepresent the two stiff-\nness fields which characterize the torque exerted on the\nmagnetization when it is tipped along the x- ory-axis\n[25]. The degree of hybridization as well as the nature\nof the mode (A or B) change the values and signs of Hx\nandHy. For the binding mode, the magneto-dipolarcou-\npling generates an elliptical precession whose long axis\nis along the two disks axis. The induced ellipticity Eis\nmaximumattheanti-crossing( x= 0), withanamplitude\nE=β−1\nβ+1Ω\nωB≈3%, with β={N1,2\nxx}/{N1,2\nyy} ≈ −2. An\nincreaseofellipticity inducesanincreaseofthelinewdith,\na behavior which is consistent with the small additional\nbroadening measured in our experiment.\nThe analytical model used above to analyze the data4\nFIG. 3. a) Variation of the amplitude of the binding (red)\nand anti-binding (blue) resonances as a function of the late ral\nposition of the probe for the two disks separated by 200 nm.\nThe solid lines correspond to thebehavior following from Eq .4\n(see text). b) Linewidth of the binding mode for the same\npair at the tuning position ( x= 0) c) Comparison with the\nmeasurement of the linewidth above a single disk.\nassumes a uniform magnetization throughout the mag-\nnetic body. To take more precisely into account the 3D\ntexture of the magnetization and the static deformation\ninduced by the probe, we have also calculated the eigen-\nfrequencies of the two lowest energy modes as a function\nofxusing SpinFlow3D, a finite element solver developed\nby In Silicio [26]. The disks are discretized with a mesh\nsize of 10 nm using a Delaunay mesh construction. At\neach position of the probe, we first calculate the equilib-\nrium configuration in the disks. The Arnoldi algorithm\nis then used to compute the lowest eigen-values of the\nproblem as well as the associated eigen-vectors. The re-\nsult is represented in red and blue in FIG.2b for the two\nlowest energy modes. The precession patterns associated\nto each mode at the anti-crossing are shown in inset.\nIn this color representation, the hue indicates the phase\n(or direction) of the oscillating magnetization, while the\nbrightness indicates its amplitude. The simulation re-\nsults confirm very nicely the interpretation made above\nin terms of amplitude and peak position.\nWe have then repeated the same procedure on the two\nother pairs of disks, with larger edge to edge separation\ns. The strength of the dynamical coupling measured by\nf-MRFM is plotted as a function of sin FIG.4. The\nmain results is that, with our experimental parameters,\nsneeds to be less than the diameter of the disks in order\nto have Ω larger than the linewidth ∆ f. The data are\nplotted along with the analytical prediction (continuous\nline) and the simulations (dashed line with small dots).\nWe observe an excellent overall agreement between the\nthree sets of results, which all exhibit a similar decay\nwiths(not a simple polynomial law [27]). Still, the ex-\nFIG. 4. Coupling strength as a function of the separation\nsbetween two disks. The plot compares the experimental\nfindings to the predicted amplitude of the magneto-dipolar\ninteraction either analytically (continuous line) or by mi cro-\nmagnetic simulations (small dots, dashed line is a guide to\nthe eye).\nperimental points are systematically slightly below the\ntheoretical expectation. This could be explained by the\nfact that the disks are slightly smaller than their nomi-\nnal value ( e.g., due to some oxidationat their periphery),\nor that the true separation between the disks is slightly\nlarger than expected, which we have represented on the\ngraph by the horizontal error bars. The agreement be-\ntween the analytical model and the simulation is very\ngood until s= 0.1µm. The discrepancy for very small\nsis due to a significative change in the static magnetic\ntexture. These changes are not taken into account by the\nanalytical model. The effect of the static coupling is to\nproduce a static magnetization along the x-direction. As\nshown by the simulations, this deformation enhances the\nstrength of the dynamical coupling.\nIn conclusion, we have shown that f-MRFM enables a\ndetailed investigation of the dynamical dipolar coupling\nbetween two nearby magnetic objects, owing to the pos-\nsibility of the technique to study both the tuned and de-\ntuned regime on the same object. It has been applied to\nstudy the collective SW dynamics in pairs nano-disks of\nFe-V, an ultra-low damping material. Several signatures\nof the collective behavior have been experimentally ev-\nidenced and quantitatively explained: the anti-crossing,\nthe hybridization of the modes and the effects on the\nlinewidth. Moreover, we have found that in order to\nhaveafrequencysplittinglargerthanthelinewidth ofthe\nmodes, the edgeto edge separationbetween ourdisks has\nto be smaller than their diameter, due to the fast decay\nof the magneto-dipolar interaction. 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M.Gonzalez, andK.Y.Guslienko,\nApplied Physics Express 4, 065003 (2011)"    },    {        "title": "1501.07672v1.Microscopic_properties_of_degradation_free_capped_GdN_thin_films_studied_by_Electron_Spin_Resonance.pdf",        "content": "arXiv:1501.07672v1  [cond-mat.mtrl-sci]  30 Jan 2015Microscopic properties of degradation-free capped GdN thi n films studied by Electron\nSpin Resonance\nTokuro Shimokawa,1Yohei Fukuoka,2Masashi Fujisawa,3Weimin Zhang,4Susumu Okubo,4Takahiro Sakurai,5\nHitoshi Ohta,4,∗Reddithota Vidyasagar,6Hiroaki Yoshitomi,6Shinya Kitayama,6and Takashi Kita6\n1Center for Collaborative Research and Technology Developm ent,\nKobe University, 1-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan\n2Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan\n3Research Center for Low Temperature Physics, Tokyo Institu te of Technology,\n2-12-1 Ohokayama, Meguro-ku, Tokyo 152-8551, Japan\n4Molecular Photoscience Research Center, Kobe University,\n1-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan\n5Center for Supports to Research and Education Activities,\nKobe University, 1-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan\n6Department of Electrical and Electronic Engineering, Grad uate School of Engineering,\nKobe University, 1-1 Rokkodai, Kobe 657-8501, Japan\n(Dated: August 13, 2018)\nThe microscopic magnetic properties of high-quality GdN th in films have been investigated by\nelectron spin resonance (ESR)andferromagnetic resonance (FMR) measurements. Detailed temper-\nature dependence ESR measurements have shown the existence of two ferromagnetic components\nat lower temperatures which was not clear from the previous m agnetization measurements. The\ntemperature, where the resonance shift occurs for the major ferromagnetic component, seems to\nbe consistent with the Curie temperature obtained from the p revious magnetization measurement.\nOn the other hand, the divergence of line width is observed ar ound 57 K for the minor ferromag-\nnetic component. The magnetic anisotropies of GdN thin films have been obtained by the analysis of\nFMR angular dependence observed at 4.2 K. Combining the X-ra ydiffraction results, the correlation\nbetween the magnetic anisotropies and the lattice constant s is discussed.2\nI. INTRODUCTION\nFerromagnetic semiconductors are expected to be a key material for the future spintronics.1,2GdN is one of these\nferromagnetic semiconductors and it is particularly interesting due to its partially filled 4f and 5d orbitals with\nsaturation moment of 7 µB/Gd3+.3Therefore, GdN has been the object of a series of theoretical an d experimental\nstudies since more than a half century.4–18\nHowever, it is well known from a number of studies of bulk GdN in the 60 ’s and 70’s4,16–21that it is very difficult to\nobtain the high-quality bulk GdN because nitrogen vacancies and oxy gen can damage it very easily. For example, it\nwas reported that there is a strong decrease in the magnetic mome nt of bulk GdN even with few percent of oxygen.18\nThe decrease in Curie temperature of bulk GdN was confirmed with a r ange of oxygen concentration.17Cutleret al\nalso reported that the nitrogen vacancies make the hysteresis eff ect and the remenance much smaller.17Understanding\nthe properties of ”pure” GdN is still challenging because of the difficu lty to produce high quality single crystals.\nHowever,thesituationhaschangedowingtotheadvancedtechno logyofthethin filmsynthesissincethe2000s.7,13,22\nThese thin films are capped by AlN on GdN to make surface smooth and to restrict the oxygen contamination. These\nearly studies of GdN thin film reported that the properties of the Gd N thin film, contrary to bulk one, are very\nsensitive to their epitaxial strain, structural distortion and surf ace effect for nanocrystalline films. From the view\npoint of the development in spintronics devices, therefore, it is ver y important to understand these parameters and\nhow they affect the magnetic and optical properties of the high-qu ality GdN thin film.\nMore recently, H. Yoshitomi et al. and R. Vidyasagar et al. have studied the optical and magnetic properties\nin epitaxial AIN/GdN/AIN double heterostructures grown by reac tive radio-frequency (rf) sputtering under ultra-\npure conditions.23–27For example, their high-quality GdN thin film of 95 nm showed the indirec t and direct optical\ntransitions, and the considerable size effects of the optical band g ap were observed with a decrease in the GdN\nthickness. They also investigated the saturation magnetization an d Curie temperature estimated by Arrott plots as a\nfunction ofthe thicknessofGdN.13,23However,few cases, except for the ferromagneticresonance( FMR) measurement\nby K. Khazen et al22, studied the microscopic magnetic properties of these GdN thin films .\nIn this study, we investigate the microscopic magnetic properties o f high-quality GdN thin films by the detailed\ntemperature dependence of electron spin resonance (ESR), and the angular dependence of FMR at 4.2 K.\nII. EXPERIMENTAL DETAILS\nWe investigate the micro magnetic properties of three GdN samples. One has the thickness of 95nm whose optical\nand macro magnetic properties has been investigated by H. Yoshito miet al.23We call this sample “08GdN” in this\npaper. The other samples has the thicknesses of 29 nm and 97 nm, r espectively; we call these two samples “10GdN”\nin this paper. All samples were grown on c-sapphire (0001) substra tes at 500◦C by reactive radio-frequency(RF)\nmagnetron sputtering28in an ultrahigh vacuum chamber. The input RF power was 250 W. AlN/Gd N/AlN double\nheterostructureswereusedtoavoidoxidation.18ThegrowthchamberequippedwithmultitargetsforAlNandGdNwas\nseparated from the substrate introduction chamber to avoid oxid ation of the target when introducing the substrate.\nAl(99.99%) and Gd(99.9%) were used as metal targets. We used an u ltrapure (99.9999%) gas mixture of argon and\nnitrogen for reactive growth. For the synthesis of 08GdN thin film, the partial pressure ratio of argon and nitrogen\nwas even, and the total sputtering pressure was 5 Pa. For the sy ntheses of 10GdN thin films, on the other hand, the\npartial pressure ration was 9:6, and the total sputtering pressu re was 6 Pa for the purpose to decrease the number\nof nitrogen vacancies. However, the transmission and absorption spectral measurement showed that the number of\nfree carrier in 10GdN is more than that in 08GdN. It’s not known exac tly why the number of nitrogen vacancy in\n10GdN is more than that in 08GdN.29The X-ray diffraction measurement showed that the lattice consta nt along\n(111) direction for 08GdN sample is a= 0.507 nm. The 29nm and 97nm thin films of 10GdN have a= 0.506 nm\nanda= 0.507 nm, respectively. These lattice constants are longer than the bulk value a= 0.4998 nm. In addition,\nwe also confirmed that the lattice constant along (200) direction is s maller than the bulk value a= 0.250 nm; for\nexample, the value for 95 nm thickness of 08GdN is a= 0.249 nm. Therefore, our GdN thin films have uniaxial lattice\ndistortion.\nOur ESR/FMR measurements were performed by the Bruker X-ban d ESR spectrometer EXM081 at Center for\nSupports to Research and Education Activities, Kobe University, w ith 100kHz field modulation using a TE103 rect-\nangular cavity in the temperature range of 4.2 K to 300 K. We show th e geometry of the FMR measurements in\nFig. 1, which is the same condition as that in Khazen’s paper.22The GdN samples lie in the x-zplane, and the y-axis\nis parallel to the growth face direction [111] of our thin films. The out -of plane variation of the external magnetic\nfield is in the xyplane.3\nYZ[\n.\n)П)В)В\nП\nFigure 1. (Color online) The geometry of the FMR measurement s.MandHare spontaneous magnetization and external\nmagnetic field, respectively. θ(θH),ϕ(ϕH) are polar and azimuthal angles for M(H) vector.\nIII. EXPERIMENTAL RESULTS\nFirstly, we show the temperature dependence of the resonance fi eld. We successfully obtained the result of Fig. 2\n(a) from the spectrum fitting, using the following double lorentzian e quation with respect to the ESR spectrum at\neach temperature. Here, we would like to point out that the observ ed ESR signal is the differential curve. Total\nintensity Ias a function of external magnetic field His written by\nI(H) =−16I′\nm1[(H−H1)/(∆HPP1/2)]\n[3+{(H−H1)/(∆HPP1/2)}2]2\n−16I′\nm2[(H−H2)/(∆HPP2/2)]\n[3+{(H−H2)/(∆HPP2/2)}2]2\n+m+m′H, (1)\nwhereI′\nmiis the intensity, Hiis the resonance field, ∆ HPPiis the line width for component iandmandm′are\nbackground. These parameters are determined by the fitting exp erimental data. As an example, we show the ESR\nspectrum and the fitting result at T=40 K for 95 nm thickness of 08GdN in the inset of Fig. 2 (a). Note tha t\nthe external magnetic field was applied to in the plane ( ϕH= 0). Fig. 2 (a) shows the temperature dependence\nof resonance field for 95 nm thickness of 08GdN and 97nm thickness of 10GdN. We estimate the g-factorg∼1.96\nby using the resonance field value 3524.9 G at the highest temperatu re 260 K. This gvalue is consistent to that of\nLand´ eg-factorgL= 2 of Gd3+whose total orbital angular momentum Lis 0. At the low temperature region, we\ncan see clearly two kinds of phases not only for 08GdN but also for 10 GdN which has the larger number of nitrogen\nvacancy. Therefore, the origin of the phase separation is not com ing from the nitrogen vacancy. We have already\nconfirmed the existence of such two kinds of phases in the other Gd N samples.24The Curie temperature ( Tc) for the\n95 nm thickness of 08GdN has been reported about 37 K by using Arr ott plot analysis23, therefore, the shift in the\nresonance field around 40 K (res. 2 in Fig. 2 (a)) comes from a domina nt part of the magnetization of GdN thin\nfilm in the ferromagnetic phase. In the present study, we also confi rmed that the Tcvalue for the 97 nm thicknesses\nof 10GdN is about 29 K by using Arrott plot analysis. The difference of theTcbetween 08GdN and 10GdN comes\nfrom the number of nitrogen vacancy.17Careful observation of Fig. 2 (a) tells us that the resonance shift of 10 GdN\n(res. 2) begins at lower temperature than that of 08 GdN, and whic h is consistent that the Curie temperature for 10\nGdN is lower than that of 08 GdN. Therefore, the shift in the resona nce field around 30 K (res. 2) for 10GdN also\ncomes from a dominant part of the magnetization in the ferromagne tic phase. The second shift at higher temperature\nside (∼70K (res. 1)) is originated from another ferromagnetic phase, wh ich cannot be ascribed to the short-range\ncorrelation of spins at T > T cbecause without the phase separation we cannot observe two ESR ’s in the intermediate\ntemperature region (25 ∼57 K). The high-Tc phase may come from the interface because the contribution of res. 1 to\nthe static magnetization is less than 1% for 08GdN at 50 K where the s hift of res. 1 is close to the saturation while\nthe shift of res. 2 has just started.24However it requires further investigation. Here we would like to emph asize that\nthe observation of the two kinds of phases as in the case of Fig. 2 (a ) suggests ESR measurements can easily detect4\n\tB\n\tC\n \n\tD\n0 2000 4000 6000 8000\nB  (G)–10000 –500005000 10000 Intensity (arb. units) \n0 100 200 300 \nT (K ) 01ʷ10 92ʷ10 93ʷ10 9Integrated Intensity 08GdN 95nm \nS1=I’ m1 (∆ H PP1 )2:\nS2=I’ m2 (∆ H PP2 )2:0 100 200 300 \nT (K ) 5001000 1500 2000 L ine width(G)08GdN 95nm\n∆HPP1 :\n∆HPP2:0 100 200 300 \nT  (K )02000 4000 R esonance Field (G)08GdN 95nm (res.1:  ) \n10GdN 97nm (res.1:  ) (res.2:  ) (res.2:  ) \nFigure 2. (Color online) (a) Temperature dependences of the resonance field for 95 nm thickness of 08GdN and 97 nm thicknes s\nof 10GdN. The word of “res. 1(2)” means the resonance shift at higher (lower) temperature region. The inset is the fitting\nresult at T= 40 K for 95 nm thickness of 08 GdN by using double Lorentzian e quation (1). (b) and (c) are temperature\ndependences of the line width and of integrated intensity fo r 95 nm thickness of 08GdN, respectively.\nthe microscopic properties such as the phase separation which is diffi cult to be observed by macro measurements.\nWe also note that it was difficult to measure the resonance field for 10 GdN samples at higher temperature region\nbecause the nitrogen vacancy provides carriers at higher temper ature which causes the decrease of Q-factor in ESR\nmeasurements.\nNext, we investigated the temperature dependence of the line widt h for 95 nm thickness of 08GdN. Fig. 2(b) shows\nthe results about the line width. Owing to ESR measurements, we con firm the decreasing behaviors of the line\nwidth ∆HPP1and ∆HPP2below 57 K which corresponds to the mid-point of res. 1 resonance s hift in Fig. 2(a). The\ndecreasing behavior with decreasing temperature in the ferromag netic phase is well known as a typical property in\nthe ferromagnetic region.30It is also interesting that the divergence behavior of ∆ HPP1can be observed clearly near\nat 57 K. This divergence of the line width indicates the presence of sp in fluctuations near Tc.31The similar increase\nbehavior near above Tchas been also confirmed roughly in typical FM thin films.31However, no such divergence\nbehavior is observed for ∆ HPP2suggesting the different spin dynamics for res. 1 and res. 2. We also investigated\nthe temperature dependence of the integrated intensity as show n in Fig. 2(c). Here, integrated intensity Sifor each5\ncomponent ican be calculated by the line width ∆ HPPiand intensity I′\nmi. It is well known in ESR that Sican be\nestimated by I′\nmi(∆HPPi)2. According to this result, we can determine which of two separated phases is smoothly\nconnected to the dominant part in the ferromagnetic ground stat e. Therefore, we can say that “res. 2” in Fig. 2 (a)\nand ∆HPP2in Fig. 2(b) correspond to the dominant part in ferromagnetic stat e.\nNext, we investigated the angular dependence of FMR. For example , Fig. 3 shows the angular dependence of the\nFMR for the 08GdN at 4.2K where ϕHis varied from −90◦to 90◦. Here, we measured the FMR in the interval of\n10 degrees. We confirm that the resonance field is very sensitive to the angular variation. Applying the equation (1)\nto analyze data in Fig. 3 we get the angular dependence of the line widt h in Fig. 4. Owing to the closely-spaced\nangulars we took, we see the peak structures clearly at ϕH∼90◦in Fig. 4. We also gain the angular dependence of\nthe resonance field. Fig. 5 presents the angular dependence of th e resonance field for our three samples obtained by\nthe Lorentzian fitting. In order to investigate the film thickness de pendence, we add the data for 29 nm thickness of\n10GdN sample. All sample shows that the resonance field was maximize d when external magnetic field was applied\nin the direction of out-of-plane, and minimized when in the direction of in-plane. This result consistent with typical\nFMR spectra32, and with behaviors of Khazen’s samples22. A careful observation of Fig. 5 enable us to confirm the\nresonance field value for 29 nm thickness of 10GdN is lower than thos e for the other samples. This behavior means\nthat the magnetic anisotropy for 29 nm thickness of 10GdN is differe nt from the other samples.\nFinally, we analyzed the magnetic anisotropy for our GdN thin film from the angular dependence of the FMR\nspectra. We use “Smit-Beljer formalism”33for our analysis which is applicable to thin film of cubic symmetry\nallowing for a possible uniaxial deformation, because our GdN samples also have uniaxial anisotropy in the process\nof synthesis. In this case, the energy density Ecan be written by\nE=−MH+K1(α2\n1α2\n2+α2\n2α2\n3+α2\n3α2\n1)+(2πM2−Ku)α2\n2, (2)\nwhich represent the Zeeman interaction, the magnetic anisotropic energy, and the demagnetization energy. Here,\nK1is the fourth order cubic magnetocrystalline anisotropy constant , andKuis the second order uniaxial anisotropy\nconstant. αiis the direction cosines of the magnetization Mrelative to the cubic crystal axes, and His the applied\nfield (See Fig. 1). In order to analyze the results in Fig. 5, we can gen erally fix θ=θH=π/2 in Fig. 1 and use the\nfollowing equations. One is the static equilibrium orientation of the mag netization\nHsin(ϕH−ϕ) = (4πM−2Ku\nM)sin(ϕ)cos(ϕ)+K1\n2Msin(4ϕ), (3)\nand the other is the resonance field equation\n(ω\nγ)2= [Hcos(ϕH−ϕ)+K1\nM(2−sin22ϕ)−(4πM−2Ku\nM)sin2ϕ]\n×[Hcos(ϕH−ϕ)+2K1\nMcos4ϕ+(4πM−2Ku\nM)cos2ϕ]. (4)\nThese two fitting equations for FMR measurements are derived fro m the following three resonance conditions (Smit-\nBeljers equatons33):\n∂E\n∂θ=∂E\n∂ϕ= 0, (5)\n(ω\nγ)2=1\nM2sin2θ[∂2E\n∂2θ∂2E\n∂2ϕ−(∂2E\n∂ϕ∂θ)2]. (6)\nIn Fig. 5 the example ofthe fitting result for 08GdNsample is shownby the black line. The fitting is rathersuccessfull.\nSmall deviations between the data and the fitting close to ±90◦may be due to the subtle misalignment ot the sample\nto the applied magnetic field. We also performed the same fitting to th e obtained data for 10GdN 29 nm and 97 nm\nsamples where the fitting lines are not shown in Fig. 5 to avoid the comp lication in Fig. 5. However, the fittings are\nalso rather successful. Table 1 shows our analysis results for magn etic anisotropy constants. Here the magnetization\nMfor each sample is obtained from the paper.27These crystal anisotropies K1andKuare much different from\nthe Khazen’s results22. OurK1value is almost one third, and Kuis two or three times of each value of Khazen’s\nbulk sample, respectively. The reason of the difference comes from the difference in the crystal growth process. The\nKhazen’s GdN samples were deposited on (100) oriented Si substra te and these films were polycrystalline. On the\nother hand, our samples were grown along to the (111) direction of GdN on c-sapphire (0001) substrates by reactive6\nFigure 3. Angular dependence of FMR measurement at T=4.2 K for 95 nm thickness of 08GdN.\nradio-frequency magnetron sputtering in an ultrahigh vacuum cha mber. More concretely, the lattice constant of the\nKhazen’s extended film was increased 2.4% uniformly not along to a spe cific direction, and the K1value of 2.4 %\nincreased samples are larger than that of Khazen’s bulk sample. The lattice constant along (200) direction of our\nsamples were decreased although the lattice constant along (111) direction were increased. In fact, the lattice constant\nalong (111) direction for the samples of the 95 nm thickness of 08Gd N and 97 nm thickness of 10GdN is is a=0.507\nnm which is larger than the reported bulk value a=0.4998 nm.The lattice constant along (200) direction for 97 nm\nthickness of 08GdN is a=0.249 nm and this value is smaller than the bulk v alue ofa=0.250 nm. In addition, the\ncoefficient of thermal expansion of our substrate AlN is 4.4 ×10−6K−1which is larger than that of the Si substrates,\n2.4×10−6K−1. Therefore, we can naturally accept the difference between our a nd Khanzen’s samples. We also note\nthat the K1value for 29 nm thickness of 10GdN is larger than those of our other samples. The largeness comes from\nthe small lattice constant for 29 nm thickness of 10GdN along (111) direction, a=0.506 nm. In other words, the\nK1values has the tendency to come close to the bulk value when the latt ice constant approaches to the bulk value\na=0.4998 nm. We should be careful to the fact that these lattice con stant values measured by X-ray diffraction are\njust average values. Therefore, we can not discuss about the se cond order uniaxial anisotropy Kufrom the view point\nof the lattice constant because the Kuvalues are mainly affected by the interfacial surface of crystal. It is naturally\nexpected that our Kuvalues are more sensitive than Khazen’s sample because Khazen’s film s are polycrystalline. This\nis the origin that our Kuvalues are larger than those of Khazen’s. The Kuvalue of 95 nm thickness for 08 GdN is\nslightly largerthan that of 97 nm thickness for 10GdN. We speculate that it may come from the nitrogen vacancy, that\nmeans, the strain at the interfacial surface for 10GdN samples wa s relaxed by the large number of nitrogen vacancy.\nWe also comment about the characteristic which our thinner sample o f 29 nm thickness for 10 GdN has the largest\nKuvalue. It is characteristic of ferromagnetic thin films that the thinn er thickness sample has the larger value of Ku.\nThis behavior is well known theoretically and experimentally, for exam ple, Fe/MgO multilayered films.32According\nto these obtained results, the FMR analysis is very useful to obtain the microscopic properties.7\n–100 0 100 ПH (degree)010000 20000 L ine Width (G) 08GdN 95nm \nFigure 4. Angular dependence of line width at T=4.2 K for 95 nm thickness of 08GdN.\n–50 0 50 \nθ (degree) 01000 2000 3000 4000 R easonance Field (G) 08GdN 95nm:\n10GdN 29nm:\n10GdN 97nm:\nFigure 5. (Color online) Angular dependences of resonance fi eld of FMR for our all samples at T= 4.2 K. The external\nmagnetic field was applied out-of-plane at −90◦and 90◦. The black line is the fitting result for 08GdN sample.\n4πM(Oe)2Ku/M(Oe)2K1/M(Oe)Ku(erg/cm3)K1(erg/cm3)T(K)\n08GdN 95nm 24167 11167 404 1.07×1073.88×1054.2in this paper\n10GdN 29nm 28660 14760 394 1.68×1074.49×1054.2in this paper\n10GdN 97nm 24027 10527 380 1.01×1073.63×1054.2in this paper\nBulk film 22220 5759 1292 5.09×1061.14×1064.0ref. 23\n2.4 % extended film 15620 2897 2252 1.8×1061.4×1064.0ref. 23\nTable I. Spontaneous magnetization M, the fourth order cubic magnetocrystalline anisotropy con stantK1and the second order\nuniaxial anisotropy constant Kufor our samples and Khazen’s.\nIV. CONCLUSION\nWe have investigated microscopic properties of high-quality GdN thin films. Detailed temperature dependence\nESR measurements have been performed for the first time and the y showed the existence of two ferromagnetic\ncomponents at lower temperatures. 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Rep. 10, 1113 (1955)."    },    {        "title": "1204.5342v1.Nonlocal_feedback_in_ferromagnetic_resonance.pdf",        "content": "Nonlocal feedback in ferromagnetic resonance\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany\u0003\n(Dated: April 27, 2022)\nAbstract\nFerromagnetic resonance in thin films is analyzed under the influence of spatiotemporal feedback\neffects. The equation of motion for the magnetization dynamics is nonlocal in both space and time\nandincludesisotropic, anisotropicanddipolarenergycontributionsaswellastheconservedGilbert-\nand the non-conserved Bloch-damping. We derive an analytical expression for the peak-to-peak\nlinewidth. It consists of four separate parts originated by Gilbert damping, Bloch-damping, a mixed\nGilbert-Bloch component and a contribution arising from retardation. In an intermediate frequency\nregimetheresultsarecomparablewiththecommonlyusedLandau-Lifshitz-Gilberttheorycombined\nwith two-magnon processes. Retardation effects together with Gilbert damping lead to a linewidth\nthe frequency dependence of which becomes strongly nonlinear. The relevance and the applicability\nof our approach to ferromagnetic resonance experiments is discussed.\nPACS numbers: 76.50.+g; 76.60.Es; 75.70.Ak; 75.40.Gb\n\u0003thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1arXiv:1204.5342v1  [cond-mat.mes-hall]  24 Apr 2012I. INTRODUCTION\nFerromagnetic resonance enables the investigation of spin wave damping in thin or ul-\ntrathin ferromagnetic films. The relevant information is contained in the linewidth of the\nresonance signal [1–3]. Whereas the intrinsic damping included in the Gilbert or Landau-\nLifshitz-Gilbert equation [4, 5], respectively, predicts a linear frequency dependence of the\nlinewidth [6], the extrinsic contributions associated with two-magnon scattering processes\nshow a nonlinear behavior. Theoretically two-magnon scattering was analyzed for the case\nthat the static external field lies in the film plane [7, 8]. The theory was quantitatively\nvalidated by experimental investigations with regard to the film thickness [9]. Later the\napproach was extended to the case of arbitrary angles between the external field and the\nfilm surface [10]. The angular dependence of the linewidth is often modeled by a sum of\ncontributions including angular spreads and internal field inhomogeneities [11]. Among oth-\ners, two-magnon mechanisms were used to explain the experimental observations [12–17]\nwhereas the influence of the size of the inhomogeneity was studied in [18]. As discussed in\n[3, 14] the two-magnon contribution to the linewidth disappears for tipping angles between\nmagnetization and film plane exceeding a critical one \bcrit\nM=\u0019=4. Recently, deviations from\nthis condition were observed comparing experimental data and numerical simulations [17].\nSpin pumping can also contribute to the linewidth as studied theoretically in [19]. How-\never, a superposition of both the Gilbert damping and the two-magnon contribution turned\nout to be in agreement very well with experimental data illustrating the dependence of the\nlinewidth on the frequency [16, 20–23]. Based on these findings it was put into question\nwhether the Landau-Lifshitz-Gilbert equation is an appropriate description for ferromag-\nnetic thin films. The pure Gilbert damping is not able to explain the nonlinear frequency\ndependence of the linewidth when two-magnon scattering processes are operative [3, 24].\nAssuming that damping mechanisms can also lead to a non-conserved spin length a way\nout might be the inclusion of the Bloch equations [25, 26] or the the Landau-Lifshitz-Bloch\nequation [27, 28] into the concept of ferromagnetic resonance.\nAnother aspect is the recent observation [29] that a periodic scattering potential can alter\nthe frequency dependence of the linewidth. The experimental results are not in agreement\nwith those based upon a combination of Gilbert damping and two-magnon scattering. It\nwas found that the linewidth as function of the frequency exhibits a non monotonous be-\n2havior. The authors [29] suggest to reconsider the approach with regard to spin relaxations.\nMoreover, it would be an advantage to derive an expression for the linewidth as a measure\nfor spin damping solely from the equation of motion for the magnetization.\nTaking all those arguments into account it is the aim of this paper to propose a gener-\nalized equation of motion for the magnetization dynamics including both Gilbert damping\nand Bloch terms. The dynamical model allows immediately to get the magnetic susceptibil-\nity as well as the ferromagnetic resonance linewidth which are appropriate for the analysis\nof experimental observations. A further generalization is the implementation of nonlocal\neffects in both space and time. This is achieved by introducing a retardation kernel which\ntakes into account temporal retardation within a characteristic time \u001cand a spatial one\nwith a characteristic scale \u0018. The last one simulates an additional mutual interaction of\nthe magnetic moments in different areas of the film within the retardation length \u0018. Re-\ncently such nonlocal effects were discussed in a complete different context [30]. Notice that\nretardation effects were already investigated for simpler models by means of the Landau-\nLifshitz-Gilbert equation. Here the existence of spin wave solutions were in the focus of the\nconsideration [31]. The expressions obtained for the frequency/damping parameters were\nconverted into linewidths according to the Gilbert contribution which is a linear function\nof the frequency [31, 32]. In the present approach we follow another line. The propagating\npart of the varying magnetization is supplemented by the two damping terms due to Gilbert\nand Bloch, compare Eq. (9). Based on this equation we derive analytical expressions for the\nmagnetic susceptibility, the resonance condition and the ferromagnetic resonance linewidth.\nDue to the superposition of damping and retardation effects the linewidth exhibits a non-\nlinear behavior as function of the frequency. The model is also extended by considering\nthe general case of arbitrary angles between the static external field and the film surface.\nMoreover the model includes several energy contributions as Zeeman and exchange energy\nas well as anisotropy and dipolar interaction. The consequences for ferromagnetic resonance\nexperiments are discussed.\nII. DERIVATION OF THE EQUATION OF MOTION\nIn order to define the geometry considered in the following we adopt the idea presented\nin [10], i.e. we employ two coordinate systems, the xyz-system referring to the film surface\n3ΘMey\nex,eX\nezMS\neZeY\nΘHH0\n/Bullet\n/Bullet\n/BulletξM(z1)\nM(z2)\nM(z3)hrf\nd\nlxlzFIG. 1. (Color online) The geometry referring to the film and the magnetization. Further descrip-\ntion in the text.\nand the XYZ-system which is canted by an angle \u0002Mwith respect to the film plane. The\nsituation for a film of thickness dis sketched in Fig. 1. The angle \u0002Mdescribing the direction\nof the saturation magnetization, aligned with the Z-axis, originates from the static external\nfieldH0which impinges upon the film surface under an angle \u0002H. Therefore, it is more\nconvenient to use the XYZ-system for the magnetization dynamics. As excitation source\nwe consider the radio-frequency (rf) magnetic field hrfpointing into the x= X-direction. It\nshould fulfill the condition hrf\u001cH0. To get the evolution equation of the magnetization\nM(r;t),r= (x;y;z )we have to define the energy of the system. This issue is well described\nin Ref. [10], so we just quote the most important results given there and refer to the cited\nliterature for details. Since we consider the thin film limit one can perform the average along\nthe direction perpendicular to the film, i.e.\nM(rk;t) =1\ndZd=2\n\u0000d=2dyM(r;t); (1)\nwhere rk= (x;0;z)lies in the film plane. In other words the spatial variation of the\nmagnetization across the film thickness dis neglected. The components of the magnetization\npoint into the directions of the XYZ-system and can be written as [33]\nM(rk;t) =MX(rk)eX+MY(rk)eY+\u0012\nMS\u0000M2\nX(rk) +M2\nY(rk)\n2MS\u0013\neZ:(2)\n4Typically the transverse components MX;Yare assumed to be much smaller than the satu-\nration magnetization MS. Remark that terms quadratic in MX;Yin the energy will lead to\nlinear terms in the equation of motion. The total energy of the system can now be expressed\nin terms of the averaged magnetization from Eq. (1) and reads\nH=Hz+Hex+Ha+Hd: (3)\nThe different contributions are the Zeeman energy\nHz=\u0000Z\nd3rH0sin (\u0002 H\u0000\u0002M)MY(rk)\n\u0000Z\nd3rH0cos (\u0002 H\u0000\u0002M)\u0012\nMS\u0000MX(rk)2+MY(rk)2\n2MS\u0013\n;(4)\nthe exchange energy\nHex=D\n2MSZ\nd3r\u0002\nrMX(rk)\u00032+\u0002\nrMY(rk)\u00032; (5)\nthe surface anisotropy energy\nHa=HSMSV\n2sin2(\u0002M) +HS\n2sin(2\u0002 M)Z\nd3rM Y(rk)\n+HS\n2MScos(2\u0002 M)Z\nd3rM Y(rk)2\u0000sin2(\u0002M)Z\nd3rM X(rk)2;(6)\nand the dipolar energy\nHd=2\u0019M2\nSVsin2(\u0002M) +\u0019Z\nd3r\u001a\n2MSsin(2\u0002 M)MY(rk)\n+\u0012dk2\nz\nkksin2(\u0002M)\u0000(dkk\u00002) cos2(\u0002M)\u00002 sin2(\u0002M)\u0013\nMY(rk)2\n+\u0012dk2\nx\nkk\u00002 sin2(\u0002M)\u0013\nMX(rk)2\u00002dkxkz\nkksin(\u0002 M)MX(rk)MY(rk)\u001b\n:(7)\nIn these expressions V=lxlzdis the volume of the film, Ddesignates the exchange stiffness\nandHS/d\u00001represents the uniaxial out-of-plane anisotropy field. If HS<0the easy axis\nis perpendicular to the film surface. The in-plane anisotropy contribution to the energy is\nneglected but it should be appropriate for polycrystalline samples [16]. Moreover kk=jkkj\nis introduced where kk=kxex+kzezis the wave vector of the spin waves parallel to the\nfilm surface. Eqs. (3)-(7) are valid in the thin film limit kkd\u001c1. In order to derive Hdin\nEq. (7) one defines a scalar magnetic potential and has to solve the corresponding boundary\n5value problem inside and outside of the film [34]. As result [10] one gets the expressions in\nEq. (7).\nIn general if the static magnetic field is applied under an arbitrary angle \u0002Hthe mag-\nnetization does not align in parallel, i.e. \u0002M6= \u0002 H. The angle \u0002Mcan be derived from\nthe equilibrium energy Heq=H(MX= 0;MY= 0). Defining the equilibrium free energy\ndensity asfeq(\u0002M) =Heq=Vaccording to Eqs. (3)-(7) one finds the well-known condition\nsin(\u0002 H\u0000\u0002M) =4\u0019M S+HS\n2H0sin(2 \u0002 M) (8)\nby minimizing feqwith respect to \u0002M. We further note that all terms linear in MYin\nEqs. (3)-(7) cancel mutually by applying Eq. (8) as already pointed out in Ref. [10].\nThe energy contributions in Eqs. (3) and the geometric aspects determine the dynamical\nequation for the magnetization. The following generalized form is proposed\n@\n@tM(rk;t) =ZZ\ndr0\nkdt0\u0000(rk\u0000r0\nk;t\u0000t0)(\n\r\u0002\nHeff(r0\nk;t0)\u0002M(r0\nk;t0)\u0003\n+\u000b\u0014\nM(r0\nk;t0)\u0002@\n@t0M(r0\nk;t0)\u0015\n\u00001\nT2M?(r0\nk;t0))\n;(9)\nwhere\r=g\u0016B=~is the absolute value of the gyromagnetic ratio, T2is the transverse\nrelaxation time of the components M?=MXeX+MYeYand\u000bdenotes the dimensionless\nGilbertdampingparameter. Thelatterisoftentransformedinto G=\u000b\rM Srepresentingthe\ncorresponding damping constant in unit s\u00001. The effective magnetic field Heffis related to\nthe energy in Eqs. (3)-(7) by means of variational principles [35], i.e. Heff=\u0000\u000eH=\u000eM+hrf.\nHere the external rf-field hrf(t)is added which drives the system out of equilibrium.\nRegarding the equation of motion presented in Eq. (9) we note that a similar type was\napplied in [12] for the evaluation of ferromagnetic resonance experiments. In this paper\nthe authors made use of a superposition of the Landau-Lifshitz equation and Bloch-like\nrelaxation. Here we have chosen the part which conserves the spin length in the Gilbert form\nand added the non-conserving Bloch term in the same manner. That the combination of\nthesetwodistinctdampingmechanismsissuitablefortheinvestigationofultrathinmagnetic\nfilms was also suggested in [24]. Since the projection of the magnetization onto the Z-axis is\nnot affected by T2this relaxation time characterizes the transfer of energy into the transverse\ncomponents of the magnetization. This damping type is supposed to account for spin-spin\nrelaxation processes such as magnon-magnon scattering [33, 36]. In our ansatz we introduce\n6another possible source of damping by means of the feedback kernel \u0000(rk\u0000r0\nk;t\u0000t0). The\nintroduction of this quantity reflects the assumption that the magnetization M(rk;t2)is\nnot independent of its previous value M(rk;t1)providedt2\u0000t1< \u001c. Here\u001cis a time\nscale where the temporal memory is relevant. In the same manner the spatial feedback\ncontrols the magnetization dynamics significantly on a characteristic length scale \u0018, called\nretardation length. Physically, it seems to be reasonable that the retardation length differs\nnoticeably from zero only in z-direction which is shown in Fig. 1. As illustrated in the figure\nM(x;z1;t)is affected by M(x;z2;t)while M(x;z3;t)is thought to have negligible influence\nonM(x;z1;t)sincejz3\u0000z1j>\u0018. Therefore we choose the following combination of a local\nand a nonlocal part as feedback kernel\n\u0000(rk\u0000r0\nk;t\u0000t0) =\u0000 0\u000e(rk\u0000r0\nk)\u000e(t\u0000t0)\n+\u00000\n4\u0018\u001c\u000e(x\u0000x0) exp\u0014\u0000jz\u0000z0j\n\u0018\u0015\nexp\u0014\u0000(t\u0000t0)\n\u001c\u0015\n; t>t0:(10)\nThe intensity of the spatiotemporal feedback is controlled by the dimensionless retardation\nstrength \u00000. The explicit form in Eq. (10) is chosen in such a manner that the Fourier-\ntransform \u0000(kk;!)!\u00000for\u0018!0and\u001c!0, and in case \u00000= 1the ordinary equation\nof motion for the magnetization is recovered. Further,R\ndrkdt\u0000(rk;t) = \u0000 0<1, i.e. the\nintegral remains finite.\nIII. SUSCEPTIBILITY AND FMR-LINEWIDTH\nIf the rf-driving field, likewise averaged over the film thickness, is applied in X-direction,\ni.e.hrf(rk;t) =hX(rk;t)eX, the Fourier transform of Eq. (9) is written as\n\u0014i!\n\r\u0000(kk;!)+1\n\rT2+H21(kk)\u0015\nMX(kk;!) =\u0000\u0014\nH1(kk) +i\u000b!\n\r\u0015\nMY(kk;!);\n\u0014i!\n\r\u0000(kk;!)+1\n\rT2+H12(kk)\u0015\nMY(kk;!) =\u0014\nH2(kk) +i\u000b!\n\r\u0015\nMX(kk;!)\u0000MShX(kk;!):\n(11)\n7The effective magnetic fields are expressed by\nH1(kk) =H0cos(\u0002 H\u0000\u0002M) + (4\u0019M S+HS) cos(2 \u0002 M)\n+ 2\u0019dkkMS \nk2\nz\nk2\nksin2(\u0002M)\u0000cos2(\u0002M)!\n+Dk2\nk\nH2(kk) =H0cos(\u0002 H\u0000\u0002M)\u0000(4\u0019M S+HS) sin2(\u0002M)\n+ 2\u0019dM Sk2\nx\nkk+Dk2\nk;(12)\nand\nH12(kk) = 2\u0019dM Skxkz\nkksin(\u0002 M) =\u0000H21(kk): (13)\nThe Fourier transform of the kernel yields\n\u0000(kk;!) =\u00000(1 + i!\u001c) + \u0000 1\n2 (1 + i!\u001c)(!2\u001c2\u001c1)'\u00000+ \u0000 1\n2\u0000i\n2\u00001!\u001c;\n\u00001=\u00000\n1 +\f2; \f =\u0018kz;(14)\nwhere the factor 1=2arises from the condition t > t0when performing the Fourier trans-\nformation from time into frequency domain. In Eq. (14) we discarded terms !2\u001c2\u001c1.\nThis condition is fulfilled in experimental realizations. So, it will be turned out later the\nretardation time \u001c\u001810 fs. Because the ferromagnetic resonance frequencies are of the order\n10:::100 GHz one finds!2\u001c2\u001810\u00008:::10\u00006. The retardation parameter \f=\u0018kz, introduced\nin Eq. (14), will be of importance in analyzing the linewidth of the resonance signal. With\nregard to the denominator in \u00001, compare Eq. (14), the parameter \fmay evolve ponderable\ninfluence on the spin wave damping if this quantity cannot be neglected compared to 1.\nAs known from two-magnon scattering the spin wave modes can be degenerated with the\nuniform resonance mode possessing wave vectors kk\u0018105cm\u00001. The retardation length \u0018\nmay be estimated by the size of inhomogeneities or the distance of defects on the film sur-\nface, respectively. Both length scales can be of the order \u001810:::1000 nm, see Refs. [18, 29].\nConsequently the retardation parameter \fcould reach or maybe even exceed the order of 1.\nLet us stress that in case \f= 0,\u001c= 0,\u00000= 1and neglecting the Gilbert damping,\ni.e.\u000b= 0, the spin wave dispersion relation is simply \rp\nH1(kk)H2(kk)\u0000H2\n12(kk). This\nexpression coincides with those ones given in Refs. [7] and [10].\nProceeding the analysis of Eq. (11) by defining the magnetic susceptibility \u001fas\nM\u000b(kk;!) =X\n\f\u001f\u000b\f(kk;!)h\f(kk;!);f\u000b;\fg=fX;Yg;(15)\n8whereh\fplays the role of a small perturbation and the susceptibility \u001f\u000b\fexhibits the\nresponse of the system. Eq. (15) reflects that there appears no dependence on the direction\nofkk.\nSince the rf-driving field is applied along the eX-direction it is sufficient to focus the\nfollowing discussion to the element \u001fXXof the susceptibility tensor. From Eq. (11) we\nconclude\n\u001fXX(kk;!) =MSh\nH1(kk;!) +i\u000b!\n\ri\nh\nH1(kk;!) +i\u000b!\n\rih\nH2(kk;!) +i\u000b!\n\ri\n+h\ni!\n\r\u0000(kk;!)+1\n\rT2i2:(16)\nBecause at ferromagnetic resonance a uniform mode is excited let us set kk= 0in Eqs. (12)-\n(13). Considering the resonance condition we can assume \f=\u0018kz= 0. For reasons men-\ntioned above we have to take \f=\u0018kz6= 0when the linewidth as a measure for spin damping\nis investigated. Physically we suppose that spin waves with non zero waves vectors are not\nexcited at the moment of the ferromagnetic resonance. However such excitations will evolve\nduring the relaxation process. In finding the resonance condition from Eq. (16) it seems to\nbe a reasonable approximation to disregard terms including the retardation time \u001c. Such\nterms give rise to higher order corrections. In the same manner all the contributions orig-\ninated from the damping, characterized by \u000bandT2, are negligible. Let us justify those\napproximation by quantitative estimations. The fields H1,H2and!=\rare supposed to\nrange in a comparable order of magnitude. On the other hand one finds \u000b\u001810\u00003:::10\u00002,\n!T2\u001810\u00002and!\u001c\u001810\u00004. Under these approximations the resonance condition reads\n\u0012!r\n\r\u00132\n= \u00002\n0H1(kk= 0)H2(kk= 0): (17)\nThisresultiswellknownforthecasewithoutretardationwith \u00000= 1. Althoughtheretarda-\ntion time\u001cand the retardation length \u0018are not incorporated in the resonance condition, the\nstrength of the feedback may be important as visible in Eq. (17). Now the consequences for\nthe experimental realization will be discussed. To address this issue the resonance condition\nEq. (17) is rewritten in terms of the resonance field Hr=H0(!=!r)leading to\nHr=1\n2 cos(\u0002 H\u0000\u0002M)8\n<\n:s\n(4\u0019M S+HS)2cos4(\u0002M) +\u00121\n\u000002!r\n\r\u00132\n\u0000(4\u0019M S+HS)(1\u00003 sin2(\u0002M))9\n=\n;:(18)\n9ΘM[deg]\nΘH[deg]Γ0= 0 .7\nΓ0= 1 .0\nΓ0= 1 .3FIG. 2. (Color online) Dependence of the magnetization angle \u0002Mon the angle \u0002Hunder which the\nstatic external field is applied for !r=(2\u0019) = 10 GHz . The parameters are taken from [16]: 4\u0019MS=\n16980 G,HS=\u00003400 G;\r= 0:019 GHz=G.\nThe result is arranged in the in the same manner as done in [16]. The difference is the\noccurrence of the parameter \u00000in the denominator. In [16] the gyromagnetic ratio \rand\nthe sum (4\u0019M S+HS)were obtained from \u0002H-dependent measurements and a fit of the\ndata according to Eq. (18) with \u00000= 1under the inclusion of Eq. (8). If the saturation\nmagnetization can be obtained from other experiments [16] the uniaxial anisotropy field HS\nresults. Thus, assuming \u000006= 1the angular dependence \u0002M(\u0002H)and the fitting parameters\nas well would change. In Fig. 2 we illustrate the angle \u0002M(\u0002H)for different values of \u00000and\na fixed resonance frequency. If \u00000<1the curve is shifted to larger \u0002Mand for \u00000>1to\nsmaller magnetization angles. To produce Fig. 2 we utilized quantitative results presented\nin [16]. They found for Co films grown on GaAs the parameters 4\u0019M S= 16980 G ,HS=\n\u00003400 Gand\r= 0:019 GHz=G. As next example we consider the influence of HSand denote\nH(0)\nS=\u00003400 Gthe anisotropy field for \u00000= 1andH(R)\nSthe anisotropy field for \u000006= 1. The\nabsolute value of their ratio jH(R)\nS=H(0)\nSj, derived from Hr(H(0)\nS;\u00000= 1) =Hr(H(R)\nS;\u000006= 1),\nisdepictedinFig.3forvariousfrequencies. Inthisgraphweassumedthatallotherquantities\nremain fixed. The effect of a varying retardation strength on the anisotropy field can clearly\nbeseen. Thechangeinthesignoftheslopeindicatesthattheanisotropyfield H(R)\nSmayeven\nchange its sign. From here we conclude that the directions of the easy axis and hard axis\nare interchanged. For the frequencies 4 GHzand10 GHzthis result is not observed in the\nrange chosen for \u00000. Moreover, the effects become more pronounced for higher frequencies.\n10/vextendsingle/vextendsingle/vextendsingleH(R)\nS/H(0)\nS/vextendsingle/vextendsingle/vextendsingle\nΓ04 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHzFIG. 3. (Color online) Effect of varying retardation strength on the uniaxial anisotropy field for\nvarious frequencies and \u0002M=\u0019=3.4\u0019MS= 16980 G ,HS=\u00003400 G;\r= 0:019 GHz=G, see [16].\nIn Fig. 3 we consider only a possible alteration of the anisotropy field. Other parameters like\nthe experimentally obtained gyromagnetic ration were unaffected. In general this parameter\nmay also experiences a quantitative change simultaneously with HS.\nLet us proceed by analyzing the susceptibility obtained in Eq. (16). Because the following\ndiscussion is referred to the energy absorption in the film, we investigate the imaginary part\nofthesusceptibility \u001f00\nXX. SinceexperimentallyoftenaLorentziancurvedescribessufficiently\nthe resonance signal we intend to arrange \u001f00\nXXin the form A0=(1 +u2), whereA0is the\nabsolute value of the amplitude and uis a small parameter around zero. The mapping to a\nLorentzian is possible under some assumptions. Because the discussion is concentrated on\nthe vicinity of the resonance we introduce \u000eH=H0\u0000Hr, whereHris the static external\nfield when resonance occurs. Consequently, the fields in Eq. (12) have to be replaced by\nH1;2!H(r)\n1;2+\u000eHcos(\u0002 H\u0000\u0002M). Additionally, we take into account only terms of the order\np\n\u000f\u0015in the final result for the linewidth where f\u000f;\u0015g/f!=\r[\u000b+!\u001c] + 1=(\rT2)g. After a\nlengthy but straightforward calculation we get for \u000eH=H(r)\n1;2\u001c1and using the resonance\ncondition in Eq. (17)\n\u001f00\nXX(!) =A0\n1 +h\nH0\u0000Hr\n\u0001Ti2; A0=MS\n(1 +\u0014) cos(\u0002 H\u0000\u0002M) \u0001T; \u0014=H(r)\n2\nH(r)\n1:(19)\nHere we have introduced the total half-width at half-maximum (HWHM) \u0001Twhich can be\n11brought in the form\n\u0001T=1\ncos(\u0002 H\u0000\u0002M)q\n\u00012\nG+ \u00012\nB+ \u00012\nGB+ \u00012\nR: (20)\nThe HWHM is a superposition of the Gilbert contribution \u0001G, the Bloch contribution \u0001B,\na joint contribution \u0001GBarising from the combination of the Gilbert and Bloch damping\nparts in the equation of motion and the contribution \u0001Rwhich has its origin purely in the\nfeedback mechanisms introduced into the system. The explicit expressions are\n\u0001G=!\n\rs\n\u000b\u0014\n\u000b\u000016p\u0014\n(1 +\u0014)\u00000\u00001!\u001c\n(\u00000+ \u0000 1)3\u0015\n; (21a)\n\u0001B=4 \u00000\n(\u00000+ \u0000 1)p\u0014\n(1 +\u0014)s\n1\n(\rT2)2\u00004 \u00001\n(\u00000+ \u0000 1)2!\n\r!\u001c\n\rT2; (21b)\n\u0001GB=s\n8\u00000\n(\u00000+ \u0000 1)p\u0014\n(1 +\u0014)\u000b!\n\r2T2; (21c)\n\u0001R=8p\u0014\n(1 +\u0014)!\n\r\u00000\u00001!\u001c\n(\u00000+ \u0000 1)3: (21d)\nThe parameter \u00001is defined in Eq. (14). If the expressions under the roots in Eqs. (21a)\nand (21b) are negative we assume that the corresponding process is deactivated and does\nnot contribute to the linewidth \u0001HT. Typically, experiments are evaluated in terms of the\npeak-to-peak linewidth of the derivative d\u001f00\nXX=dH0, denoted as \u0001H\u0011. One gets\n\u0001H\u0011=2p\n3\u0001\u0011; (22)\nwhere the index \u0011stands for G(Gilbert contribution), B(Bloch contribution), GB(joint\nGilbert-Bloch contribution), R(pure retardation contribution) or Tdesignating the total\nlinewidth according to Eq. (20) and Eqs. (21a)-(21d). Obviously these equations reveal a\nstrong nonlinear frequency dependence, which will be discussed in the subsequent section.\nIV. DISCUSSION\nAs indicated in Eqs. (20) - (22) the quantity \u0001H\u0011consists of well separated distinct\ncontributions. Thebehaviorof \u0001H\u0011isshowninFigs.4-6asfunctionofthethreeretardation\nparameters, the strength \u00000, the spatial range \fand the time scale \u001c. In all figures the\nfrequencyf=!=(2\u0019)is used. In Fig. 4 the dependence on the retardation strength \u00000is\n12∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nΓ0∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 4. (Color online) Influence of the retardation strength \u00000on the peak-to-peak linewidth \u0001HT\nfor various frequencies (top graph) and on the single contributions \u0001H\u0011forf= 70 GHz (bottom\ngraph). \u0001B= 0is this frequency region. The parameters are: \u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,\nT2= 5\u000210\u00008s;\u001c= 1:7\u000210\u000014s. The other parameters are 4\u0019MS= 16980 G ,HS=\u00003400 G;\r=\n0:019 GHz=G, compare [16].\nshown. As already observed in Figs. 2 and 3 a small change of \u00000may lead to remarkable\neffects. Hence we vary this parameter in a moderate range 0:5\u0014\u00000\u00142. The peak-to-peak\nlinewidth \u0001HTas function of \u00000remains nearly constant for f= 4 GHz andf= 10 GHz ,\nwhereas for f= 35 GHz a monotonous growth-up is observed. Increasing the frequency\nfurther tof= 50 GHz and70 GHzthe curves offers a pronounced kink. The subsequent\nenhancement is mainly due to the Gilbert damping. In the region of negative slope we\nset\u0001HG(\u00000) = 0, while in that one with a positive slope \u0001HG(\u00000)>0grows and tends\nto2\u000b!=(p\n3\r)for\u00000!1. The other significant contribution \u0001HR, arising from the\nretardation decay, offers likewise a monotonous increase for growing values of the retardation\nparameter \u00000. This behavior is depicted in Fig. 4 for f= 70 GHz . Now let us analyze the\ndependence on the dimensionless retardation length \f=\u0018kz. Because\fis only nonzero if\n13∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nβ∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 5. (Color online) Influence of the dimensionless retardation length \f=\u0018kzon the total\npeak-to-peak linewidth \u0001HTfor various frequencies (top graph) and on the single contributions\n\u0001H\u0011forf= 70 GHz (bottom graph); \u0001B= 0in this range. The parameters are: \u0002H= \u0002 M= 0,\n\u00000= 1:1,\u000b= 0:01,T2= 5\u000210\u00008s;\u001c= 1:7\u000210\u000014s. The other parameters: 4\u0019MS= 16980 G ,\nHS=\u00003400 Gand\r= 0:019 GHz=Gare taken from [16].\nkz6= 0this parameter \u0018accounts the influence of excitations with nonzero wave vector. We\nargue that both nonzero wave vector excitations, those arising from two-magnon scattering\nand those originated from feedback mechanisms, may coincide. Based on the estimation\nin the previous section we consider the relevant interval 10\u00002\u0014\f\u001410. The results are\nshown in Fig.5. Within the range of \fone recognizes that the total peak-to-peak linewidths\n\u0001HTforf= 4 GHz andf= 10 GHz offer no alteration when \fis changed. The plotted\nlinewidths are characterized by a minimum followed by an increase which occurs when \f\nexceeds approximately 1. This behavior is the more accentuated the larger the frequencies\nare. The shape of the curve can be explained by considering the single contributions as\nis visible in the lower part in Fig. 5. While both quantities \u0001HG(\f)and\u0001HR(\f)remain\nconstant for small \f,\u0001HG(\f)tends to a minimum and increases after that. The quantity\n14∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nτ[fs]∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 6. (Color online) Influence of the retardation time \u001con the total peak-to-peak linewidth\n\u0001HTfor various frequencies (top graph) and on the single contributions \u0001H\u0011forf= 70 GHz\n(bottom graph). \u0001B= 0in this region. The parameters are \u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,\nT2= 5\u000210\u00008s;\u00000= 1:1; the other parameters are taken from [16]: 4\u0019MS= 16980 G ,HS=\n\u00003400 G;\r= 0:019 GHz=G.\n\u0001HR(\f)develops a maximum around \f\u00191. Thus, both contributions show nearly opposite\nbehavior. The impact of the characteristic feedback time \u001con the linewidth is illustrated\nin Fig. 6. In this figure a linear time scale is appropriate since there are no significant\neffects in the range 1 fs\u0015\u001c\u00150. The total linewidth \u0001HT(\u001c)is again nearly constant\nforf= 4 GHz andf= 10 GHz . In contrast \u0001HT(\u001c)reveals for higher frequencies two\nregions with differing behavior. The total linewidth decreases until \u0001HG(\u001c)becomes zero.\nAfter that one observes a positive linear slope which is due to the retardation part \u0001HR(\u001c).\nThis linear dependency is recognizable in Eq. (21d), too. Below we will present arguments\nwhy the feedback time \u001cis supposed to be in the interval 0< \u001c < 100 fs. Before let us\nstudy the frequency dependence of the linewidth in more detail. The general shape of the\ntotal linewidth \u0001HT(!)is depicted in Fig. 7. Here both the single contribution to the\n15∆Hη[G]\nf[GHz]∆HG\n∆HB\n∆HGB\n∆HR\n∆HTFIG. 7. (Color online) Frequency dependence of all contributions to the peak-to-peak linewidth for\n\u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,T2= 5\u000210\u00008s,\u001c= 1:7\u000210\u000014sand\u00000= 1:2. Parameters taken\nfrom Ref. [16]: 4\u0019MS= 16980 G ,HS=\u00003400 Gand\r= 0:019 GHz=G. The Bloch contribution\n\u0001HBis shown in the inset.\nlinewidth and the total linewidth are shown. Notice that the total linewidth is not simply\nthe sum of the individual contributions but has to be calculated according to Eq. (20). One\nrealizes that the Bloch contribution \u0001HBis only nonzero for frequencies f\u00146 GHzin the\nexamples shown. Accordingly \u0001HB= 0in Figs. 4-6 (lower parts) since these plots refer to\nf= 70 GHz . The behavior of the Gilbert contribution deviates strongly from the typically\napplied linear frequency dependence. Moreover, the Gilbert contribution will develop a\nmaximum value and eventually it disappears at a certain frequency where the discriminant\nin Eq. (21a) becomes negative. Nevertheless, the total linewidth is a nearly monotonous\nincreasing function of the frequency albeit, as mentioned before, for some combinations of\nthe model parameters there might exist a very small frequency region where \u0001HGreaches\nzero and the slope of \u0001HTbecomes slightly negative. The loss due to the declining Gilbert\npart is nearly compensated or overcompensated by the additional line broadening originated\nbytheretardationpartandthecombinedGilbert-Blochterm. Thelatteroneis \u0001HGB/pf\nand\u0001HR/f2, see Eqs. (21c)-(21d). In the frequency region where \u0001HG= 0only \u0001HGB\nand\u0001HRcontribute to the total linewidth, the shape of the linewidth is mainly dominated\nby\u0001HR. Thispredictionisanewresult. Thebehavior \u0001HR/f2, obtainedinourmodelfor\nhigh frequencies, is in contrast to conventional ferromagnetic resonance including only the\nsum of a Gilbert part linear in frequency and a two-magnon contribution which is saturated\n16at high frequencies. So far, experimentally the frequency ranges from 1 GHzto225 GHz,\nsee [21]. Let us point out that the results presented in Fig. 7 can be adjusted in such a\nmanner that the Gilbert contribution will be inoperative at much higher frequencies by the\nappropriate choice of the model parameters. Due to this fact we suggest an experimental\nverification in more extended frequency ranges. Another aspect is the observation that\nexcitations with a nonzero wave vector might represent one possible retardation mechanism.\nRegarding Eqs. (21a)-(21d) retardation can also influence the linewidth in case kz= 0\n(i.e.\f= 0and\u00001= \u0000 0). Only if\u001c= 0the retardation effects disappear. Therefore let us\nconsider the time domain of retardation and its relation to the Gilbert damping. The Gilbert\ndamping and the attenuation due to retardation can be considered as competing processes.\nSo temporal feedback can cause that the Gilbert contribution disappears. In the same\nsense the Bloch contribution is a further competing damping effect. In this regard temporal\nfeedback has the ability to reverse the dephasing process of spin waves based on Gilbert and\nBloch damping. On the other hand the retardation part \u0001Rin Eq. (21d) is always positive\nfor\u001c > 0. Thus, the retardation itself leads to linewidth broadening in ferromagnetic\nresonance and consequently to spin damping. Whether the magnitude of retardation is able\nto exceed the Gilbert damping depends strongly on the frequency. With other words, the\nfrequency of the magnetic excitation ’decides��� to which damping mechanisms the excitation\nenergyistransferred. Ourcalculationsuggeststhatforsufficienthighfrequenciesretardation\neffects dominate the intrinsic damping behavior. Thus the orientation and the value of the\nmagnetization within the retardation time \u001cplays a major role for the total damping.\nGenerally, experimental data should be fit according to the frequency dependence of the\nlinewidth in terms of Eqs. (20)-(22). To underline this statement we present Fig. 8. In this\ngraph we reproduce some results presented in [7] for the case \u0002H= \u0002 M= 0. To be more\nspecific, we have used Eq. (94) in [7] which accounts for the two-magnon scattering and\nthe parameters given there. As result we find a copy of Fig. 4 in [7] except of the factor\n2=p\n3. Further, we have summed up the conventional Gilbert linewidth /fwith the Gilbert\ndamping parameter \u000b1= 0:003. This superposition yields to the dotted line in Fig. 8. The\nresult is compared with the total linewidth resulting from our retardation model plotted as\nsolid line. To obtain the depicted shape we set the Gilbert damping parameter according\nto the retardation model \u000b2= 0:0075, i.e. to get a similar behavior in the same order of\nmagnitude of \u0001HTwithin both approaches we have to assume that \u000b2is more than twice\n17∆HT[G]\nf[GHz]retardation model\nGilbert+2-magnonFIG. 8. (Color online) Comparison with the two-magnon model. Frequency dependence of the total\npeak-to-peak linewidth \u0001HTfor\u0002H= \u0002 M= 0,\f= 0:5,\u000b1= 0:003,\u000b2= 0:0075,T2= 5\u000210\u00008s,\n\u001c= 1:22\u000210\u000014sand\u00000= 1:2. Parameters taken from [7]: 4\u0019MS= 21000 G ,HS=\u000015000 Gand\nfrom [37]:\r= 0:018 GHz=G(derived from g= 2:09for bulk Fe). The dotted line is a superposition\nof Fig. 4 in [7] reflecting the two-magnon contribution and the Gilbert contribution (denoted as\n\u000b1in the text) linear in the frequency.\nas large compared to \u000b1.\nFinally we discuss briefly the \u0002H-dependence of the linewidth which is shown in Fig. 9.\nIn the upper part of the figure one observes that \u0001HT(\u0002H)exhibits a maximum which is\nshifted towards lower field angles as well as less pronounced for increasing frequencies. The\nlower part of Fig. 9, referring to f= 10 GHz , displays that the main contribution to the total\nlinewidth arises from the Gilbert part \u0001HG. This result for f= 10 GHz is in accordance\nwith the results discussed previously, compare Fig. 7. For higher frequencies the retardation\ncontribution \u0001HRmay exceed the Gilbert part.\nV. CONCLUSIONS\nA detailed study of spatiotemporal feedback effects and intrinsic damping terms offers\nthat both mechanisms become relevant in ferromagnetic resonance. Due to the superposi-\ntion of both effects it results a nonlinear dependence of the total linewidth on the frequency\nwhich is in accordance with experiments. In getting the results the conventional model in-\ncluding Landau-Lifshitz-Gilbert damping is extended by considering additional spatial and\n18linewidth ∆ HT[G]\n4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHzlinewidth ∆ Hη[G]\nΘH[deg]∆HB\n∆HR\n∆HGB\n∆HG\n∆HTf= 10 GHzFIG. 9. (Color online) Angular dependence of the total peak-to-peak linewidth \u0001HTfor various\nfrequencies (top graph) and all contributions \u0001H\u0011forf= 10 GHz (bottom graph) with \f= 0:5,\n\u000b= 0:01,T2= 5\u000210\u00008s,\u001c= 1:7\u000210\u000014sand\u00000= 1:1. The parameters are taken from\n[16]: 4\u0019MS= 16980 G ,HS=\u00003400 Gand\r= 0:019 GHz=G.\ntemporal retardation and non-conserved Bloch damping terms. Our analytical approach\nenables us to derive explicit expressions for the resonance condition and the peak-to-peak\nlinewidth. We were able to link our results to such ones well-known from the literature.\nThe resonance condition is affected by the feedback strength \u00000. The spin wave damping is\nlikewise influenced by \u00000but moreover by the characteristic memory time \u001cand the retar-\ndation length \u0018. As expected the retardation gives rise to an additional damping process.\nFurthermore, the complete linewidth offers a nonlinear dependence on the frequency which\nis also triggered by the Gilbert damping. From here we conclude that for sufficient high\nfrequencies the linewidth is dominated by retardation effects. Generally, the contribution of\nthedifferentdampingmechanismstothelinewidthiscomprisedofwellseparatedrateswhich\nare presented in Eqs. (20)-(22). Since each contribution to the linewidth is characterized\nby adjustable parameters it would be very useful to verify our predictions experimentally.\n19Notice that the contributions to the linewidth in Eqs. (20)-(22) depend on the shape of\nthe retardation kernel which is therefore reasonable not only for the theoretical approach\nbut for the experimental verification, too. One cannot exclude that other mechanisms as\nmore-magnon scattering effects, nonlinear interactions, spin-lattice coupling etc. are likewise\nrelevant. Otherwise, we hope that our work stimulates further experimental investigations\nin ferromagnetic resonance.\nWe benefit from valuable discussions about the experimental background with Dr. Khali\nZakeri from the Max-Planck-Institute of Microstructure Physics. 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Lett. 49, 658\n(2000)\n22"    },    {        "title": "0805.4068v1.Self_sustained_magnetoelectric_oscillations_in_magnetic_resonant_tunneling_structures.pdf",        "content": "arXiv:0805.4068v1  [cond-mat.mes-hall]  27 May 2008Self-sustained magnetoelectric oscillations in magnetic resonant tunneling structures\nChristian Ertler∗and Jaroslav Fabian\nInstitute for Theoretical Physics, University of Regensbu rg,\nUniversit¨ atsstrasse 31, D-93040 Regensburg, Germany\nThe dynamic interplay of transport, electrostatic, and mag netic effects in the resonant tunneling\nthrough ferromagnetic quantum wells is theoretically inve stigated. It is shown that the carrier-\nmediatedmagnetic orderintheferromagnetic region notonl yinduces, butalsotakespartinintrinsic,\nrobust, and sustainable high-frequency current oscillati ons over a large window of nominally steady\nbias voltages. This phenomenon could spawn a new class of qua ntum electronic devices based on\nferromagnetic semiconductors.\nFerromagnetism of diluted magnetic semiconductors\n(DMSs), such as GaMnAs [1], depends strongly on the\ncarrier density [2, 3, 4, 5, 6]. The possibility to tailor\nspace charges in semiconductors by bias or gate fields\nnaturally suggests similar tailoring of magnetic proper-\nties of DMSs. While early experiments have indeed suc-\nceededingeneratingferromagnetisminDMSselectrically\nor optically [7, 8], ramifications of the strong carrier-\nmediated ferromagnetism in the transport through DMS\nheterostructures are largely unexplored.\nIn resonanttunneling througha quantum well not only\nthe tunneling current, but also the carrier density in the\nwell are sensitive to the alignment of the electronic spec-\ntra in the leads and in the well. If the quantum well\nis a paramagnetic DMS, the resulting transport is influ-\nenced by the spin splitting of the carrier bands in the\nwell, as observed experimentally [9]. The magnetic res-\nonant diodes are prominent spintronic devices [10, 11],\nproposed for spin valves and spin filtering [12, 13], or for\ndigital magnetoresistance [14, 15].\nIf the quantum well is made of a ferromagnetic DMS\n[16, 17], resonant tunneling conditions should influence\nmagnetic ordering as well. It has already been pre-\ndicted that the critical temperature Tcof the well can\nbe strongly modified electrically [6, 18, 19, 20]. Here we\nshow that the magnetic ordering affects back the tun-\nneling current, in a peculiar feedback process, leading to\ninteresting dynamic transport phenomena.\nConventional nonmagnetic RTDs can exhibit subtle\nintrinsic bistability and terahertz current oscillations\n[21, 22, 23, 24, 25] resulting from the nonlinear feedback\nofthestoredchargeinthequantumwell. Interestingphe-\nnomena occur also in multiple quantum wells and super-\nlattices, in which electric field domains form whose dy-\nnamicsleadstocurrentoscillationsinthe kHz-GHzrange\n[26]. This effect has been exploited for spin-dependent\ntransport by incorporating paramagnetic quantum wells\n[27, 28].\nIn this article we introduce a realistic model of a\nself-consistently coupled transport, charge, andmagnetic\n∗email: christian.ertler@physik.uni-regensburg.de\ncollector emitter well\nEcut\ndedc\nw\nRewell\nCe CcUwVappl(a) \n(b) µeµc\nRc\nFIG. 1: (Color online) (a) Schematic scheme of the band pro-\nfile of the magnetic double barrier structure. The exchange\ninteraction of the magnetic ions is mediated by the carriers\ntunneling in and out of the well. Here, the cut off of the emit-\nter tunneling rate is realized by a cascaded left barrier. (b )\nEquivalent circuit model of the resonant tunneling structu re\nintroducing the emitter and collector capacitances Ce,Ccand\nresistances Re,Rc, respectively.\ndynamics and apply it to generic asymmetric resonant\ndiodes with a ferromagneticquantum well to predict self-\nsustained, stable high-frequency oscillations of the elec-\ntric current and quantum well magnetization. We for-\nmulate a qualitative explanation for the appearance of\nthese magnetoelectric oscillations. In essence, ferromag-\nnetic quantum wells exhibit strong nonlinear feedback\nto the electric transport since the ferromagnetic order,\nwhich gives rise to the exchange splitting of the quantum\nwell subbands, is itself mediated by the itinerant carriers.\nThis together with the Coulomb interaction which effec-\ntively modifies the single electron electrostatic potential\nin the well, leads to a strong coupling of the transport,\nelectric, as well as magnetic properties of ferromagnetic\nresonant tunneling structures.\nOur model resonant tunneling structure with a ferro-\nmagnetic well made of a DMS-material, e.g., GaMnAs, is\nsketched in Fig. 1(a). To exhibit magnetoelectric oscilla-\ntions the structure needs a built-in energy cut-off, Ecut,\nof the emitter tunneling rate. Such an energy cut-off\nmight be realized by a cascaded left barrier, as shown in2\nFig. 1(a), by which the tunneling for carriers with ener-\ngiessmallerthan Ecutis exponentially suppressed, due to\nthe increased barrier width. Other possibilities to realize\nthe cutoff are discussed below.\nAs we aim to understand the most robust features of\nthe ferromagnetic resonant tunnel structures, we present\na minimal theoretical model which captures the essen-\ntial physics. The longitudinal transport through the sys-\ntem can be described in terms of sequential tunneling,\nas the high densities of magnetic impurities residing in\nthe ferromagnetic well will likely cause decoherence of\nthe propagating carries. Based on the transfer Hamil-\ntonian formalism a master equation for the semiclassical\nparticle distribution in the well can be derived, as de-\nscribed elsewhere [10, 29]. We assume that there is only\na single resonant well level E0in the energy range of\ninterest, allowing us to write the rate equations for the\nspin-resolved,time dependent quantumwell particleden-\nsitiesnσ(t),(σ=↑,↓=±1/2) as,\ndnσ\ndt= Γe(Eσ)ne,σ+Γc(Eσ)nc,σ−Γ(Eσ)nσ\n−nσ−n0,σ\nτs. (1)\nHere, Γ {e,c}denotes the energy-dependent tunneling rate\nfrom the emitter (e) and the collector (c), Γ = Γ e+ Γc\ndenotes the total tunneling rate, τsstands for the spin\nrelaxationtime in the well, n0,σdenotes the quasiequilib-\nrium particle spin density, and nσ,{e,c}are the densities\nof particles in the emitter and collector reservoir, respec-\ntively, having the resonant longitudinal energy Eσ. The\nspin-split resonant energies are\nEσ=E0+Uw−σ∆, (2)\nwithUwbeing the electrostatic well potential and ∆ de-\nnoting the subband exchange splitting. The total en-\nergy of the particles Etotis then given by the sum of\nthe longitudinal energy Eσand the in-plane kinetic en-\nergy:Etot=Eσ+Tin. The physical meaning of the right\nside of Eq. (1) is as follows: the first two terms are the\ngainterms, describingtunneling fromtheemitterandthe\ncollector into the well; the third term describes all loss\nprocesses due to the tunneling out of the well, and the\nlast term models the spin relaxation in the well. Con-\nsidering the Fermi-Dirac distributions in the emitter and\nthe collector, the particle densities ni,σare,\nni,σ=D0kBTln{1+exp[(µi−Eσ)/kBT]}i=e,c,\n(3)\nwithkBdenoting Boltzmanns’ constant; Tis the lead\ntemperature, µiare the emitter and collector chemical\npotentialswith µc=µe−eVapplwhereVapplistheapplied\nbias, andD0=m/2π/planckover2pi12is the two-dimensional density\nof states per spin for carriers with the effective mass m.\nThetunneling ratesareessentiallygivenbytheoverlapofthe lead and well wave functions according to Bardeen’s\nformula [30]. For high barriers the rate becomes propor-\ntional to the longitudinal momentum pzof the particles\n[29], i.e., Γ e,c∝(Ez)1/2withEzdenoting the longitudi-\nnal energy.\nIn the framework of a mean field model for the carrier\nmediated ferromagnetism in heterostructure systems [2,\n5, 6, 10] the steady state exchange splitting of the well\nsubbands is determined by\n∆0=Jpd/integraldisplay\ndznimp(z)|ψ0(z)|2\n×SBS/bracketleftBigg\nSJpds(n↓−n↑)|ψ0(z)|2\nkBT/bracketrightBigg\n.(4)\nHere,Jpddenotes the coupling strength between the im-\npurity spin and the carrier spin density (in case of GaM-\nnAs p-like holes couple to the d-like impurity electrons),\nzis the longitudinal (growth) direction of the structure,\nnimp(z) is the impurity density profile, ψ0(z) labels the\nwell wave function, BSdenotes the Brillouin function\nof orderS, andSands= 1/2 are the impurity and\nparticle spin, respectively. (In the case of Mn impuri-\nties S = 5/2.) The expression shows that the well spin-\nsplitting depends basically on the particle spin polariza-\ntionξ=n↓−n↑in the well and the overlap between the\nwavefunction andthe impuritybandprofile. Forsimplic-\nity, we consider here a homogenous impurity distribution\nin the well, which makes ξthe determining factor for ∆ 0.\nAfter a sudden change of the well spin polarization\nthe magnetic impurities need some time to respond until\nthe corresponding mean field value ∆ 0is established. In\nthe case of GaMnAs, experimental studies of the mag-\nnetization dynamics revealed typical response times of\nabout 100 ps [31]. We model the magnetization evolu-\ntion within the relaxation time approximation, d∆ /dt=\n−(∆−∆0)/τ∆, withτ∆denoting the well spin-splitting\nrelaxation time.\nFinally, to take into account the nonlinear feedback of\ntheCoulombinteractionofthe wellcharges,weintroduce\nemitter-well and collector-wellcapacitances, C=Ce+Cc\naccording to the equivalent circuit model of a resonant\ntunneling diode, as shown in Fig. 1(b). The capacitances\nCeandCcare determined by the geometrical dimensions\nof the barriers and the well [29]. The electrostatic well\npotential can then be written as\nUw=1\nC/bracketleftbig\ne2(n−nback)−CceVappl/bracketrightbig\n(5)\nwithedenoting the elementary charge, nbackis the pos-\nitive background charge (from magnetic donors) in the\nwell. AlltheequationsarenonlinearlycoupledviaEq.(2)\nfor the resonant well levels Eσ, making a numerical solu-\ntion indispensable.\nIn the numerical simulations we use generic parame-\nters assuming a GaMnAs well: m= 0.5m0,εr= 12.9,3\n0 10 20 300510x 1015\nVoltage (mV)j (a.u.)\n  jmax\njmin\nI\nII(a)\n0 10 20 3005101520\nVoltage (mV)Δ (meV)\n  \nΔmax\nΔmin(b)\nII\nI\n501001502000510x 1015\n  \nTime (t*)j (a.u.)jtot\nj↑\nj↓(c)\n501001502002468101214x 1011\nTime (t*)n (1/cm2)\n  ntot\nn↑\nn↓(d)\nFIG. 2: (Color online) (a) Current-voltage (IV)-character istic\nof the investigated structure at the energy cut-off Ecut= 85\nmeV. In the unsteady region between 11 to 22 mV the current\ndoes not reach a steady state value; instead intrinsic curre nt\noscillations appear, in which both the maximum and mini-\nmum values of the oscillations are indicated by the dashed\nand solid lines, respectively. Two different dynamic modes I\nand II can be identified. (b) Well splitting ∆ versus applied\nvoltage. In the unsteady region the maximum (dashed) and\nminimum (solid) values of the magnetization oscillations a re\nplotted. (c) and (d) Transients of the spin-resolved curren tj\nand the well particle density nat the applied voltage V= 12\nmV and Ecut= 85 meV. The time is measured in units of t∗\n(typically some picoseconds) as explained in the text.\nde= 50˚A,dc= 20˚A,w= 10˚A,µe= 100 meV,\nnimp= 1.5×1020cm−3,Jpd= 0.06 eV nm3,τs= 1t∗,\nτ∆= 10t∗, wherede,dcandware the emitter barrier,\ncollector barrier and quantum well widths, m0denotes\nthe free electron mass, and εris the relative permittiv-\nity of the well. The characteristic time scale t∗is the\ninverse emitter tunneling rate at the emitters’ Fermi en-\nergy, [t∗= 1/Γe(µe)], being of the order of picoseconds.\nForthe well backgroundchargeweconsiderthat in GaM-\nnAs the actual carrier density is only about 10% of the\nnominal Mn doping density [32]: nback= 0.1nimp. Our\ncalculations are performed at T= 4.2 K, which is well\nbelowthecriticaltemperatureofourspecificwell Tc≈10\nK, where we estimated Tcby using the mean-field result\ngiven in Ref. [20].\nFigure 2(a) shows the current-voltage (IV) character-\nistic of the structure. Up to about V = 11 mV the\ntypical peaked IV-curve of a resonant tunneling diode\nis obtained. However, in the voltage range of 11 to 22\nmeV, which we will call hereafter the “unsteady” re-\ngion, the current does not settle down to a steady state\nvalue; instead stable high-frequency oscillations occur,\nas shown in Fig. 2(c) for the applied voltage of V= 12\nmV. The currentis alwaysevaluated at the collectorside:\njc,σ= Γc(Eσ)(nσ−nc,σ). Along with the current, oscil-\nlations of the well magnetization as well as of the spin(a) (b) \n(c) Ecut\nξ > 0 Δ > 0 \nΔ < 0 \nξ < 0 n < 0 \nUw< 0 \n(d) Uw> 0 \nn > 0 \nFIG. 3: (Color online) Explanation for the occurrence of sel f-\nsustained oscillations in the case of mode I. For mode II the\narguments are completely analogous but there the spin down\nlevel is crossing the cut-off energy Ecut. (a) According to\nthe in-tunneling spin-up carriers the spin polarization ξis\nincreased, giving rise to an increasing well splitting ∆. (b )\nWhen the spin up level falls below the cutoff energy the total\nparticle number decreases and, hence, also the electrostat ic\npotential Uwdoes. (c) This pushes the spin-up level even\ndeeper into the cut-off region leading to a fast decrease of th e\nspin polarization and consequently of the well splitting, w hich\nbrings the spin up level back to the emitter’s supply region\n(d), restarting the whole cycle.\ndensities appear, as shown in Fig 2(b) and (d). Those\nmagneto-electric oscillations are the main results of this\npaper.\nThe IV-curve in the unsteady region suggests the ex-\nistence of two qualitatively different dynamic modes. In-\ndeed, comparing the transients in these two voltage re-\ngions reveals that in region I the spin up level is recur-\nringlycrossingthe emitter energycut off Ecut, whereasin\nregion II this is done by the spin down level, as schemat-\nically illustrated in Fig. 3. This insight offers the follow-\ning explanation for the the occurrence of self-sustained\noscillations. Take mode I; the arguments for mode II are\nsimilar. The dropping of the spin up level below the cut-\noff energy (due to the increasing exchange splitting) as\ntwo implications: (i) the supply of the emitter spin up\nelectrons sharply decreases. Hence, the total well par-\nticle density n=n↑+n↓decreases because the spin up\nelectrons residing in the well are tunneling out to the col-\nlector. A decreased particle density leads to a decreased\nelectrostatic potential according to Eq. (5), which effec-\ntively drives the spin up level even deeper into the cut-off\nregion. (ii) Since the spin up electrons are the majority\nspins in the well, a decrease of n↑implies a decreasing\nspin polarization ξin the well. This causes, via Eq. (4), a\nrapid decrease of the subband exchange splitting, bring-\ning the spin up level back to the emitter supply region.\nThe spin up electrons can then tunnel again into the well\nand the whole process starts from the beginning, produc-\ning the calculated cycles.\nThe occurrence of these oscillations needs the concur-\nrentinterplayof boththeelectricandmagneticfeedbacks:4\nfrequency (1/t*)\nVoltage (mV)Ecut (meV)\n  \n10 20 3080859095\n0.10.20.30.4(a)\nII\nI\nEcut (meV)period (t*)\n  \nVoltage (mV)10 20 3080859095\n0200400(b)\nIII\nFIG. 4: (Color online) Contour plots of the frequency (a) and\nthe period (b) of the intrinsic oscillations as a function of\nthe applied voltage and the cut-off energy Ecut. The two is-\nlandsofhigh-frequencyoscillations correspondtothedyn amic\nmodes I and II. They are separated by a crossover region III\nof high-period oscillations, which becomes most evident in\nperiod contour plot (b).\nthe electrostaticfeedbackactslikean“inertia”forthe os-\ncillations, allowing the spin level to get deeper into the\ncut-off region, whereas the magnetic feedback is needed\nto bring the level back into the emitter’s supply region.\nFrom the above discussions it also follows that a steep\ndescent in the tunneling rate at the cut-off energy is nec-\nessary. This is confirmed in our simulations, where we\nassumed an exponential decay of Γ EforEz<Ecut. Ac-\ntually, this is also the reason why no oscillations are oc-\ncurring at the emitter’s conduction band edge ( Ez= 0),\nwhich is an otherwise natural emitter’s cut-off, because\nnear the band edge the tunneling rate is already almost\nvanishing according to Γ E∝(Ez)1/2.\nThe typical timescale for the oscillationsis made up by\ntwo contributions: (i) the evolution of the well splitting\n∆ in the emitter’s supply region, i.e., the time needed\nfor the well splitting ∆ to become large enough that one\nof the spin levels crosses the cut off energy, and (ii) the\ndynamics of ∆ in the cut-off region. The timescale of\ncontribution (i) can be estimated to be proportional to\nτ∆(r−1), whereris the ratio of the well splitting when\nthe cut-off energy is reached to the initial splitting at the\nbeginning of the cycle, whereas the dynamics of contri-\nbution (ii) is mostly governed by a fast rearrangement\nof the spin densities in the well, which happens on the\norder of the tunneling time t∗. This is in line with the\nnumerical result that the oscillation frequency increases\nwith decreasing τ∆. In contrast, a decreasing spin relax-\nation time, which diminishes more and more effectively\nthespinpolarizationinthewelland, hence, thewellsplit-\nting, gives rise to a decrease of the frequency.\nThe fastest magnetoelectric oscillations in our simu-\nlations for the generic parameters used exhibit periods\nof about 2.5×t∗≈5 ps. Important, the oscillation fre-\nquencycanbemodifiedbyvaryingtheappliedvoltage,as\ncan be seen in Fig. 4, which displays contour plots of the\nfrequency and the period versus the applied voltage and\nthe cut-off energy. Two “islands” can be distinguished,\ncorresponding to the two regions of the dynamic modesI and II, respectively (see Fig. 2). Mode II appears at\nhigher voltages, where the collector chemical potential\nis already below the cut-off energy. In this case the de-\ncrease of the well particle density is much stronger when\nthe spin up level crosses the cut-off energy, as illustrated\nin Fig. 3(b), since none of the out-tunneling electrons are\n“Pauli-blocked”by the collectorelectrons. Thisgivesrise\nto a much stronger decrease of the electrostatic potential\nas compared to mode I, pushing boththe spin up and\ndown levels below the cut-off energy. In the following\noscillations this makes the spin down level recurringly\ncrossing the cut-off energy instead of the spin up level as\nis the case for mode I.\nBoth modes are separated by a crossover region III, in\nwhich the oscillations have much longer periods than on\nthe “islands”, as can be seen in Fig. 4(b). The reason for\ntheselow-frequencyoscillationsisthattheinitialspin po-\nlarization and, hence, the well splitting at the beginning\nof each cycle is small, yielding a large ratio r. Therefore,\nin region III it can take 10-100 times longer that the well\nsplitting reaches the cut-off energy as for modes I or II.\nThe possibility of controlling the frequency of the mag-\nnetoelectric oscillations by electrical means is especially\ninteresting from the applications point of view.\nFor the experimental observation of the self-sustained\noscillations the emitter needs to emit electrons in a\nsharply defined energy window. In the model above we\npropose an asymmetric barrier, as in Fig. 1(a). Another\npractical way would be using an auxiliary resonant tun-\nneling structure in the emitter itself, providing a spectral\nfilter allowing only resonant electrons to pass through.\nYet another possibility would be employing a hot elec-\ntron emitter, say using a Schottky barrier.\nWe have shown that the charge and magnetization\ndynamics in ferromagnetic tunneling heterostructures\nare influenced by the highly nonlinear feedback of both\nCoulomb and magnetic couplings on the tunneling trans-\nport. The feedback results in high-frequency self-\nsustained intrinsic oscillations suggesting applications of\nferromagnetic QWs in tunable high-power current oscil-\nlators.\nThis work has been supported by the Deutsche\nForschungsgemeinschaft SFB 689.\n[1] H. Ohno, Science281, 951 (1998).\n[2] T. Dietl, A. Haury, and Y. M. d’Aubign´ e, Phys. Rev. B\n55, R3347 (1997).\n[3] T. 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B67, 155201 (2003)."    },    {        "title": "0706.1733v1.Magnetic_fingerprint_in_a_ferromagnetic_wire__Spin_torque_diode_effect_and_induction_of_the_DC_voltage_spectrum_inherent_in_the_wire_under_application_for_RF_current.pdf",        "content": "  1/1 Magnetic fingerprint in a ferromag netic wire: Spin torque diode \neffect and induction of the DC voltage spectrum inherent in the \nwire under application for RF current \n \n     \nA. Yamaguchi1, T. Ono2, Y. Suzuki3, S. Yuasa4, and H. Miyajima1 \n \n \n1Department of Physics, Faculty of Science a nd Technology, Keio University, 3-14-1 Hiyoshi, \nYokohama, Kanagawa 223-8522, Japan \n \n2Institute for Chemical Research, Kyoto Univ ersity, Gokasho, Uji, Kyoto 611-0011, Japan \n \n3Graduate School of Engineering and Science, Osaka University, Machikaneyama 1-3, \nToyonaka, Osaka 560-8531, Japan \n \n4Nanoelectronics Research Institute, National Ins titute of Advanced Industrial Science and \nTechnology, Tsukuba 305-8568, Japan \n   2/2  [Abstract] \nWe report the rectifying effect of a constant-w ave radio frequency (RF) current by a magnetic \ndomain wall (DW) on a single-layered ferromagnetic  wire. A direct-current (DC) voltage is \ngenerated by the spin torque diode effect, wh ich is a consequence of magnetoresistance \noscillation due to the resonant spin wave excita tion induced by the spin-polarized RF current. \nThe DC voltage spectrum strongly depends on the internal spin structure in the DW, which \ncorresponds to the magnetic fingerprint of th e spin structure in the ferromagnetic wire. \n \n \n             \n \n          \n   3/3 A magnetic domain wall (DW) is the magnetizatio n spatial transition region between magnetic \ndomains, in which the magnetic moments undergo re orientation. Most of the magnetic changes \nunder the action of weak and m oderate magnetic fields occur at DWs and understanding of DW \nbehavior is hence one of the essentials in d escribing the magnetizing process. There are many \nstudies on planer displacement of rigid DWs under the action of the magnetic field [1 – 3] and electrical current [4 – 13]. In general, the DW  is subjected to magnetic potential energy in a \nferromagnet. The potential energy varies thr oughout the ferromagnet since some defects \nexisting in the material provide minima of  the potential energy, while the region of \ninhomogeneous microstress, associated with dislo cations, provides either energy minima or \nmaxima depending on the sign of the stress a nd magnetostriction coefficient [1 – 3]. The \nfingerprint appropriate for the random potential in  magnetic materials has been studied from the \nviewpoint of the Barkhausen effect in Fe-N i-Co alloy [14] and from the magnetic field \ndependence of conductivity fluctuations in a va riety of mesoscopic systems [15, 16, 17].  \nRecently, Vila et al . have reported that DWs make a reproducible spin contribution to the \nuniversal conductance fluctuation and that they also lead to hysteric behavior with \nirreproducible fluctuations observed in the minor loop [17]. The shape of the potential is   4/4 irregular in the space and acts on the internal sp in structure of the DW in the absence of a \nmagnetic field. \nWhen a spin-polarized electrical current flows through some spin structure with \nvariation such as ferromagnetic multilayers [18 – 21] , the DW [4 – 9], or a magnetic vortex [22], \nmagnetization switching and magnetic excitation are i nduced by the spin-transfer effect [23, 24], \nwhich is a consequence of the spin angular momentum transfer occurring the interaction \nbetween the spin-polarized current and the magne tic moment. One of the interesting properties \nis the spin-torque diode effect found in the ma gnetic tunnel junction (MTJ) [24], the spin-valve \ngiant magnetoresistive device (SV-GMR) [25], a nd single-layered ferromagnetic wires [27]. It \nshould be noted that direct-current (DC) volta ge is produced when a radio-frequency (RF) \ncurrent flows through a nanometer-scale MTJ, a SV-GMR, or a ferromagnetic submicron wire. \nThis DC voltage generation originates in the magnetoresistance oscillation due to the \nferromagnetic resonance (FMR) generated by the sp in-transfer effects [25, 26]. As reported in a \nprevious paper [27], the spin torque diode effect  in the single-layered ferromagnetic wire is the \nconsequence of the magnetoresistance oscillation generated by the spin-transfer torque. The \ntorque is a combined effect of the spatial non-uniformity of the magnetization and the flow of   5/5 the spin-polarized current.  In the other words, the spin torque diode effect is highly sensitive \nto the spin structure in nano-scale magnets. \n In this study, we present the spin torque diode effect in a single-layered ferromagnetic \nwire without an external magnetic field. The DC  voltage spectrum depends on the internal spin \nstructure at the DW, which corresponds to the magnetic fingerprint reflected by the variation of the potential energy in the wire. \n \nTwo kinds of wire, boomerang-shaped and se micircular-shaped ferromagnetic wires of Ni 81Fe19, \nwere fabricated on MgO substrates using elect ron beam lithography and the liftoff method. \nFigure 1(a) shows an optical micrograph of the boomerang-shaped wire and the present RF \nelectric circuit.  The size and shape of the wires prepared in this study are listed in Table 1. \nThe measurement system is almost same as that  described in the previous paper [27]. A \nmicrowave probe is connected by the wire, and the bias tee circuit detects the DC voltage difference induced by the RF current flowing th rough the wire, as schematically illustrated in \nFig. 1 (a). A vector network analyzer (frequenc y range: 45 MHz – 67 GHz) or a signal generator \n(frequency range: 10 MHz – 40 GHz) injects the RF current into the wires. All experiments are \nperformed at room temperature in the absence of an external magnetic field.   6/6 In the soft ferromagnetic Ni 81Fe19 narrow wire, the magnetization is directed along the \nwire axis owing to the presence of strong uniaxial shape anisotropy. Therefore, we are able to \nintroduce or remove a single DW in the followi ng manner: A DW is introduced (removed) by \napplying a magnetic field, H=1 kOe, at an angle of 90θ=D ( 0θ=D) so as to saturate the \nmagnetization and decrease the field to zero. The magnetic structures of the head-to-head DW \nand no DW in the wire calculated by a microm agnetics simulator (OOMMF) [28] are shown in \nFig. 1 (b) and (c), respectively. The parameters  used for the calculation are as follows: a unit \ncell size of 5 nm × 5 nm with a constant thickness of 50 nm, a magnetization of 1.08 T, and a \ndamping constant of 0.1α= . The size of the simulation model is same as sample SC1 apart \nfrom the wire length. \nThe dynamical properties of the DW magnetization are probed using the vector \nnetwork analyzer in the microwave reflection mode. In order to remove the effect of cables, on-\nwafer calibration is performed using the open- short-load calibration procedure [29]. We \ncalibrate reflection signal S 11 of sample SC1 with the DW, and measure signal S 11 after the DW \nis removed in the previously described manner. In  Fig. 1 (d), the red dashed line and the black \nsolid line correspond to S 11 with and without a DW, respectively. The eigenmodes of the \nmagnetization in the wire with and without a DW are determined by free damping applying a   7/7 pulsed magnetic field along the z-direction (perpend icular to the plane). After this excitation, \nthe magnetization in the wire performs damped free oscillations. Figures 1(e) and 1(f) show the \ntime variation of the z-componet of the magnetization and the Fourier spectrum, respectively. \nAs shown in Fig. 1(f), the wire without a DW has an intrinsic frequency of 9 GHz, while the \nwire with a DW takes a frequency of 1, 2,  and 8 GHz. The frequency of the magnetization \noscillation in the wire with a DW is smaller than  that in the wire without a DW. The dips in S 11 \nof the wire without a DW at frequency of 9.5 and 10.4 GHz correspond to the magnetization \noscillation in the wire with and without  a DW, respectively. The decrease in S 11 of the wire \nwithout a DW in the range lower than 5 GHz also corresponds to the magnetization oscillation \ncalculated in the wire with a DW. \nFigure 2 shows the RF current frequency (RF power: -5 dBm) dependence of the DC \nvoltage difference due to the spin torque diode effect in the absence of a magnetic field. The \nblack dashed line and the red solid line correspond to the spectrum without a DW and with a \nDW, respectively. As shown in Fig. 2, the obser ved resonance spectrum of the wire with a DW \nis more complicated than that of the wire w ithout a DW. This indicates that the complicated \nresonance spectrum is a consequence of the spa tial non-uniformity of the magnetization due to \nthe existence of a DW. In other words, the e ffective magnetic field is homogenous in the wire   8/8 without a DW, but the effective field depends on the position of the wire with a DW due to the \ninternal spin structure. The spatial inhom ogeneity of the effective field produces the \ncomplicated spectrum. That is, the spin torque  diode signal shows a pattern inherent in each \nwire, or a magnetic fingerprint, reflecting the spa tial variation of the spin structure with high \nsensitivity compared with S 11 measurement. \nFigure 3 shows the RF power and frequency dependences of the induced DC voltage \ndifference for samples SC1 and SC2 with a DW. The amplitude of the DC voltage spectrum \nincreases with increasing RF current, but the spectru m structure is conserved. This indicates that \nthe spin-transfer torque or the Joule heating due to the RF current is not as large as the magnetic \nspatial structure disarranged. That is, there is spectral reproducibility [14 – 17]. \nWe repeated erasing the DW, introducing the DW again, and measuring the rectifying \neffect of the RF current in the wire in the absence of an external  magnetic field. The DC voltage \nspectra of samples SC2 and SC3 are shown in Fig.  4 (a) and (b), respectively, in which the \nnumber following the symbol # means the number of measurement times. As is seen in Fig. 4 (a), three kinds of spectrum are observed; the sp ectra for #2, #3, #4, and #6 are the same as each \nother but different from those of #1 and #5, while the spectra for #1 and #6 are the same but \ndifferent from those of #2, #3, #4, and #5, as shown in Fig. 4 (b). After the DW is erased and   9/9 introduced, a small change in the spin configura tion in the wire slightly modifies the magnetic \nfingerprint of the DW. This is attributed to the internal spin structure w ithin the DW influenced \nby the pinning potential associated with certain  defects such as dislocations, impurities, the \nmicro-structural inhomogeneties, or edge roughness in the wire. As shown in Fig. 4, after the \nDW is reintroduced into the wire, a new spin configuration including the DW, which is influenced by the random pinning potential sites in  the wire, results in a change in the magnetic \nfingerprint. As shown in Fig. 3, the spin struct ure once made is unchanged by the spin-transfer \ntorque or the thermal fluctuation caused by Joul e heating due to the RF current that is not as \nlarge as the magnetic spatial structure disarranged.  As a result, the magnetic fingerprint due to \nthe spin torque diode effect provides information of  the variation of the pinning potential sites in \nthe wire. \nIn conclusion, the DC voltage is induced by the spin torque diode effect in a single-\nlayered ferromagnetic wire without an external  magnetic field. The frequency spectrum of the \nDC voltage is influenced by the microscopic sp in configuration inherent in the random pinning \nsites in the wire, which corresponds to the magne tic fingerprint. The magnetic fingerprint is a \nhighly sensitive measurement of spin structure in nano-scale magnets. \n   10/10 The present work was partly supported by MEXT Grants-in-Aid for Scientific \nResearch in Priority Areas, JSPS Grants-in-Aid for Scientific Research, an Industrial \nTechnology Research Grant Program in ‘05 from NEDO of Japan, and the Keio Leading-edge \nLaboratory of Science and Technology project 2006. \n  \n \n \n  \n \n  \n \n  \n \n \n \n  \n \n   11/11  [References] \n[1] A. Hubert and R. Schäfer, Magnetic Domain s: The Analysis of Magnetic Microstructures, \n(Springer-Verlag Berlin Heidelberg New York, 2000). \n[2] A. P. Malozemoff and J. C. Slonczewski, Magnetic Domain Walls in Bubble Materials \n(Academic, New York, 1979). \n[3] D. Jiles, Introduction to Magnetism and Magnetic Materials (Cha pman & Hall, 1998), 2nd ed. \n[4] P. P. Freitas and L. Berger, J. Appl. Phys. 57, 1266 (1985); C. Y. Hung and L. Berger, J. \nAppl. Phys. 63, 4276 (1988). \n[5] L. Gan et al., IEEE Trans. Magn. 36, 3047 (2000); H. Koo et al., Appl. Phys. Lett. 81, 862 \n(2002). \n[6] A. Yamaguchi et al., Phys. Rev. Lett. 92, 077205 (2004); A. Yamaguchi et al., Jpn., J. Appl. \nPhys. 45, 3850 (2006). \n[7] N. Vernier et al., Europhys. Lett. 65, 526 (2004).  \n[8] M. Kläui et al., Phys. Rev. Lett. 94, 106601 (2005); M. Kläui et al., Phys. Rev. Lett. 95, \n026601 (2005). [9] G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 97, \n057203 (2006). \n[10] L. Berger, J. Appl. Phys. 55, 1954 (1984); L. Berger, J. Appl. Phys., 71, 2721 (1992). \n[11] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). \n[12] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). \n[13] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). \n[14] J. S. Urbach, R. C. Madison, and J. T. Markert, Phys. Rev. Lett. 75, 4694 (1995). \n[15] S. Washuburn and R. A. Webb, Rep. Prog. Phys. 55, 1311 (1992); M. B. Weissman, Rev. Mod. Phys. 65, 829 (1993). \n[16] P. G. N. de Vegvar, L. P. Lévy, and T. A. Fulton, Phys. Rev. Lett. 66, 2380 (1991). \n[17] L. Vila et al., Phys. Rev. Lett. 98, 027204 (2007).   12/12 [18] E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 \n(1999); J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. \n84, 3149 (2000); S. I. Kiselev et al., Nature 425, 380 (2003). \n[19] J. Grollier et al., Appl. Phys. Lett. 78, 3663 (2001). \n[20] S. Urazhdin, N. O. Birge, W. P. Pratt, and J. Bass, Phys. Rev. Lett. 91, 146803 (2003). \n[21] W. H. Rippard, M. R. Pufall, S. Kaka, S.  E. Russek, and T. J. S ilva, Phys. Rev. Lett. 92, \n027201 (2004). [22] S. Kasai, Y. Nakatani, K. Kobayashi,  H. Kohno, and T. Ono, Phys. Rev. Lett. 97, 107204 \n(2006). \n[23] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). \n[24] L. Berger, Phys. Rev. B 54, 9353 (1996). \n[25] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa, Nature 438, 339 (2005). \n[26] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. \nRalph, Phys. Rev. Lett. 96, 227601 (2006). \n[27] A. Yamaguchi et al., to be published: A. Yamaguchi, H. Miyajima, T. Ono, Y. Suzuki, S. \nYuasa, A. Tulapurkar, and Y. Nakatani, Appl. Phys. Lett. 90, 182507 (2007).  \n[28] \nhttp://math.nist.gov/oommf/  \n[29] R. B. Marks, IEEE Trans. Microwave Theory Tech. 39, 1205 (1991). \n \n \n \n      13/13  \n[Figure captions] \n \nFigure1 \n(a) The radio-frequency (RF) electric circuit and the optical micrograph (top view) of the \nsystem consisting of electrodes and a magnetic na nowire. The magnetic field is applied in the \nsubstrate plane at an angle of θ to the longitudinal axis of the nanowire. The direct-current (DC) \nvoltage across the wire is measured by using a bias tee. The in-plane magnetization distribution \n(b) with and (c) without a domain wall (DW) at the bottom of the wire loop visualized by \nOOMMF. (d) Reflection signal S 11 as a function of the RF current frequency.  (e) The temporal \nevolution of Mz during free damping. (f) The averaged power spectrum for Mz.  \n \nFigure 2 The DC voltage induced by the spin torque di ode effect as a function of the RF current \nfrequency. The red solid line and the black d ashed line correspond to the DC voltage spectrum \nwith and without DW, respectively. \n \nFigure 3 The DC voltage spectra for SC1 and SC2 as a func tion of RF power during the existence of the \nDW. \n Figure 4 \nThe magnetic fingerprint of (a) SC2 and (b) SC3.  Each spectrum is measured after the DW is \nerased and then introduced again.   14/14 \nBias-tee\nVSignal\ngenerator V olt meter50 nm-thick Ni81Fe19\nH\nθ10 µm(a)\nθ =0 : 1 kOe    ޓ  0 Oe(b) (c)\nθ =90 : 1 kOe ޓ     0 Oe300 nm 300 nm\n-80-70-60-50S11 (dB)\n20 15 10 5 0\nRF frequenc y (GHz)no DWDW\n0 Oe\n-10 dBm(d)\nAmplitude (arb. unit)\nTime (ns)1 5\nFiled offno DW\n DWAmplitude (arb. unit)\n20 15 10 5 0\nFrequency (GHz)4DW no DW4(e)\n(f)\n \n \nFig. 1 \n \n   15/15 \nDC voltage ( mV)\n15 10 5 0\nRF current frequency (GHz)5 mVSC1\nSC2\nSC3\nSC4DW\nDW\nDW\nDW0 Oe\n \n \nFig. 2 \n \n               16/16 \nDC voltage ( mV)\n15 10 5 0\nRF current frequency (GHz)5 mVSC1\nSC2-5 dBm-10 dBm-15 dBm\n \n \nFig. 3 \n \n                   17/17 \nDC voltage ( mV)\n15 10 5 0\nRF current frequency (GHz)5 mV#1\n#2\n#3\n#4\n#5\n#6\nno DWSC2 (a)\n \n \nDC voltage ( mV)\n15 10 5 0\nRF current frequency (GHz)5 mV#1\n#2\n#3\n#4\n#5\n#6\nno DWSC3 (b)\n \n \nFig. 4 \n \n    18/18 Table I. Summary of wire shapes and sizes used in the present study  \nSample       Type          Width          Thickness          Radius   \n                        ( n m )              ( n m )              ( µm)  \nSC1      Boomerang       300               50               3 \n    SC2      Semicircular      300               30              10     SC3      Semicircular      500               30              10     SC4      Semicircular     1000               30              10 \n "    },    {        "title": "1412.0777v1.A_rigorous_two_dimensional_model_for_the_stripline_ferromagnetic_resonance_response_of_metallic_ferromagnetic_films.pdf",        "content": " 1 A rigorous two-dimensional model for the stripline ferromagnetic \nresonance response of metallic ferromagnetic films \n \nZ. Lin 1,2  and M. Kostylev 1* \n \n1.  School of Physics, The University of Western Austra lia, Crawley 6009, \nAustralia \n2.  University of Science and Technology of China, Hefe i, China \n \n \nAbstract.  In this work we constructed a two-dimensional nume rical model for \ncalculation of the stripline ferromagnetic resonanc e (FMR) response of metallic \nferromagnetic films. We also conducted numerical ca lculations by using this software. \nThe calculations demonstrated that the eddy current  contribution to the FMR response \ndecreases with a decrease in the stripline width. T he most important manifestations of \nthe conductivity (eddy current) effect are excitati on of the higher-order standing spin \nwaves across the film thickness in the materials fo r which the standing spin wave \npeaks would be absent in cavity FMR measurements an d strong dependence of the \noff-resonance series conductance of the stripline o n the stripline width. Whereas the \ncontribution of the eddy currents to the stripline FMR response can be very significant, \nbecause wide striplines (100nm+) are conventionally  used for the FMR measurements, \nit is negligible in the case of excitation of spin waves, just because very narrow \nstripline transducers (0.5-5micron wide) are requir ed in order to excite spin waves in \nmetallic ferromagnetic films in a noticeable freque ncy/applied field range.    \n \n \n1. Introduction \nThe microwave conductivity contribution to the stri pline broadband \nferromagnetic resonance (FMR) response of highly-co nducting (metallic) magnetic \nmultilayers and nanostructures of sub-skin-depth th icknesses has attracted significant \nattention in recent years [1-8]. It has been shown that these effects are important every \ntime the microwave magnetic field is incident on on ly one of the two surfaces of a \nplanar metallic material [3,9-11].  \nThe geometry of a stripline ferromagnetic resonance  experiment [12-15,4] is \ncharacterized by such single-surface incidence of t he microwave magnetic field on the \nsample surface [3]. This experiment usually employs  a macroscopic coplanar (CPW) \nor microstrip (MSL) stripline through which a micro wave current flows (Fig. 1).  A \nsample - a film or a nanostructure - sits on top of  this line separated by an insulating \nspacer (not shown in the figure) in order to avoid an electrical contact of the sample \nwith the stripline. The stripline with the sample i s placed in a static magnetic field \napplied along the stripline. The microwave current in the stripline drives \nmagnetization precession in the ferromagnetic mater ial. The complex transmission \ncoefficient S21 of the stripline is measured either  as a function of microwave \nfrequency f for a given applied magnetic field H=e zH (“frequency resolved FMR”), or \nas a function of H for given f (“field-resolved FMR”) to produce FMR traces. The \nFMR absorption by the material is seen as a deep in  the Re(S21) vs. H or f trace.   \nIn the stripline FMR traces the effect of eddy curr ents is seen as anomalously \nlarge absorption amplitudes of the higher-order sta nding spin wave modes [3]. For \n                                                 \n*Corresponding author: mikhail.kostylev@uwa.edu.au  2 single-layer ferromagnetic metallic films with unpi nned surface spins on both film \nsurfaces the standing spin waves modes are not seen  at all in the cavity FMR spectra \nfor symmetry reasons [16]. However, as an existing theory shows, in the case of the \nstripline FMR they may develop amplitudes up to 1/3  of the amplitude of the \nfundamental peak [17].  For the multilayered films lacking inversion symmetry in the \ndirection of the stack thickness this effect may be  much more dramatic [4].  \nM. Bailleul [1] considered the electrodynamics of a  thin metallic non-\nmagnetic layer with a sub-skin-depth thickness on t op of a coplanar stripline. He \nshowed that there are two types of shielding by mic rowave eddy currents for the \nstripline configuration – electric and magnetic. Th e electric shielding is due to \ncapacitive  coupling of the layer to both signal and ground li nes of CPW. Because a \nmicrowave voltage is applied between the signal lin e and the ground lines, this \ncoupling results in a microwave current flowing fro m the signal line to the ground \nlines, i.e. in the direction perpendicular  to the CPW line. (Essentially, the metallic \nlayer acts a short-circuiting plug.)   \nIt is clear, that the Oersted field induced by this  current is along H and hence \ncannot contribute to the FMR response of a metallic  ferromagnetic layer. However, in \naddition to electric shielding magnetic shielding m ay simultaneously exist. This type \nof shielding is due to inductive  coupling of the metallic layer to the signal line of the \nCPW. The inductive coupling results in a microwave current flowing in the layer in \nthe direction opposite  to the direction of the current in the signal line . The origin of \nthe current in the metallic layer is the same as in  the case of the incidence of a plane \nelectromagnetic wave on a metal surface – an eddy c urrent is induced in the material. \nThe direction of the eddy current is such that the total microwave magnetic field is \nenhanced in front of the sample (i.e strong back-re flection of the electromagnetic \nwave takes place) and gradually drops inside the me tal with a distance from its \nsurface (skin effect).   \nFor the metallic layers with sub-skin-depth thickne sses the skin effects takes \nan unusual form – the microwave magnetic field insi de the layer drops linearly  from a \nmaximum at the surface exposed to the microwave rad iation to zero  at the far film \nsurface. For the ferromagnetic metallic films this was first theoretically demonstrated \nby solving a simplified one-dimensional problem [3, 11]. Later this effect was \nconfirmed with more rigorous two-dimensional numeri cal modelling. The 2D study \nwas carried out for non-magnetic  films [1,18,17].  \nThe simplified quasi-analytical 1-dimensional appro ach from [3] is valid for \nstriplines whose width w can be assumed as infinite – MSLs with large width s of the \nstrip and CPWs with large widths of the signal line . The 2D calculations mentioned \nabove assume realistic geometry of the stripline cr oss-section – they account for the \nfinite widths of the strip of MSL and of the signal  line of CPW as well as for the \npresence of the ground lines in the CPW case. It ha s been shown that the strength of \nshielding by the eddy currents strongly depends on w. The amplitude of the \nmicrowave magnetic field at the far film surface is  practically zero for w in the range \n0.3 to 1.5mm. For smaller w-values it gradually decreases with w and becomes almost \nequal to the field value at the front film surface for w=50 micron or so and for \nrealistic values of all other parameters of the cal culation [18,17].  \nIn the present work we construct a numerical model and solve the 2-\ndimensional electromagnetic problem numerically for  ferromagnetic  films. In \naddition to the film conductivity and electromagnet ic interactions, the exchange and \nZeeman interactions are taken into account. We focu s specifically on the case of the \nMSL lines. A simple estimation demonstrates that th e electric current shielding is  3 negligible for microstrip lines due to a large (ele ctrical) distance between the sample \nand the ground plane of the MSL (Fig. 1) resulting in an almost negligible capactitive \nconductance between the film and the ground plane.  \nOur numerical results show that the FMR absorption amplitude for the 1 st  \nstanding spin wave mode is large for strips of macr oscopic sizes and gradually drops \nto zero with a decrease in w. This happens due to the decrease in the microwave  \nshielding effect, as previously found for non-magne tic films (see above). We also see \na strong increase in the amplitude of the fundament al FMR mode with the decrease in \nw. This increase is accompanied by a growth in the off -resonance characteristic \ninductance of the MSL loaded by the ferromagnetic f ilm. For w=3 micron the \ninductance becomes equal to one of the MSL without a film on its top. The latter fact \ndemonstrates that for small w values the film becomes effectively insulating and  its \npresence does not alter the transmission characteri stics of the MSL off-resonance.  \nAll these findings explain, why strong contribution  of the microwave shielding \neffect to the broadband stripline FMR response of m etallic ferromagnetic films has \nbeen observed in a number of experiments, but all e xisting experimental data (see e.g. \n[19-22]) do not show any indication of an eddy curr ent impact on excitation of \ntravelling spin waves in ferromagnetic metallic fil ms. This happens just because for \nthe FMR measurements wide macroscopic MLS are typic ally employed, but in order \nto efficiently excite travelling spin waves in a br oad frequency range one needs much \nnarrower transducers - 0.5 µm< w<5  µ m or so.   \n \n2. Numerical model \n \n \nFigure 1  (a) Sketch of the modelled geometry. 1: The ground  plane of the microstrip \nline. 2: substrate of the microstrip line of thickn ess d. 3: infinitely thin strip of width w \ncarrying a microwave current. 4 ferromagnetic film of thickness  L  in the direction y \nand of width ws in the direction  x .  (b) Equivalent circuit of the microstrip line wi th \nthe ferromagnetic film on its top (“loaded MSL”). T he sketch shows one unit cell of  4 the periodic chain. (c) Equivalent circuit of the m icrostrip line without film on its top \n(“unloaded MSL”). \n \n  \n To solve the problem we use the numerical approach  first suggested in [8]. We \nconsider the model in which the y-axis is perpendicular to the surfaces of the \nconducting magnetic film (Fig. 1(a)).  We assume th at the film size ws in the direction \nx is larger than 2 w. Because the microwave field of a microstrip line is well localized \nwithin the area of the width 2 w below the strip of the microstrip line, this allow s us to \nconsider the sample as of an infinite width in the direction x in the following (but we \nwill use the finite value of ws to post-process the results of the numerical simul ations). \n  Thus, the film of thickness L is assumed to be continuous in the x- and z-directions. \nThe external magnetic field zH=H u is applied in the positive direction of the z-axis. \nAll the dynamic variables depend harmonically on ti me - exp( ) i t ω, where ω is \nmicrowave frequency. In order to include the magnet ization dynamics in the \nferromagnetic layer into the model, we employ the l inearized Landau-Lifshitz \nequation \n0 | | ( ) z eff i t M ω γ = − × + × m m H u h  (1) \nIn (1) the dynamic magnetization vector m has only two non-vanishing components \n(mx, m y) that are perpendicular to the static magnetizatio n 0zMu, where 0M is \nsaturation magnetization for the ferromagnetic film  and zu is a unit-vector in the \ndirection z. The dynamic effective field heff  has two components: the exchange field \nhex  given by \n \n2 2 2 2 ( / / ) exch x y α= ∂ ∂ + ∂ ∂ h m  (2) \n \nand the dynamic magnetic field h. In (2) the coefficient α  is the exchange constant.  \n The calculation of the dynamic magnetic field in t he presence of the eddy currents \nrepresents the main difficulty of the problem.  For  the exchange-free case ( α=0) a \nsophisticated analytical solution exists [23], but for α≠0 constructing a numerical \ncode is much more appropriate.  \n \nThe dynamic magnetic field is solution of Maxwell e quations: \n σ ∇× = h e ,  (3)  \n0( ) iωµ ∇× = − + e h m ,  (4)  \n∇ = −∇ h m ,  (5)  \n \nwhere 0µ is the magnetic permittivity of vacuum and h = (hx, h y). Because the \nmicrowave magnetic field of MSL is quasi-static and  also because the ferromagnetic \nfilm is metallic we neglected the term involving th e electric permittivity on the right-\nhand side of Eq.(3) and take into account only the term involving the film \nconductivity σ.  \n We also assume that all dynamical variables do not  depend on z (quasi-static \napproach to description of microwave transmission l ines), therefore our problem is \ntwo-dimensional. The standard real-space methods fa ce serious difficulties [1,18] \neven in two dimensions because of incompatibility o f the length scales in Fig. 1(a) – \ncompare L with w and d.  To get around this problem and ultimately to acc elerate the  5 numerical solution we take advantage of the transla tional symmetry of the sample in \nthe direction x (recall, we assume that the film is continuous in this direction).  To this \nend we Fourier-transform Eqs.(3-5) and (1,2) with r espect to x. This will allow us to \nremove the areas y<0 and y>L from the numerical model and thus to get around th e \nscale incompatibility problem. Furthermore, equatio ns for different Fourier wave \nnumbers k separate. In principle, this allows solving the co nstructed numerical \nproblem in parallel for large number of k values, and hence parallel computations.   \n To implement the Fourier transformation we assume that  \n, , exp( ) k k iky dy ∞\n−∞ = − ∫m h m h ,  \nwhere \n, 1/ (2 ) , exp( ) k k iky dy π∞\n−∞ =∫m h m h . \n This procedure results in a system of equations:  \n/x y z h y ikh e σ ∂ ∂ + = −  , (6)  \n0 / ( ) z x x e y i h m ωµ ∂ ∂ = − + ,  (7)  \n0( ) z y y ke h m ωµ = − + ,   (8) \n  / / x y y x ikh h y m y ikm − + ∂ ∂ = −∂ ∂ +  .   (9)  \n \nHere and at many places below we drop the subscript  “ k” to simplify notations.  \n We differentiate (6) and substitute (7) into the r esulting differentiated equation and \nthen make use of (9). This gives:  \n \n2 2 2 2 ( / ) / 0 x x y y K h K m ik m y ∂ ∂ − − − ∂ ∂ = ,  (10) \n \nwhere 2 2 \n0 K k i σωµ = + .  \n In this work Eqs. (10) and (9) will be solved nume rically to produce the dynamic \nmagnetic field. To obtain the numerical solution we  need the electromagnetic \nboundary conditions relating the electromagnetic fi elds inside and outside the film. \nFurthermore, to accelerate the numerical computatio n it is useful to exclude the areas \noutside the film from the discrete model. This is p ossible, since an analytic solution \nfor the space outside the film exists [3,8].   \n The microwave magnetic field outside the film is g iven by the same Eq.(10), but \nfor 0 σ= = m . Let us first consider the area behind the film y>L. From (10) and the \ncondition of the vanishing of the microwave magneti c field at y=+ ∞ one easily finds \nthat outside the film ( y>L)  \n | | \ny x kh i h k= − ,  (11) \nand at the film surface ( y=L)  \n \n| | ( ) 0 yi yi xi kh m ih k+ + = . (12) \nIn this expression the subscript “ i” indicates that these field components are taken a t \nthe film surface from inside  the film. Eq.(12) represents the electromagnetic b oundary \ncondition at y=L which excludes the area y>L from consideration.  6  A similar boundary condition can be also obtained for the area in front of the film. \nThis area contains the strip and the ground plane o f the MSL. We model the strip as \nan infinitely thin sheet of microwave current zIu with uniform current density \n/ j I w =  (Fig. 1). The width of the sheet along the x-axis is w. The sheet is infinite in \nthe z direction to ensure continuity of the current. The  Fourier image of the current \ndensity reads: \n \nsin( / 2) \n2 / 2 kI kw jkw π=  (13). \n \nFor simplicity we place the current directly on the  surface of the film ( y=0). This \nallows one employing the electro-dynamic boundary c ondition which includes a \nsurface current. This boundary condition reads (her e we write it down in the Fourier \nspace): \n \nxi xe k h h j = −  , (14) \n \nwhere the subscript “ e” denotes the  area outside the film.   \n The MSL ground plain is modelled as a surface of a  metal with infinite \nconductivity (“ideal metal”). It is located at y= −d. At the ideal-metal surface ez=0. By \nsolving (10) for −d<y<0 analytically together with this boundary conditi on and the \nboundary condition (14) one obtains a boundary cond ition at y=0 as follows: \n \n | | | | ( )coth(| | ) yi yi xi k k k h m k d i h i j k k + − = . (15) \n \nOne sees that Eq.(15) includes only components insi de the film and hence excludes \nthe area in front of the film from consideration in  the numerical model.  \n The system of equations (1), (10), (9), (12) and ( 15) will be solved numerically to \nobtain the dependence mk(y). This system is inhomogeneous, since Eq.(15) cont ains \nan excitation term given by its right-hand side.  I n the next section we will give details \nof the numerical method. Before we proceed to the n ext section we need to introduce \nequations which will be used to post-process result s of the numerical calculation.  \n In particular, we will be interested in the comple x surface impedance of the film Zr. \nThis quantity is a measure of the microwave magneti c absorption by the film [10].  It \ncan be defined as follows: \n \n rUZI= − , (16) \n \nwhere the linear voltage U (measured in V/m) is the mean value of the total e lectric \nfield induced at the surface of the strip of MSL:  \n \n/2 \n/2 1 sin( / 2) ( 0) ( 0) / 2 w\nz zk \nwkw U e y dx e y dk w kw ∞\n− −∞ = = = = ∫ ∫.  (17) \n \nSince no electric field is externally applied to th e MSL in our model, the source of \nthis voltage is the dynamic fields of the film. Hen ce the negative sign in Eq.(16).  7  The electric field zk e can be easily obtained with Eq.(6), once mk and hk have been \ncalculated numerically.    \n Once Zr has been computed, it is a straightforward procedu re to calculate S21 [3]. \nWe make use of the equivalent circuits for a micros trip line with the film on its top \n(“loaded MSL”, Fig. 1(b)) and for an MSL without th e film (“unloaded MSL”, Fig. \n1(c)). The characteristic impedance for the loaded MSL is  \n \n0/ ( ) f r c Z Z Y Y = + ,  (18) \n \nwhere 0Y is the intrinsic (i.e. in the absence of the sampl e) parallel capacitive \nconductance of MLS  and cY is the parallel capacitive conductance due to the electric \nshielding effect (see introduction). It is obvious that far off FMR Z r should become \npurely imaginary and reduce to the in-series induct ive impedance of MSL with a thin \nfilm on its top.  Furthermore, for σ=0 the latter quantity should further reduce to the  \nintrinsic in-series inductive impedance of MSL Z0 (Fig. 1(c )). Indeed, our numerical \ncalculations will show that off resonance Zf(σ=0) converges to Z0. This justifies, why \nwe do not include Z0 into (18).  \n Z0 can be calculated using the standard formalism for microstrip transmission lines \n[24].  For Zr=Z0 and 0cY= Eq.(18) reduces to the expression for the intrinsi c \ncharacteristic impedance of MSL: \n \n0 0 /cZ Z Y =   (19). \n \nThe intrinsic propagation constant for MSL is given  by the formula as follows: \n \n0 0 0 eff i γ ω ε ε µ = , (20) \n  \nwhere eff ε is the effective electric permittivity of MSL \n \n1 1 1\n2 2 1 12 / s s \neff d w ε ε ε+ − = + \n+  , (21) \n \nsε is the relative dielectric permittivity of the MLS  substrate and 0ε is the dielectric \npermittivity of vacuum.  The expression for cZcan be found in any textbook on \nmicrowaves (see e.g. [24]), therefore we do not sho w it here. Importantly, once cZ \nand 0γ have been calculated, Z0 and Y0 can be obtained by solving (19) together with  \n \n0 0 0 Z Y γ=   (22). \n \n As shown in [3], the complex transmission coeffici ent of a section of MSL with the \nfilm on its top is given by     \n2\n2121 exp( ) exp( ) f s f s Sl l γ γ Γ − =Γ − − , (23) \n  8 where sl is the length of the film in the direction z, Γ is the complex reflection \ncoefficient from the front edge of the section of M SL covered by the film \n \nf c \nf c Z Z \nZ Z −Γ = + (24) \n \nand  \n \n0 ( ( ) f r c Z Y Y γ= +   (25) \n \nis the propagation constant for MSL loaded by the f ilm. To make use of (23) we also \nneed cY. Derivation of a suitable expression for cY is given in Appendix 1. \n \n \n3. The discrete model  \n \n To solve the system of equations (1), (10) and (9)  numerically we use the finite \ndifference approach. These equations are differenti al equations of only one spatial \nvariable – y. Accordingly, we employ a one-dimensional mesh. Th e number of mesh \npoints is n and they all are located inside the film, i.e. bet ween y=0 and y=L. The \nequations are discretized by using the standard 3-p oint formula for numerical \ncalculation of derivatives. The boundary conditions  (12) and (15) are included into the \ndiscrete version of (10) and (9) for the mesh point s at the film surfaces y=∆/2 and \ny=L−∆ /2, where ∆=L/n  is the mesh step.  In the same way we discretize the exchange \noperator (2) and include the exchange boundary cond itions for dynamic magnetization \ninto it. For simplicity we assume that m is unpinned at both film surfaces y=0 and y= \nL: \n \n/ 0 y ∂ ∂ = m . (26) \n \nThe method of inclusion of the boundary conditions into the differential operators was \nexplained in Appendix to [3], therefore we do not r epeat its description here.    \n The dicretization of (10) simultaneously with (9) results in a vector-matrix \nequation in the form \n \nˆ ˆ ˆ | | | k k mw mwk H h M m H h > + >= > . (27) \n \nSimilarly, Eqs.(1) and (2) are cast in the form \n \nˆ | | k k h G m >= > . (28)  \n \nIn these expressions |kh> and |km> are column vectors containing values of x and y \ncomponents of hk and mk respectively at the mesh points.  The vector |mwk h > \ncontains only one non-vanishing element. It is the x-component at y=∆/2. It is equal to \njk. This is consistent with Eq.(14) which relates the  x-component of the dynamic field \nat the film surface to the current density in the m icrostrip.  9  Combining (27) and (28) one obtains  \n \nˆˆ ˆ ˆ | | | k k mw mwk HG m M m H h > + >= > . (29) \n \nIn this work we solve Eq.(29) for |km> numerically for I=1A and given values of H,  \nand ω using the standard numerical methods of linear alg ebra.  The result is then \nutilized to compute ( 0) zk e y = with (28) and (6).  This is done for a large numbe r of k-\nvalues for which jk (Eq.(13)) is non-negligible. Then we use the “modi fied” inverse \nFourier transform (17) in order to calculate U. Zr and S21 are then obtained with (16) \nand (23) respectively.   \n \n4. Discussion \n This algorithm has been implemented as a MathCAD w orksheet. Therefore the \ncomputation time is rather long – 12 hours for one program run during which \n( 0) zk e y = is computed for 4096 k-values and 80 H-values for a given frequency and \nn=100. However, our Fourier-space formulation of the  problem should make it \nperfectly suitable for parallel computing. Indeed, using multiple processors large \nnumber of independent solutions of (29) for differe nt values of k may be obtained in \nparallel. This should drastically decrease the comp utation time. Furthermore, \nrewriting the code in a language which is more appr opriate for lengthy computations \nthan the programming tool built in into MathCAD sof tware will make computations \nmuch faster, even without parallelization.   \n Several tests of the produced code were run in ord er to make sure that the software \ndelivers correct results. The first test was to sol ve (27) assuming | 0 km>=  and w is \nlarge ( w=1.5mm). For L=100nm and  σ=4.5x10 6Sm/m which is the typical \nconductivity of Permalloy (Ni 80 Fe 20 ), we found that hx(y=0, x=0)= j and hx(y=L,x=0)=0. \nThis is consistent with previous theories of the mi crowave shielding for thin non-\nmagnetic metallic films ([3,18]).  For a σ−value close to zero we found that Ζr=Ζ0. \nThe latter (i.e. Ζ0) was calculated analytically using (20) and (22) a nd the standard \nexpression for the characteristic impedance of MSL [24].  \n Also, for α=0 an analytic solution of (1), (9) and (10) exists . It is similar to one in \nRef. [23] and will be reported elsewhere.  Our solu tion of the full system (29) (i.e. for \n| 0 km>≠ ) was checked against this analytic solution and sh owed excellent agreement.  \n   Below we demonstrate some important results of o ur computations. Figure 2 \ndisplays Z r(H) for different values of w. The frequency is 15 GHz and the Permalloy \nfilm thickness is either 50nm or 100nm. Since we ke ep the frequency constant and \nvary the applied field, this simulation mimics the conditions of the field-resolved \nFMR experiment.  10  (d) \nApplied field (Oe) 0 500 1000 1500 2000 2500 3000 Re( Zr) (Ohm/cm) \n35 36 37 \nIm( Zr) (Ohm/cm) \n234(e) Re( Zr) (Ohm/cm) \n200 240 280 \nIm( Zr) (Ohm/cm) \n140 180 220 \n(f) Re( Zr) (Ohm/cm) \n140 160 180 \nIm( Zr) (Ohm/cm) \n65 85 105 \n(g) Re( Zr) (Ohm/cm) \n56 58 60 62 64 66 \nIm( Zr) (Ohm/cm) \n12 14 16 18 20 22 \n(h) \nApplied field (Oe) 1500 2000 2500 3000 Re( Zr) (Ohm/cm) \n18 19 20 21 22 \nIm( Zr) (Ohm/cm) \n-1 0123(a) Re( Zr) (Ohm/cm) \n220 240 260 \nIm( Zr) (Ohm/cm) \n260 280 300 \n(b) Re( Zr) (Ohm/cm) \n180 190 200 \nIm( Zr) (Ohm/cm) \n170 180 190 \n(c) Re( Zr) (Ohm/cm) \n94 96 98 100 \nIm( Zr) (Ohm/cm) \n42 44 46 48 0.0 0.5 1.0 \n0.0 0.5 1.0 \n0.0 0.5 1.0 \n0.0 0.5 1.0 \n \n \nFig. 2. Surface impedance of the film for different  values of the strip width w. (a)-(d): \nfilm thickness L=50nm. (e)-(h): L=100nm. (a) and (e): strip width w=50 µm; (b) and \n(f): w=100 µm; (c) and (g): w=350 µm; (d) and (h): w=1500 µm.  \n Insets to Panels (a)-(d) show respective profiles hx(y) of the microwave magnetic \nfield across the film thickness at the symmetry pla ne of the strip x=0. They are \ncalculated for the maximums of the resonance line f or the fundamental mode. The \nhorizontal axes of the graphs in the insets are not  labelled, in order not to overload the \npanels. Left edges of the graphs correspond to y=0 and the right edges to y=L.  \n Parameters of calculation. Microwave frequency: 15  GHz; saturation \nmagnetization µ0M0=1T; magnetic loss parameter for the ferromagnetic film \n∆H=6mT. The microstrip substrate thickness d and its electric permittivity εs are \nchosen such that the intrinsic characteristic imped ance of the microstrip line Zc=50 \nOhm for all panels: (a) and (e): d=100 µm, εs =16; (b) and (f): d=100 µm, εs =9; (c) and \n(g): d=160 µm, εs=3.55; (d) and (h): d=700 µm, εs=3.55. Film conductivity \nσ=4.5x10 6Sm/m. Solid lines: Re( Zr); dashed lines: Im( Zr). \n \n The first observation from this figure is that the  amplitude of first higher-order \nstanding spin wave mode (1 st  SSWM, located at 1290 Oe for L=50nm and at 2060 Oe \nfor 100nm) is significant, although the dynamic mag netization vector is unpinned at \nboth film surfaces. This mode is excited due to the  eddy-current contribution to FMR \nresponse.  For w=1.5mm the relative amplitude of this mode is the s ame as given by  11  the 1-dimensional theory from [3]. It becomes almos t negligible with the decrease in \nw. The latter result is the main funding of this wor k.  \n As previously shown in [18], the microwave shieldi ng effect gradually disappears \nwith a decrease in w. This is confirmed by the present calculation for the \nferromagnetic films (see insets to Fig. 2). As Fig.  2 demonstrates, this leads to \nsuppression of excitation of the 1 st  SSWM.  The possibility of the stripline FMR to \nexcite the 1 st  SSWM is very important for characterization of mag netic films, since its \nfrequency and field position with respect to the fu ndamental mode carries information \nabout the value of the exchange constant for the ma terial (see Eq.(2)). The cavity \nFMR is unable to excite this mode, unless significa nt pinning of magnetisation is \npresent at one of the film surfaces. This is becaus e the eddy currents vanish in the \nconditions of the cavity FMR for symmetry reasons ( see e.g. [17]). As follows from \nFig. 2, the stripline FMR has an important advantag e from this point of view, \nprovided one uses wide striplines or CPWs.  \n The next observation from this figure is that the amplitude of the fundamental \nFMR peak (the larger peak in each panel) grows sign ificantly with the decrease in w.  \nThis fact is well-known to experimentalists. Theref ore they tend to use narrow \nstriplines (100 to 300nm wide) for their measuremen ts (see e.g. [25]). However, in \nthis way, they unintentionally decrease the respons e of the 1 st  SSWM, as it is clear \nfrom this figure.   \n  From Fig. 3 one also sees that the off-resonance values of Re( Zr) and Im( Zr) \ngrow with the decrease in w. This effect can be also attributed to the decreas e in the \nmicrowave shielding effect. Loading of a wide MSL b y a metallic film decreases the \nin-series inductive impedance of MSL with respect t o Ζ0. The decrease is larger for \nwider stripes. For instance, for w=1.5mm Ζ0=270 Ohm/cm, but the off-resonance \nvalue of Im( Zr)  for L=100nm and w=1.5mm in Fig. 2 is about 1 Ohm/cm. \n For w=3 µm, however, the Im(Z r) off resonance is 540 Ohm/cm (as measured at \nH=800 Oe, Fig. 3). This is close to Ζ0 for this value of w  - 620 Ohm/cm. To \ndemonstrate the disappearance of shielding, in Fig.  3 we also show the result of \ncalculation for a negligible value of film conducti vity (5 Sm/m). One sees that the two \nresults are very close to each other. This shows th at the film becomes effectively \ninsulating for small stripline widths.   \n \n(a) Re( Zr) (Ohm/cm) \n0200 400 600 800 \nIm( Zr) (Ohm/cm) \n300 500 700 900 1100 \n(b) \nApplied field (Oe) 1000 1500 2000 2500 3000 Re( Zr) (Ohm/cm) \n0200 400 600 800 \nIm( Zr) (Ohm/cm) \n400 600 800 1000 1200 \n \nFig. 3. Surface impedance for a narrow strip - w=3 µm. (a) Film thickness L=100nm. \n(b) L=50nm. d=10 µm and εs =16 for both panels. All other calculation paramet ers are  12  the same as for Fig. 2. (The unrealistically small value of d was chosen in order to \nobtain Z0=50 Ohm, and hence to allow comparison of Figs. 2 a nd 3.) Thick lines: \nσ=4.5x10 6Sm/m; thin lines: σ=5 Sm/m.  Solid lines: Re( Zr); dashed lines: Im( Zr). \n \n Another important observation from Fig. 3 is the t otal disappearance of the 1 st  \nSSWM peak and appearance of small waviness of Re( Zr) for smaller applied fields. \nThe total absence of the SSWM peak is in agreement with the idea of the complete \ndisappearance of the shielding effect. The appearan ce of the waviness, however, does \nnot relate to the film conductivity, since it is th e same for both conducting and \ninsulating films in the upper panel of Fig. 3. The waviness has to be considered in \nconjunction with significant broadening of the fund amental resonance peak seen in \nFig. 3. This broadening is due to contribution of t ravelling spin waves to the stripline \nFMR response [26]. For L=100nm and  w=3micron the width of the strip is \nsignificantly smaller than the free propagation pat h of spin waves – tens of micron. \nThe spin waves are efficiently excited by this MSL “antenna” and leave the area of \nlocalization of the driving field. They carry energ y away from the antenna which is \nseen as additional resonance losses leading to the broadening of the resonance line. \n An alternative explanation to the spin wave contri bution is as follows. Z r(k) (where \nk  has now sense of spin wave wave number) scales ro ughly as square of   sin( / 2) \n/ 2 kw \nkw . \nThis is seen from Eqs.(13) and (17). The first lobe  of this function is located between \nk=0 and k=2 π/w. Most of microwave power goes into energy of spin waves in this \nwave number range. This wave number range correspon ds to an applied-field range \nroughly equal to Vg2π/(| γ|w), where Vg is the spin wave group velocity. When the \nintrinsic resonance linewidth (intrinsic magnetic l oss parameter) for the material ∆H \nis larger than Vg2π/(| γ|w), the spin wave contribution to the stripline FMR linewidth is \nnegligible.  On the contrary, when ∆H<V g2π/(| γ|w), non-negligible the spin wave \ncontribution will exist. The group velocity of the Damon-Eschbach spin wave [27] for \nL=50 nm is twice smaller than for L=100nm. Therefore the peak width in Fig. 3 for \nL=50nm is narrower by approximately two times. \n The waviness in Fig. 3 is reflection of the fact t hat Z r(k)∼ 2sin( / 2) \n/ 2 kw \nkw   \n    . \nTherefore, because for small spin wave wave numbers the spin wave dispersion law is \npractically linear, the shape of Z r(H)  is close to the shape of the sin 2(x)/ x2 function. \n The very small difference in the shapes and amplitu des of the results for σ=0 and \nσ≠0 in Fig. 3(a) explains why conductivity effects ha ve never been observed in the \nexperiments on excitation of travelling spin waves in ferromagnetic metals with \nstripline antennas [19-21]. Indeed, in order to exc ite spin waves in a noticeable \nfrequency or wave number range, very narrow antenna s are needed (see Fig. 3). For \nthe narrow antennas the conductivity effects are ne gligible, as follows from above.       13  (a) Re(S21) \n0.388 0.392 0.396 0.400 0.404 \nIm(S21) \n-0.484 -0.480 -0.476 -0.472 -0.468 \n(b) Re(S21) \n-0.27 -0.26 -0.25 -0.24 -0.23 \nIm(S21) \n-0.26 -0.25 -0.24 -0.23 -0.22 \n(c) \nApplied field (Oe) 1000 1500 2000 2500 3000 Re(S21) \n0.94 0.95 0.96 0.97 0.98 0.99 \nIm(S21) \n-0.13 -0.12 -0.11 -0.10 -0.09 -0.08 -0.07 \n(f) \nApplied field (Oe) 1000 1500 2000 2500 3000 Re(S21 f)\n0.94 0.95 0.96 0.97 0.98 0.99 1.00 \nIm(S21 f)\n-0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 (e) Re(S21 f)\n0.31 0.32 0.33 0.34 0.35 \nIm(S21 f)\n-0.13 -0.12 -0.11 -0.10 -0.09 (d) Re(S21 f)\n0.46 0.47 0.48 \nIm(S21 f)\n-0.41 -0.40 -0.39 \n \nFig. 4. (a-c): Unprocessed S21 data. (d-f): process ed S21 data. (a) and (d): L=100nm, \nw=1500 µm; (b) and (e): L=50nm and w=100 µm;  (c) and (f): L=100nm, w=3 µm. All \nother parameters of calculation are the same as for  Figs. 2 and 3. Solid lines: Re(S21); \ndashed lines: Im(S21). (d) φ=− 110 ο. ( e) φ=− 60 ο. ( g) φ= 0ο. \n \n \n   In Fig. 4(a-c) we demonstrate the S21( H) dependence calculated with the data \nfrom Figs. 2(b), 2(h) and 3(a). One sees that we ob tain realistic values of S21 and the \nshapes of its real and imaginary parts are very rea sonable. Interestingly, similar to Fig. \n2(b) in Fig. 4(b) one observes different shapes for  the fundamental and the 1st  SSWM \npeaks. The fundamental peak in Re(S21) has a shape close to Lorentzian, which is the \ncharacteristic shape for the imaginary part of the complex microwave magnetic \npermeability [28], but the response of the 1 st  SSWM has a shape characteristic to the \nreal part of the complex permeability. This effect of difference in the peak shapes is \noften seen in the experiment (see e.g. Fig. 2(c) in  [4]).    \n Another observation from Fig 4(a-c) is that the sh apes of Re(S21) and Im(S21) \nchange from panel to panel. However, if one divides  the “raw S21 data” from Fig. \n4(a-c) by S21 0=S21(Z f=Z 0,γf=γ0)exp(- iφ), where φ is phase correction, one obtains \ntraces of “standard” shapes, as seen in panels (d-f ). The latter panels display \nS21 f=S21/S21 0 for values of φ which were carefully chosen to produce the most \nsymmetric (anti-symmetric) shapes of Re(S21) (Im(S2 1)) respectively.  Previously a \nsimilar procedure was employed to process experimen tal data [4].  Traces of S21 \ntaken without a sample on top of the microstrip lin e were used as S21 0 in that \npublication. This procedure also resulted in traces  of the “standard” shape (see Fig. \n2(a-d) in that paper).  \n \nConclusion \n In this work we constructed a two-dimensional nume rical model for calculation of \nthe stripline ferromagnetic resonance response of m etallic ferromagnetic films. We  14  also conducted numerical calculations by using this  software. The calculations \ndemonstrated that the eddy current contribution to the FMR response decreases with a \ndecrease in the stripline width. The most important  manifestations of the conductivity \n(eddy current) effect are (1) excitation of the hig her-order standing spin waves across \nthe film thickness in the materials for which the s tanding spin wave peaks would be \nabsent in cavity FMR measurements and (2) strong de pendence of the off-resonance \nseries conductance of the stripline on the striplin e width. \n The model is suitable for description of both stri pline ferromagnetic resonance \nexperiments and experiments on excitation of travel ling spin waves. It also \ndemonstrates gradual transition from the former reg ime to the latter one with a \ndecrease in the width of the stripline. Whereas the  contribution of the eddy currents to \nthe stripline FMR response can be very significant,  because wide striplines (100nm+) \nare conventionally used for the FMR measurements, i t is negligible in the case of \nexcitation of spin waves, just because very narrow stripline transducers (0.5-5micron \nwide) are required in order to excite spin waves in  metallic ferromagnetic films in a \nnoticeable frequency/applied field range.    \n For simplicity we considered magnetization dynamic s in uniform ferromagnetic \nfilms. However, it is known that for films lacking inversion symmetry in the direction \nof the film thickness the impact of eddy currents o n the stripline FMR response may \nbe even more pronounced. Examples are bi-layer film s [4] and single-layer metallic \nfilms with unequal strengths of magnetization pinni ng at the film surfaces [17]. The \nstripline FMR response of these films strongly depe nds on the film orientation (e.g. \nlayer ordering) with respect to the stripline [5]. The results of the present study \nindicate that for these materials one may also expe ct a gradual decrease in the strength \nof the eddy current effects with a decrease in the stripline width and hence a decrease \nin the difference in the FMR responses for the two possible film orientations with \nrespect to the stripline.    \n \nAcknowledgment \nSupport by the Australian Research Council, the Uni versity of Western Australia \n(UWA) and the UWA’s Faculty of Science is acknowled ged. Z.L. acknowledges the \nstudent exchange program between USTC and UWA. We a lso thank I.S. Maksymov \nfor fruitful discussions. \n \nAppendix: Derivation of an expression for the paral lel capacitive conductance Yc. \n Here we derive an expression which allows estimati on of Yc (Fig. 1(b)). This \nparameter is a measure of the strength of the elect ric shielding [1]. We calculate Yc \nbased on the equivalent circuit in Fig. A1. We assu me that the capacitance between \nthe strip and the film is infinite. This is a valid  assumption, since the film is located \ndirectly on the strip surface (Fig. 1(a)). Due to t he electric shielding effect a \nmicrowave current flows along the film in the direc tion x (Figs. 1(a) and A1) and then \nfrom the film to the stripline ground plane due to the distributed capacitance between \nthe film and the stripline C∆x∆z =εsε0 ∆ x∆z /d (Fig A1). The resistance of a unit area \n∆x∆z of the film R∆x=∆x/(σL∆z). The total length of the R-C chain is ( ws−w)/2 and it \nis terminated by an open end (the edge of the film) . There are two such “chains”: to \nthe right and to the left from the strip.  \n The input impedance Zin  of each chain can be calculated by using the same \nformalism of Eqs. (19-25). The in-series linear imp edance Z=R∆x and the linear \nparallel conductance is Y=iωC∆x. The characteristic impedance of a chain is then   15  0 / / ( ) cc s Z Z Y z d i L ε ε σ ω = = ∆ . The propagation constant is \n0/ ( ) cc s ZY i Ld γ ε ε ω σ = = . For a chain terminated by an open end, Zin  is given by \na formula as follows: coth( ( ) / 2) in cc s Z Z w w γ = − . Then, taking into account that \nthere are two identical chains (Fig. A1), we obtain  the final formula \n \n 0 0 2 / 2 ( ) / tanh[ / ( )( ) / 2] c in s s s Y z Z i L d i Ld w w ε ε σ ω ε ε ω σ = ∆ = − .  (A1). \n \n Calculations show that Yc is usually significantly smaller than Y0. For instance, for \nf=15GHz, L=100nm, w=1.5mm, ws=5mm, d=0.7mm and εs=3.55 one obtains Y0=10.1 i \nOhm/m and Yc=6.1x10 −4(1+ i).  \n \n \n \nFig. A1. Equivalent circuit for estimating Yc. \n \n \nReferences \n1. M Bailleul,  Appl. Phys. Lett . 103 , 192405 (2013).  \n2. W.E. Bailey, C.Cheng, R. Knut, O. Karis, S. Auff ret, S. Zohar, D. Keavney, P. \nWarnicke, J.S. Lee, and D.A. Arena, Nat. Commun.  4, 2025 (2013). \n3. M. Kostylev, J. Appl. Phys . 106 , 043903 (2009). \n4. K. J. Kennewell, M. Kostylev, N. Ross, R. Magara ggia, R.L. Stamps, M. Ali, A.A. \nStashkevich, D. Greig D and B.J. Hickey, J. Appl. Phys.  108 , 073917 (2010). \n5. M. Kostylev, A.A. Stashkevich, A.O. Adeyeye, C. Shakespeare, N. Kostylev, N. \nRoss, K. Kennewell, R. Magaraggia, Y. Roussigné, an d R.L. Stamps,  J. Appl. Phys . \n108 , 103914 (2011). \n6. R.Bali, M. Kostylev, D. Tripathy, A.O. Adeyeye a nd S. Samarin “, Phys. Rev. B  85 , \n104414 (2012). \n7. M. Kostylev, J. Appl. Phys . 113 , 053907 (2013). \n8. I. Maksymov and M. Kostylev, J. Phys. D: Appl. Phys . 46 , 495001 (2013). \n9. J. Chen, B. Zhang, D. Tang, Y. Yang, W. Xu, and H. Lu, J. Magn. Magn. Mate r., \n302 ,  368 (2006). \n10. N. Chan, V. Kamberský, D. Fraitová,  J. Magn. M agn. Mater. 214 , (2000). \n11. M. Kostylev, J. Appl. Phys.  112 , 093901 (2012). \n12. S.S. Kalarickal, P. Krivosik, M. Z.Wu, C.E. Pat ton, M.L. Schneider, P. Kabos, T.J. \nSilva, J.P. Nibarger, J. Appl. Phys.  99 , 093909 (2006).  \n13. J.M. Shaw, H.T. Nembach, T.J. Silva, C.T. Boone , J. Appl. Phys ., 114 , 243906 \n(2013). \n14. F. Zighem, A.E. Bahoui, J. Moulin, D. Faurie, M . Belmeguenai, S. Mercone and \nH. Haddadi, J. Appl. Phys . 116 , 123903 (2014). \n15. X. M. Liu, H.T. Nguyen, J. Ding, M.G. Cottam, a nd A. O. Adeyeye, Phys. Rev. B  \n90 , 064428 (2014). \n16. C. Kittel, Phys. Rev.  110 , 1295 (1958).  16  17. I. S. Maksymov and M. Kostylev, “Broadband stri pline ferromagnetic resonance \nspectroscopy”, invited critical review, under revie w in Physica E .  \n18. I.S. Maksymov and M. Kostylev , J. Appl. Phys . 116 , 173905 (2014). \n19. M. Bailleu, D. Olligs, C. Fermon and S.O. Demok ritov, Europhys. Lett , 56 , 741 \n(2001). \n20. V.E. Demidov, S. Urazhdin, S.O. Demokritov,  Ap pl. Phys. Lett. 95 262509 \n(2009). \n21. K. Vogt, H. Schultheiss, S. Jain, J.E. Pearson,  A. Hoffmann,  S.D. Bader, and B. \nHillebrands, Appl. Phys. Lett. 101, 042410 (2012). \n22. C.S.Chang, M. Kostylev, E. Ivanov, J. Ding and A. O. Adeyeye, Appl. Phys. Lett . \n104 , 032408 (2014). \n23. N. S. Almeida and D. L. Mills, Phys. Rev. B , 53  12232 (1996).   \n24. I. J. Bahl and D. K. Trivedi, \"A Designer's Gui de to Microstrip Line\", Microwaves , \nMay 1977, 174-182 (1977).  \n25. J. M. Shaw, H. T. Nembach, and T. J. Silva, Phys. rev. B  85 , 054412 (2012). \n26. G. Counil, J.-V. Kim, T. Devolder, and C. Chapp ert, J. Appl. Phys . 95 , 5646 \n(2004). \n27. R.W. Damon, and J.R. Eshbach, J. Phys. Chem. Sol .,19, 308 (1961). \n28. A.G. Gurevich and G.A. Melkov, “Magneiization o scillations and waves” Boca \nRaton: CRC Press 1996. \n \n \n \n \n \n \n  \n \n "    },    {        "title": "1009.1553v1.The_temperature_dependent_magnetization_profile_across_an_epitaxial_bilayer_of_ferromagnetic_La2_3Ca1_3MnO3_and_superconducting_YBa2Cu3O7_d.pdf",        "content": "The temperature- dependent magnetization profile across an \nepitaxial bilayer of ferromagnetic La2/3Ca 1/3MnO 3 and \nsuperconducting YBa 2Cu 3O7-δ \n \nS. Brück1,2, S. Treiber2, S. Macke2, P. Audehm2, G. Christiani3, S. Soltan3, H.-U. \nHabermeier3, E. Goering2 and J. Albrecht4 \n1University of Würzburg, Physikalisches Institut, Am Hubland, D -97074 Würzburg, \nGermany.  \n2Max Planck Institute for Metals Research, Heisenbergstr. 3, D -70569 Stuttgart, \nGermany.  \n3Max Planck Institute for Solid State Research, Heisenbergstr. 1, D -70569 Stuttgart, \nGermany.  \n4Hochschule Aalen, Beethovenstr. 1, D -73430 Aalen, Germany.  \n \nE-mail: seb@brueck -online.com  \n \nAbstract . Epitaxial bilayers of ferromagnetic La 2/3Ca1/3MnO 3 (LCMO) and \nsuperconducting YBa 2Cu3O7-δ (YBCO) have been g rown on single -crystalline SrTiO 3 \n(STO) substrates by pulsed laser deposition. The Manganese magnetization profile \nacross the FM layer has been determined with high spatial resolution at low \ntemperatures by X -ray resonant magnetic reflectivity (XRMR). It i s found that not \nonly the adjacent superconductor but also the substrate underneath influences the \nmagnetization of the LCMO film at the interfaces at low temperatures. Both effects \ncan be investigated individually by XRMR.  \n \n1. Introduction  \nProperties of heterostructures made of high -temperature superconductors such as YBCO and spin -\npolarized ferromagnets such as LCMO are strongly influenced by coupling phenomena at the \ninterfaces. In general, superconductivity and ferromagnetism are incompatible ordering principles and \ntheir combination leads to substantial competition between the two ordered ground states [1]. In the \ncase of hybrids of high-temperature superconductors and oxide ferromagnets, this usually leads to the \nsuppression of both transition temperatures  [2,3] . Several mechanisms have been discussed to explain \nthis observation, including oxygen diffusion [4] ,  charge transfer  [5] due to a discontinuity in the \nchemical potential, a redistribution of the orbital occup ancy  [6], diffusion of spin- polarized \nquasiparticles [2,7] , and finally dipolar magnetic coupling via stray fields  [8,9] . In addition to lowered \nordering temperatures, magnetic coupling phenomena can be observed in particular, in superlattices. \nPeriodic structure s of superconductors and ferromagnets with many layers, magnetic coupling between \nsecond nearest neighbours  can be found indicating magnetic coupling via a superconducting \nlayer  [10] . This, however, could be experimentally shown only in superlattices where the \nsuperconducting part consist s of Y 0.6Pr0.4Ba2Cu3O7-x rather than YBa 2Cu3O7-x. \nTo obtain fundamental understanding of possible magne tic coupling phenomena in hybrid or \nheterostructures  of YBCO and LCMO we focus in this paper on the simplest possible structure, a \nbilayer of LCMO and YBCO on a single -crystalline substrate. This system was investigated by \nXRMR, a technique which provides structural and element selective magnetic depth prof iles with very \nhigh spatial resolution [11-13]. Determining the magnetization profile across the ferromagnetic layer by XRMR we can directly \ninvestigate the interaction at the LCMO/YBCO and the LCMO/substrate interface. Both results can act \nas a prerequisite f or the understanding of coupling phenomena in LCMO/YBCO superlattices.  \n \n2. Experiment al \nA 5x5 mm² SrTiO 3 (STO) single crystal with (100) cut and polished surface is used as a substrate. The \nsubstra te has been chemically treated  [14] to provide a TiO 2 terminated (100) surface; it was then \nmounted in a growth c hamber with a base pressure of 5· 10-4 Pa. Bilayers of LCMO and YBCO with  a \nnominal thickness of 400 Å each have been deposited by PLD at a growth temperature of 730 ° C at an \noxygen pressure of 50 Pa. Full oxygenation has been achieved by annealing the films at 530°  C in an \noxygen atmosphere of 105 Pa for ½ hour and then slowly cooling  down to room  temperature.  These \nparameters lead to epitaxial growth of LCMO on STO and the resulting LCMO single layer film with \na nominal thickness of 400  Å is a ferromagnetic metal with a TCurie of 240 K. Subsequently a 400 Å \nlayer of optimally doped YBCO was grown on top of the LCMO. The superconducting transition \ntemperature was experimentally derived from an identical sample using SQUID magnetometry and a \nvalue of T C=88 K is found which verifies the high quality and optimal  doping of the layer.  \nThe XRMR and XMCD measurements were carried out at the undulator beamline UE56/2 -PGM1 at \nBESSY II in Berlin, Germany. We use a dedicated soft X -ray magnetic reflectometry endstation built \nat the MPI for Metals Research  [15]. This setup allows measuring XRMR curves for angles of \nincidence of  Ɵ=0° to 70°. Additionally high- quality X -ray absorption (XAS) and X -ray magnet ic \ncircular dichroism (XMCD) can be measured by total electron yield (TEY) and fluorescence yield \n(TFY) providing in- situ spectroscopy data necessary for reflectometry simulations.  \nIn general it is not possible to obtain the optical profile directly from a  measured reflectivity curve. \nInstead, an iterative approach is used to indirectly obtain the optical profile by fitting a simulated \ncurve to the measurement. Using the simulation software ReMagX, which has been developed \nspecially for this purpose at the MPI for Metals Research  [16] , we can simulate magnetic reflectivity \nand asymmetry curves and fit them to the measured reflectivity data. This task is further complicated \nby the fact that XRMR has to be measured at a magnetic active resonant absorption edge, i.e. a \ntransition which shows circular dichroism ( usually L/M edges in the soft X -ray regime). The optical \nproperties at resonant edges show dramatic changes with the energy and the electronic configuration \nof the elements, e.g. oxidation state  [17] . It is therefore  imperative to experimentally obtain the optical \nand magneto- optical constants of the investigated sampl e [18] . By measuring the near -edge X -ray \nabsorption fine structure (NEXAFS) and also the XMCD for the relevant energy range, the optical and \nmagneto -optical constants can be derived (see e.g. in  [19,20] ). This information is then used for the \nsimulation of the XRMR curves while the structural (roughness/thickness) parameters are free \nparameters for the fit ting process. This method ultimately provides element selective structural and \nmagnetic depth profiles of a layered sample with very high spatial resolution  [11-13]. \n3. Results  \n3.1 Magnetometry & Magneto -Optics  \nSQUID magnetometry has been used to characterize the bulk magnetic properties of the \nLCMO/YBCO bilayer at T=100  K, i.e. above the superconducting transition. To avoid damaging the \nsurface of the sample intended for the XRMR investigation, a sister sample simultaneously prepared in \nthe same run for the deposition has been used for this measurement. The sample was mounted in a \nQuantum Design MPMS -XL 7 T SQUID, cooled from room temperature to T=100  K in zero field, and \na magnetic hysteresis was recorded with the magn etic field oriented parallel to the sample surface, i.e. \nin in-plane geometry . Figure 1 shows the resulting T=100 K hysteresis loop of the sample  which has \nbeen corrected for the diamagnetic contribution from the STO substrate .  \nFigure  1. Magnetic in -plane hysteresis measured at T=100K using a SQUID magnetometer . \nA clear ferromagnetic hysteresis with a coercive field of H C=25 mT and a saturation magnetization of \nMSat=(1.1±0.6)·10-4 emu is found. Using a value of  390 Å  for the thickness of the LCMO layer as \nrevealed by reflectometry (see below) and taking the lateral film size of 5x5  mm² (substrate size) into \naccount, the saturation magnetization is 𝑀𝑆𝑎𝑡=113  emu /cm3 which corresponds to a magnetic \nmoment per Mn atom of 0. 7𝜇𝐵. This value is much smaller compared to the bulk magnetization value \nof 3.7𝜇𝐵 for La 2/3Ca1/3MnO 3 [21] . From previous investigations of LCMO/YBCO bilayers it is known \nthat the magnetic moment of LCMO is affected by the presence and thickness of the YBCO layer  [2]. \nThe maximum saturation magnetization of LCMO decreases for increasing YBCO layer thickness and \nour results are compar able to th ose of Ref  [2]. \nThe superconducting properties of the YBCO layer were characterized using the magneto -optical \nFaraday Effect . This method uses an iron garnet indicator film to image the flux line density in the \nYBCO layer and allows determining the critical current density of the film  [22] . Figure 2 shows \nimage s of the magnetic flux density distribution (left) and critical current density distribution  (right)  in \nthe YBCO layer at a temperature of T=8  K measured in remanence [23] . The critical current density \ndistribution was from the magnetic flux density distribution derived using the above described \ntechn ique. \n \nFigure  2. Magnetic flux  density distribution (left) and calculated c urrent density distribution (right) in \nthe YBCO layer measured at T=8K in remanence. The average critical current density in the film is \n𝟕∙𝟏𝟎𝟏𝟎A/m² as  derived from the horizontal line profile (bold black line in the right image) . \nA maximal critical current density of 𝑗 𝑠=7∙1010 A/m² is found. Compared to bulk YBCO, this value \nis reduced by nearly  one order of magnitude which is related to the presence of the ferromagnet  [24] . \n3.2 Optical Constants  \nThe energy dependent optical properties of the individual layers, i.e. the complex indices of refraction \n𝑁𝑗(𝐸)=1−𝛿(𝐸)+𝑖𝛽(𝐸) where j is the layer index, are a necessary prerequisite for the \ndetermination of the structural and magnetic depth profile. While the imaginary (absorptive ) part, β , is \ndirectly accessible from a soft x -ray absorption measurement by fitting the data to tabulated \nvalues [25] , the real (dispersive) part, δ, can be calculated from β using the Kramers- Kronig \nrelation  [19]. \nAn X -ray absorption spect rum (XAS) was measured by total electron yield (TEY)  [26]  in the energy \nrange E=600  eV to E=700  eV. Linear polarized X -rays were used fo r the measurement and the sample \nwas set to an angle of incidence of Ɵ=45°. The resulting spectrum was then adapted to tabulated X -ray \nscattering factors [25]  to obtain the imaginary part of the scattering factor f 2 including the sample \nspecific L 2,3 edge  resonances. A Kramers- Kronig transform similar to the one described in [19] was \nused to calculate the real part f 1. Finally the optical theorem  [27]  links the optical constants δ and β  to \nthe forward scattering cross sections f1 and f 2 according to:  𝛿(𝐸)=𝑁𝐴𝜌\n𝐴𝑟0𝜆2\n2𝜋𝑓1(𝐸) and 𝛽(𝐸)=\n𝑁𝐴𝜌\n𝐴𝑟0𝜆2\n2𝜋𝑓2(𝐸). Here, r0 is the classical electron radius, N A is the Avogadro constant, A is the atomic \nmass number in (mass/mol), ρ is the mass density in (mass/volume) and λ is the wavelength of the \nincident X -rays.  The results are shown i n figure  3 for the energy range close to the Mn L 2,3 edges.  \n \nFigure  3. Optical constants for La 2/3Ca1/3MnO 3 in the vicinity of the Mn L edges derived from \nmeasuring a XAS and applying a Kramers- Kronig transform (see details in the text) . The indicated \nenergy of E=638 eV was used for the XRMR investigation. As can be seen, the dispersion dominates \nat this energy while the imaginary part of the index of refraction is  still small.  \nFor the calculation of δ and β from f 1 and f 2 it was assumed that the stoichiometry of the LCMO layer \nis exactly La 2/3Ca1/3MnO 3. The validity of this assumption will later be confirmed by the  high quality \nof the structural reflectometry fits presented  below . The indicated energy of E=638 eV is used for all \nsubsequent XRMR measurements. This particular energy provides a large scattering cross section (δ is \nmaximal) and nearly no resonant enhancement of the absorption. The optical constants of the YBCO \nlayer and the STO substrate were directly retrieved from tabulated values [25] . This is justified since \nall resonant edges in these materials (Cu, Ba, and O)  are far away from the energy where the XRMR is \nmeasured . \n3.3 X -ray Resonant  Reflectometry & Structural Characterisation  \nX-ray resonant reflectivity and XRMR are used to investigate the structural properties, i.e. layer \nthicknesses and roughness, of the bilayer system and the magnetic depth profile of the LCMO layer. \nThe structura l/chemical depth profile of the sample is obtained by measuring non- magnetic resonant \nreflectivity (XRR) curves, i.e. resonant reflectivity without XMCD contrast, and fitting a simulation to \nthem.  It is favourable to measure XRR and XRMR at the same energy  because it allows using the \nresults from XRR directly as input for the XRMR.  The structural depth profile is determined by the \nlayer thicknesses, the interface roughness and the chemical composition. The chemical compos ition \nwas used as a fixed input parameter  (nominal compositions) , while the roughness and thicknesses of \nall layers and the substrate were set as free parameters of the fitting process.  \nWe use the dedicated software tool “ReMagX” [16]  for simulation and fitting of the reflectivity data. \nReMagX  uses a 4x4 matrix formalism  [28]  which allows simulating the reflectivity from an arbitrary \nmagnetic layer stack . While standard reflectometry for optical isotropic media uses the Parratt  \nformalism  [29]  for simulation this technique is not ap plicable to the resonant magnetic data presented \nhere since since it neglects the optical anisotropic nature of the XMCD effect and therefore e. g. the \nitensity transfer from  σ to π polarization and vice versa  inside the stack . In contrast ReMagX \ncalculates  the full polarization dependent eigenfunctions  for each layer  and thus accounts for  the \nanisotropy and any intensity transfer. Interface r oughn ess and local variations of the magneto -optical \nproperties and hence the magnetic moment is incorporated into the matrix description by slicing the \nsample [16,30] . The l ocal variations of the magneto -optical constants can be introduc ed by  Gaussian \nprofiles with a var iable position, FWHM and height. This either m odifies an existing magnetic \nmoment, i.e. magneto- optical contribution or it introduces a new, artificial magnetic signal . More \ndetails and also restrictions of the XRMR data processing and can be found e.g. in  references [17-\n19,31] . \nFigure 4 shows the resonant non- magnetic reflectivity measured at T=110  K at an X -ray energy of \n𝐸ℎ𝜈=638 eV using linear horizontally polarized light along with the best simulation result (red solid \ncurve) obtained from fitting the data.  \n \nFigure  4. Resonant X -ray reflectivity curve without magnetic contrast measured at T=110  K using an \nX-ray energy of 638 eV and linear polarized light. The red curve is the best fit result. The inset shows \na close up of the curve for q z=0.0625 to 0.2 Å-1 together with the results obtained at T=40  K and \nT=300  K. Note that the curves have been shifted vert ically for better readability. No significant change \nof the non- magnetic reflectivity and therefore also the underlying chemical structure is found.  \nThe simulation perfectly reproduces the measured reflectivity curve in the range from q z=0.08 to 0.275  \nÅ-1 where the relevant, structure determined, features are located. The deviation for small angles is \nrelated to the incomplete illumination of the sample. The non- magnetic reflectivity was also measured \nfor T=300  K and T=40  K to verify that the chemical stru cture does not change with temperature. A \nclose -up of all three measured curves in the range from q z=0.0625 to 0.2 Å-1 is shown as inset in  figure  \n4 together with the corresponding simulation results.  \nFrom the simulation shown i n figure  4, the optical depth profile , which corresponds to the structural \ndepth profile, is derived and the result is shown  in figure  5. \n \nFigure  5. Optical, i.e. structural, depth profile of the sample measured at an X -ray energy of E=638 \neV and a temperature of T=110 K.  \nA thickness of (390±2)  Å and (403±5)  Å is found for the LCMO and YBCO layer, respectively. Both \ninterfaces are very smooth showing a RMS roughness of (9.5±0.1)  Å for the LCMO/YBCO interface \nand only 1 Å for the STO/LCMO interface. The errors were derived from comparing the three \ntemperatures measured.  \n3.4 X -ray Resonant Magnetic Reflectometry  \nA series of temperature dependent XRMR curves were measured for Ɵ ranging from Ɵ=0° to 27.75° \n(corresponding to q z=0 to 0.3  Å-1) in an applied  external magnetic field of H ext=130 mT parallel and \nantiparallel to the sample surface (in -plane geometry). X -rays with an energy of E=638 eV (Mn L 3 \nedge) with positive helicity were used and the external magnetic field was flipped at every angle value \nfrom positive to negative and the two specular reflected intensities were measured to obtain the \ncircular dichroic reflectivity curves. Subsequently the measurement was repeated for negative helicity \nof the X -rays.  \nTo extract the magnetic signal from the two  dichroic curves, the asymmetry ratio between the two is \ncalculated according to 𝐴=(𝑅↑↑−𝑅↑↓)/(𝑅↑↑+𝑅↑↓). Here the first arrow indicates the helicity of the \nX-rays ( ↑= positive and ↓ = negative circular polarized light) and the second arrow indicates the \ndirection of the external magnetic field ( ↑= positive and ↓ = negative) with respect to the direction of \nthe incident X -rays.  \nFirst the magnetic asymmetry signal associated with Mn was measured at a temperature of T=110  K \nwhich is well above the superc onducting transition temperature . Figure 6 shows the resulting \nasymmetry ratio together with the best simulation result obtained from fitting the data.  \n \nFigure  6. Element  selective Mn asymmetry ratio  between R↑↑ and R↑↓ measured at an energy of E=638 \neV. The curve is the average of the positive and negative helicity measurements. The red solid curve \nrepresents the best fit result based on the magneto -optical profile shown in  figure  7. \nA small asymmetry ratio on the order  of 2.5% is found originating from ferromagnetic Mn in the \nLCMO layer. This asymmetry ratio exhibits two major features: a short range oscillation which is mostly related to the YBCO thickness and a long range modulation which should originate from a \nsmall er structure in the magneto -optical profile. Analyzing the short range oscillation yields \nΔq\nz=0.0079 Å-1 which corresponds to a thickness of 397 Å. This value  corresponds , within the error, \nto the structural properties of the YBCO layer (compare  figure  4). To obtain the magneto- optical \nprofile connected with the asymmetry signal, the data has been fitted  [16] and the fit result is also \nshown in the plot as a red solid curve. The fit reproduces the major features of the asymmetry very \ngood. The fitting process consists of two steps: first magneto -optical constants for LCMO were \nderived from XMCD data of ferroma gnetic LCMO and rescaled to account for the reduced magnetic \nmoment of Mn in this sample (see Sec. III). In a second step the magneto -optical constants were used \nto fit the asymmetry signal while allowing local variations (enhancement or suppression) of th e \nmagneto -optical profile  and hence the saturation magnetization . The result is shown  as inset  in figure  \n7. \nThe overall shape of the asymmetry ratio, especially the long range modulation can only be \nreproduced if a layer with reduced or completely suppress ed magnetization close to the YBCO is \ninserted into the magneto- optical profile. Compared to the overall thickness of the LCMO film, this \nfeature is quite narrow . Therefor e, figure  7 shows a close- up of the interface region between LCMO \nand YBCO while the full magneto -optical profile is shown as inset in the same figure . \n \nFigure  7. Magnified structural and magnetic profile close to the LCMO/YBCO interface. Note both \ngraphs were normalized to the bulk value of the LCMO. The magnetic profile sets in roughly 10 Å \nbeneath the structural transition. The magnetic moment close to the inter face is strongly reduced. The \ninset to the upper right shows the absolute overall magneto- optical profile for the LCMO layer.  \nHere the Mn density (blue) and the magnetic signal (red open circles) are plotted in percent with \nrespect to the bulk layer proper ties. The magnetic profile is shifted compared to the structural interface \nby 9.6 Å which means that the LCMO directly at the interface to the YBCO has a reduced or zero \nmagnetic moment. Note, that the small asymmetry makes it difficult to derive the exact magnetization \nat the interface. Therefore w e cannot exclude a small remaining mom ent for the LCMO directly at the \ninterface; the magnetic signal i n figure  7 can still refer to a ferromagnetic ordered state but with a \nstrongly suppressed magnetization value.  \nThe existence of a layer of suppressed LCMO magnetization at the interface to Y BCO  was already  \nreported for some FM/SC superlattices [32,33] . Our res ults show that a suppressed layer already \noccurs at a single FM/SC interface and does not depend on the superconducting state of the YBCO in \nthe bilayer system since it is already present at T=110  K. This latter observation is consistent with \nmagnetization  measurements of YBCO/LCMO bilayers which clearly show a YBCO thickness related \ndecrease of the LCMO saturation moment already well above the superconducting transition  (see \nfigure  6 in reference [2]). \nIn contrast to the results presented here and  in [32,33] , a uniform magnet ization of the LCMO layer, \ni.e. no suppression at the interface, was also found in  superlattices  [6,10] . A possible explanation for \nthe contradictory results might be related to  the roughness of the LCMO/YBCO interface. There  is a \nstriking coincidence of the interface roughness ( 𝜎𝑟𝑚𝑠 =9.5 Å) as revealed by X -ray reflectometry and \nthe shift of the structural and magnetic profile (see figure 7). This interface roughness – substantially \nlarger than  in the YBCO/LCMO superlattices [6,10]  – could be accounted for the suppression of the \nmagnetization at the interface . This interpretation is further stressed by r ecent high -resolution \ntransmission electron microscopy (HRTEM) result s of a YBCO/LCMO/YBCO trilayer revealed that \nthe interface properties and also the roughness strongly  depend on whether YBCO is grown on top of \nLCMO or vice versa [34] . In our case we find a larger interface roughness for YBCO grown on \nLCMO with reduced interface magnetization in the LCMO. The second explanation is related to the \nthickness of the YBCO layers. As shown in  figure  6 of Ref.  [2], the magnetic moment in LCMO \ndecreases for larger YBCO thickness, revealing suppression of the FM.  While the superlat tices in \nRef. [6,10]  were prepared for thin YBCO (100  Å) the samples investigated here have four times higher \nthickness, consistent to the observation of the reduced FM.  \n \nFigure  8. Magnetic asymmetry ratio measured at T=40  K and an X -ray energy of 638 eV.  \nFigure 8 depicts  the analogue measurement to  figure  6, now at a temperature of T=40 K, much below \nthe superconducting transition of the YBCO layer. Similar to the high temperature data, we can \nsimulate the asymmetry nearly perfectly by our algorithm.  \nIn addition to that, we find a striking difference between the results at T=110  K and T=40 K, \nrespectively. The asymmetry curve at T=40  K exhibits a clear increase at q z values of q z=0.2 Å-1 and \nabove. A wide maximum seems to occur for high momentum transfer. Since the observe d \nreflectometry oscillations  refer to inverse length scales as in reciprocal space, a wide maximum refers \nto a narrow structure in the magnetization profile. Moreover, the fact that it occurs at large q z values \ndirectly indicates that this feature has to b e buried in a large depth under the surface. \nOur algorithm again allows the determination of the corresponding magnetic profile which is depicted \nin figure 9 . At a temperature of T=40  K a new feature of the magnetization profile across the LCMO \nlayer is found. A sharp peak where the local magnetization is roughly doubled is found close to the \ninterface to the substrate. Obviously there is an interaction between the LCMO film and the STO \nsubstrate a low temperatures leading to an additional magnetic effect.  \nThe presence of a magnetization increase close to the substrate in a YBCO/LCMO superlattice was \nalready reported from neutron scattering  [35]. We confirm this finding and show in the following an \nanalysis of the temperature dependence of the effect providing deeper insights into its origin.  \n \nFigure  9. MO profile for the absorptive part of the index of refraction in the vicinity of the \nLCMO/STO interface obtained for the T=40  K asymmetry. The MO constants are both increased by \nmore than 100% which indicates an increase of the Manganese magnetic moment per atom to 𝟏.𝟒𝟒𝝁𝑩 \nwith respect to the SQUID derived bulk value.  \nTo discuss this finding systematically, we p erformed the analogue analysis to  figure  6 and figure  8 at \nadditional temperatures. For a quantitative statement, the asymmetry curves at T=40  K, T=65 K, T=80  \nK, and T=110 K have been fitted and the result is plotted in a new graph which is shown in figure  10. \n \nFigure  10. Temperature dependence of the increased Mn magnetic moment close to the STO \ninterface. The lower part shows all measured asymmetry curves along with the corresponding fit \nresults. The upper section shows the percent increase o f the Mn magnetization at the STO interface \n(see figure  9) derived from the simulation curves.  \nAll four temperatures can be fitted using the same general shape of the magneto -optical profile close \nto the STO interface (compare figure  9). The increased noise level of the T=65  K and T=80  K \nmeasurements is related to a reduced integration time for these measurements. Nevertheless the \ngeneral trend is clearly visible from the curves and can be fitted successfully. We clearly see that the \neffect continuously  decreases with increasing temperature and seems to van ish around T=100K.  \nThe observed temperature- dependence of the LCMO/substrate interaction is tentatively considered to \nbe related to the structural properties of STO single -crystals at low temperatures.  Below T=105K a \nstructural phase transition from cubic to tetragonal occurs which in our case directly leads to strain at \nthe LCMO/STO interface [36,37] . Since the temperature dependence of the additional magnetization \npeak looks similar to the structural data in the same temperature range (see e.g. in  [37] ) it is \nreasonable to imply a direct correlation of the magnetization increase and the tetragonal distortion of \nthe STO.  \nThis result is contradictory to other investigations about the effect of compressive strain on the \nmagnetic properties of LCMO which show that the magnetization is usually decreased in such \nfilms  [38- 41]. Howe ver these investigations used samples with a static strain e.g. by using different \nsubstrates or by varying the layer thickness. Moreover they were obtained by using standard magnetometry which measures  the whole film and not a local variation of it. The i nvestigation \npresented here is very different in two important points: the XRMR technique detects even small \nlocalized changes of the magnetization profile. In the present case the observed increase at the \nLCMO/STO interface is less than 1% (for T=40  K) of  the total LCMO magnetic moment. Such a \nsmall signal is hard to detect with standard , bulk sensitive  magnetometry. The second important \ndifference is that the increase of the magnetization observed in the present work is reversible. The \nmeasurement sequence was 40  K  110 K  64 K  80 K and the effect is visible at T=40  K, gone \nat T=110 K and comes back when cooling again to 64 K (see figure  10). This implies that the strain \nwhich is responsible for the effect is reversible. Therefore it might not be possi ble to directly compare \nthe current results with investigations on growth induced strain.  \nInstead we propose a scenario where the growth induced strain from the substrate  [41]  and strain \ncaused by the presence of the YBCO layer  [2] create a thickness dependent state of stress in the \nLCMO layer which is responsible for the reduced magnetic moment in the bulk  observed here . The \ntetragonal distortion of the STO below T=105 K  disturbs this balance and allows a relaxation of the \nfirst few monolayers of the LCMO which causes a local increase of the saturation moment. This \nscenario would also explain why the effect was not yet observed for pure LCOM layers (see e.g. in \n [42] ). \n4. Conclusion  \nIn conclusion, we have found that the magnetization profile across ferromagnetic LCMO layers in bilayers of LCMO/YB CO on STO substrates is strongly affected by the interfaces to the adjacent \nmaterials. At all temperatures we find a suppression of the LCMO magnetization close to the YBCO \nlayer on a length scale of 10Å, the transition to superconductivity seems not to in fluence this finding. \nA second effect occurs at the LCMO/STO interface at low temperatures. Owing to the structural phase transition in STO single crystals at T=105 K a peak -like magnetization increase develops inside the \nLCMO layer close to the substrate interface which has its origin in strain arising from the STO \nsubstrate.  \nAcknowledgement  \nThe authors are indebted to  B. Keimer, E. Benckiser, and V. Hinkov for helpful  discussions . Support \nby B. Zada , W. Mahler and the whole BESSY II team during the beamtime  is gratefully \nacknowledged . \nReferences  \n[1] C. A. R. Sá de Melo, Phys. Rev. Lett. 79, 1933 (1997).  \n[2] S. Soltan, J. Albrecht, and H. -U. Habermeier, Phys. Rev. B 70, 144517 (2004).  \n[3] Z. Sefrioui, D. Arias, V. Peña, J. E. Villegas, M. Varela, P. Prieto, C. León, J. L. 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B 80 , 052403- 4 (2009).  \n "    },    {        "title": "2402.01179v1.Mechanism_of_ferromagnetism_enhancement_in_a_La___2_3___Sr___1_3___MnO__3__membrane_released_from_epitaxial_strain.pdf",        "content": "1 Mechanism of ferromagnetism enhancement in a La2/3Sr1/3MnO3 membrane released from epitaxial strain Takahito Takeda,1* Takuma Arai,1 Kohei Yamagami,2 Le Duc Anh,1,3 Masaaki Tanaka,1,3,4 Masaki Kobayashi,1,3 and Shinobu Ohya1,3,4† 1Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Japan Synchrotron Radiation Research Institute (JASRI), 1-1-1 Kouto, Sayo, Hyogo, 679-5198, Japan 3Center for Spintronics Research Network, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 4Institute for Nano Quantum Information Electronics (NanoQuine), The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo153-8505, Japan Corresponding authors: *takeda@cryst.t.u-toky.ac.jp, †ohya@cryst.t.u-toky.ac.jp Abstract Recent studies have shown that the magnetic properties of the ferromagnetic perovskite oxide La2/3Sr1/3MnO3 (LSMO) grown on an SrTiO3 (STO) substrate, such as its magnetic moment and Curie temperature, can be improved by releasing the film from the substrate. However, the microscopic origin of this enhancement is not yet well understood. In this study, we use synchrotron radiation measurements to investigate the mechanism of ferromagnetism enhancement in an LSMO membrane released from an STO substrate by dissolving a water-soluble Sr4Al2O7 buffer layer. Using resonant photoemission spectroscopy on the as-grown LSMO film and LSMO membrane, we elucidate that the strain release from the STO substrate enhances the itineracy of the Mn-3d electrons via p–d hybridization, and this strengthens the double-exchange interaction. The reinforcement of the double-exchange interaction, in turn, improves the ferromagnetism of LSMO. 1. Introduction Perovskite oxides have attracted significant attention due to their rich properties and potential device applications [1–4]. In particular, La1−xSrxMnO3 is a promising ferromagnet due to it having a Curie temperature (TC) above 300 K and exhibiting half-metallicity [5,6]. The ferromagnetism of La1−xSrxMnO3 is based on the double-exchange interaction within Mn3+-O-Mn4+ [7]: Mn 3d electrons hop to the neighbor Mn ions through the O ion between them and align the spin direction of the Mn ions. The Mn 3d electrons are responsible for the transport and magnetic properties. Some experimental studies have demonstrated the excellent performance of La1−xSrxMnO3-based spintronic devices; specifically, an extremely high tunnel magnetoresistance ratio of ⁓1900% has been found in La2/3Sr1/3MnO3 (LSMO)-based magnetic tunnel junctions [8,9], and a giant spin-valve ratio of 140% has been reported in La1−xSrxMnO3-based lateral spin-MOSFET devices [10]. However, these outstanding features appear only at low temperatures since the magnetization and conductivity of La1−xSrxMnO3 are significantly reduced near 2 interfaces with other perovskite oxides [11,12]. This interface layer is called the “dead layer.” The origin of dead layers has been attributed to various interfacial effects such as strain and oxygen octahedral rotation [13–15]. Recently, epitaxial lift-off of La1−xSrxMnO3 thin films from a substrate using a water-soluble sacrificial buffer layer has attracted substantial interest as a promising approach to solving the dead-layer problem [16–22]. Some studies have reported that the magnetization and/or TC values is improved  [17,19,21,22] and that the dead layer is reduced [19,22] by the epitaxial lift-off. For device applications, it is essential to understand the mechanism of the enhancement of ferromagnetism by epitaxial lift-off. In this study, we investigate the relationship between the magnetism and electronic structures of an as-grown LSMO film and an LSMO membrane to consider the mechanism of ferromagnetism enhancement in the LSMO membrane. Resonant photoemission spectroscopy (RPES) is used to measure the occupied Mn-3d partial density of states (PDOS) [23–25], and x-ray absorption spectroscopy (XAS) is used to study the unoccupied Mn-3d states [26,27]. The RPES spectra of the LSMO membrane show more itinerant features of the Mn-3d electrons than those of the as-grown LSMO film. Our findings suggest that strain release as a result of the epitaxial lift-off enhances the p–d hybridization and improves the itineracy of the Mn-3d electrons, which in turn reinforces the double-exchange interaction. 2. Experiments We epitaxially grew a heterostructure composed of LSMO (30 unit cells ~12 nm) / Sr4Al2O7 (17 nm) on an SrTiO3 (STO) (001) substrate by molecular beam epitaxy [Fig. 1(a)]. The detailed growth conditions were the same as those described elsewhere [22]. To transfer the LSMO layer onto a p-Si substrate [16], we attached a polydimethylsiloxane (PDMS) film to the as-grown LSMO surface and immersed the sample in deionized water for one day. After dissolving the Sr4Al2O7 layer, we adhered the LSMO surface to the p-Si substrate by heating it for 10 min at 70°C. By releasing the PDMS film, the LSMO membrane was transferred onto the p-Si substrate, as shown in the microscope image of Fig. 1(b). The as-grown LSMO film is fully tensile-strained, and the in-plane lattice constants are the same as those of the STO substrate; in contrast, the LSMO membrane is epitaxial-strain-free and the lattice constants are the same as those reported for an LSMO single crystal [22]. The saturation magnetization at 5 K and the TC value of the LSMO membrane (as-grown LSMO film) are 3.2 µB/Mn (2.0 µB/Mn) and 323 K (312 K), respectively [22]. We performed XAS and RPES experiments at beamline BL25SU of SPring-8 [28,29]. The measurements were conducted under an ultrahigh vacuum below 2.0 × 10−8 Pa at 77 K. The angle between the hemispherical analyzer and the incident x-rays was set at 90°. For the measurements, the sample was irradiated at an angle of 7° from its surface with a circularly polarized x-ray beam with its photon energy (hν) varied from 630 to 680 eV. The total energy resolution, including thermal broadening, was about 100 meV . We determined the Fermi level (EF) positions in the RPES spectra by measuring the Fermi-edge position of a deposited Au film in electrical contact with each of the samples. Mn L3 x-ray absorption spectroscopy (XAS) spectra were obtained in the total-electron-yield mode. Note that the transfer process does not dramatically change the electronic structure near the film surface [30]. 3 3. Results and discussion We studied the Mn-3d electronic structures of the as-grown LSMO film and LSMO membrane using the XAS and RPES measurements. The XAS spectra at the Mn L3 absorption edge originate from electron excitation from the Mn-2p3/2 states to the unoccupied Mn-3d states. The obtained XAS spectra for the as-grown film [right panel of Fig. 2(a)] and membrane [right panel of Fig. 2(b)] were found to have similar spectral shapes to those previously reported for LSMO films [31–35]; those shapes reflect the mixed valence states between Mn3+ and Mn4+. To study the occupied Mn-3d electronic structures, we measured the hν dependence of the RPES spectra at the Mn L3 absorption edge [left panels in Figs. 2(a) and 2(b)], where EB represents binding energy. The used hν values are expressed by the horizontal broken lines drawn on the XAS spectra in the right panels of Figs. 2(a) and 2(b). Resonance enhancement of the RPES signal occurs at above hν = ~639 eV, while the spectra taken at hν = 637.5 eV for the as-grown film and 637.4 eV for the membrane are not enhanced. Here, we refer to those spectra (taken with hν = 637.5 and 637.4 eV , black bold curves) as the off-spectra. The RPES spectra have peaks whose positions depend on hν (shown with triangles as guides to the eye) and those whose positions do not depend on hν (black dashed vertical lines). The former and latter represent the Auger components and Mn-3d-related states, respectively. The Mn-3d-related components taken at hν = 641 eV of the membrane are different from those of the as-grown film, which are discussed later. To emphasize the resonantly enhanced components, we plot the second-derivative mappings of the difference spectra between the off-spectra and the RPES spectra taken at various hν values, as shown in Figs. 3(a) and 3(b). The mappings show several Mn-3d-related components (green dashed lines) and two Auger components (red dashed lines). The Auger components, especially the component represented by the red dashed bold line, are a little more distinct in the LSMO membrane than in the as-grown film. Generally, the Auger components are observed when electrons that are excited from Mn-2p states to unoccupied 3d states relax to different states before compensating for the Mn-2p hole formed by the excitation [36–38]. Thus, the Auger components reflect the degree of the delocalization of photoexcited Mn-3d electrons. Our results suggest that the Mn-3d electrons in the membrane are more itinerant than those in the as-grown film; as such, the enhancement of the itineracy is consistent with the improvement of the conductivity by the epitaxial lift-off [16]. To strengthen the evidence for this scenario, we examined the Mn-3d-related states of the as-grown film [Fig. 4(a)] and membrane [Fig. 4(b)] using the difference spectra between the off-spectrum and RPES spectra (at hν = 641 and 642.5 eV). There are five structures, labeled A–E, assigned to the Mn-3d eg band, the Mn-3d t2g band, the O-2p-derived mixing state composed of O 2p and Mn 3d, the Mn-3d-t2g–O-2p bonding state, and the Mn-3d-eg–O-2p bonding band [23,39]. At hν = 642.5 eV, the difference spectrum of the membrane has a spectral shape similar to that of the as-grown film. In contrast, the difference spectra taken at hν = 641 eV substantially differ in shape; the membrane has larger intensities at structures C and D than at structures B and E, while the as-grown film shows the opposite trend. Since structure C mainly derives from the O-2p states mixed with the Mn-3d states [40], the observed difference indicates that the membrane has a stronger mixing of Mn-3d and O-2p electrons through p–d hybridization than the as-grown film. 4 To analyze the behavior of the resonance enhancement, we plot the constant-initial-state (CIS) spectrum, which is a plot of the RPES intensity at constant EB as a function of hν. Figures 5(a) and 5(b) show the CIS spectra near EF (EB = 0.25 eV, see the arrows in Fig. 4) and at structure C (EB = 3.4 eV for the as-grown film, and EB = 3.5 eV for the membrane). Since the resonance enhancement is roughly proportional to the intensity of the super Coster–Kronig (SCK) decay involved in the x-ray-absorption process [41], the CIS spectra have similar spectral shapes to the XAS spectra. In Figs. 5(a) and 5(b), the peak positions of the CIS spectra (around 640.2, 641, and 642.5 eV, see the broken vertical lines) are close to the shoulder and peak positions of the Mn L3 XAS spectra (black and gray curves). The CIS spectra near EF and at structure C have similar shapes to the Mn3+ and Mn4+ L3 XAS spectra, respectively [31,42]. Thus, in the Mn L3 XAS spectra, the shoulder at hν = ~640.2 eV and peak at hν = ~642.5 eV mainly originate from Mn3+, and the shoulder at hν = ~641 eV is induced by Mn4+. The Mn4+-like feature of the CIS spectrum at structure C can be attributed to the stronger O-2p–Mn-3d hybridization of Mn4+ than that of Mn3+ due to the negative charge-transfer energy between Mn4+ and O2− ions [43–45]. The Mn3+-like shape of the CIS spectra near EF is explained by the PDOS near EF derived from the eg electrons of Mn. The distance between the peaks of the CIS spectrum near EF is 2.4 eV, as illustrated by the red arrow in Fig. 5(b). This value is consistent with the energy difference (2–3 eV) between the eg↑ and t2g↓-derived states, where ↑ and ↓ denote the majority and minority spins, respectively [40,46,47]. Thus, the resonant enhancements at hν = ~640.2 eV and 642.5 eV occur in the following processes (i) and (ii), respectively: (i) Mn 2p63d4(t2g↑3eg↑1) + hν → Mn 2p63d3(t2g↑3) + e− (direct photoemission) Mn 2p63d4(t2g↑3eg↑1) + hν → Mn 2p53d5(t2g↑3eg↑2) → Mn 2p63d3(t2g↑3) + e− (SCK decay) (ii) Mn 2p63d4(t2g↑3eg↑1) + hν → Mn 2p63d3(t2g↑3) + e− (direct photoemission) Mn 2p63d4(t2g↑3eg↑1) + hν → Mn 2p53d5(t2g↑3eg↑1 t2g↓1) → Mn 2p63d3(t2g↑3) + e− (SCK decay) The influence of strain release and the associated band-structure change are apparent in the CIS spectra near EF in Figs. 5(a) and 5(b), in which it can be seen that the membrane has sharper structures than the as-grown film. The as-grown film is under tensile strain, and the membrane has the same lattice constants as single-crystal LSMO [22]. In single-crystal LSMO, the lattice is under rhombohedral distortion and has D3d symmetry without the Jahan–Teller (JT) effect [5,48]. Similarly, the LSMO membrane has D3d symmetry, splitting the t2g state into the a1g and twofold-degenerate 𝑒!\" states, and the eg state remains twofold degenerate [right panel of Fig. 6(a)]. Meanwhile, for the as-grown LSMO film, due to the tensile strain (and the JT effect for Mn3+), LSMO has D4h symmetry, splitting the eg state into the a1 (𝑑#$!%&!) and b1 (𝑑'!%(!) states, and the t2g state to the e and b2 states [left panel of Fig. 6(a)]. The CIS spectra near EF reflect the unoccupied Mn-3d states. Thus, for the as-grown film, the splitting of the unoccupied states makes the spectrum broader [see Figs. 5(a) and 6(a)]. The shape of the CIS spectrum at structure C for the as-grown film is relatively similar to that for the membrane. In the O-2p–Mn-3d hybridized states, the broadness of the O-2p states, whose energy scale is much larger than the band modulation by strain, probably leads to the observed insensitivity to epitaxial strain. Due to strain release, the epitaxial lift-off shifts the EF position from the b1 (and a1) state for the as-grown film to the eg state for the membrane [Fig. 6(a)] [27,49,50]. Thus, 5 in the membrane, 𝑑'!%(! and 𝑑#$!%&! electrons contribute to the double-exchange interaction through p–d hybridization with O-2p electrons, more specifically between the 𝑑'!%(! and px/py orbitals and between the 𝑑#$!%&! and pz orbitals due to the orbital symmetry. In contrast, in the as-grown film, the 𝑑'!%(! electrons of the b1 state mainly cause the double-exchange interaction via p–d hybridization with the px/py orbitals. The tensile strain increases the bonding length between Mn and O ions in the (001) plane and weakens the p–d hybridization, as shown in Fig. 6(b). Therefore, the epitaxial lift-off strengthens the p–d hybridization. The disappearance of the tensile strain in the membrane enhances the double-exchange interaction. 4. Summary Using RPES, we have unveiled the mechanism of ferromagnetism enhancement in an LSMO membrane released from an STO substrate. Our RPES measurements on the as-grown LSMO film and LSMO membrane demonstrate that the release from the tensile strain caused by the STO substrate enhances the itineracy of the Mn-3d electrons via the strengthened p–d hybridization between the Mn-3d and O-2p electrons. The itinerant feature of the Mn-3d electrons enhances the ferromagnetism via the double-exchange interaction within Mn3+-O-Mn4+ in the LSMO membrane. Acknowledgments This work was supported partly by Grants-in-Aid for Scientific Research (Nos. 23H03802, 22H04948, 22K18293, 20H05650), CREST (No. JPMJCR1777), ERATO (JPMJER2202) of the Japan Science and Technology Agency, Spintronics Research Network of Japan (Spin-RNJ). Supporting experiments at SPring-8 were approved by the Japan Synchrotron Radiation Research Institute (JASRI) Proposal Review Committee (Proposal No. 2023B1357). 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Sample structures of the as-grown LSMO film and LSMO membrane. (a) Schematic of the heterostructure of the as-grown LSMO film. (b) Microscope image of the LSMO membrane transferred to a p-Si substrate.   \nSTO sub.Sr4Al2O717 nmLSMO 12 nm\n1.0 mm(a)(b)p-SiLSMO11  FIG. 2. RPES (left panels) and XAS (right panels) spectra at the Mn L3 absorption edge of (a) the as-grown film and (b) the membrane. The vertical broken lines and triangles are guides of the Mn 3d-related components and the two Auger components, respectively. The dotted horizontal lines on the XAS spectra of the right panels represent the hν values that were used for measuring the RPES spectra.   650645640635hν (eV)XAS (arb. units)650645640635hν (eV)XAS (arb. units)Intensity (arb. units)1050EB (eV)Intensity (arb. units)1050EB (eV)(a)\n(b)As-grown LSMO thin film\nLSMO membraneAs-grown\nMembraneMn L3XAS\nMn L3XASRPES\nRPES12  FIG. 3. Second-derivative mappings of the difference spectra between the off-spectra and RPES spectra [see Figs. 2(a) and 2(b)] for (a) the as-grown film and (b) the membrane. The dashed green and red lines represent the Mn 3d-related components and Auger components, respectively.   \n646644642640638Photon energy (eV)1050EB (eV)1050EB (eV)(a)(b)\nAs-grownMembrane\n1.00.50LowHigh13  FIG. 4. Difference and off-spectra of (a) the as-grown film and (b) the membrane. The purple and green curves were obtained by subtracting the off-spectra from the purple (hν = 642.5 eV) and green (hν = 641 eV) RPES spectra in Figs. 2(a) and 2(b), respectively. The off-spectrum of the membrane is multiplied by 3.   Intensity (arb. units)Intensity (arb. units)86420EB (eV)AAB\nBCCDDE\nE(a) As-grown(b)Membrane Off 641 eV 642. 5 eVX 3 Off 641 eV 642. 5 eV14  FIG. 5. Comparison between the XAS and CIS spectra at the Mn L3 absorption edge of (a) the as-grown film and (b) the membrane. The blue and yellow CIS spectra were taken at structure C and near EF (EB = 0.25 eV), respectively. The black and gray curves represent the Mn L3 XAS and the second derivative of the XAS spectra, respectively.   10.50Intensity (arb. units)10.50Intensity (arb. units)646644642640638636hν (eV)(a) As-grown\n(b) Membrane Mn L3 XAS 2nd Deriv. XAS CIS of C CIS near EF\nShoulder of Mn3+Mn4+15  FIG. 6. Schematic illustrations of the influence of the tensile strain on (a) the Mn-3d electronic structures and (b) p–d hybridization. (a) Mn-3d states of the as-grown film (tensile-strained) in the left panel and the membrane (epitaxial-strain-free) in the right panel, in which the solid and dotted black curves represent the majority- and minority-spin states, respectively. In the (epitaxial-strain-free) membrane, Mn-3d states split into the eg, a1g, and 𝑒!\" states. In the as-grown (tensile-strained) film, the tensile strain and Jahan–Teller effect split the eg state into the a1 and b1 states, and the t2g state is similarly split into the e and b2 states. (b) p–d hybridization in the zx plane of tensile-strained (left panel) and epitaxial-strain-free (right panel) LSMO. The red and blue circles represent the Mn and O ions. The transparent red and blue areas represent the distribution of the Mn-3d (𝑑#$!%&!, 𝑑'!%(!) and O-2p (px, pz) orbitals, respectively. The dotted red curves represent the unoccupied 𝑑#$!%&! state. Under tensile strain, the bond lengths along the x (z) axis are increased (decreased). DOSeg↑a1g↑EFDOSe↑b2↑b1↑a1↑EFBinding energyEpitaxial lift-off(a)\n(b)As-grown (tensile strain)Membrane (epitaxial-strain-free)\neg↑πzxpzOMndx2-y2pxd3z2-r2zxpzpxdx2-y2OMnd3z2-r2a1↓e↓b2↓b1↓eg↓a1g↓eg↓π"    },    {        "title": "0806.4612v1.Field_dependent_AC_susceptibility_of_itinerant_ferromagnets.pdf",        "content": "arXiv:0806.4612v1  [cond-mat.str-el]  27 Jun 2008Field-dependent AC susceptibility of itinerant\nferromagnets\nM D Vannette\nR Prozorov\nAmes Laboratory and Iowa State University, Dept. of Physics & As tronomy, Ames,\nIA 50011\nE-mail:vannette@iastate.edu\nE-mail:prozorov@ameslab.gov\nAbstract.\nWhereas dc measurements of magnetic susceptibility, χ, fail to distinguish between\nlocal and weak itinerant ferromagnets, radio-frequency (rf) me asurements of χin the\nferromagnetic state show dramatic differences between the two. We present sensitive\ntunnel-dioderesonatormeasurementsof χinthe weakitinerantferromagnetZrZn 2ata\nfrequencyof23MHz. BelowCurietemperature, TC≈26K,thesusceptibilityisseento\nincrease and pass through a broad maximum at approximately 15 K in z ero applied dc\nmagnetic field. Application of a magnetic field reduces the amplitude of the maximum\nand shifts it to lower temperatures. The existence and evolution th is maximum with\napplied field is not predicted by either the Stoner or self-consistent renormalized(SCR)\nspin fluctuations theories. For temperatures below TCboth theories derive a zero-field\nlimit expression for χ. We propose a semi-phenomenological model that considers\nthe effect of the internal field from the polarized fraction of the co nduction band on\nthe remaining, unpolarized conduction band electrons. The develop ed model accurate\ndescribes the experimental data.\nPACS numbers: 75.40.Gb, 75.50.Cc, 75.30.Cr\nSubmitted to: J. Phys.: Condens. MatterField-dependent AC susceptibility of itinerant ferromagn ets 2\nDC measurements of magnetic susceptibility fail to distinguish betwe en local and\nitinerantferromagnets. Acommonmethodofdetermining χdcismeasuringthemagnetic\nmoment and then dividing by the applied field, H. However, this is only applicable\nprovided the magnetization is linear in HfromH= 0 up to the measurement field.\nFor magnetically soft or small moment ferromagnets, this criterion may not be satisfied.\nPerhaps a more careful method is to measure Min two slightly different magnetic fields\nand then calculate ∆ M/∆Has shown in Fig. 1. Further, due to limited sensitivity,\nDC measurements are usually conducted in significant magnetic fields , on the order of\n1-10 Oe. In exceptionally soft materials these fields may be sufficient to smear certain\nzero-field features.\nIn itinerant systems the situation is even more complicated. In orde r to deduce the\nsize of a magnetic moment per ion, one has to be in a saturation regime by applying a\nlarge field. However, this tells nothing about the magnitude of this mo ment in zero field\nthat, in itinerant systems, is field - dependent. Yet, even with a con ventional definition,\nobservation of a magnetic moment that is a fraction of a Bohr magne ton per ion is\nnot bulletproof evidence for the itinerant nature of magnetism. Fo r example, the local-\nmoment metallic rare-earth compound CeAgSb 2[1] and the insulating titinate YTiO 3\n[2] both possess a fractional magnetic moment per ion. Possible exp lanations of such\nfractional local moments could be a canted antiferromagnetic str ucture as may occur in\nYTiO3[3], or crystalline electric field effects as has been proposed for CeAg Sb2[4].\nUnlike dc measurements, ac susceptibility can be measured in much low er magnetic\nfields. It became a valuable technique in studying magnetic materials. However,\ninterpretation of the results, especially in itinerant systems is comp licated. Forexample,\nlow frequency ac susceptibility measurements on the insulating two d imensional\nferromagnet K 2CuF4[5] is strikingly similar to that measured in palladium slightly\ndoped with manganese [6] as well as in an Fe-Ni-B-Si alloy [7]. While it is clea r that the\ninsulating compound is a local moment, the nature of the latter two is open to debate.\nIt is suggested in these works that the similarities in the data is due to a combination\nof demagnetization effects and domain wall motion.\nIn this contribution we report radio frequency temperature depe ndent ac\nsusceptibility of the well-studied commonly accepted itinerant ferro magnet ZrZn 2,\nand examine its evolution with an applied magnetic field. We then presen t a semi-\nphenomenological model that describes our data. Low frequency ac susceptibility\nmeasurements on ZrZn 2have been reported [8], however, as the focus of that work\nwas not the ferromagnetism of the compound, the only presented data is for T <2 K.\nRecently, it has been shown [9] [10] that rf measurements of ac mag netic\nsusceptibility, χac, seem to distinguish between local moment and itinerant\nferromagnetism. Further, in Ref. [9] an explanation ruling out the d emagnetzation or\nmagnetic domains effects was presented. It is clear from Fig. 2 that the rf susceptibility\nof the weak itinerant ferromagnet ZrZn 2[11] is distinctly different from that of the 4 f\nlocalmomentCeAgSb 2[1]. Thepurposeofthisworkistopresent aneffectiveWeiss-type\nmodel to describe the data derived from itinerant ferromagnets.Field-dependent AC susceptibility of itinerant ferromagn ets 3\nThe design and operation of a tunnel-diode resonator (TDR) are de scribed in detail\nelsewhere [12] [13] [14]. The device is built around a tunnel diode, a sem iconducting\ndevice with a voltage bias region of negative differential resistance. Biasing to this\nvoltage regionallows thetunnel diodeto drive an LCtank circuit at its natural resonant\nfrequency. In magnetic measurements, a sample is placed in the coil of the tank circuit\nthereby changing the total inductance, and, hence, the tank cir cuit’s resonant frequency.\nIt can be shown [15] that the frequency shift of the tank circuit is d irectly proportional\nto the magnetic susceptibility, χ, of the sample in the coil as\n∆f\nf0≈ −1\n2Vs\nVc4πχm. (1)\nHereVsandVcare the volumes of the sample and coil, respectively, and χmis the\nmeasured susceptibility of the sample. Careful design and constru ction allows one to\nresolve changes in resonant frequency induced by the sample on th e order of 1-10 mHz.\nThe resulting tuned circuit, operating at 10-20 MHz, gives frequen cy sensitivity on the\norder of a few parts per billion. This translates to a typical sensitivit y of 10−7−10−8\nchange in χinduced by temperature or magnetic field. Due to the operating fre quency\nthe measured susceptibility is composed of two parts. The first is du e to the magnetic\nmoments in the sample, and may be either para- or diamagnetic depen ding on the\nmaterial studied. The second is due to the screening of an rf field via the normal skin\neffectinmetals. Thisscreening isadiamagneticcontributionandisame asureofchanges\nin resistivity [16].\nRadio frequency susceptibility data presented herein were collecte d in a TDR\noperating at 23 MHz mounted in a4He cryostat. The design is similar to that presented\nin Ref. [9]. The temperature of the sample can be varied from 3 to 100 K and a dc\nmagnetic field up to 2.5 T coaxial with the rf excitation field ( ∼20 mOe) can be\napplied with a superconducting magnet. The magnet is mounted inside the vacuum can\nof the cryostat resulting in no trapped magnetic field at the beginnin g of each run. As\nthe effects studied herein are completely suppressed by fields on th e order of 500 Oe,\nany such trapped magnetic field could affect the data. Single crysta l samples were used\nin this study. The CeAgSb 2sample was prepared as described in Ref. [1], while the\nZrZn2sample was prepared as described in Ref. [17].\nFigure 1 compares temperature dependent dc susceptibility for th e local moment\nCeAgSb 2with that for the itinerant ZrZn 2as measured in a Quantum Design MPMS-\n5. Two different techniques were used to determine these suscept ibilities. Panel (a)\nshows the usual χdcwhere the magnetic moment is measured in a small applied field\n(H= 20Oe)and M/Hiscalculated. While thismethodisappropriatefortemperatures\nwell above TCwhere magnetization is linearly dependent on field over a fairly large\nfield range, it should be expected to fail in the ferromagnetic state becauseMis not\nnecessarily linear in Hall the way down to H= 0. Bearing this in mind, a delta\nmeasurement of χdcwas performed (results in panel (b)) as follows. Magnetic moment\nversus temperature was measured first in a 17 Oe field and then in a 2 2 Oe field. The\ndifference in the resulting moment was divided by the 5 Oe difference in a pplied fields toField-dependent AC susceptibility of itinerant ferromagn ets 4\n/s49/s46/s48\n/s48/s46/s56\n/s48/s46/s54\n/s48/s46/s52\n/s48/s46/s50\n/s48/s46/s48/s68/s77/s47/s68/s72/s32/s40/s110/s111/s114/s109/s46/s41\n/s49/s46/s52 /s49/s46/s50 /s49/s46/s48 /s48/s46/s56 /s48/s46/s54\n/s84/s47/s84/s67/s32/s32/s32/s32/s32/s32/s68/s77/s47/s68/s72\n/s32/s67/s101/s65/s103/s83/s98/s50\n/s32/s90/s114/s90/s110/s50\n/s72/s97/s118/s103/s32/s61/s32/s50/s48/s32/s79/s101\n/s68/s72/s32/s32/s32/s61/s32/s53/s32/s79/s101\n/s32/s40/s98/s41/s49/s46/s48\n/s48/s46/s56\n/s48/s46/s54\n/s48/s46/s52\n/s48/s46/s50\n/s48/s46/s48/s77/s47/s72/s32/s40/s110/s111/s114/s109/s46/s41\n/s49/s46/s52 /s49/s46/s50 /s49/s46/s48 /s48/s46/s56 /s48/s46/s54\n/s84/s47/s84/s67/s32/s32/s32/s32/s32/s32/s32/s32/s32/s77/s47/s72\n/s67/s101/s65/s103/s83/s98/s50\n/s32/s32/s90/s114/s90/s110/s50\n/s32/s72/s32/s61/s32/s50/s48/s32/s79/s101\n/s40/s97/s41\nFigure 1. (Color online) Comparison of normalized measurements of dc suscep tibility\nof the 4flocal moment ferromagnetic CeAgSb 2(black circles, TC≈9.8 K) and the\nweak itinerant ferromagnet ZrZn 2(red squares, TC≈27 K) on a reduced temperature\nscale. Panel (a) is the conventional dc susceptibility, M/H. Panel (b) is the result\nof dc delta measurements as explained in the text. Both sets of mea surements were\ncarried out with Havg= 20 Oe. Normalization is relative to the 5 K value of M/H\nand performed for each sample separately.\ndetermine ∆ M/∆H. The advantage of this method is that it only requires approximate\nlinearity in M(H) over the 5 Oe window defined by the upper and lower fields. Thus, it\ncan be expected to approximate χ=dM\ndHmore closely. Obviously, a smaller Hwindow\nis more likely to conform to the linear M(H) approximation.\nWhile the delta measurement results in a lower susceptibility in both sam ples,\nboth dc techniques produce quite similar χ(T) curves. This is to be contrasted with the\nresults of zero field radio frequency susceptibility versus tempera ture as shown in Fig. 2.\nWhereas the local moment system shows a sharp, well defined peak inχacat the Curie\ntemperature, the itinerant system exhibits a broad maximum well be lowTC.\nConventional theories of itinerant ferromagnetism fail to predict the behavior seen\nin the TDR data of ZrZn 2. The development of the Stoner theory [18] was driven by a\ndesiretounderstandhowafractionalBohrmagnetonmagneticmo mentcouldbecreated\nin nickel. The failures of Stoner theory, i.e.Curie temperatures that are too high andField-dependent AC susceptibility of itinerant ferromagn ets 5\n/s49/s49/s48/s48\n/s56/s48/s48\n/s53/s48/s48\n/s50/s48/s48\n/s45/s49/s48/s48/s99/s97/s99/s32/s40/s97/s114/s98/s46/s41\n/s51/s48 /s50/s53 /s50/s48 /s49/s53 /s49/s48 /s53\n/s84/s32/s40/s75/s41/s90/s114/s90/s110/s50\n/s72/s32/s61/s32/s48/s84/s67\n/s45/s49/s53/s48/s48/s45/s53/s48/s48/s53/s48/s48/s49/s53/s48/s48/s99/s97/s99/s32/s40/s97/s114/s98/s46/s41\n/s49/s48/s46/s50 /s57/s46/s56 /s57/s46/s52 /s57/s46/s48\n/s84/s32/s40/s75/s41/s67/s101/s65/s103/s83/s98/s50\n/s72/s32/s61/s32/s48/s84/s67\nFigure 2. (Color online) Radio frequency susceptibility of the weak itinerant\nferromagnet ZrZn 2(top) and the 4 flocal moment ferromagnet CeAgSb 2(bottom)\nin zero applied field. TCmarks the feature at the Curie temperature for each material.\nThe general decrease in the measured χ(aboveTCfor ZrZn 2, and below for CeAgSb 2)\nis caused by a decrease in resistivity. Note the different temperatu re scales in each\nplot.\nthe lack of a Curie-Weiss susceptibility above TC, were impetus for the development of\nthe spin fluctuation theory of Moriya and Kawabata [19]. While spin fluc tuation theory\ndoes indeed predict a Curie-Weiss type paramagnetic state and larg ely correct the Curie\ntemperatures, neither it nor the Stoner theory adequately desc ribe the broad maximum\nseen in the rf data. Indeed, both theories derive a strictly zero fie ld limit of χjust below\nTCof the same form\nχ(T < T C) =χ0/parenleftbigg\n1−/parenleftbiggT\nTC/parenrightbiggn/parenrightbigg−1\n. (2)\nThe difference between the two theories is the value of n. For Stoner theory n= 2 while\nfor spin fluctuations n= 1 [20]. The Stoner theory does predict a nonzero χatT= 0\nandH= 0 [21], however there is no prescription for how the susceptibility sh ould evolve\nfromTCto 0 K.\nA zero field limit calculation, however, is not representative of a ferr omagnetic\nsystem below TC. As the system begins to order there is a non-zero field in theField-dependent AC susceptibility of itinerant ferromagn ets 6\nsample from the ordered moments themselves. In itinerant system s this mean field\nshould continue to increase as Tdecreases and the fraction of spin polarized conduction\nelectrons increases. The increase in the itinerant mean field may be e xpected to be more\ndramatic than in a local moment mean field because in the former ther e are physically\nmore magnetic carriers as temperature decreases, while in the latt er there is merely\nless thermal randomization of the moment directions. To account f or the self-field we\npropose a modified Brillouin function for a spin-1/2 system in a magnet ic field. We\nchoose a spin-1/2 system because in itinerant systems it is the single electron spin that\nis of interest.\nm∗(t,h) =m∗\n0tanhh\n1−tn. (3)\nIn the above equation, t=T\nT∗whereT∗is a characteristic temperature not\nnecessarily equal to TCandhis a dimensionless field term. It should be noted that\nthis form does not represent the magnetization of the sample. Rat her,m∗may be taken\nas some measure of the unpolarized fraction of the conduction ban d. Differentiating\nwith respect to hgives\nχ(t,h) =χ0\n1−tncosh−2h\n1−tn. (4)\nIn the limit h→0, this reduces to Eq. 2 if T∗→TC, despite the fact that we are not\nbeginning with the proper form for the magnetization.\nTo account for the resistivity contribution to the measured susce ptibility, data\ncollected in a dc field of 1 kOe were subtracted from lower field runs. T his field was\nsufficiently large to completely suppress the maximum in χacwhile being small enough\nthat it is not expected to result in a significant magnetoresistance.\nData fits to the model were attempted for various values of n. It was found that\nn= 1, corresponding to the spin fluctuations theory, gave the best agreement. Figure 3\nshows best fits of Eq. 4 to the TDR data for ZrZn 2. The resulting values of the fitting\nparameters are shown in Fig. 4. χ0decreases with applied H.T∗is constant within the\nerrors and it is approximately equal to TC. The value of his approximately constant up\nto applied fields of about 125 Oe and thereafter grows monotonically asHis increased.\nIn weak itinerant ferromagnets, like ZrZn 2, the susceptibility in the ferromagnetic\nstate is dominated by contributions from the polarized fraction of c onduction electrons\nand the unpolarized fraction. This second contribution comes from a band with a\nlarge Stoner enhancement, so it should be expected to have a corr espondingly large\nsusceptibility. We suggest that our model accounts for the behav ior of the unpolarized,\nfluctuating portion of the conduction band.\nThe operating frequency in this study is commensurate with domain- wall resonance\ntechniques [22]. However, in ZrZn 2dc magnetization measurements suggest that single\ncrystals are forced into a single domain in fields as small as 30 Oe [23] eff ectively ruling\nout any domain wall motion.\nIn conclusion, we have presented rf susceptibility measurements o n the weak\nitinerant ferromagnet ZrZn 2in various applied dc fields. The zero field limit expressionsField-dependent AC susceptibility of itinerant ferromagn ets 7\n/s49/s52/s48/s48\n/s49/s50/s48/s48\n/s49/s48/s48/s48\n/s56/s48/s48\n/s54/s48/s48\n/s52/s48/s48\n/s50/s48/s48\n/s48/s99/s97/s99/s32/s40/s97/s114/s98/s46/s41\n/s50/s53 /s50/s48 /s49/s53 /s49/s48 /s53\n/s84/s32/s40/s75/s41/s32/s32/s65/s112/s112/s108/s105/s101/s100/s32/s72 /s32/s40/s79/s101/s41/s32/s32\n/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32\n/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s55/s48\n/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s49/s50/s53\n/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s50/s48/s48\n/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s51/s48/s48\nFigure 3. (Color online) Comparison of data (points) and fits (solid lines) from t he\nmodel presented in Eq. 4 with n= 1. For clarity only every seventh data point is\nshown. In all fits, R2values were greater than 0.98.\n/s48/s46/s55\n/s48/s46/s53\n/s48/s46/s51/s104\n/s51/s48/s48 /s50/s53/s48 /s50/s48/s48 /s49/s53/s48 /s49/s48/s48 /s53/s48 /s48\n/s72/s97/s112/s112/s32/s40/s79/s101/s41/s49/s50/s48/s48\n/s49/s48/s48/s48\n/s56/s48/s48\n/s54/s48/s48/s99/s48\n/s51/s48\n/s50/s53\n/s50/s48/s84/s42/s32/s40/s75/s41\nFigure 4. (Color online) Values of fitting parameters χ0,T∗, andh(top to bottom)\nfor ZrZn 2single crystal TDR data as functions of applied field ( Happ) derived from\nEq. 4. Errors were determined by individually varying the fit paramet ers until the R2\nvalue dropped below 0.95.\nforχpredicted by existing theories were shown to be insufficient to explain the data.\nA Weiss-like model based on the assumption that the rf response of the sample in\nthe ferromagnetic state is dominated by the fluctuating, unpolariz ed fraction of the\nconduction band was shown to fit the data quite well. It is hoped that this data will\nspur theoretical effort in understanding the dynamic properties o f the ferromagnetic\nstate of such systems.Field-dependent AC susceptibility of itinerant ferromagn ets 8\nAcknowledgments\nWe thank P. C. Canfield and G. Lapertot for supplying the samples, a nd J. Schmalian,\nP. C. Canfield, S. L. Bud’ko, G. Samolyuk, and V. Antropov for fruit ful discussions.\nWork at the Ames Laboratory was supported by the Department o f Energy, Office of\nBasic Energy Sciences under Contract No. DE-AC02-07CH11358. R. P. acknowledges\nsupport from the Alfred P. Sloan foundation.\nReferences\n[1] Myers K D, Bud’ko S L, Fisher I R, Islam Z, Kleinke H, Lacerda A H and Canfield P C 1999 J.\nMag. Mag. Mat. 20527–52\n[2] Garrett J D, Greedan J E and MacLean D A 1981 Mat. Res. Bull. 16145–148\n[3] Ulrich C, Kahliullin G, Okamoto S, Reehuis M, Ivanov A, He H, Taguchi Y, Tokura Y and Keimer\nB 2002Phys. Rev. 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